| Russian Mathematical Surveys |
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Criteria for M. Ya. Mazalovab, P. V. Paramonovcb, K. Yu. Fedorovskiycbd a National Research University "Moscow Power Engineering Institute", Smolensk b Saint-Petersburg State University c Lomonosov Moscow State University, Faculty of Mechanics and Mathematics d Lomonosov Moscow State University, Moscow Center for Fundamental and Applied Mathematics
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    1. Introduction and setups for approximation problemsWe investigate necessary and sufficient conditions for the approximation of functions by solutions of homogeneous second-order elliptic partial differential equations with constant complex coefficients. Approximations are carried out on compact sets in the spaces $\mathbb R^N$ for $N\geqslant2$, in the norm of $C^m$ for $m\in[0,2)$. Approximation criteria in these problems are stated in terms of $ \operatorname{\textit{Lip}} ^m$-$\mathcal L$- and $C^m$-$\mathcal L$-capacities, which are analytic characteristics of subsets of $\mathbb R^N$ related to the differential operator $\mathcal L$ specifying the equation solutions of which are used for approximations. For these capacities, for all values of $m$ in the above interval we present a complete description in metric and/or integro-geometric terms. Note that the case $m\geqslant2$ is well known and was investigated thoroughly by O’Farrell (1979) and Verdera (1987); in this case approximability criteria are stated in slightly different terms, but we also present them for completeness. All these problems go back to classical questions of approximation for holomorphic and harmonic functions, which were of interest to classical authors on complex analysis and approximation theory throughout the 20th century. The detailed history of the question up to 2012 was presented in our paper [1]. In this survey we also present some classical results for completeness and the reader’s convenience, and also to demonstrate the evolution of ideas, methods, and approaches in the transition from approximation by holomorphic or harmonic function to approximation by solutions of more general elliptic equations and systems. During the last 12 years the authors and their co-authors established Vitushkin-type capacitary criteria for $C^m$-approximation of functions by solutions of homogeneous second-order elliptic equations with constant complex coefficients on compact subsets of $\mathbb R^N$ in all dimensional $N\in\{2,3,\dots\}$ and for all smoothness exponents $m\in[0,2)$. These criteria are stated for individual functions. They yield directly the corresponding criteria for classes of functions established previously by Mateu, Orobitg, Netrusov, and Verdera (1996, apart from the cases $m=0$ and $m=1$). Another significant result established in that period of time was an integro-geometric description of all capacities arising in these criteria in the cases $m=0$ (Mazalov, 2024) and $m=1$ (Tolsa, 2021). In particular, these capacities were shown to be subadditive. For $m\in(0,1)\cup(1,2)$ the capacities were previously described by Verdera (1987) in terms of the relevant Hausdorff contents. Let
$$
\begin{equation*}
\begin{gathered} \, L(\boldsymbol{x})=\sum_{n_1,n_2=1}^N c_{n_1n_2}x_{n_1}x_{n_2}, \\ \boldsymbol{x}=(x_1,\dots,x_N)\in\mathbb R^N,\qquad N\in\{2,3,\dots\}, \end{gathered}
\end{equation*}
\notag
$$
be a quadratic form with constant complex coefficients $c_{n_1n_2}=c_{n_2n_1}$ that satisfies the condition of ellipticity:
$$
\begin{equation*}
L(\boldsymbol{x})\ne 0\quad \text{for}\ \ \boldsymbol{x}\ne0.
\end{equation*}
\notag
$$
We can associate with $L(\boldsymbol{x})$ the second-order elliptic operator
$$
\begin{equation*}
\mathcal L=\sum_{n_1,n_2=1}^Nc_{n_1n_2} \frac{\partial^2}{\partial x_{n_1}\,\partial x_{n_2}}\,.
\end{equation*}
\notag
$$
Examples here are the Laplace operator $\mathcal L=\Delta$ in $\mathbb R^N$ and the Bitsadze operator $\mathcal L=\overline\partial{}^2\equiv \dfrac{\partial^2}{\partial\overline{z}^2}$ in $\mathbb R^2$, where $z=x_1+ix_2 \in {\mathbb C}$ is the complex variable. Unless otherwise stated, in what follows we let $\mathcal L$ denote an arbitrary operator of the above form. For an open set $D$ in $\mathbb R^N$ let
$$
\begin{equation*}
\mathcal A_{\mathcal L}(D)=\{f\in C^2(D)\colon\mathcal Lf(\boldsymbol{x})=0\ \forall\,\boldsymbol{x}\in D\}.
\end{equation*}
\notag
$$
We say that functions in this class are $\mathcal L$-analytic in $D$ (by Weyl’s lemma $\mathcal A_{\mathcal L}(D)\subset C^{\infty}(D)$). In this survey we state and discuss Vitushkin-type criteria in terms of capacities for the $C^m$-approximability of functions by $\mathcal L$-analytic functions on compact subsets of $\mathbb R^N $, where $N\in\{2,3,\dots\}$ and $m\geqslant0$. We consider all cases. The capacities arising are described in metric (or, for $m=1$, integro-geometric) terms. We focus on our own results and the ones due to our collaborators and established in the period of the last 12 years. For $s\in\mathbb Z_+:=\{0,1,2,\dots\}$ we denote by $BC^s=BC^s(\mathbb R^N)$ the space of (complex) functions $f\in C^s(\mathbb R^N)$ (where we set $C^0(\mathbb R^N)=C(\mathbb R^N)$) such that
$$
\begin{equation*}
\|f\|_s:=\max_{|\beta|\leqslant s}\|\partial^\beta f\|<+\infty,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\partial^\beta f(\boldsymbol{x})=\frac{\partial^{|\beta|}f(\boldsymbol{x})} {\partial x_1^{\beta_1}\cdots\partial x_N^{\beta_N}}\,,\quad \beta=(\beta_1,\dots,\beta_N)\in\mathbb Z_+^N,\quad |\beta|=\beta_1+\cdots+\beta_N,
\end{equation*}
\notag
$$
and $\|g\|=\sup_{\boldsymbol{x}\in\mathbb R^N}|g(\boldsymbol{x})|$. In the last equality $g$ can be a (bounded) function $g\colon{\mathbb R}^N \to {\mathbb C}$, a vector-valued function $g\colon {\mathbb R}^N \to {\mathbb C}^N$, or even a (vector-valued) function in $L^\infty(\mathbb R^N)$, where we use the same notation for the standard norm in $L^\infty(\mathbb R^N)$. For a bounded (vector-valued) function $g$ in $\mathbb R^N$ and $\mu\in[0,1]$ we define the $\mu$-modulus of continuity of $g$ on $\mathbb R^N$ by
$$
\begin{equation*}
\omega^{\mu}(g,\delta)=\sup\frac{|g(\boldsymbol{x})-g(\boldsymbol{x}')|} {|\boldsymbol{x}-\boldsymbol{x}'|^\mu}\,,\qquad \delta\in(0,+\infty],
\end{equation*}
\notag
$$
where the above supremum is taken over all $\boldsymbol{x}$ and $\boldsymbol{x}'$ such that $0<|\boldsymbol{x}-\boldsymbol{x}'|<\delta$. For brevity we set $\omega(g,\delta)=\omega^0(g,\delta)$. Let $\|g\|'_\mu=\omega^\mu(g,+\infty)$, so that $\|g\|'_0$ agrees with the norm $\|g\|_0=\|g\|$ in $BC^0(\mathbb R^N)= BC$ and the standard norm in $L^\infty(\mathbb R^N)$, and the norm $\|g\|'_1$ in $BC^1(\mathbb R^N)$ is equal to $\|\nabla g\|$. For $s\in\mathbb Z_+$ and $\mu\in(0,1)$ let
$$
\begin{equation*}
BC^{s+\mu}=\{f\in BC^s \colon \omega_s^\mu(f,+\infty)<+\infty,\ \omega_s^\mu(f,\delta)\to0\text{ as }\delta\to0+\},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\omega_s^\mu(f,\delta)=\max_{\{\beta\colon|\beta|=s\}} \omega^\mu(\partial^{\beta}f,\delta).
\end{equation*}
\notag
$$
We also set $\|f\|'_{s+\mu}=\omega_s^\mu(f,+\infty)$. The norm in $BC^{s+\mu}$ is as follows: Let $ \operatorname{\textit{BLip}} ^0=L^\infty(\mathbb R^N)$ with the standard norm. For $s\in\mathbb Z_+$ and $\mu\in(0,1]$ the spaces $ \operatorname{\textit{BLip}} ^{s+\mu}$ are defined similarly to the spaces $BC^{s+\mu}$ (with the same norm), except that there is no condition
$$
\begin{equation*}
\omega_s^\mu(f,\delta)\to 0\quad\text{as}\ \ \delta\to 0+;
\end{equation*}
\notag
$$
here the quantities $\omega_s^1(f,\delta)$, $\|f\|'_{s+1}$ and $\|f\|_{s+1}$, not defined formally yet, have the same definitions as for $\mu\in(0,1)$. It is well known that the spaces $BC^m$ embed in $ \operatorname{\textit{BLip}} ^m$ for all $m\geqslant0$, and the corresponding norms agree (up to positive multiplicative constants depending on $N$ alone). We denote the closure of the support of a function (a distribution) $g$ by $\operatorname{Supp}(g)$. Set
$$
\begin{equation*}
C^m_0=C^m_0(\mathbb R^N)=\{g\in BC^m \colon \operatorname{Supp}(g) \text{ is a compact subset of } \mathbb R^N\}.
\end{equation*}
\notag
$$
Also let $ \operatorname{\textit{Lip}} ^m= \operatorname{\textit{Lip}} ^m(\mathbb R^N)$ consist of all functions $h$ on $\mathbb R^N$ such that $h\varphi\in \operatorname{\textit{BLip}} ^m(\mathbb R^N)$ for each function $\varphi\in C_0^m$. In a similar way we define the spaces $C^m=C^m(\mathbb R^N)$. A Fréchet space structure is introduced on $ \operatorname{\textit{Lip}} ^m$ and $C^m$ in a natural way. For non-integer $m=s+\mu$ ($s=[m]$) and $f\in BC^m$ we need the regularized moduli of continuity
$$
\begin{equation*}
\Omega^m(f,\delta)=\int_1^{+\infty} \frac{\omega^\mu_s(f,t\delta)}{t^{2-\mu}}\,dt,\qquad \delta\in(0,+\infty];
\end{equation*}
\notag
$$
furthermore, for these $m$ and $f\in BC^m$ we have
$$
\begin{equation*}
\omega^\mu_s(f,\delta)\leqslant \Omega^m(f,\delta)\to0\quad\text{as}\ \ \delta\to0.
\end{equation*}
\notag
$$
Finally, set $\Omega^0(f,\delta)=\omega(f,\delta)$ and $\Omega^1(f,\delta)=\omega(\nabla f,\delta)$. We turn to the statements of the problems discussed here. Consider the class of functions
$$
\begin{equation*}
\begin{aligned} \, \mathcal A^m_{\mathcal L}(X)&=\bigl\{f\in BC^m \colon \exists\{f_n\}_{n=1}^{+\infty}, \ f_n\in BC^m\cap\mathcal O_{\mathcal L}(X), \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad \|f-f_n\|_m\to0 \text{ as } n\to+\infty\bigr\}, \end{aligned}
\end{equation*}
\notag
$$
where we write $g\in\mathcal O_{\mathcal L}(X)$ to indicate that the function $g$ is defined and $\mathcal L$-analytic in some neighbourhood (depending on $g$) of the compact set $X$. Our first and central problem is as follows. Given a compact set $X$ in $\mathbb R^N$, a real number $m\geqslant 0$ and a function $f\in BC^m$, find necessary and sufficient conditions ensuring that $f$ belongs to the class $\mathcal A^m_{\mathcal L}(X)$, that is, can to any accuracy be approximated in the $C^m$-norm by functions each of which is $\mathcal L$-analytic in an (own) neighbourhood of $X$. This statement is connected with the Whitney space $C^m_{\rm jet}(X)$ and the corresponding Whitney theorem [2] on $C^m$-extensions from compact subsets to the whole of $\mathbb R^N$ (also see [3], Chap. 6, and [4], § 2). It is easy to show that
$$
\begin{equation*}
\mathcal A^m_{\mathcal L}(X)\subset C^m_{\mathcal L}(X):= BC^m\cap\mathcal A_{\mathcal L}(X^{\circ}),
\end{equation*}
\notag
$$
so condition $f\in C^m_{\mathcal L}(X)$ is called the simplest necessary condition for approximation. Here and throughout, we let $X^{\circ}$ denote the interior of the set $X$, and $\partial X$ is its boundary. In this way the second problem considered here arises naturally, the problem of approximation for classes of functions: for what compact sets $X\subset\mathbb R^N$ do we have the equality $\mathcal A^m_{\mathcal L}(X)=C^m_{\mathcal L}(X)$? Finally, another significant and difficult problem is to give a metric description of the capacities in terms of which the corresponding approximation criteria are stated. We point out straight away that we introduce some notation and give some definitions repeatedly in different sections. This is for the reader’s convenience as the text is quite lengthy. 2. Requisite definitions and statements of resultsFirst we discuss the following natural question: what are $N$, $\mathcal L$, and $m$ such that $\mathcal A^m_{\mathcal L}(X)=C^m_{\mathcal L}(X)$ for all compact subsets $X$ of $\mathbb R^N$? The answer is due to Mazalov [5]. Theorem 2.1. This property holds only for $N=2$ and $m=0$, in the case of non-strongly elliptic operators $\mathcal L$ (that is, of operators with bounded fundamental solutions). The following result of O’Farrell [6] and Verdera [7] also has a quite natural and simple form. Theorem 2.2. Let $X$ be a compact set in $\mathbb R^N$, and let $m\geqslant 2$ and $f\in BC^m$. Then the following conditions are equivalent: In particular, $\mathcal A^m_{\mathcal L}(X)=C^m_{\mathcal L}(X)$ if and only if $\overline{X^{\circ}}=X$. Throughout what follows (except Theorem 9.1) we discuss exponents of smoothness $m\in[0,2)$. Our immediate aim is to state the criteria solving the central problem from a unified standpoint. They are direct analogues of Vitushkin’s criteria [8] (for uniform holomorphic approximations), which set the standards (both in their form and in the methods of research) for all the topics considered here. Let $\varPhi_{\mathcal L}(\boldsymbol{x})$ denote the standard fundamental solution for the operator $\mathcal L$. Its explicit form and main properties are discussed in the next section. For example,
$$
\begin{equation*}
\varPhi_\Delta(\boldsymbol{x})=-\frac{1}{4\pi|\boldsymbol{x}|} \quad\text{in } \mathbb R^3\quad\text{and}\quad \varPhi_{\overline\partial{}^2}(z)= \frac{\overline{z}}{\pi z}\quad\text{in } \mathbb R^2.
\end{equation*}
\notag
$$
For $ m\in[0,2)$ and $N\in\{3,4,\dots\}$ we define the $ \operatorname{\textit{Lip}} ^m$-$\mathcal L$-capacity of the non-empty bounded subset $E$ of $ \mathbb R^N$ by
$$
\begin{equation}
\gamma^m_{\mathcal L}(E)=\sup_T\bigl\{\vert\langle T,1\rangle \vert\colon \operatorname{Supp}(T)\subset E,\ g=\varPhi_{\mathcal L}*T\in \operatorname{\textit{BLip}} ^m(\mathbb R^N),\ \|g\|'_m\leqslant 1\bigr\},
\end{equation}
\tag{1}
$$
where the supremum is taken over all distributions $T$ as indicated; we let $\langle T,\varphi\rangle$ denote the action of $T$ on the function $\varphi$ in the class $C^\infty$, and $*$ is convolution. For non-strongly elliptic operators $\mathcal L$ in $\mathbb R^2$ (see the next section) and $m\in(0,2)$ the definition (1) of the capacity $\gamma^m_{\mathcal L}(E)$ remains the same. However, for strongly elliptic operators $\mathcal L$ in $\mathbb R^2$ and $m\in(0,2)$ the definition of $\gamma^m_{\mathcal L}(E)$ must be corrected slightly: the condition $g=\varPhi_{\mathcal L}*T\in \operatorname{\textit{BLip}} ^m(\mathbb R^N)$ in (1) must be replaced by $g\in \operatorname{\textit{Lip}} ^m(\mathbb R^N)$, that is, in fact, from (1) we exclude the assumption that $g$ is bounded in a punctured neighbourhood of $\infty$. This is related to the unboundedness of $\varPhi_{\mathcal L}(\boldsymbol{x})$ close to $\infty$ in the case of a strongly elliptic operator $\mathcal L$ in $\mathbb R^2$. Under the above assumptions ($N\geqslant3$, or $N=2$ and $m\in(0,2)$) we can define in a similar way the $C^m$-$\mathcal L$-capacity $\alpha^m_{\mathcal L}(E)$ of the non-empty bounded set $E\subset\mathbb R^N$: we must just add to (1) the additional assumption For strongly elliptic operators the cases when both $N=2$ and $m=0$ are considered separately (see § 6, formulae (81) and (83)). We discuss the metric (or, for $m=1$, integro-geometric) description of these capacities in §§ 4–6 below. Given a form $L(\boldsymbol{x})= \displaystyle\sum_{n_1,n_2=1}^Nc_{n_1n_2}x_{n_1}x_{n_2}$, a function $f\in C(\mathbb R^N)$, and an open ball $B=B(\boldsymbol{a},r)$ with centre $\boldsymbol{a}$ and radius $r>0$, we define the so-called $L$-oscillation (or $\mathcal L$-oscillation) of $f$ on $B$ by
$$
\begin{equation}
\mathcal O^L_B(f)=\frac{1}{\sigma(\partial B)}\int_{\partial B} f(\boldsymbol{x})\frac{L(\boldsymbol{x}-\boldsymbol{a})}{r^2}\, d\sigma(\boldsymbol{x})-\frac{\sum_{n=1}^Nc_{nn}}{N|B|}\int_B f(\boldsymbol{x})\,d\boldsymbol{x}.
\end{equation}
\tag{2}
$$
Here $|B|$ is the Lebesgue measure (volume) of the ball $B$ in $\mathbb R^N$, and $\sigma$ is the surface Lebesgue measure on the sphere $\partial B$. For example, if $L(\boldsymbol{x})=\displaystyle\sum_{j=1}^Nx_j^2$ (that is, $\mathcal L=\Delta$), then we have
$$
\begin{equation}
\mathcal O^\Delta_B(f)=\frac{1}{\sigma(\partial B)}\int_{\partial B} f(\boldsymbol{x})\,d\sigma(\boldsymbol{x})- \frac{1}{|B|}\int_B f(\boldsymbol{x})\,d\boldsymbol{x}.
\end{equation}
\tag{3}
$$
The next statement is the cumulative formulation of our results in [9]–[22]. The most complicated case $m=0$ was considered in [22]. Theorem 2.3. Let $X$ be a compact subset of $\mathbb R^N$, $N\in\{2,3,\dots\}$; let $m\in[0,2)$ ($m \ne 0$ for $N=2$) and $f\in C^m_0$. Then the following conditions are equivalent: An analogue of this theorem for $N=2$ and $m=0$ is presented in § 7 below (see Theorem 7.3). Some consequences of these results, which are relevant approximation criteria for classes of functions, are presented in § 8. We also discuss there some questions relating to the description of sets of removable singularities of $\mathcal L$-analytic functions in the classes of $C^m$- and $ \operatorname{\textit{Lip}} ^m$-smooth functions. These are null sets for the corresponding $C^m$-$\mathcal L$- and $ \operatorname{\textit{Lip}} ^m$-$\mathcal L$-capacities, and this description allows one to see once more that the capacities under consideration measure the ‘mass’ of the singularities of $\mathcal L$-analytic functions in the corresponding classes. Note that the metric properties of these capacities depend significantly on $m$. 3. Fundamental solutions $\varPhi_{\mathcal L}(\boldsymbol{x})$To study the approximation problems stated above we must know the properties and explicit forms of fundamental solutions of the equations $\mathcal Lu=0$. Although for a large class of such equations the requisite explicit formulae are well known and can be found, for instance, in [23], § 6.2, the general case is virtually not described in the literature to our knowledge. It was considered in the recent paper [20]. In this section we present some results we need and their proofs. The main of these are Theorem 3.1 (and an explicit formula for fundamental solutions) and Corollary 3.2 to it, on two-sided bounds for fundamental solutions of equations from a large class (for instance, for all equations under consideration in the case of dimensions $N=3$ and $4$). Elliptic quadratic forms with complex coefficients in $\mathbb R^N$Let $N\geqslant2$ be a fixed integer and $C$ be a symmetric $ N\times N $ matrix with complex entries $c_{mn}=c_{nm}$, $1\leqslant m,n\leqslant N$. Let $Q$ be the quadratic form in $\mathbb R^N$ defined by $C$, that is,
$$
\begin{equation*}
Q(\boldsymbol{x})=\boldsymbol{x}^{\top} C\boldsymbol{x}= \sum_{m,n=1}^{N}c_{mn}x_mx_n
\end{equation*}
\notag
$$
for $\boldsymbol{x}=(x_1,\dots,x_N)^{\top}\in\mathbb R^N$, where $(\,\cdot\,)^\top$ indicates transposition of matrices. In what follows we also write $Q_N$ in place of $Q$, to stress the dimension $N$ of the space on which $Q$ acts. Note that it is only in this section that the vectors $\boldsymbol{x}$, $\boldsymbol{y}$, $\boldsymbol{z}$, $\boldsymbol{0}$, and so on are treated as columns. Definition 3.1. We say that the quadratic form $Q_N$ is elliptic if $Q_N(\boldsymbol{x})\ne0$ for all $\boldsymbol{x}\in\mathbb R^N_*$. Here and throughout, $ \mathbb R^N_*=\mathbb R^N\setminus\{\boldsymbol{0}\}$ and $\mathbb C^n_*=\mathbb C^n\setminus\{\boldsymbol{0}\}$, $n\in\mathbb N$. Also let $\mathbb T=\{z\in\mathbb C\colon |z|=1\}$ be the unit circle in $\mathbb C$ with the standard parametrization $\mathbb T=\{\varGamma_1(t)=e^{2\pi it}\colon t\in[0,1]\}$. We look closer at the concept of an elliptic quadratic form, especially since a number of important and useful properties of such forms are not yet fully covered in the literature. For instance, while constructing fundamental solutions in his classical monograph [23] (see § 6.2 there), Hörmander merely postulated without proofs the properties of the corresponding quadratic forms that we state in Lemmas 3.4 and 3.5 below. We begin our investigations of elliptic quadratic forms from $N= 2$. Here the main definition can be reformulated as follows: a form
$$
\begin{equation*}
Q_2\bigl((x_1,x_2)^\top\bigr)=c_{11}x_1^2+2c_{12}x_1x_2+c_{22}x_2^2
\end{equation*}
\notag
$$
in $\mathbb R^2$ is elliptic if and only if the roots $\lambda_1$ and $\lambda_2$ of the corresponding characteristic equation $c_{11}\lambda^2+2c_{12}\lambda+c_{22}=0$ are not real. In this case $Q_2$ is said to be strongly elliptic if $\lambda_1$ and $\lambda_2$ lie in the distinct half-planes
$$
\begin{equation*}
\mathbb C_+ =\{z\in\mathbb C\colon \operatorname{Im}{z}>0\}\quad\text{and}\quad \mathbb C_- =\{z\in\mathbb C\colon \operatorname{Im}{z}<0\}
\end{equation*}
\notag
$$
of the complex plane $\mathbb C$ (relative to the real axis). In the two-dimensional case we can conveniently view $Q_2$ as a function of the complex variable $z$, that is, $Q_2(z)=Q_2((\operatorname{Re}{z},\operatorname{Im}{z})^\top)$. Recall that $\Delta_{\varGamma}\operatorname{Arg}{g}$ denotes the increment of the angle argument of the complex function $g(z)$ along the curve $\varGamma$ in $\mathbb C$. Lemma 3.1. Let $Q_2$ be an elliptic quadratic form in $\mathbb R^2$. Then $Q_2$ is strongly elliptic if and only if This is equivalent to $Q_2(\mathbb T)$ lying in some (open) half-plane in $\mathbb C$ whose boundary contains zero. Proof. Let $\lambda_1\in\mathbb C_+$ and $\lambda_2\in\mathbb C_-$. Then it is easy to verify that Since $Q_2(z)=c_{11}(x_1-\lambda_1x_2)(x_1-\lambda_2x_2)$ for $z=x_1+ix_2$, we have $\Delta_{\mathbb T}\operatorname{Arg}{Q_2}=0$. Conversely, if both characteristic roots $\lambda_1$ and $\lambda_2$ lie in $\mathbb C_+$ or $\mathbb C_-$, then $\Delta_{\mathbb T}\operatorname{Arg}{Q_2}=\pm4\pi$. This proves the first assertion of the lemma.
To prove the second note that which is easy to verify by direct calculations in the polar variables. If $c_{11}=c_{22}$ or $c_{12}=\tau(c_{11}-c_{22})$ for some $\tau\in\mathbb R$ (both cases can occur in the strongly elliptic case), then the above parametric expression describes a straight-line segment (lying away from zero and traversed four times). Otherwise this expression describes an ellipse, traversed twice and encircling or not encircling zero depending on whether we have non-strong or strong ellipticity, respectively. This is a direct consequence of the argument in the proof of the first assertion. $\Box$Corollary 3.1. In the case when $Q_2$ is not strongly elliptic (and only then) and there exist points $\boldsymbol{x}\in\mathbb R^2_*$ and $\boldsymbol{y}\in\mathbb R^2_*$ such that $Q_2(\boldsymbol{x})=-Q_2(\boldsymbol{y})$. It also follows from Lemma 3.1 that the properties of ellipticity and strong ellipticity of quadratic forms are preserved by non-degenerate linear transformations of $\mathbb R^2$. Now let $\mathbb S^{N-1}=\{\boldsymbol{x}\in\mathbb R^N\colon x_1^1+\dots+x_N^2=1\}$ be the unit sphere in $\mathbb R^N$ (then $\mathbb T$ coincides with $\mathbb S^1\subset\mathbb R^2$ as a set, although the definition of $\mathbb T$ also implies that some orientation is fixed). Lemma 3.2. Let $Q_2$ be an elliptic quadratic form in $\mathbb R^2$, and let $N\in\{3,4,\dots\}$. Then an elliptic quadratic form $Q_N$ in $\mathbb R^N$ such that exists if and only if $Q_2$ is strongly elliptic. Proof. Let $Q_2$ be a strongly elliptic quadratic form in $\mathbb R^2$, and let $V=\{t\boldsymbol{y}\colon t\in\mathbb R_+:=(0,+\infty),\ \boldsymbol{y}\in Q_2(\mathbb S^1)\}$. Fix some $c_{nn}\in V$, $n\in\{3,\dots,N\}$. Then it follows directly from Lemma 3.1 that the form $Q_N$ in $\mathbb R^N$ defined by is elliptic in $\mathbb R^N$.
On the other hand let $Q_2$ be a non-strongly elliptic form in $\mathbb R^2$. Suppose that $Q_2$ is the restriction to $\mathbb R^2$ of an elliptic quadratic form $Q_N$ in $\mathbb R^N$, $N>2$. Let $\varGamma$ be a homotopy on $\mathbb S^{N-1}\subset\mathbb R^N$ that takes the circle $\varGamma_1\subset\mathbb R^2_{(x_1,x_2)^\top}$ to a point $\boldsymbol{a} \in\mathbb S^{N-1}$ (we identify $\varGamma_1$ with the set $\{\boldsymbol{x}\in\mathbb S^{N-1}\colon x_3=\dots=x_N=0\}$). Then the composition $Q_N\circ\varGamma$ is a homotopy in $\mathbb R^2_*$ which takes the cycle $\varGamma_2=Q_N\circ\varGamma_1=Q_2\circ\varGamma_1$ to the point $Q_N(\boldsymbol{a})$. However, there can be no such homotopy because $\Delta_{\varGamma_2}\operatorname{Arg}z= \Delta_{\varGamma_1}\operatorname{Arg}Q_2=\pm4\pi$ by Corollary 3.1. $\Box$ Lemma 3.3. Let $Q_N$ be a quadratic form in $\mathbb R^N$, $N\geqslant3$. Then $Q_N$ is elliptic if and only if the set $Q_N(\mathbb S^{N-1})\subset\mathbb C_*$ lies in an open half-space in $\mathbb C$ with boundary containing zero. This is equivalent to $V_N=Q_N(\mathbb R^N_*)$ being a closed sector of opening $\vartheta_{Q_N}<\pi$ in $\mathbb C_*$ which is punctured at the vertex at the origin. Proof. Let $Q_N$ be an elliptic form. It is sufficient to show that there exist no points $\boldsymbol{x}\in\mathbb R^N_*$ and $\boldsymbol{y}\in\mathbb R^N_*$ such that $Q_N(\boldsymbol{x})=-Q_N(\boldsymbol{y})$. In fact, otherwise the restriction of $Q_N$ to the plane passing through the origin and the points $\boldsymbol{x}$ and $\boldsymbol{y}$ is not a strongly elliptic form by Corollary 3.1, in contradiction to Lemma 3.2. The converse assertion is obvious. $\Box$ It follows from Lemma 3.3 that for each elliptic quadratic form $Q_N$ in $\mathbb R^N$, $N\geqslant3$, there exist $\tau\in(0,1)$, $\vartheta\in(-\pi,\pi]$, and $\vartheta_{Q_N}\in (0,\pi)$ such that the form $Q^{\vartheta}(\boldsymbol{x})=e^{i\vartheta}Q_N(\boldsymbol{x})$ satisfies the conditions
$$
\begin{equation}
|{\arg(Q^{\vartheta}(\boldsymbol{x}))}|\leqslant \frac{\vartheta_{Q_N}}2 < \frac{\pi}{2}\quad\text{and}\quad \operatorname{Re}Q^{\vartheta}(\boldsymbol{x})\geqslant \tau|Q^{\vartheta}(\boldsymbol{x})|\geqslant \tau^2|\boldsymbol{x}|^2,\qquad \boldsymbol{x} \in \mathbb R^N_*.
\end{equation}
\tag{4}
$$
Below we assume that $Q_N$ is an elliptic quadratic form with matrix $C$ in $\mathbb R^N$. We also set $A:=\operatorname{Re}{C}$ and $B:=\operatorname{Im}{C}$, so that $C=A+iB$. Proof. Suppose that $\det{C}=0$. Then there exists $\boldsymbol{z}\in\mathbb C^N$, $\boldsymbol{z}\ne\boldsymbol{0}$, such that $C\boldsymbol{z}=\boldsymbol{0}$. Let $\boldsymbol{z}=\boldsymbol{x}+i\boldsymbol{y}$. The condition $C\boldsymbol{z}=(A+iB)(\boldsymbol{x}+i\boldsymbol{y})=\boldsymbol{0}$ is equivalent to the pair of conditions $A\boldsymbol{x}=B\boldsymbol{y}$ and $A\boldsymbol{y}=-B\boldsymbol{x}$, hence
$$
\begin{equation*}
Q_N(\boldsymbol{x}) =\boldsymbol{x}^\top(A+iB)\boldsymbol{x}= \boldsymbol{x}^\top B\boldsymbol{y}-i\boldsymbol{x}^\top A\boldsymbol{y}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
Q_N(\boldsymbol{y}) =\boldsymbol{y}^\top (A+iB)\boldsymbol{y}= -\boldsymbol{y}^\top B\boldsymbol{x}+i\boldsymbol{y}^\top A\boldsymbol{x}.
\end{equation*}
\notag
$$
Since the matrices $A$ and $B$ are symmetric, it follows from the last condition that $Q_N(\boldsymbol{y})=-Q_N(\boldsymbol{x})$ (in particular, $\boldsymbol{x}\ne\boldsymbol{0}$ and $\boldsymbol{y}\ne\boldsymbol{0}$). However, this contradicts Lemma 3.3. $\Box$ Note that the condition $N\geqslant3$ is essential in Lemma 3.4: the elliptic quadratic form $Q_2(x_1,x_2)=(x_1+ix_2)^2/4$ in $\mathbb R^2$ has the matrix
$$
\begin{equation*}
C_2=\frac{1}{4}\begin{pmatrix} 1 & i \\ i & -1 \end{pmatrix},
\end{equation*}
\notag
$$
and $\det{C_2}=0$. Lemma 3.5. Let $N\geqslant 3$, let $Q_N$ and $C$ be as indicated above, and let $Q'_N$ be the quadratic form with matrix $C^{-1}$. Then $Q'_N$ is also elliptic. Proof. By Lemma 3.4 we have $\det{C}\ne 0$. Suppose that there exists $\boldsymbol{a}\in\mathbb R^N_*$ such that $\boldsymbol{a}^\top C^{-1}\boldsymbol{a}=\boldsymbol{0}$. Let $\boldsymbol{z}=C^{-1}\boldsymbol{a}$, so that $\boldsymbol{z}\in\mathbb C^N_*$. Then $\boldsymbol{a}=C\boldsymbol{z}$ and because $C$ is a symmetric matrix.
As before, let $\boldsymbol{x}:=\operatorname{Re}{\boldsymbol{z}}$, $\boldsymbol{y}:=\operatorname{Im}{\boldsymbol{z}}$, $A:=\operatorname{Re}{C}$, and $B:=\operatorname{Im}{C}$. Since $C\boldsymbol{z}=\boldsymbol{a}$, we have $\operatorname{Im}((A+iB)(\boldsymbol{x}+i\boldsymbol{y}))=\boldsymbol{0}$, so that $B\boldsymbol{x}=-A\boldsymbol{y}$. Now, as
$$
\begin{equation*}
\boldsymbol{0}=\boldsymbol{z}^\top C\boldsymbol{z}= (\boldsymbol{x}^\top+i\boldsymbol{y}^\top)(A+iB)(\boldsymbol{x}+i\boldsymbol{y})= (\boldsymbol{x}^\top+i\boldsymbol{y}^\top)(A\boldsymbol{x}-B\boldsymbol{y}),
\end{equation*}
\notag
$$
we have the equalities $\boldsymbol{x}^\top A\boldsymbol{x}=\boldsymbol{x}^\top B\boldsymbol{y}$ and $\boldsymbol{y}^\top A\boldsymbol{x}=\boldsymbol{y}^\top B\boldsymbol{y}$. These three equalities show that
$$
\begin{equation*}
\boldsymbol{x}^\top B\boldsymbol{x}=(B\boldsymbol{x})^\top \boldsymbol{x}= (-A\boldsymbol{y})^\top \boldsymbol{x}=-\boldsymbol{y}^\top A\boldsymbol{x}= -\boldsymbol{y}^\top B\boldsymbol{y}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\boldsymbol{y}^\top A\boldsymbol{y}=(A\boldsymbol{y})^\top \boldsymbol{y}= (-B\boldsymbol{x})^\top \boldsymbol{y}=-\boldsymbol{x}^\top B\boldsymbol{y}= -\boldsymbol{x}^\top A\boldsymbol{x}.
\end{equation*}
\notag
$$
However, then
$$
\begin{equation*}
Q_N(\boldsymbol{y})=\boldsymbol{y}^\top C\boldsymbol{y}= \boldsymbol{y}^\top A\boldsymbol{y}+i\boldsymbol{y}^\top B\boldsymbol{y}= -\boldsymbol{x}^\top A\boldsymbol{x}-i\boldsymbol{x}^\top B\boldsymbol{x}= -Q_N(\boldsymbol{x}),
\end{equation*}
\notag
$$
which contradicts Lemma 3.3. $\Box$ The explicit form of fundamental solutions for second-order elliptic operators in $\mathbb R^N$ with constant complex coefficientsLet $N\geqslant2$ be a fixed integer and $Q$ be an elliptic quadratic form in $\mathbb R^N$ defined by a (symmetric) matrix $C$ with complex entries $c_{mn}$, $1\leqslant m,n\leqslant N$. This form specifies an elliptic differential operator of the second order with constant complex coefficients
$$
\begin{equation*}
\mathcal L=\sum_{m,n=1}^{N}c_{mn} \frac{\partial^2}{\partial x_m\,\partial x_n}\,,
\end{equation*}
\notag
$$
which we call the operator associated with $Q$. In turn, we call $Q$ the symbol of the operator $\mathcal L$. We have already considered the main examples, the Laplace operator
$$
\begin{equation*}
\Delta_N=\sum_{n=1}^{N}\frac{\partial^2}{\partial x_n^2}
\end{equation*}
\notag
$$
in $\mathbb R^N$, which is associated with $x_1^2+\dots+x_N^2$, and the Bitsadze operator (squared Cauchy–Riemann operator) in $\mathbb R^2$, defined by
$$
\begin{equation*}
\frac{1}{4}\biggl(\frac{\partial}{\partial x_1}+ i\frac{\partial}{\partial x_2}\biggr)^2= \frac{\partial^2}{\partial\overline{z}^2}\,.
\end{equation*}
\notag
$$
We have also already encountered its associated form $Q_2(x)=(x_1+ix_2)^2/4$. As mentioned above, all elliptic quadratic forms in $\mathbb R^2$ fall into two classes: the class of strongly elliptic forms and the class of forms that are not strongly elliptic. Accordingly, we call a second-order elliptic operator $\mathcal L$ in $\mathbb R^2$ strongly elliptic if its symbol is a strongly elliptic form, and we call it non-strongly elliptic otherwise. For example, the Laplace operator $\Delta_2$ in $\mathbb R^2$ is strongly elliptic, while the Bitsadze operator is not. Thus, we cannot ‘lift’ the latter to an elliptic operator in $\mathbb R^N$ for any $N>2$. Note also that the quadratic form equal to the symbol of the Bitsadze operator has a matrix providing the above example of a singular matrix of an elliptic quadratic form. It is important to note that dividing the elliptic operators of the second order in $\mathbb R^2$ into strongly and non-strongly elliptic is fundamental in nature and is based on quite different properties of these operators. The profound difference between strongly and non-strongly elliptic operators appears in problems of the description of sets of removable singularities of solutions of the corresponding equations $\mathcal Lf=0$, in problems of approximation of functions by solutions of these equations and in conditions for the solvability and uniqueness of the solution of classical boundary-value problems for these equations. We present in what follows examples of results highlighting this difference. Furthermore, Lemma 3.2 above is also an interesting example of the significant difference between these two classes of operators. Let us establish an explicit formula expressing the fundamental solution for an arbitrary elliptic operator $\mathcal L$ of the second order in $\mathbb R^N$, $N\geqslant3$. Let $\mathcal L$ be associated with a quadratic form $Q$ with matrix $C$. By Lemma 3.4 this matrix is non-singular: $\det{C}\ne0$. Consider the matrix $D=C^{-1}$ and the corresponding quadratic form
$$
\begin{equation}
\varLambda(\boldsymbol{x})=\boldsymbol{x}^\top D\boldsymbol{x},\qquad \boldsymbol{x}\in\mathbb R^N.
\end{equation}
\tag{5}
$$
By Lemma 3.5 $\varLambda$ is an elliptic form. Let $\vartheta_{\varLambda}$ be the opening of the sector $\varLambda(\mathbb R^N_*)$, so that $\vartheta_{\varLambda}<\pi$. Let $S(z)$ be a holomorphic branch of the two-valued function $\sqrt{z}$ which is defined for $\{z\in\mathbb C \colon -\pi<\arg{z}<\pi\}$, for instance, $S(z)=\sqrt{|z|}\, e^{i\arg(z)/2}$. By Lemma 3.3 (see (4)) there exists $\vartheta\in(-\pi,\pi]$ such that $|\arg(e^{i\vartheta}\Lambda(x))|\leqslant\vartheta_{\varLambda}/2<\pi/2$, which means that $\operatorname{Re}(e^{i\vartheta}\Lambda(\boldsymbol{x}))>0$ for all $\boldsymbol{x}\in\mathbb R^N_*$. Thus, the function
$$
\begin{equation*}
\varPsi_*(\boldsymbol{x})=S(e^{i\vartheta}\varLambda(\boldsymbol{x}))
\end{equation*}
\notag
$$
is real analytic in $\mathbb R^N_*$, homogeneous of order $1$, and, in addition,
$$
\begin{equation}
|{\arg(\varPsi_*(\boldsymbol{x}))}|\leqslant\frac{\vartheta_{\varLambda}}{4}< \frac{\pi}{4}\quad \forall\, \boldsymbol{x}\in\mathbb R^N_*.
\end{equation}
\tag{6}
$$
The following result is checked by direct calculations. We present it for the completeness of presentation. Lemma 3.6. Let $\mathcal L$ and $\varPsi_*$ be as indicated above, and let for $\boldsymbol{x}\in\mathbb R^N_*$. Then $\mathcal L\varPsi(\boldsymbol{x})=0$ for all $\boldsymbol{x}\in\mathbb R^N_*$. Proof. We can assume without loss of generality that $\vartheta=0$. Let $p=(2-N)/2$, so that $\varPsi(\boldsymbol{x})=\varLambda(\boldsymbol{x})^p_*$, where $\varLambda(\,\cdot\,)$ is defined by (5) and $*$ indicates that we consider the principal branch $w^p_*=\exp(p\log_*w)$ of the multivalued function $w^p$ in $\mathbb C\setminus (-\infty,0]$ (here $\log_*w=\log|w|+i\arg w$ is the principal branch of the multivalued logarithm in $\mathbb C\setminus(-\infty,0]$).
Since $(w^p_*)'_w=\bigl(\exp(p\log_*w)\bigr)'_w=pw^p_*\big/w$, it follows that
$$
\begin{equation*}
\frac{\partial\varPsi(\boldsymbol{x})}{\partial x_m}= p\,\frac{\varPsi(\boldsymbol{x})}{\varLambda(\boldsymbol{x})}\, \frac{\partial\varLambda(\boldsymbol{x})}{\partial x_m}\,,
\end{equation*}
\notag
$$
and therefore
$$
\begin{equation*}
\frac{\partial^2\varPsi(\boldsymbol{x})}{\partial x_m\,\partial x_n}= p\,\frac{\varPsi(\boldsymbol{x})}{\varLambda(\boldsymbol{x})^2} \biggl((p-1)\,\frac{\partial\varLambda(\boldsymbol{x})}{\partial x_m}\, \frac{\partial\varLambda(\boldsymbol{x})}{\partial x_n}+ \varLambda(\boldsymbol{x})\,\frac{\partial^2\varLambda(\boldsymbol{x})} {\partial x_m\,\partial x_n}\biggr)
\end{equation*}
\notag
$$
for $1\leqslant m,n\leqslant N$.
Since $p-1=-N/2$, to prove the equality $\mathcal L\varPsi(\boldsymbol{x})=0$ for $\boldsymbol{x}\in\mathbb R^N_*$ it is sufficient to show that
$$
\begin{equation}
R(\boldsymbol{x})=\sum_{m,n=1}^{N}c_{mn}\biggl(-\frac{N}{2}\, \frac{\partial\varLambda(\boldsymbol{x})}{\partial x_m}\, \frac{\partial\varLambda(\boldsymbol{x})}{\partial x_n}+ \varLambda(\boldsymbol{x})\,\frac{\partial^2\varLambda(\boldsymbol{x})} {\partial x_m\,\partial x_n}\biggr)=0
\end{equation}
\tag{8}
$$
for all $\boldsymbol{x}\in\mathbb R^N_*$, where the $c_{mn}$ are (as before) the entries of $C$. Let $d_{mn}$ ($1\leqslant m,n\leqslant N$) denote entries of $D$. Taking the symmetry of $D$ into account we obtain
$$
\begin{equation*}
\frac{\partial\varLambda(\boldsymbol{x})}{\partial x_m}= 2\sum_{j=1}^{N}d_{jm}x_j\quad\text{and}\quad \frac{\partial^2\varLambda(\boldsymbol{x})} {\partial x_m\,\partial x_n}=2d_{mn}.
\end{equation*}
\notag
$$
Now the quantity $R(\boldsymbol{x})$ in (8) assumes the following form: We denote the coefficients of the quadratic form $R(\boldsymbol{x})$ by $r_{jk}$, $1\leqslant j,k\leqslant N$. Then (as $D$ is symmetric) It remains to observe that $(d_{j1},\dots,d_{jN})C(d_{1k},\dots,d_{Nk})^\top $ is the entry $d_{jk}$ of the matrix $D=DCD$ and $\displaystyle\sum_{m,n=1}^{N}c_{mn}d_{mn}=N$ because $DC=I$ (the identity matrix). Thus, $r_{jk}=-2Nd_{jk}+2Nd_{jk}=0$. $\Box$Lemma 3.6 enables us to deduce an explicit formula for the fundamental solution of $\mathcal L$. Recall that a distribution $\varPhi_{\mathcal L}$ is called a fundamental solution for $\mathcal L$ if $\mathcal L\varPhi_{\mathcal L}= \delta_{\boldsymbol{0}}$, where the symbol $\delta_{\boldsymbol{0}}$ denotes the Dirac delta function with support at the origin. Theorem 3.1. In the notation of Lemma 3.6, for a suitable constant $c_{\mathcal L}\in\mathbb C_*$ the function is a fundamental solution for $\mathcal L$. Proof. Let
$$
\begin{equation*}
f_n(\boldsymbol{x})=\sum_{j=1}^{N}c_{nj}\, \frac{\partial\varPsi}{\partial x_j}\,,\qquad n\in\{1,\dots,N\},
\end{equation*}
\notag
$$
where the above equality can be treated in the usual sense (for $\boldsymbol{x}\ne \boldsymbol{0}$) and in the sense of distributions (in the whole of $\mathbb R^N$) alike. Each $f_n$ is a locally integrable odd homogeneous function of degree $-N+1$ in the class $C^\infty(\mathbb R^N_*)$. Then (in the sense of distributions)
$$
\begin{equation*}
\chi(\boldsymbol{x})\equiv\mathcal L\varPsi(\boldsymbol{x})= \sum_{n=1}^{N}\frac{\partial f_n(\boldsymbol{x})}{\partial x_n}\,,
\end{equation*}
\notag
$$
$\chi(\boldsymbol{x})=0$ in $\mathbb R^N_*$, and by the definition of distributional derivatives, for each function $g\in C_0^\infty(\mathbb R^N)$ (with support in some ball $B_r=B(\boldsymbol{0},r)$) we have
$$
\begin{equation*}
\begin{aligned} \, \langle\mathcal L\varPsi,g\rangle&= -\sum_{n=1}^{N}\bigg\langle f_n,\frac{\partial g}{\partial x_n}\bigg\rangle= -\int_{B_r}\sum_{n=1}^{N}f_n(\boldsymbol{x}) \frac{\partial g(\boldsymbol{x})}{\partial x_n}\,d\boldsymbol{x} \\ &=-\lim_{\varepsilon\to0}\int_{B_r\setminus B_{\varepsilon}} \sum_{n=1}^{N} f_n(\boldsymbol{x}) \frac{\partial g(\boldsymbol{x})}{\partial x_n}\,d\boldsymbol{x}. \end{aligned}
\end{equation*}
\notag
$$
We transform the last integral (setting $D=B_r\setminus B_{\varepsilon}$) by using the Gauss–Ostrogradskii formula (of integration by parts):
$$
\begin{equation*}
-\int_D\sum_{n=1}^{N} f_n(\boldsymbol{x}) \frac{\partial g(\boldsymbol{x})}{\partial x_n}\,d\boldsymbol{x}= \int_D g(\boldsymbol{x})\chi(\boldsymbol{x})\,d\boldsymbol{x}- \int_{\partial D}(gF,\nu)\,d\sigma= \int_{\partial B_{\varepsilon}}(gF,\nu_+)\,d\sigma,
\end{equation*}
\notag
$$
where $F=\{f_1,\dots,f_N\}$ (a vector field), $\nu$ is the unit outward normal on $\partial D$ (while $\nu_+=-\nu$ is the unit outward normal to $\partial B_{\varepsilon}$), and $\sigma$ is the surface measure on $\partial D$. Since the field $F$ is homogeneous of degree $-N+1$, the integral
$$
\begin{equation*}
\int_{\partial B_{\varepsilon}}\big(g(\boldsymbol{0})F,\nu_+\big)\,d\sigma= g(\boldsymbol{0})\int_{\partial B_1}(F,\nu_+)\,d\sigma=: d_{\mathcal L}g(\boldsymbol{0})
\end{equation*}
\notag
$$
is independent of $\varepsilon$. Moreover, it is clear that so that we finally obtain As it is obvious that $d_{\mathcal L}\ne0$, we have $c_{\mathcal L}=1/d_{\mathcal L}$. $\Box$ For instance, for $\mathcal L=\Delta_N$ (the Laplace operator in $\mathbb R^N$) we have
$$
\begin{equation*}
\varLambda(x)=Q(x)=|x|^2,\qquad \varPsi(x)=|x|^{2-N},\quad\text{and}\quad F(x)=\nabla\varPsi(x)=-\frac{(N-2)x}{|x|^N}\,,
\end{equation*}
\notag
$$
which means that $d_{\mathcal L}=-(N-2)\sigma_N$ and
$$
\begin{equation*}
\varPhi_{\Delta}(\boldsymbol{x})= -\frac{1}{\sigma_N (N-2) |\boldsymbol{x}|^{N-2}}\,,
\end{equation*}
\notag
$$
where $\sigma_N$ is the surface ($(N-1)$-dimensional) Lebesgue measure of the sphere $\partial B(\boldsymbol{0},1)$ in $\mathbb R^N$. The following result is a direct consequence of (6) and (7). Corollary 3.2. Let $N=3$ or $4$. Then for each operator $\mathcal L$ there exist $\lambda=\lambda_{\mathcal L}\in(-\pi,\pi]$ and $A=A_{\mathcal L}\geqslant 1$ such that for all $\boldsymbol{x}\in\mathbb R^N_*$. A similar result holds for operators $\mathcal L$ in $\mathbb R^N$, $N\geqslant 5$, that satisfy the additional condition In addition, as a direct consequence of (7), for $N\geqslant3$ the estimate
$$
\begin{equation*}
\frac{1}{A|\boldsymbol{x}|^{N-2}}\leqslant |\varPhi_{\mathcal L}(\boldsymbol{x})|\leqslant \frac{A}{|\boldsymbol{x}|^{N-2}}
\end{equation*}
\notag
$$
holds for all operators $\mathcal L$. Now we present another property of fundamental solutions, which can be of use not only in connection with the questions of approximation under consideration here, but also in other problems. Proposition 3.1. Let $N\geqslant 3$. Then for each operator $\mathcal L$ with fundamental solution $\varPhi=\varPhi_{\mathcal L}$ and each $R>0$ where $B_R=B(\boldsymbol{0},R)$ and $\sigma$ denotes the surface Lebesgue measure on the corresponding sphere. Proof. We assume without loss of generality that $\mathcal L$ satisfies (4) for $\vartheta=0$. Since $\varPhi$ is a homogeneous function of degree $2-N$, the following equalities are easy to establish:
$$
\begin{equation}
\int_{\partial B_r}\varPhi(\boldsymbol{x})\,d\sigma(\boldsymbol{x})= \frac{r}{R}\int_{\partial B_R} \varPhi(\boldsymbol{x})\,d\sigma(\boldsymbol{x})\ \ \text{and}\ \int_{B_R}\varPhi(\boldsymbol{x})\,d\boldsymbol{x}=\frac{R}{2} \int_{\partial B_R}\varPhi(\boldsymbol{x})\,d\sigma(\boldsymbol{x}).
\end{equation}
\tag{13}
$$
Hence it is sufficient to verify only the second inequality in (12) for $R=1$. We need a well-known property of fundamental solutions, which can be found, for instance, in [3], Chap. 3, § 3.5 (where formula (15) below and Theorem 6 in [3], § 3.5, for the operators $T_1(f)=(\Delta\varPhi)*f$ and $T_2(f)=(\mathcal L\varPhi_{\Delta})*f$ are used), and is as follows:
$$
\begin{equation}
\Delta\varPhi=\Delta\varPhi_{\mathcal L}= \lambda_0\delta_{\boldsymbol{0}}+\varPsi_0\quad\text{and}\quad \mathcal L\varPhi_{\Delta}=\lambda\delta_{\boldsymbol{0}}+\varPsi,
\end{equation}
\tag{14}
$$
where $\lambda_0,\lambda\in\mathbb C$, and $\varPsi_0$ and $\varPsi$ are appropriate standard Calderón–Zygmund kernels in $\mathbb R^N$. Equalities (14) must be treated in the sense of distributions, and the functions $\varPsi_0$ and $\varPsi$ have the following properties: they belong to the class $C^\infty(\mathbb R^N_*)$, are homogeneous of degree $-N$, and
To both sides of the equality $\mathcal L\varPhi=\delta_{\boldsymbol{0}}$ we apply the Fourier transform $T\mapsto\widetilde{T}$ acting in the space $\mathcal S'$ of tempered distributions $T$. By the standard properties of the Fourier transform $-Q(\boldsymbol{y})\widetilde{\varPhi}(\boldsymbol{y})=(2\pi)^{-N/2}$, which yields
$$
\begin{equation*}
\widetilde{\varPhi}(\boldsymbol{y})= -\frac{(2\pi)^{-N/2}}{Q(\boldsymbol{y})}\,,
\end{equation*}
\notag
$$
and therefore
$$
\begin{equation}
\widetilde{\Delta\varPhi}(\boldsymbol{y})= -|\boldsymbol{y}|^2\widetilde{\varPhi}(\boldsymbol{y})= (2\pi)^{-N/2}\frac{|\boldsymbol{y}|^2}{Q(\boldsymbol{y})}\quad\text{and}\quad \widetilde{\mathcal L\varPhi_{\Delta}}(\boldsymbol{y})= (2\pi)^{-N/2}\frac{Q(\boldsymbol{y})}{|\boldsymbol{y}|^2}
\end{equation}
\tag{15}
$$
are infinitely differentiable homogeneous functions of degree 0 on $\mathbb R^N_*$, which is equivalent to (14) by [3], Chap. 3, § 3.5.
On the other hand, from (14) we obtain
$$
\begin{equation}
\widetilde{\Delta\varPhi}(\boldsymbol{y})= \lambda_0(2\pi)^{-N/2}+\widetilde{\varPsi_0}(\boldsymbol{y}),
\end{equation}
\tag{16}
$$
where (taking (15) into account again) the function $\widetilde{\varPsi_0}\in C^{\infty}(\mathbb R^N_*)$ is homogeneous of degree $0$. From the definition of the Fourier transform of a distribution we obtain $\langle\widetilde{\varPsi_0},\varphi\rangle= \langle \varPsi_0,\widetilde{\varphi}\rangle$. Substituting in $\varphi(\boldsymbol{x})=\exp(-|\boldsymbol{x}|^2/2)$ we see that
$$
\begin{equation*}
\int_{\mathbb R^N}\widetilde{\varPsi_0}(\boldsymbol{y}) \exp\biggl(-\frac{|\boldsymbol{y}|^2}2\biggr)\,d\boldsymbol{y}= \operatorname{(p.v.)}\int_{\mathbb R^N}\varPsi_0 (\boldsymbol{x}) \exp\biggl(-\frac{|\boldsymbol{x}|^2}2\biggr)\,d\boldsymbol{x}=0.
\end{equation*}
\notag
$$
Thus, since $\widetilde{\varPsi_0}$ is, as mentioned above, homogeneous of degree 0, it follows straight away that
$$
\begin{equation*}
\int_{\partial B_1}\widetilde{\varPsi_0}(\boldsymbol{y})\, d\sigma(\boldsymbol{y})=0.
\end{equation*}
\notag
$$
In particular, from (15) and (16) we obtain
$$
\begin{equation}
\lambda_0=\frac{1}{\sigma(\partial B_1)}\int_{\partial B_1} \frac{|\boldsymbol{y}|^2}{Q(\boldsymbol{y})}\,d\sigma(\boldsymbol{y})\ne0,
\end{equation}
\tag{17}
$$
because $\operatorname{Re}Q(\boldsymbol{y})>0$ for $\boldsymbol{y}\in\mathbb R^N_*$ (as follows directly from (4)).
Now fix a radial function $\varphi(\boldsymbol{x})=\varphi(|\boldsymbol{x}|)\in C^\infty_0(B_1)$ such that $\varphi(0)\ne0$. Then
$$
\begin{equation*}
\begin{aligned} \, \langle\Delta\varPhi,\varphi\rangle&=\int_{B_1}\varPhi(\boldsymbol{x}) \Delta\varphi(\boldsymbol{x})\,d\boldsymbol{x}= \langle\lambda_0\delta_{\boldsymbol{0}}+\varPsi_0,\varphi\rangle \\ &=\lambda_0\varphi(0)+\int_{B_1}\varPsi_0(\boldsymbol{y}) (\varphi(\boldsymbol{y})-\varphi(0))\,d\boldsymbol{y}= \lambda_0\varphi(0)\ne0. \end{aligned}
\end{equation*}
\notag
$$
Thus, $\displaystyle\int_{B_1}\varPhi(\boldsymbol{x}) \Delta\varphi(\boldsymbol{x})\,d\boldsymbol{x}\ne0$. Now, because $\varphi(\boldsymbol{x})=\varphi(r)$, $r=|\boldsymbol{x}|$ is radially symmetric, from the formula $\Delta\varphi(\boldsymbol{x})=\varphi''(r)+ ((N-1)/r)\varphi'(r)$, as $\varPhi$ is homogeneous and we have (13), we obtain
$$
\begin{equation*}
\int_{B_1}\varPhi(\boldsymbol{x}) \Delta\varphi(\boldsymbol{x})\,d\boldsymbol{x}= \int_{\partial B_1}\varPhi(\boldsymbol{x}')\, d\sigma(\boldsymbol{x}') \int_0^1r\biggl(\varphi''(r)+\frac{N-1}{r}\,\varphi'(r)\biggr)\,dr\ne0,
\end{equation*}
\notag
$$
so it is immediate that (12) holds. $\Box$ The reader can find simple explicit formulae for fundamental solutions of second-order elliptic operators $\mathcal L$ in $\mathbb R^2$, for instance, in [24]. It is clear from these formula that an analogue of Proposition 3.1 fails in $\mathbb R^2$ (for example, for the Bitsadze operator). 4. The case $m\in(0,1)\cup(1,2)$. ${\mathcal L}$-oscillations. Reduction to harmonic ($N>2$) and bianalytic ($N=2$) approximationsRecall that the $p$-dimensional ($p\in(0,N]$) Hausdorff content of a bounded set $E$ in $\mathbb R^N$ is the quantity where the infimum is taken over all at most countable ball covers $\{B_j\}$ of the set $E$ (each family $\{B_j\}$ is an at most countable cover of $E$ in $\mathbb R^N$ by balls $B_j$ of radii $r_j$).The following metric description of $ \operatorname{\textit{Lip}} ^m$-$\mathcal L$-capacities for $m\in(0,1)\cup(1,2)$ was established in [7], Lemma 3.1: for each fixed $m\in(0,1)\cup(1,2)$, each $N\in\{2,3,\ldots\}$, and each operator $\mathcal L$ in $\mathbb R^N$ there exists a constant $A=A(N,L,m)\in(1,+\infty)$ such that the following estimate holds for all $\sigma$-compact bounded sets $E$ in $\mathbb R^N$:
$$
\begin{equation}
A^{-1}\mathcal M^{N-2+m}(E)\leqslant\gamma^m_{\mathcal L}(E)\leqslant A\mathcal M^{N-2+m}(E).
\end{equation}
\tag{18}
$$
Similar estimates hold for $\gamma^m_{\mathcal L}(E)$ replaced by $\alpha^m_{\mathcal L}(E)$ and $\mathcal M^{N-2+m}(E)$ replaced by the Hausdorff lower content $\mathcal M^{N-2+m}_*(E)$ (see below for the corresponding definition and estimates (27) and (28)). The concept of $\mathcal L$-oscillation introduced in (2) requires a comment. Lemma 4.1. For $\boldsymbol{a}\in\mathbb R^N$ and $r\in(0,+\infty)$ let $\psi_{\boldsymbol{a}}^r(\boldsymbol{x})=(r^2-|\boldsymbol{x}- \boldsymbol{a}|^2)\big/(2N|B|)$ in $B=B(\boldsymbol{a},r)$ and $\psi_{\boldsymbol{a}}^r(\boldsymbol{x})=0$ outside $B(\boldsymbol{a},r)$. Then for each function $\varphi\in C^\infty(\mathbb R^N)$ that is, the result of the action $\langle\mathcal L\psi_{\boldsymbol{a}}^r,\varphi\rangle$ of the distribution $\mathcal L\psi_{\boldsymbol{a}}^r$ on the function $\varphi$ is equal to $\mathcal O^L_B(\varphi)$; this action can be extended by continuity to all functions $\varphi\in C(\mathbb R^N)$. Proof. Clearly, we can limit ourselves to $\boldsymbol{a}=\boldsymbol{0}$. Set $\psi_{\boldsymbol{0}}^r(\boldsymbol{x})=\psi(\boldsymbol{x})$, and fix $i,j\in\{1,\dots,N\}$. Then
$$
\begin{equation}
\biggl\langle\psi, \frac{\partial^2\varphi}{\partial x_i\,\partial x_j}\biggr\rangle= -\biggl\langle\frac{\partial\psi}{\partial x_i}\,, \frac{\partial\varphi}{\partial x_j}\biggr\rangle= \frac{1}{2N|B|}\int_B 2x_i \frac{\partial\varphi}{\partial x_j}\,d\boldsymbol{x}.
\end{equation}
\tag{20}
$$
Let $B_j'=\{\boldsymbol{x}=(x_1,\dots,x_N)\in B \colon x_j=0\}$ and $\boldsymbol{x}'_j=(x_1,\dots,x_{j-1},0,x_{j+1},\dots,x_N)$ (of course, the last formula must be changed for $j=1$ and $j=N$). For $\boldsymbol{x}=(x_1,\dots,x_N)\in B$ let $\boldsymbol{x}_j^+=(x_1,\dots,x_{j-1},x_j^+,x_{j+1},\dots,x_N)$ (respectively, $\boldsymbol{x}_j^-=(x_1,\dots,x_{j-1},x_j^-,x_{j+1},\dots,x_N)$) be defined (uniquely) by the conditions $\boldsymbol{x}_j^+\in\partial B$ and $x_j^+\geqslant0$, (respectively, by the conditions $\boldsymbol{x}_j^-\in\partial B$ and $x_j^-\leqslant0$). Integrating repeatedly (beginning with $x_j$) on the right-hand side of (20) (and, in particular, integrating by parts for $i=j$) and taking the equality $d\boldsymbol{x}_j'=|x_j^{\pm}|\,d\sigma(\boldsymbol{x}_j^{\pm})/r$ into account, for $i\ne j$ we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber \frac{1}{N|B|}\int_B x_i \frac{\partial\varphi}{\partial x_j}\,d\boldsymbol{x}&= \frac{1}{N|B|}\int_{B_j'} x_i\bigl(\varphi(\boldsymbol{x}_j^+)- \varphi(\boldsymbol{x}_j^-)\bigr)\,d\boldsymbol{x}_j' \\ &=\frac{1}{N|B|r}\int_{\partial B} x_ix_j \varphi(\boldsymbol{x})\,d\sigma(\boldsymbol{x}), \end{aligned}
\end{equation}
\tag{21}
$$
while for $i=j$ we have It remains to multiply these equalities by the (appropriate) $c_{ij}$ and sum with respect to $i$ and $j$, bearing in mind that $\sigma(\partial B)=N|B|/r$. $\Box$ The proof of Theorem 2.3 for $m\in(0,1)\cup(1,2)$The approach used here and related to the invariance of Calderón–Zygmund operators in Lipschitz spaces, is not new (for instance, see [7], [25], and the bibliography in these papers). The proof is based on the reduction of a general operator $\mathcal L$ to the harmonic case for $N>2$ and to the bianalytic case for $N= 2$. Let $\mathcal L$ be one of the operators under consideration in $\mathbb R^N$, with fundamental solution $\varPhi(\boldsymbol{x})=\varPhi_{\mathcal L}(\boldsymbol{x})$. Set $\mathcal L_0=\Delta$ for $N\geqslant3$ and $\mathcal L_0=\overline\partial^2$ for $N=2$; we introduce the notation $\varPhi_0=\varPhi_{\mathcal L_0}$. Fix $m\in(0,1)\cup(1,2)$ and an arbitrary function $f\in BC^m$. We assume without loss of generality (see Lemma 4.2 below) that $f\in C^m_0=C^m_0(\mathbb R^N)$. Let $T=\mathcal Lf$ and $f_0=\varPhi_0*T$, that is, let $\mathcal L_0f_0=T=\mathcal Lf$. Then where $E=\mathcal L\varPhi_0=\lambda\delta_{\boldsymbol{0}}+\varPsi$ (in the notation of formula (14); the case $N=2$ is considered similarly); also see [7], Lemma 1.1.It is known [26] (see Lemma 4.3 below for details) that the operator $f\mapsto E*f$ is locally continuous in $BC^m$ (a similar result also holds for the operator $\mathcal L_0\varPhi=\lambda_0\delta_{\boldsymbol{0}}+\varPsi_0$). Since it also takes functions from the class $\mathcal A_{\mathcal L}(U)$ (for any open set $U$) to functions in $\mathcal A_{\mathcal L_0}(U)$ (because $\mathcal L_0f_0=\mathcal Lf$), we can reduce problems of $C^m$-$\mathcal L$-approximation directly to problems of $C^m$-$\mathcal L_0$-approximation. For $\mathcal L_0=\Delta$, $N\geqslant3$ and $m\in(0,1)$, or for $\mathcal L_0=\overline\partial^2$, $N=2$, and $m\in(0,1)\cup(1,2)$ these problems were considfered earlier (see [9], Theorem 1 and Remark 1.3, and [11], Theorem 1(b), respectively). We consider the case of the operator $\mathcal L_0=\Delta$ for $N\geqslant3$ and $m\in(1,2)$ below in this section. One must only bear in mind that by Lemma 4.1
$$
\begin{equation}
\mathcal O^L_B(f)=\langle\mathcal L\psi_{\boldsymbol{a}}^r,f\rangle= \langle\mathcal L_0\psi_{\boldsymbol{a}}^r,f_0\rangle= \mathcal O^{L_0}_B(f_0),
\end{equation}
\tag{23}
$$
which means that for $f$ and $f_0$ the estimates in part (c) of Theorem 2.3 are ‘the same’. In other words,
$$
\begin{equation*}
f\in\mathcal A^m_{\mathcal L}(X)\quad\Longleftrightarrow\quad f_0\in\mathcal A^m_{\mathcal L_0}(X).
\end{equation*}
\notag
$$
For a thorough implementation of this plan of the proof we need the following two lemmas (under the assumptions of Theorem 2.3). But first we recall that for a non-empty compact subset $K$ of $ \mathbb R^N$ and $m\geqslant0$ we can define the space $C^m(K)=BC^m\big|_K$ with the norm
$$
\begin{equation*}
\|h\|_{m,K}=\inf\bigl\{\|H\|_m \colon H\in BC^m,\ H\big|_K=h\bigr\}.
\end{equation*}
\notag
$$
Lemma 4.2. Let $X$ be a compact set in $\mathbb R^N$, $B=B(\boldsymbol{0},R)$ be a ball of radius $R$ such that $X\subset B(\boldsymbol{0},R-1)$, and let $g\in C^m=C^m(\mathbb R^N)$. Then the following conditions are equivalent: Proof. It is obvious that (c) $\Rightarrow$ (a) $ \Rightarrow$ (b). It remains to prove the implication (b) $\Rightarrow$ (c). In condition (b) we fix some $\varphi\in C_0^\infty(B)$ (so that $\varphi=0$ outside $B$) such that $\varphi=1$ on $B(\boldsymbol{0},R-1)$ and set $h_3=g(1-\varphi)+h_2\varphi$ (so that $h_3=g$ outside $B$). It remains to observe that $h_3\in C^m$ and Recall that $\mathcal L\varPhi_0=\lambda\delta_{\boldsymbol{0}}+\varPsi$, where $\varPsi$ is the standard Calderón–Zygmund kernel in $\mathbb R^N$. Lemma 4.3. Let $B=B(\boldsymbol{0},R)$ be a ball in $\mathbb R^N$, and let $m\in(0,1)\cup(1,2)$, $s=[m]$, and $m=s+\mu$. Then for each function $g\in C^m_0(B)$ where $A$ depends on $N$, $\mathcal L$, $m$, and $R$. Proof. Results of this type are usually regarded as folklore, but we could not find the statement we need here in the literature (for instance, in the paper [26], which is referred to most often, the authors look at the periodic case and do not introduce the $m$-moduli of continuity $\omega^{\mu}_s$ which are under consideration here). So for the reader’s convenience we present the proof of this lemma. First recall (see [7], Lemma 1.1) that
$$
\begin{equation*}
\begin{aligned} \, \varPsi*g(\boldsymbol{x})&=\operatorname{(p.v.)}\int\varPsi(\boldsymbol{x}- \boldsymbol{y})\bigl(g(\boldsymbol{y})- g(\boldsymbol{x})\bigr)\,d\boldsymbol{y} \\ &=\lim_{\varepsilon\to0}\int_{\varepsilon<|\boldsymbol{y}- \boldsymbol{x}|<1/\varepsilon}\varPsi(\boldsymbol{x}- \boldsymbol{y})\bigl(g(\boldsymbol{y})- g(\boldsymbol{x})\bigr)\,d\boldsymbol{y}. \end{aligned}
\end{equation*}
\notag
$$
We establish the bound $\omega^m(G,r)\leqslant A\Omega^m(g,r)$ for $m\in(0,1)$. If $m\in(1,2)$, then the equality can be used, after which $\omega^{\mu}_1(G,r)$ is estimated in a similar way.
This way to estimates has long been known. Let $\omega(r)=\omega^m(g,r)$, $r\geqslant0$. Fix $\boldsymbol{x}_1\ne\boldsymbol{x}_2\in\mathbb R^N$, and set $|\boldsymbol{x}_1-\boldsymbol{x}_2|=r>0$, $\boldsymbol{x}_0=(\boldsymbol{x}_1+\boldsymbol{x}_2)/2$, and
$$
\begin{equation*}
\begin{alignedat}{2} D_1&=B\biggl(\boldsymbol{x}_1,\frac{r}{2}\biggr),&\qquad D_2&=B\biggl(\boldsymbol{x}_2,\frac{r}{2}\biggr), \\ D_3&=B(\boldsymbol{x}_0,r)\setminus(D_1\cup D_2),&\qquad D_4&=\mathbb R^N\setminus B(\boldsymbol{x}_0, r). \end{alignedat}
\end{equation*}
\notag
$$
Then we have
$$
\begin{equation*}
\begin{aligned} \, &|G(\boldsymbol{x}_1)-G(\boldsymbol{x}_2)|\leqslant \sum_{l=1}^4 J_l \\ &\qquad:=\sum_{l=1}^4\biggl|\operatorname{(p.v.)} \int_{D_l}\bigl(\varPsi(\boldsymbol{x}_1-\boldsymbol{y}) \bigl(g(\boldsymbol{y})-g(\boldsymbol{x}_1)\bigr)- \varPsi(\boldsymbol{x}_2-\boldsymbol{y})\bigl(g(\boldsymbol{y})- g(\boldsymbol{x}_2)\bigr)\bigr)\,d\boldsymbol{y}\biggr|. \end{aligned}
\end{equation*}
\notag
$$
Taking the inequalities $|\varPsi(\boldsymbol{x})|\leqslant A/|\boldsymbol{x}|^{N}$ and $|g(\boldsymbol{y})-g(\boldsymbol{x})|\leqslant \omega(|\boldsymbol{y}- \boldsymbol{x}|)|\boldsymbol{y}-\boldsymbol{x}|^\mu$ into account, integrating in the spherical coordinates ($\rho=|\boldsymbol{y}-\boldsymbol{x}_l|$ for $J_l$, $l\in\{1,2\}$), we obtain For an estimate of $J_4$ note that $\operatorname{(p.v.)}\displaystyle\int_{D_4} \varPsi(\boldsymbol{x}_0-\boldsymbol{y})\,d\boldsymbol{y}=0$ and for $\boldsymbol{y}\in D_4$ we have $|\varPsi(\boldsymbol{x}_l-\boldsymbol{y})-\varPsi(\boldsymbol{x}_0- \boldsymbol{y})|\leqslant Ar/|\boldsymbol{y}-\boldsymbol{x}_0|^{N+1}$ ($l\in\{1,2\}$). Hence, adding the function $\varPsi(\boldsymbol{x}_0- \boldsymbol{y})(g(\boldsymbol{y})-g(\boldsymbol{x}_2))$ and subtracting $\varPsi(\boldsymbol{x}_0-\boldsymbol{y})(g(\boldsymbol{y})- g(\boldsymbol{x}_1))$ under the integral sign in $J_4$ and then integrating again in the spherical variables ($\rho=|\boldsymbol{y}-\boldsymbol{x}_0|$) we obtain where we have set $\rho=rt$. It remains to check that $\omega(r)\leqslant\Omega(r)$, which is elementary. $\Box$ We finish the proof of Theorem 2.3 for $m\in(0,1)\cup(1,2)$ by using the reduction arguments mentioned above. Fix a ball $B=B(\boldsymbol{0},R)$ such that $X\subset B(\boldsymbol{0},R-1)$, and let $f\in BC^m$ (by Lemma 4.2 we can assume that $\operatorname{Supp}(f)\subset B(\boldsymbol{0},R-1)$). Set $f_0=\varPhi_0*(\mathcal Lf)=(\lambda\delta_{\boldsymbol{0}}+\varPsi)*f$. By Lemma 4.3
$$
\begin{equation*}
f_0\in BC^m\quad\text{and}\quad \omega^m(f_0,r)\leqslant A\Omega^m(f,r).
\end{equation*}
\notag
$$
Let $f\in\mathcal A^m_{\mathcal L}(X)$, so that there exists a sequence $\{f_n\}^{+\infty}_{n=1}\subset BC^m$ such that each function $f_n$ is $\mathcal L$-analytic in an (own) neighbourhood of $X$ and $\|f-f_n\|_m\to0$ as $n\to+\infty$ (by Lemma 4.2 we can assume that $\operatorname{Supp}(f_n)\subset B(\boldsymbol{0},R)$). Then the functions $f_{0n}=\varPhi_0*{\mathcal L}f_n$ are $\mathcal L_0$-analytic in some neighbourhoods of $X$ and $\|f_0-f_{0n}\|_m\to0$ as $n\to+\infty$, that is, $f_0\in\mathcal A^m_{\mathcal L_0}(X)$. Conversely, let $f_0\in\mathcal A^m_{\mathcal L_0}(X)$, and let $f_{0n}$ (for $n\in\mathbb N$) be $\mathcal L_0$-analytic functions in some neighbourhoods of $X$ such that $\|f_0-f_{0n}\|_m\to0$ as $n\to+\infty$. Fix $\varphi$ as in the proof of Lemma 4.2, and let $f_0^*=\varphi f_0$ and $f_{0n}^*=\varphi f_{0n}$. Since it is obvious that $\|f^*_0-f^*_{0n}\|_m\to0$ as $n\to+\infty$, so that $f^*_0\in\mathcal A^m_{\mathcal L_0}(X)$, we can show (by reversing our calculations) that
$$
\begin{equation*}
f^*=\varPhi*(\mathcal L_0f^*_0)\in\mathcal A^m_{\mathcal L}(X).
\end{equation*}
\notag
$$
Now it is sufficient to take account of the equality $\mathcal L(f-f^*)=\mathcal L_0(f_0-f^*_0)=\mathcal L_0(f_0(1-\varphi))=0$ in a neighbourhood of $X$, so that $f\in\mathcal A^m_{\mathcal L}(X)$. Thus, we have performed the reduction of Theorem 2.3 for $m\in(0,1)\cup(1,2)$ and an arbitrary operator $\mathcal L$ to the harmonic or bianalytic case. To complete the proof of the theorem it remains to consider the case when $\mathcal L=\Delta$, $m\in(1,2)$, and $N\geqslant3$. The proof of Theorem 2.3 for $\mathcal L=\Delta$, $m\in(1,2)$, and $N\geqslant3$We establish the corresponding result where in (c) the quantity $\Omega^{m}(f,r)$ is replaced by $\omega^{\mu}(\nabla f,r)$. We need several auxiliary results and constructions. Let $\varPhi$ denote the fundamental solution for the Laplace operator $\Delta$ in $\mathbb R^N$ (the value of $N\in\{3,4,\dots\}$ is fixed). For $\varphi\in C_0^\infty(\mathbb R^N)$ we define the Vitushkin-type localization operator [8], [7] in the context of harmonic functions:
$$
\begin{equation*}
f\mapsto V_\varphi(f)=\varPhi*(\varphi\Delta f),\qquad f\in C(\mathbb R^N).
\end{equation*}
\notag
$$
We must establish a few new properties of this operator, related to its possible extension to a wider class of ‘subscripts’ $\varphi$ under certain smoothness-type constraints on the function $f$. For a bounded (vector-valued) function $g$ on an (at least two-point) set $E\subset\mathbb R^N$, for $\mu\in(0,1)$ and $\delta\in(0,+\infty]$ we consider the moduli of continuity
$$
\begin{equation*}
\omega^\mu_{E}(g,\delta)= \sup\biggl\{\frac{|g(\boldsymbol{x})-g(\boldsymbol{y})|} {|\boldsymbol{x}-\boldsymbol{y}|^\mu} \colon \boldsymbol{x}\in E,\ \boldsymbol{y}\in E,\ 0<|\boldsymbol{x}-\boldsymbol{y}|<\delta\biggr\}
\end{equation*}
\notag
$$
and the seminorms $\|g\|'_{\mu,E}=\omega^\mu_{E}(g,+\infty)$. For $E=\mathbb R^N$ and $m=1+\mu\in(1,2)$ we set
$$
\begin{equation*}
\omega^\mu_{f}(\delta)=\omega^\mu_{\mathbb R^N}(f,\delta)\quad\text{and}\quad \omega^m_{f}(\delta)=\omega^\mu_{\mathbb R^N}(\nabla f,\delta);
\end{equation*}
\notag
$$
this agrees with the notation $\omega^\mu_1(f,\delta)$ introduced above. Lemma 4.4. Let $B=B(\boldsymbol{a},r)$ and $\psi(\boldsymbol{x})=\psi_{\boldsymbol{a}}^r(\boldsymbol{x})$ (see Lemma 4.1), and let $m=1+\mu\in(1,2)$. Then for $f\in BC^m$ the function $V_\psi(f)\in BC^m$ is well defined, harmonic on the sets where $f$ is harmonic and harmonic outside $\overline{B}$, and, moreover, The reader can compare this statement with the standard results in the area: see Lemma 2.1 in [7]. Proof. It is sufficient to look at $\boldsymbol{a}=\boldsymbol{0}$. Moreover, in view of regularization we can also assume that $f\in C_0^\infty$. Then it is easy to show that the function $V_\psi(f)\in BC^m$ is well defined. Now we express $V_\psi(f)$ and the components of the vector $\nabla(V_\psi(f))$ in the form more suitable for extending their definitions to $f\in BC^m$ (in fact, even $f\in C^m(\overline{B})$ can be sufficient):
$$
\begin{equation*}
V_\psi(f)=\varPhi*\biggl(\psi\sum_{i=1}^N \frac{\partial f_i}{\partial x_i}\biggr)=\sum_{i=1}^N \bigl(\varPhi_i*(\psi f_i)-\varPhi*(f_i\psi_i)\bigr),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
f_i=\frac{\partial f}{\partial x_i}- \frac{\partial f}{\partial x_i}\bigg|_{\boldsymbol{0}},\quad \psi_i=\frac{\partial \psi}{\partial x_i}\quad\text{and}\quad \varPhi_i=\frac{\partial\varPhi}{\partial x_i}\,.
\end{equation*}
\notag
$$
The above sum presents a consistent expression for $V_\psi(f)$ in the case of an arbitrary $f\in C^m(\overline{B})$, as all integrals involved are absolutely convergent. Now,
$$
\begin{equation*}
\frac{\partial}{\partial x_j}V_\psi(f)= \varPhi*(\psi_j\Delta f+\psi\Delta f_j)=:I_1+I_2,
\end{equation*}
\notag
$$
where (we set $\psi_{ji}=\partial\psi_j/\partial x_i$)
$$
\begin{equation*}
I_1 =\varPhi*\biggl(\psi_j\sum_{i=1}^N \frac{\partial f_i}{\partial x_i}\biggr)= \sum_{i=1}^N\varPhi_i*(\psi_jf_i)- \varPhi*\biggl(\,\sum_{i=1}^Nf_i\psi_{ji}\biggr)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, I_2&=\varPhi*\bigl(\Delta(\psi f_j)-2\nabla\psi\nabla f_j-f_j\Delta\psi\bigr) \\ &=\psi f_j-\varPhi*(f_j\Delta\psi)-2\sum_{i=1}^N\varPhi_i*(\psi_i f_j)+ 2\varPhi*(f_j\Delta\psi). \end{aligned}
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
\frac{\partial}{\partial x_j}V_\psi(f)= \psi f_j+\sum_{i=1}^N\varPhi_i*(\psi_j f_i-2\psi_i f_j)+ \varPhi*\biggl(f_j\Delta\psi-\sum_{i=1}^N f_i\psi_{ji}\biggr).
\end{equation*}
\notag
$$
As shown above (see (20)–(22)), in the sense of distributions we have (for $B=B(\boldsymbol{0},r)$)
$$
\begin{equation*}
\psi_{ij}(\boldsymbol{x})=\frac{1}{N|B|r}\,x_i x_j\, d\sigma(\boldsymbol{x})\big|_{\partial B},\qquad i\ne j,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\psi_{jj}(\boldsymbol{x})=\frac{1}{N|B|r}\,x_j^2\, d\sigma(\boldsymbol{x})\big|_{\partial B}-\frac{1}{N|B|}\,d\boldsymbol{x}\big|_B.
\end{equation*}
\notag
$$
Hence it is sufficient to estimate $\omega^{\mu}(h,\delta)$ for functions $h$ of the following form:
$$
\begin{equation*}
\begin{gathered} \, h_1=\psi f_j,\qquad h_2(\boldsymbol{x})=\int_B\varPhi_i(\boldsymbol{x}-\boldsymbol{y}) \psi_j(\boldsymbol{y})f_k(\boldsymbol{y})\,d\boldsymbol{y}, \\ h_3(\boldsymbol{x})=\frac{1}{N|B|}\int_B\varPhi(\boldsymbol{x}- \boldsymbol{y})f_j(\boldsymbol{y})\,d\boldsymbol{y}, \end{gathered}
\end{equation*}
\notag
$$
and, finally, All these functions are regular, that is, the integrals in their definitions converge absolutely for each $f\in C^m(\overline{B})$.
We estimate $\omega^{\mu}(h_1,\delta)$. Let $0<|\boldsymbol{x}_1-\boldsymbol{x}_2|<\delta$. The estimate for points $\boldsymbol{x}_1$ and $\boldsymbol{x}_2$ in an arbitrary position can easily be reduced to $\boldsymbol{x}_1\in\overline{B}$ and $\boldsymbol{x}_2\in \overline{B}$, $\delta\leqslant 2r$. Then, since
$$
\begin{equation*}
\|\psi\|\leqslant \frac{A}{r^{N-2}}\,,\qquad \|\nabla\psi\|\leqslant \frac{A}{r^{N-1}}\,,\quad \omega^\mu_{\overline{B}}(f_j,\delta)\leqslant \omega^\mu(\delta):= \omega^\mu_{\overline{B}}(\nabla f,\delta),
\end{equation*}
\notag
$$
and we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber \frac{|h_1(\boldsymbol{x}_1)-h_1(\boldsymbol{x}_2)|} {|\boldsymbol{x}_1-\boldsymbol{x}_2|^\mu} &\leqslant \frac{\psi(\boldsymbol{x}_1)|f_j(\boldsymbol{x}_1)-f_j(\boldsymbol{x}_2)|+ |f_j(\boldsymbol{x}_2)|\,|\psi(\boldsymbol{x}_1)- \psi(\boldsymbol{x}_2)|}{|\boldsymbol{x}_1-\boldsymbol{x}_2|^\mu} \\ &\leqslant \frac{A\omega^\mu(\delta)}{r^{N-2}}+ \frac{A\omega^\mu(r)(\delta/r)^{1-\mu}}{r^{N-2}}\,, \end{aligned}
\end{equation}
\tag{24}
$$
so that $\omega^\mu(h_1,\delta)\to0$ as $\delta\to0$ (that is, $h_1\in C^m(\mathbb R^N)$) and $\|h_1\|'_\mu\leqslant A\omega^\mu(r)/r^{N-2}$.
The corresponding estimates for $h_2$ and $h_3$ are deduced in a similar way, so we only discuss $h_2$. We used already this approach to estimates in the proof of Lemma 4.3. Let $|\boldsymbol{x}_1-\boldsymbol{x}_2|=\delta>0$, $b=\min\{\delta,r\}$, and $\boldsymbol{x}_0=(\boldsymbol{x}_1+\boldsymbol{x}_2)/2$. Set
$$
\begin{equation*}
\begin{alignedat}{2} D_1&=B\biggl(\boldsymbol{x}_1,\frac{\delta}{2}\biggr),&\quad D_2&=B\biggl(\boldsymbol{x}_2,\frac{\delta}{2}\biggr), \\ D_3&=B(\boldsymbol{x}_0,\delta)\setminus (D_1\cup D_2),\quad\text{and}&\quad D_4&=\mathbb R^N\setminus B(\boldsymbol{x}_0,\delta). \end{alignedat}
\end{equation*}
\notag
$$
Then we have
$$
\begin{equation*}
|h_2(\boldsymbol{x}_1)-h_2(\boldsymbol{x}_2)|\leqslant \sum_{l=1}^4\int_{D_l\cap B}|\varPhi_i(\boldsymbol{x}_1-\boldsymbol{y})- \varPhi_i(\boldsymbol{x}_2-\boldsymbol{y})|\, |\psi_j(\boldsymbol{y})f_k(\boldsymbol{y})|\,d\boldsymbol{y}=:\sum_{l=1}^4J_l.
\end{equation*}
\notag
$$
Taking the inequalities
$$
\begin{equation*}
|\varPhi_i(\boldsymbol{x})|\leqslant \frac{A}{|\boldsymbol{x}|^{N-1}}\,,\quad \|\psi_j\|\leqslant \frac{A}{r^{N-1}},\quad\text{and}\quad \|f_k\|_{\overline{B}}\leqslant\omega^\mu(r)r^\mu
\end{equation*}
\notag
$$
into account, it is easy to show that
$$
\begin{equation*}
\begin{aligned} \, J_1+J_2&\leqslant \frac{A\omega^\mu(r)r^\mu}{r^{N-1}} \int_{D_1\cap B}(|\boldsymbol{x}_1-\boldsymbol{y}|^{-N+1}+ \delta^{-N+1})\,d\boldsymbol{y} \\ &\leqslant \frac{A\omega^\mu(r)r^\mu}{r^{N-1}} \int_{B(\boldsymbol{0},b)}(|\boldsymbol{y}|^{-N+1}+ \delta^{-N+1})\,d\boldsymbol{y} \\ &\leqslant \frac{A_1\omega^\mu(r)r^\mu b}{r^{N-1}} \leqslant \frac{A_1\omega^\mu(r)b^\mu (b/r)^{1-\mu}}{r^{N-2}}\,, \end{aligned}
\end{equation*}
\notag
$$
which corresponds to (24). The estimate for $J_3$ is trivial and also corresponds to (24). To estimate $J_4$ note that for $\boldsymbol{y}\in D_4$ we have $|\varPhi_i(\boldsymbol{x}_1-\boldsymbol{y})- \varPhi_i(\boldsymbol{x}_2-\boldsymbol{y})|\leqslant A\delta\big/|\boldsymbol{y}-\boldsymbol{x}_0|^N$, and therefore
$$
\begin{equation*}
J_4\leqslant \frac{A\omega^\mu(r)r^\mu}{r^{N-1}} \int_{B(\boldsymbol{0},\delta+\varepsilon)\setminus B(\boldsymbol{0},\delta)} \delta|\boldsymbol{y}|^{-N}\,d\boldsymbol{y},
\end{equation*}
\notag
$$
where $(\delta+\varepsilon)^N-\delta^N=r^N$, so that $\varepsilon<r$. Here the case $\delta>r/2$ is trivial (yields immediately an estimate of the type of (24)). Now let $\delta\leqslant r/2$; then integrating in the spherical coordinates we obtain
$$
\begin{equation}
J_4\leqslant\frac{A\omega^\mu(r)r^\mu}{r^{N-1}}\,\delta\log\frac{r}{\delta} \leqslant \frac{A_1\omega^\mu(r)}{r^{N-2}}\,\delta^\mu \biggl(\frac{\delta}{r}\biggr)^{1-\mu}\log\frac{r}{\delta}\,,
\end{equation}
\tag{25}
$$
which also gives us the required result since for $\delta\leqslant r/2$ the expression $(\delta/r)^{1-\mu}\log(r/\delta)$ is easy to estimate by a quantity depending only on $\mu$.
It remains to estimate $\omega^\mu(h_4,\delta)$. We have
$$
\begin{equation*}
\begin{aligned} \, |h_4(\boldsymbol{x}_1)-h_4(\boldsymbol{x}_2)|&\leqslant \frac{1}{N|B|r}\int_{\partial B}|\varPhi(\boldsymbol{x}_1-\boldsymbol{y})- \varPhi(\boldsymbol{x}_2-\boldsymbol{y})|\, |f_k(\boldsymbol{y})y_i y_j|\,d\sigma(\boldsymbol{y}) \\ &\leqslant \frac{A\omega^\mu(r)r^\mu r^2}{r^{N+1}}\int_{\partial B}| \varPhi(\boldsymbol{x}_1-\boldsymbol{y})- \varPhi(\boldsymbol{x}_2-\boldsymbol{y})|\,d\sigma(\boldsymbol{y}). \end{aligned}
\end{equation*}
\notag
$$
We must estimate the last integral
$$
\begin{equation*}
J=\int_{\partial B}\bigl|\varPhi(\boldsymbol{x}_1-\boldsymbol{y})- \varPhi(\boldsymbol{x}_2-\boldsymbol{y})\bigr|\,d\sigma(\boldsymbol{y}).
\end{equation*}
\notag
$$
Let $|\boldsymbol{x}_1-\boldsymbol{x}_2|=\delta>0$, where $\boldsymbol{x}_2$ lies no farther than $\boldsymbol{x}_1$ from $\partial B$. We look at several cases.
Case 1: the point $\boldsymbol{x}_1$ does not lie on $\partial B$ and $|\boldsymbol{x}_1-\boldsymbol{x}_2|< \operatorname{dist}(\boldsymbol{x}_1,\partial B)/2$. Then
$$
\begin{equation*}
J\leqslant A\delta\int_{\partial B}\frac{d\sigma(\boldsymbol{y})} {|\boldsymbol{x}_1-\boldsymbol{y}|^{N-1}}\,.
\end{equation*}
\notag
$$
For $\delta>r/2$ we have $J\leqslant A\delta$, and this is sufficient for our aims (just as in (24)). For $\delta\leqslant r/2$ we have
$$
\begin{equation*}
J\leqslant A\delta\int_{|\boldsymbol{x}'|<r} \frac{d\boldsymbol{x}'}{(|\boldsymbol{x}'|^2+\delta^2)^{(N-1)/2}} \leqslant A\delta\log \frac{r}{\delta}\,,
\end{equation*}
\notag
$$
which is also sufficient (as in (25)).
Case 2: again, $\boldsymbol{x}_1$ is away from $\partial B$, but $|\boldsymbol{x}_1-\boldsymbol{x}_2|\geqslant \operatorname{dist}(\boldsymbol{x}_1,\partial B)/2$. Using simple reasoning we can reduce this case to the two cases below. Case 3: $\boldsymbol{x}_1$ is away from $\partial B$, and $\boldsymbol{x}_2\in\partial B$ is the nearest point to $\boldsymbol{x}_1$ on $\partial B$. Again, for $\delta>r/2$ we have $J \leqslant Ar$, which is sufficient. For $\delta\leqslant r/2$ we have
$$
\begin{equation*}
\begin{aligned} \, J &\leqslant A\int_{\partial B\cap B(\boldsymbol{x}_2,\delta)} \frac{d\sigma(\boldsymbol{y})}{|\boldsymbol{x}_2-\boldsymbol{y}|^{N-2}}+ A\delta\int_{\partial B\setminus B(\boldsymbol{x}_2,\delta)} \frac{d\sigma(\boldsymbol{y})}{|\boldsymbol{x}_2-\boldsymbol{y}|^{N-1}} \\ &\leqslant A\delta\biggl(1+\log\frac{r}{\delta}\biggr), \end{aligned}
\end{equation*}
\notag
$$
which is quite satisfactory, as before.
Case 4: both $\boldsymbol{x}_1$ and $\boldsymbol{x}_2$ lie on $\partial B$ and $\delta\leqslant 2r$. Then for $\delta>r/4$ we have The case when $\delta \leqslant r/4$ is considered similarly to the above estimate for $h_2$ (but in dimension lower by one). $\Box$Recall the concept of the $C^m$-$\Delta$-capacity of a bounded set $E\subset\mathbb R^N$ (where $N\geqslant3$ and $m=1+\mu\in(1,2)$):
$$
\begin{equation}
\alpha^m(E)=\alpha^m_{\Delta}(E)=\sup\{\vert\langle\Delta f,1\rangle\vert \colon f\in\mathcal F^m(E)\},
\end{equation}
\tag{26}
$$
where $\mathcal F^m(E)$ is the class of $f\in BC^m(\mathbb R^N)$ satisfying the following conditions: $\|\nabla f\|'_\mu\leqslant1$, $\operatorname{Supp}(\Delta f)\subset E$ (that is, $f$ is harmonic outside $E$), and $f=\varPhi*(\Delta f)$ (that is, $f$ is ‘regular’ at $\infty$). Recall the definition of the $p$-dimensional lower Hausdorff content, $p\in(0,N]$, of a bounded set $E$ in $\mathbb R^N$:
$$
\begin{equation*}
\mathcal M^p_*(E)=\sup_h\biggl\{\inf_{\{B_j\}}\sum_jh(r_j)\biggr\},
\end{equation*}
\notag
$$
where the infimum is taken over all (at most countable) covers $\{B_j\}$ of the set $E$ by balls $B_j$ of radii $r_j$, and the supremum is taken over all non-negative continuous functions $h$ on $(0,+\infty)$ such that $h(t)\leqslant t^p$ and $h(t)t^{-p}\to 0$ as $t\to 0$. As shown in [7], p. 178, for each $\sigma$-compact bounded set $E$ in $\mathbb R^N$ we have
$$
\begin{equation}
A_0^{-1}\mathcal M_*^{N+m-2}(E)\leqslant\alpha^m(E)\leqslant A_0\mathcal M_*^{N+m-2}(E),
\end{equation}
\tag{27}
$$
where $A_0>1$ depends only on $(m,N)$. In addition, it is easy to show that for $0<p<N$, for any bounded open set $E$ in $\mathbb R^N$ we have
$$
\begin{equation}
A^{-1}\mathcal M^p(E)\leqslant \mathcal M_*^p(E)\leqslant A\mathcal M^p(E),
\end{equation}
\tag{28}
$$
where $A>1$ is an absolute constant. Now we are ready for the proof of Theorem 2.3 in the case when $\mathcal L=\Delta$, $m\in(1,2)$, and $N\geqslant3$. We begin with the implication (a) $\Rightarrow$ (c). Let $f\in\mathcal A^m_{\Delta}(X)$, and let $\{f_n\}^{+\infty}_{n=1}\subset BC^m$ be a sequence of functions $f_n$ harmonic in (own) neighbourhoods of $X$ such that $\|f-f_n\|_m\to0$ as $n\to+\infty$. Fix $B=B(\boldsymbol{a},r)$ and $\varepsilon>0$. There exists $n_\varepsilon\in\mathbb N$ such that for all $n\geqslant n_\varepsilon$ we have $\|f-f_n\|_m<\varepsilon$, and therefore $\omega^\mu_{\overline{B}}\bigl((\nabla f- \nabla f_n),r\bigr)< \varepsilon$. Hence it is sufficient to establish the estimate
$$
\begin{equation*}
\begin{aligned} \, |\mathcal O^\Delta_B(f_n)|&=\biggl|\frac{1}{\sigma(\partial B)} \int_{\partial B}f_n(\boldsymbol{x})\,d\sigma(\boldsymbol{x})- \frac{1}{|B|}\int_Bf_n(\boldsymbol{x})\,d\boldsymbol{x}\biggr| \\ &\leqslant A\omega_n(r)r^{2-N}\mathcal M^{N-2+m} \bigl(B(\boldsymbol{a},r)\setminus X\bigr), \end{aligned}
\end{equation*}
\notag
$$
where $A=A(m,N)>0$ and $\omega_n(r)=\omega^\mu_{\overline{B}}(\nabla f_n,r)$. Let $h_n=V_\psi f_n$, where $\psi(\boldsymbol{x})=\psi_{\boldsymbol{a}}^r(\boldsymbol{x})$. By Lemma 4.4
$$
\begin{equation*}
h_n\in BC^m,\qquad \|\nabla h_n\|'_\mu\leqslant A\omega_n(r)r^{2-N},
\end{equation*}
\notag
$$
and $h_n$ is harmonic outside a compact subset of $E=\overline{B}\setminus X$. By the definition (26) and Lemma 4.1,
$$
\begin{equation*}
|\langle\Delta h_n,1\rangle|= |\langle\psi_{\boldsymbol{a}}^r\Delta f_n,1\rangle|= |\langle\Delta\psi_{\boldsymbol{a}}^r,f_n\rangle|= |\mathcal O^\Delta_B(f_n)|\leqslant A\omega_n(r)r^{2-N}\alpha^m(E).
\end{equation*}
\notag
$$
It remains to take (27), the equality $\mathcal M_*^{N+m-2}(E)=\mathcal M_*^{N+m-2}(B\setminus X)$, and (28) into account. Since the implication (c) $\Rightarrow$ (b) is obvious, it is sufficient to show that (b) $\Rightarrow$ (a). Thus, let $f\in C^m_0$ be a function satisfying (b). We can assume without loss of generality that $\omega^m_f(r)\leqslant \omega(r)$ (for all $r> 0$). Fix some $\delta>0$. For $\boldsymbol{j}=(j_1,\dots,j_N)\in\mathbb Z^N$ let $\boldsymbol{b}_{\boldsymbol{j}}=(\delta j_1/N,\dots,\delta j_N/N)$ and $B_{\boldsymbol{j}}=B(\boldsymbol{b}_{\boldsymbol{j}},\delta)$, so that the balls $\{B(\boldsymbol{b}_{\boldsymbol{j}},\delta/2)\}_{\boldsymbol{j}\in \mathbb Z^N}$ cover the whole of $\mathbb R^N$. We fix a function $\varphi\in C_0^\infty(B_{(0,\dots,0)})$ such that $0\leqslant\varphi(\boldsymbol{x})\leqslant1$, $\varphi(\boldsymbol{x})=1$ for $|\boldsymbol{x}|\leqslant\delta/2$, and the family
$$
\begin{equation*}
\biggl\{\varphi_{\boldsymbol{j}}(\boldsymbol{x})= \varphi(\boldsymbol{x}-\boldsymbol{b}_{\boldsymbol{j}}) \biggl(\,\sum_{\boldsymbol{m}\in\mathbb Z^N}\varphi(\boldsymbol{x}- \boldsymbol{b}_{\boldsymbol{m}})\biggr)^{-1} \colon \boldsymbol{j}\in\mathbb Z^N\biggr\}
\end{equation*}
\notag
$$
is a partition of unity on $\mathbb R^N$ satisfying the estimates
$$
\begin{equation}
\|\nabla\varphi_{\boldsymbol{j}}\|\leqslant \frac{A}{\delta}\,,\quad \|\nabla^2\varphi_{\boldsymbol{j}}\|\leqslant \frac{A}{\delta^2}\,,\quad \text{and}\quad \|\nabla^3\varphi_{\boldsymbol{j}}\|\leqslant \frac{A}{\delta^3}\,,\quad A=A(N)>1.
\end{equation}
\tag{29}
$$
Let $\Psi(\boldsymbol{x})= N(N+2)\psi^\delta_{\boldsymbol{0}}(\boldsymbol{x})\big/\delta^2$ (see Lemma 4.1); here the coefficient $N(N+2)/\delta^2$ ensures that
$$
\begin{equation*}
\int_{B(\boldsymbol{0},\delta)}\Psi(\boldsymbol{x})\,d\boldsymbol{x}=1.
\end{equation*}
\notag
$$
Consider the new partition of unity $\bigl\{\varphi^*_{\boldsymbol{j}}= \Psi*\varphi_{\boldsymbol{j}} \colon \boldsymbol{j}\in\mathbb Z^N\bigr\}$. It also satisfies (29) (where $\varphi_{\boldsymbol{j}}$ is replaced by $\varphi^*_{\boldsymbol{j}}$). Note that $\varphi^*_{\boldsymbol{j}}\in C_0^\infty(B^*_{\boldsymbol{j}})$, where $B^*_{\boldsymbol{j}}=B(\boldsymbol{b}_{\boldsymbol{j}},2\delta)$. Set $f^*_{\boldsymbol{j}}=\varPhi*(\varphi^*_{\boldsymbol{j}}\Delta f)$. Then $\Delta f^*_{\boldsymbol{j}}=\varphi^*_{\boldsymbol{j}}\Delta f$, that is, the function $f^*_{\boldsymbol{j}}$ ‘localizes’ the singularities of $f$ inside $B^*_{\boldsymbol{j}}$. The following property of the localizations $f^*_{\boldsymbol{j}}$ was established in [7], formula (20) on p 182 (which is a simplified version of Lemma 4.4 in our paper):
$$
\begin{equation}
f^*_{\boldsymbol{j}}\in BC^m\quad\text{and}\quad \|\nabla f^*_{\boldsymbol{j}}\|'_\mu\leqslant A\omega(\delta).
\end{equation}
\tag{30}
$$
Since (for fixed $\boldsymbol{x}$)
$$
\begin{equation*}
\begin{aligned} \, f^*_{\boldsymbol{j}}(\boldsymbol{x})&=\langle\varPhi(\boldsymbol{x}- \boldsymbol{y})\varphi^*_{\boldsymbol{j}}(\boldsymbol{y}), \Delta f(\boldsymbol{y})\rangle=\langle\Delta(\varPhi(\boldsymbol{y}- \boldsymbol{x})\varphi^*_{\boldsymbol{j}}(\boldsymbol{y})), f(\boldsymbol{y})\rangle \\ &=\varphi^*_{\boldsymbol{j}}(\boldsymbol{x})f(\boldsymbol{x})+ 2\int_B\nabla\varPhi(\boldsymbol{y}- \boldsymbol{x})\nabla\varphi^*_{\boldsymbol{j}}(\boldsymbol{y}) f(\boldsymbol{y})\,d\boldsymbol{y} \\ &\qquad+\int_B\varPhi(\boldsymbol{y}- \boldsymbol{x})\Delta\varphi^*_{\boldsymbol{j}}(\boldsymbol{y}) f(\boldsymbol{y})\,d\boldsymbol{y}, \end{aligned}
\end{equation*}
\notag
$$
for ${\boldsymbol x} \notin B^*_{\boldsymbol j}$ we have
$$
\begin{equation*}
\begin{aligned} \, f^*_{\boldsymbol{j}}(\boldsymbol{x})&=\int_B\varPhi(\boldsymbol{x}- \boldsymbol{b}_{\boldsymbol{j}})\Delta\varphi^*_{\boldsymbol{j}} (\boldsymbol{y})f(\boldsymbol{y})\,d\boldsymbol{y} \\ &\qquad+\int_B\bigl(\varPhi(\boldsymbol{x}-\boldsymbol{y})- \varPhi(\boldsymbol{x}-\boldsymbol{b}_{\boldsymbol{j}})\bigr) \Delta\varphi^*_{\boldsymbol{j}}(\boldsymbol{y}) f(\boldsymbol{y})\,d\boldsymbol{y} \\ &\qquad+2\int_B\nabla\varPhi(\boldsymbol{y}- \boldsymbol{x})\nabla\varphi^*_{\boldsymbol{j}}(\boldsymbol{y}) f(\boldsymbol{y})\,d\boldsymbol{y} \\ &=\lambda_{\boldsymbol{j}}\varPhi(\boldsymbol{x}- \boldsymbol{b}_{\boldsymbol{j}})+\Theta_{\boldsymbol{j}}(\boldsymbol{x}), \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\lambda_{\boldsymbol{j}}=\langle f,\Delta\varphi^*_{\boldsymbol{j}}\rangle= -\langle\nabla f,\nabla\varphi^*_{\boldsymbol{j}}\rangle= -\langle\nabla f,(\nabla \Psi)*\varphi_{\boldsymbol{j}}\rangle
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\Theta_{\boldsymbol{j}}(\boldsymbol{x})=f^*_{\boldsymbol{j}}(\boldsymbol{x})- \lambda_{\boldsymbol{j}}\varPhi(\boldsymbol{x}-\boldsymbol{b}_{\boldsymbol{j}})= O(|\boldsymbol{x}|^{-N+1}),\qquad |\boldsymbol{x}|\to+\infty.
\end{equation*}
\notag
$$
Now, by Fubini’s theorem
$$
\begin{equation*}
\begin{aligned} \, \lambda_{\boldsymbol{j}}&=-\int\biggl(\nabla f(\boldsymbol{x}) \int\nabla\Psi(\boldsymbol{x}-\boldsymbol{y})\varphi_{\boldsymbol{j}} (\boldsymbol{y})\,d\boldsymbol{y}\biggr)\,d\boldsymbol{x} \\ &=-\int\biggl(\varphi_{\boldsymbol{j}}(\boldsymbol{y}) \int\nabla f(\boldsymbol{x})\nabla\Psi(\boldsymbol{x}- \boldsymbol{y})\,d\boldsymbol{x}\biggr)\,d\boldsymbol{y} \\ &=\int\bigl\langle\Delta\Psi(\boldsymbol{x}-\boldsymbol{y}), f(\boldsymbol{x})\bigr\rangle\varphi_{\boldsymbol{j}} (\boldsymbol{y})\,d\boldsymbol{y}. \end{aligned}
\end{equation*}
\notag
$$
By Lemma 4.1 and condition (b) we have
$$
\begin{equation*}
\begin{aligned} \, |\langle\Delta\Psi(\boldsymbol{x}-\boldsymbol{y}),f(\boldsymbol{x})\rangle|&= N(N+2)\delta^{-2}|\mathcal O^\Delta_{B(\boldsymbol{y},\delta)}(f)| \\ &\leqslant A\omega(\delta)\delta^{-N}\mathcal M^{N-2+m}(B(\boldsymbol{y}, k\delta)\setminus X), \end{aligned}
\end{equation*}
\notag
$$
which implies that
$$
\begin{equation}
|\lambda_{\boldsymbol{j}}|\leqslant A\omega(\delta)\mathcal M^{N-2+m} (B(\boldsymbol{b}_{\boldsymbol{j}},(k+1)\delta)\setminus X).
\end{equation}
\tag{31}
$$
Let $B^k_{\boldsymbol{j}}=B(\boldsymbol{b}_{\boldsymbol{j}},(k+1)\delta)$ and
$$
\begin{equation*}
\boldsymbol{J}=\bigl\{\boldsymbol{j}\in\mathbb Z^N \colon B^k_{\boldsymbol{j}}\cap\operatorname{Supp}(f)\ne\varnothing\bigr\}.
\end{equation*}
\notag
$$
Clearly, $f=\displaystyle\sum_{\boldsymbol{j}\in\boldsymbol{J}}f^*_{\boldsymbol{j}}$, and if $B^*_{\boldsymbol{j}}\subset X$, then $f^*_{\boldsymbol{j}}=0$ and we stop considering such $\boldsymbol{j}$. By the definition (26), for each $\boldsymbol{j}\in\boldsymbol{J}$ there exists a function $h_{\boldsymbol{j}}\in\mathcal F_m(B^k_{\boldsymbol{j}}\setminus X)$ such that $\nu_{\boldsymbol{j}}=\langle\Delta h_{\boldsymbol{j}},1\rangle =2^{-1}\alpha^m(B^k_{\boldsymbol{j}}\setminus X)$. Let $g_{\boldsymbol{j}}= \lambda_{\boldsymbol{j}}\nu_{\boldsymbol{j}}^{-1}h_{\boldsymbol{j}}$. Then $g_{\boldsymbol{j}}\in BC^m$ is harmonic in a neighbourhood of $X$ and outside $B^k_{\boldsymbol{j}}$, while in view of (27), (28), and (31) we have
$$
\begin{equation*}
\langle\Delta g_{\boldsymbol{j}},1\rangle= \lambda_{\boldsymbol{j}}=\langle\Delta f^*_{\boldsymbol{j}},1\rangle\quad\text{and}\quad \|\nabla g_{\boldsymbol{j}}\|'_\mu\leqslant A\omega(\delta).
\end{equation*}
\notag
$$
In particular, $\nabla\big(f^*_{\boldsymbol{j}}(\boldsymbol{x})- g_{\boldsymbol{j}}(\boldsymbol{x})\big)=O(|\boldsymbol{x}|^{-N})$ as $|\boldsymbol{x}|\to+\infty$. Setting $g=\displaystyle\sum_{\boldsymbol{j}\in\boldsymbol{J}}g_{\boldsymbol{j}}$ and using Lemma 3.2 in [7] (also see (30)) we obtain $\|\nabla(f-g)\|'_\mu\leqslant A\omega(\delta)$. Finally, let $\operatorname{Supp}(f)\cup X\subset B(\boldsymbol{0},R)$, $R>0$, and fix $\varphi_0\in C_0^{\infty}(B(\boldsymbol{0},R+1))$ (independent of $\delta$) such that $\varphi_0=1$ on $B(\boldsymbol{0},R)$. Set
$$
\begin{equation*}
G(\boldsymbol{x})=\bigl(g(\boldsymbol{x})-g(\boldsymbol{0})+ f(\boldsymbol{0})+(\nabla f(\boldsymbol{0})- \nabla g(\boldsymbol{0}))\boldsymbol{x}\bigr)\varphi_0(\boldsymbol{x}).
\end{equation*}
\notag
$$
It is easy to show that $\|f-G\|_m\to0$ as $\delta \to 0$, as required. In this way we have finished the proof of Theorem 2.3 for $m\in(0,1)\cup(1,2)$. $\Box$ Observations and comments to the proof of Theorem 2.3Note that in Lemma 4.1 we can take a function $\psi_{\boldsymbol{a}}^r(\boldsymbol{x})$ of another form; for instance, $\psi_{\boldsymbol{a}}^r(\boldsymbol{x})= A(r^2-|\boldsymbol{x}-\boldsymbol{a}|^2)^2$ in $B=B(\boldsymbol{a},r)$ and $\psi_{\boldsymbol{a}}^r(\boldsymbol{x})=0$ outside $B(\boldsymbol{a},r)$ (where $A$ is a suitable positive constant). Then $\mathcal O^L_B(f)$ and conditions (b) and (c) in Theorem 2.3 assume a slightly different form, but all main assertions remain the same (for instance, see [11], Theorem 1). The above proof of Theorem 2.3 (where $N\geqslant3$, $m\in(1,2)$, and $\mathcal L_0=\Delta$) can almost literally be used for $N=2$, $m\in(1,2)$, and $\mathcal L_0=\partial^2/\partial\overline{z}^2$, but it does not work for $C^0$- or $C^1$-approximations. The main reason is as follows: singular integral convolution operators with non-trivial kernels $\varPsi$ and $\varPsi_0$ are not locally invariant (bounded) in the spaces $BC^0$ and $BC^1$, and the relevant capacities cannot be described in terms of Hausdorff contents. We consider these cases in detail in the following sections. Finally, the cases $m\in(0,1)$ and $\mathcal L=\mathcal L_0$ (dealt with in [9], Theorem 1, and [11], Theorem 1 (b)) also need a special discussion. The general scheme of reasoning here uses Vitusjkin’s constructive scheme and is the same as for $m=0$ (see § 7 below), but one difference is fundamental: the fact that singular integral operators with Calderón–Zygmund kernels are bounded in Lipschitz spaces with exponent $m\in(0,1)$ simplifies the proof significantly. Just as for $m=0$, Theorem 2.3 reduces to a main lemma similar to Lemma 7.2 (for instance, in [9] this is Lemma 1.1). Because the integral operators mentioned above are bounded for $m\in(0,1)$, the analogue of Theorem 2 in [5] is an easy result (for instance, see the derivation of Lemma 1.1 from Lemma 2.7 in [9], § 2). Lemma 2.7 in [9] states that, based exclusively on an analogue of (86), we can equate the Laurent coefficients at all partial derivatives of the first order of the fundamental solution in the expansions of all localizations of the original function (see [9], formula (1.5)). The construction leading to the proof of Lemma 2.7 in [9] was presented in [9], § 3. It is quite interesting and uses the techniques of stopping times for dyadic cubes. However, as the integral operators in question are bounded, this construction is much simpler in many details than for $m=0$; in particular, the definition of a compatible pair of cubes takes one line (see [9], formula (3.2)), and by contrast with $m=0$ no special Lipschitz graphs are required. Note that in [25], Theorem 1, for $m\in(0,1)\cup(1,2)$ the authors obtained an approximation criterion for classes of functions, which is Corollary 8.1 to Theorem 2.3, and their proof used dual arguments in Lizorkin–Triebel spaces. Unlike [25], the proof of Theorem 2.3 that we discuss here is entirely effective for $m\in(0,1)\cup(1,2)$. 5. $C^1$-$\mathcal L$-approximations. Description of $\gamma^1_{\mathcal L}$-capacitiesWe present a detailed plan of the proof in [16] of Theorem 2.3 for $m=1$ and $N\geqslant 3$ (the simpler case $N=2$ can be extracted from [14]). We defer the description of $\gamma^1_{\mathcal L}$- and $\alpha^1_{\mathcal L}$-capacities due to Tolsa [27] to the end of this section. Throughout the section we fix $N\in\{3,4,\dots\}$ and an arbitrary operator $\mathcal L$ in $\mathbb R^N$. The positive constants with indices $A_j$ (whose existence is an essential part of the corresponding statements) are fixed throughout, while the constant $A>1$ can assume different values in different expressions. For $\varphi\in C_0^\infty(\mathbb R^N)$ we define the localization operator of Vitushkin type [8], [7] corresponding to the operator $\mathcal L$:
$$
\begin{equation}
f\mapsto V_\varphi(f)=\varPhi*(\varphi\mathcal L f),\qquad f\in C(\mathbb R^N).
\end{equation}
\tag{32}
$$
We need an additional property of this operator, related to its possible extension to a wider class of ‘subscripts’ $\varphi$ for $f\in C^1$. Lemma 5.1. Fix an arbitrary function $\varphi\in C^1(\overline{B(\boldsymbol{a},r)})$ that can be (and is) extended continuously as $\varphi=0$ outside the ball $B(\boldsymbol{a},r)$. Then the following holds for each function $f\in C^1_0(\mathbb R^N)$: A full proof in the two-dimensional case was presented in [14], Lemma 2.2. It can be repeated word for word in an arbitrary dimension (also see [28], Lemma 3.4). Since the fundamental solution $\varPhi=\varPhi_{\mathcal L}$ is real analytic in $\mathbb R^N\setminus\{\boldsymbol{0}\}$ and homogeneous of degree $2-N$, we have
$$
\begin{equation}
|\partial^\beta\varPhi(\boldsymbol{x})|\leqslant |A_1|^{|\beta|}\beta!\,|\boldsymbol{x}-\boldsymbol{a}|^{2-N-|\beta|},
\end{equation}
\tag{33}
$$
where, given an $N$-index $\beta=(\beta_1,\dots,\beta_N)\in \mathbb Z_+^N$, we set, as usual, $\beta!=\beta_1!\cdots\beta_N!$ and $\boldsymbol{x}^\beta=x_1^{\beta_1}\cdots x_N^{\beta_N}$. We need the following simplified version of the result on the expansion of $\varPhi$-potentials in Laurent-type series (for instance, see [29]). Lemma 5.2. Set $A_2=2A_1+1$. Let $T$ be a distribution with compact support in a ball $B(\boldsymbol{a},r)$, and let $g=\varPhi_{\mathcal L}*T \in C^1(\mathbb R^N)$. Then the following expansion holds in the set $\{\boldsymbol{x}\in\mathbb R^N \colon|\boldsymbol{x}-\boldsymbol{a}|>A_2r\}$: The proof follows from the above properties of the functions $\varPhi=\varPhi_{\mathcal L}$ and from estimates (36) in the lemma below (for $E=B(\boldsymbol{a},r)$). Note that in Lemma 5.2 the coefficient $c_0(g)=c_{(0,\dots,0)}(g,\boldsymbol{a})$ is independent of the point $\boldsymbol{a}$, and if $c_0(g)=0$, then the coefficients $c_{\beta}(g,\boldsymbol{a})$ are independent of $\boldsymbol{a}$ for $|\beta|=1$. For the class of functions $\mathcal I$ and $\tau\geqslant 0$ we let $\tau\mathcal I$ denote the class $\{\tau g \colon g\in\mathcal I\}$. We define the $C^1$-$\mathcal L$-capacity of a bounded set $E$ in $\mathbb R^N$ by
$$
\begin{equation}
\alpha^1(E)=\alpha^1_{\mathcal L}(E)=\sup\bigl\{|\langle\mathcal Lg,1\rangle| \colon g\subset\mathcal I_1(E)\bigr\},
\end{equation}
\tag{35}
$$
where
$$
\begin{equation*}
\mathcal I_1(E)=\bigl\{\varPhi_{\mathcal L}*T \colon \operatorname{Supp}(T)\subset E,\ \varPhi_{\mathcal L}*T\in BC^1,\ \|\nabla\varPhi*T\|\leqslant 1\bigr\}.
\end{equation*}
\notag
$$
Clearly, $\alpha^1(\,\cdot\,)$ is a monotone set function invariant under shifts and homogeneous of degree $N-1$ (with respect to dilations with positive coefficients). In particular,
$$
\begin{equation*}
\alpha^1(B(\boldsymbol{a},r))=A(N,\mathcal L)r^{N-1}.
\end{equation*}
\notag
$$
It is also easy to show that for bounded open sets $E$ in $\mathbb R^N$ we have $\alpha^1_{\mathcal L}(E)=\gamma^1_{\mathcal L}(E)$. We discuss other metric properties of these capacities at the end of the section: we do not need them in the proofs that follow. The proof of the next lemma is standard (for instance, see the proofs of Lemma 3.3 and Corollary 3.4 in [30]). Lemma 5.3. Let $E\subset B(\boldsymbol{a},r)$ and $g\in\mathcal I_1(E)$. Then Below, using the notation of Lemma 5.2, we set
$$
\begin{equation}
\boldsymbol{c}^1(g,\boldsymbol{a})=(c_{(1,0,\dots,0)}(g,\boldsymbol{a}), \dots,c_{(0,\dots,0,1)}(g,\boldsymbol{a}))= (c^1_1(g,\boldsymbol{a}),\dots,c^1_N(g,\boldsymbol{a})).
\end{equation}
\tag{40}
$$
In this section we prove some generalization of Theorem 2.3. Fix some odd function $\varphi_1$ in $C(\mathbb R^N)\cap C^1(\overline{B(\boldsymbol{0},1)})$ such that $\operatorname{Supp}(\varphi_1)\subset\overline{B(\boldsymbol{0},1)}$ and $\displaystyle\int\varphi_1(\boldsymbol{x})\,d\boldsymbol{x}=1$. Let $\varphi_r^{\boldsymbol{a}}(\boldsymbol{x})= \varphi_1((\boldsymbol{x}-\boldsymbol{a})/r)/r^N$ and $\varphi_r=\varphi_r^{\boldsymbol{0}}$. Then $\|\nabla\varphi_r^{\boldsymbol{a}}\|=r^{-N-1}\|\nabla\varphi_1\|$. Similarly to Theorem 2.2 in [31], Theorem 2.3 here is a special case of the following result. Theorem 5.1. For a compact set $X$ in $\mathbb R^N$ and $f\in C^1_0(\mathbb R^N)$ the following conditions are equivalent: It is easy to see that for $\varphi_1=N(N+2)\psi_{\boldsymbol{0}}^1$ (see Lemma 4.1) this is the same as Theorem 2.3, while for $\mathcal L=\Delta$ inequality (41) takes the following form:
$$
\begin{equation*}
\biggl|\int_{B(\boldsymbol{a},r)}\nabla f(\boldsymbol{x})(\boldsymbol{x}- \boldsymbol{a})\,d\boldsymbol{x}\biggr|\leqslant \omega(r)r^2\alpha^1_{\Delta}\bigl(B(\boldsymbol{a},kr)\setminus X\bigr),
\end{equation*}
\notag
$$
where the integral on the left is (by Fubini’s theorem) the average flow of the vector field $\nabla f$ through the spheres $\partial B(\boldsymbol{a},\rho)$ for $\rho\in(0,r)$. Proof of the implication (a) $\Rightarrow$ (c) in Theorem 5.1. Let $f$ belong to $\mathcal A^1_{\mathcal L}(X)$, and assume that there exists a sequence $\{f_m\}_{m=1}^{+\infty}\subset BC^1$ such that each function $f_m$ is $\mathcal L$-analytic in an (own) neighbourhood $U_m$ of $X$ and $\|f-f_m\|_1\to0$ as $m\to+\infty$. In view of regularization considerations we can also assume that each function $f_m$ belongs to $C^\infty(\mathbb R^N)$. Fix some $B=B(\boldsymbol{a},r)$ and $\varepsilon\in(0,r/2)$. Then there exists $m_\varepsilon\in \mathbb N$ such that for all $m\geqslant m_\varepsilon$ we have $\|f-f_m\|_1<\varepsilon$, so that $\omega(\nabla f- \nabla f_m,r)< 2\varepsilon$. Hence it is sufficient to show that
$$
\begin{equation*}
\biggl|\,\sum_{i,j=1}^Nc_{ij}\int_{B(\boldsymbol{a},r)} \frac{\partial}{\partial x_i}f_m(\boldsymbol{x}) \frac{\partial}{\partial x_j}\varphi_r^{\boldsymbol{a}} (\boldsymbol{x})\,d\boldsymbol{x}\biggr|\leqslant A\omega_m(r)r^{-N}\alpha^1\bigl(B(\boldsymbol{a},r)\setminus X\bigr)
\end{equation*}
\notag
$$
for $A=A(\mathcal L,N,\|\nabla\varphi_1\|)$ and $\omega_m(r)=\omega(\nabla f_m,r)$, and then let $\varepsilon$ tend to zero.
Let $h_m=V_\varphi f_m$, where $\varphi(\boldsymbol{x})= \varphi^{\boldsymbol{a}}_{r-\varepsilon}(\boldsymbol{x})$. By Lemma 5.1
$$
\begin{equation*}
h_m\in BC^1,\qquad \|\nabla h_m\|\leqslant A_0\omega(\nabla f_m,r)r\|\nabla\varphi\|,
\end{equation*}
\notag
$$
and $h_m$ is $\mathcal L$-analytic outside a compact subset $E$ of $ B\setminus X$. From the definition (35) and Lemma 5.1, integrating by parts we obtain
$$
\begin{equation*}
\begin{aligned} \, |\langle\mathcal Lh_m,1\rangle|&=|\langle\varphi,\mathcal Lf_m\rangle|= \biggl|\,\sum_{i,j=1}^Nc_{ij}\int_{B(\boldsymbol{a},r)} \frac{\partial}{\partial x_i}f_m(\boldsymbol{x}) \frac{\partial}{\partial x_j}\varphi(\boldsymbol{x})\,d\boldsymbol{x}\biggr| \\ &\leqslant A_0\omega(\nabla f_m,r)r\|\nabla\varphi\|\alpha^1 \bigl(B(\boldsymbol{a},r)\setminus X\bigr), \end{aligned}
\end{equation*}
\notag
$$
which completes the proof of the implication (a) $\Rightarrow$ (c) because Since the implication (c) $\Rightarrow$ (b) is obvious, we turn to the proof of the central part of Theorem 5.1. Proof of the implication (b) $\Rightarrow$ (a) in Theorem 5.1. We choose $R>0$ so that $X\subset B(\boldsymbol{0},R)$ and $f(\boldsymbol{x})=0$ for $|\boldsymbol{x}|>R$. In condition (41) we always assume that $\omega(\delta)\geqslant\omega(\nabla f,\delta)$.
We fix $\delta>0$ (but let $\delta$ tend to zero at the end of the proof) and some standard $\delta$-partition of unity $\{(\varphi_{\boldsymbol{j}},B_{\boldsymbol{j}}) \colon \boldsymbol{j}=(j_1,\dots,j_N)\in\mathbb Z^N\}$ in $\mathbb R^N$. More precisely, $B_{\boldsymbol{j}}=B(\boldsymbol{a}_{\boldsymbol{j}},\delta)$, where $\boldsymbol{a}_{\boldsymbol{j}}=(j_1\delta/N,\dots,j_N\delta/N)\in \mathbb R^N$, and we have
$$
\begin{equation*}
\varphi_{\boldsymbol{j}}\in C_0^\infty(B_{\boldsymbol{j}}),\qquad 0\leqslant\varphi_{\boldsymbol{j}}(\boldsymbol{x})\leqslant1,\qquad \|\nabla\varphi_{\boldsymbol{j}}\|\leqslant \frac{A_6}{\delta}\,,\quad\text{and}\quad \sum_{\boldsymbol{j}\in\mathbb Z^N}\varphi_{\boldsymbol{j}}\equiv1.
\end{equation*}
\notag
$$
Consider another partition of unity $\{(\psi_{\boldsymbol{j}}, B'_{\boldsymbol{j}})\}$, where $\psi_{\boldsymbol{j}}= \varphi_\delta*\varphi_\delta*\varphi_{\boldsymbol{j}}$, $B'_{\boldsymbol{j}}=B(\boldsymbol{a}_{\boldsymbol{j}},3\delta)$ (recall that $\varphi_\delta=\varphi_\delta^{\boldsymbol{0}}$). It is clear that $\psi_{\boldsymbol{j}}\in C_0^\infty(B'_{\boldsymbol{j}})$ and $\|\nabla\psi_{\boldsymbol{j}}\|\leqslant A/\delta$. Now we define the so-called localized functions $f_{\boldsymbol{j}}=\varPhi_{\mathcal L}*(\psi_{\boldsymbol{j}}\mathcal Lf)$. Lemma 5.4. The functions $f_{\boldsymbol{j}}$ have the following properties. (1) $f_{\boldsymbol{j}}\in A\omega(\nabla f,\delta) \mathcal I_1(B'_{\boldsymbol{j}}\setminus X^0)$. (2) $f=\displaystyle\sum_{\boldsymbol{j}}f_{\boldsymbol{j}}$ and this sum is finite ($f_{\boldsymbol{j}}=0$ for $B'_{\boldsymbol{j}}\cap B(\boldsymbol{0},R)=\varnothing$). (3) For $|\boldsymbol{x}-\boldsymbol{a}_{\boldsymbol{j}}|>3A_2\delta$ the following expansion holds:
$$
\begin{equation*}
f_{\boldsymbol{j}}(\boldsymbol{x})= \sum_{|\beta|\geqslant0}c_{\beta\boldsymbol{j}}\partial^\beta \varPhi(\boldsymbol{x}-\boldsymbol{a}_{\boldsymbol{j}}),
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\begin{aligned} \, \nonumber c_{\beta\boldsymbol{j}}&=(-1)^{|\beta|}(\beta!)^{-1} \langle\psi_{\boldsymbol{j}}(\boldsymbol{y})\mathcal Lf(\boldsymbol{y}), (\boldsymbol{y}-\boldsymbol{a}_{\boldsymbol{j}})^\beta\rangle \\ &=(-1)^{|\beta|}(\beta !)^{-1}\int f(\boldsymbol{y}) \mathcal L\bigl(\psi_{\boldsymbol{j}}(\boldsymbol{y})(\boldsymbol{y}- \boldsymbol{a}_{\boldsymbol{j}})^\beta\bigr)\,d\boldsymbol{y}. \end{aligned}
\end{equation}
\tag{42}
$$
Let $G_{\boldsymbol{j}}=B(\boldsymbol{a}_{\boldsymbol{j}},(k+2)\delta) \setminus X$. Then the estimates
$$
\begin{equation}
|c_{0\boldsymbol{j}}| \leqslant A\omega(\delta)\alpha^1(G_{\boldsymbol{j}})
\end{equation}
\tag{43}
$$
and
$$
\begin{equation}
|\boldsymbol{c}^1_{\boldsymbol{j}}| \leqslant A\omega(\delta)\delta\alpha^1(G_{\boldsymbol{j}})
\end{equation}
\tag{44}
$$
holds where the shorthand notation
$$
\begin{equation*}
c_{0\boldsymbol{j}}=c_0(f_{\boldsymbol{j}}) \quad\textit{and}\quad \boldsymbol{c}^1_{\boldsymbol{j}}=\boldsymbol{c}^1(f_{\boldsymbol{j}}, \boldsymbol{a}_{\boldsymbol{j}})=(c^1_{1\boldsymbol{j}},\dots,c^1_{N\boldsymbol{j}})
\end{equation*}
\notag
$$
is used (see (40)). Proof. Note that the last two estimates follow from (41) rather than (36). We adhere to the proof of Lemma 2.5 in [31].
Formula (42) follows from Lemmas 5.2 and 5.1, the definition of $f_{\boldsymbol{j}}$, and integration by parts. We prove (43). Let $\varphi_{\boldsymbol{j}}^*=\varphi_\delta*\varphi_{\boldsymbol{j}}$. Then
$$
\begin{equation*}
\varphi_{\boldsymbol{j}}^*\in C_0^\infty(B(\boldsymbol{a}_{\boldsymbol{j}},2\delta)),\qquad 0\leqslant\varphi_{\boldsymbol{j}}^*\leqslant1 \quad\text{and}\quad \psi_{\boldsymbol{j}}=\varphi_\delta*\varphi_{\boldsymbol{j}}^*.
\end{equation*}
\notag
$$
From (42), Fubini’s theorem, and inequality (41) we obtain
$$
\begin{equation*}
\begin{aligned} \, |c_{0\boldsymbol{j}}|&=\biggl|\int f(\boldsymbol{x})\mathcal L(\varphi_\delta* \varphi_{\boldsymbol{j}}^*(\boldsymbol{x}))\,d\boldsymbol{x}\biggr|= \biggl|\,\sum_{i,j=1}^Nc_{ij}\int\frac{\partial}{\partial x_i} f(\boldsymbol{x})\frac{\partial}{\partial x_j}(\varphi_\delta* \varphi_{\boldsymbol{j}}^*(\boldsymbol{x}))\,d\boldsymbol{x}\biggr| \\ &=\biggl|\int_{B(\boldsymbol{a}_{\boldsymbol{j}},2\delta)} \biggl(\,\sum_{i,j=1}^Nc_{ij}\int\frac{\partial}{\partial x_i} f(\boldsymbol{x})\frac{\partial}{\partial x_j}\varphi_\delta (\boldsymbol{x}-\boldsymbol{y})\,d\boldsymbol{x}\biggr) \varphi_{\boldsymbol{j}}^*(\boldsymbol{y})\,d\boldsymbol{y}\biggr| \\ &\leqslant A\biggl|\int_{B(\boldsymbol{a}_{\boldsymbol{j}},2\delta)} \varphi_{\boldsymbol{j}}^*(\boldsymbol{y})\omega(\delta)\delta^{-N} \alpha^1(B(\boldsymbol{y},k\delta)\setminus X)\,d\boldsymbol{y}\biggr| \leqslant A\omega(\delta)\alpha^1(G_{\boldsymbol{j}}). \end{aligned}
\end{equation*}
\notag
$$
To establish (44) note that, as follows from (42),
$$
\begin{equation*}
c^1_{n\boldsymbol{j}}=-\int f(\boldsymbol{y})\mathcal L \bigl(\psi_{\boldsymbol{j}}(\boldsymbol{y}) (y_n-a_{n\boldsymbol{j}})\bigr)\,d\boldsymbol{y},\qquad a_{n\boldsymbol{j}}=\frac{j_n\delta}{N}\,,\quad n\in\{1,\dots,N\}.
\end{equation*}
\notag
$$
Then we show that the functions $\psi_{\boldsymbol{j}}(\boldsymbol{y})(y_n-a_{n\boldsymbol{j}})$ have the form $\varphi_\delta*\chi_{n\boldsymbol{j}}$, where $\chi_{n\boldsymbol{j}}\in C_0^\infty(B(\boldsymbol{a}_{\boldsymbol{j}},2\delta))$ and $\|\chi_{n\boldsymbol{j}}\|\leqslant A\delta$. This is similar to the proof of Lemma 2.5 in [31] or to [11], Lemma 3.4, and uses the Fourier transform. Next we proceed as in the proof of (43). $\Box$ Now we describe the scheme for the approximation of $f=\displaystyle\sum f_{\boldsymbol{j}}$ by analogy with [30], [11], § 6, and [14]. Set $\boldsymbol{J}=\{\boldsymbol{j}\in\mathbb Z^N \colon B'_{\boldsymbol{j}}\cap\partial X\ne\varnothing\}$. If $\boldsymbol{j}\not\in \boldsymbol{J}$, then $f_{\boldsymbol{j}}\in\mathcal A^1_{\mathcal L}(X)$ by property (1) in Lemma 5.4, so we need not approximate these $f_{\boldsymbol{j}}$. Now let $\boldsymbol{j}\in\boldsymbol{J}$. By the definition of $\alpha^1(G_{\boldsymbol{j}})$ (recall that $G_{\boldsymbol{j}}=B(\boldsymbol{a}_{\boldsymbol{j}},(k+2)\delta) \setminus X$) and in view of (43) there exist functions $f_{\boldsymbol{j}}^*\in A\omega(\delta){\mathcal I}_1(G_{\boldsymbol{j}}) \subset\mathcal A^1_{\mathcal L}(X)$ such that $c_0(f_{\boldsymbol{j}}^*)=c_0(f_{\boldsymbol{j}})$. Let $g_{\boldsymbol{j}}=f_{\boldsymbol{j}}-f_{\boldsymbol{j}}^*$ (and let $f_{\boldsymbol{j}}^*= f_{\boldsymbol{j}}$ and $g_{\boldsymbol{j}}\equiv0$ for $\boldsymbol{j}\not\in\boldsymbol{J}$). Then
$$
\begin{equation}
\|\nabla g_{\boldsymbol{j}}\|\leqslant A\omega(\delta),\qquad c_0(g_{\boldsymbol{j}})=0.
\end{equation}
\tag{45}
$$
From (36) (where $|\beta|=1$) for $E=G_{\boldsymbol{j}}$ and $g=f_{\boldsymbol{j}}^*$ and from (44) we obtain (recall that $\boldsymbol{c}^1(g_{\boldsymbol{j}},\boldsymbol{a})= \boldsymbol{c}^1(g_{\boldsymbol{j}})$ is independent of $\boldsymbol{a}$)
$$
\begin{equation}
|\boldsymbol{c}^1(g_{\boldsymbol{j}})|\leqslant A\omega(\delta)\delta\alpha^1(G_{\boldsymbol{j}}),\qquad \boldsymbol{j}\in\boldsymbol{J}.
\end{equation}
\tag{46}
$$
Let $p=A_2(k+2)$. Using (38) and (39) for the function $g=g_{\boldsymbol{j}}$ and the set $E=B(\boldsymbol{a}_{\boldsymbol{j}},(k+2)\delta)=B^*_{\boldsymbol{j}}$, for $|\boldsymbol{x}-\boldsymbol{a}_{\boldsymbol{j}}|>p\delta$ we obtain
$$
\begin{equation}
|\nabla g_{\boldsymbol{j}}(\boldsymbol{x})|\leqslant \frac{A\omega(\delta)\delta\alpha^1(G_{\boldsymbol{j}})}{|\boldsymbol{x}- \boldsymbol{a}_{\boldsymbol{j}}|^N}+\frac{A\omega(\delta)\delta^{N+1}} {|\boldsymbol{x}-\boldsymbol{a}_{\boldsymbol{j}}|^{N+1}}\,.
\end{equation}
\tag{47}
$$
We introduce some shorthand notation. Recall that $\delta>0$ is fixed and sufficiently small. For $\boldsymbol{j}\in\boldsymbol{J}$ set $\alpha_{\boldsymbol{j}}=\alpha^1(G_{\boldsymbol{j}})$, so that all these $\alpha_{\boldsymbol{j}}$ are positive. For $\boldsymbol{I}\subset\boldsymbol{J}$ and $\boldsymbol{x}\in\mathbb R^N$ set
$$
\begin{equation*}
\begin{gathered} \, B^*_{\boldsymbol{I}}=\bigcup_{\boldsymbol{j}\in \boldsymbol{I}}B^*_{\boldsymbol{j}},\qquad G_{\boldsymbol{I}}=\bigcup_{\boldsymbol{j}\in \boldsymbol{I}}G_{\boldsymbol{j}},\qquad \alpha_{\boldsymbol{I}}=\sum_{\boldsymbol{j}\in \boldsymbol{I}}\alpha_{\boldsymbol{j}},\qquad g_{\boldsymbol{I}}=\sum_{\boldsymbol{j}\in \boldsymbol{I}}g_{\boldsymbol{j}}, \\ \boldsymbol{I}'(\boldsymbol{x})=\bigl\{\boldsymbol{j}\in\boldsymbol{I}\colon |\boldsymbol{x}-\boldsymbol{a}_{\boldsymbol{j}}|>p\delta\bigr\}, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
S'_{\boldsymbol{I}}(\boldsymbol{x})=\sum_{\boldsymbol{j}\in \boldsymbol{I}'(\boldsymbol{x})} \biggl(\frac{\delta\alpha_{\boldsymbol{j}}}{|\boldsymbol{x}- \boldsymbol{a}_{\boldsymbol{j}}|^N}+\frac{\delta^{N+1}}{|\boldsymbol{x}- \boldsymbol{a}_{\boldsymbol{j}}|^{N+1}}\biggr).
\end{equation}
\tag{48}
$$
Also set $S_{\boldsymbol{I}}(\boldsymbol{x})=S'_{\boldsymbol{I}}(\boldsymbol{x})$ for $\boldsymbol{I}=\boldsymbol{I}'(\boldsymbol{x})$ and $S_{\boldsymbol{I}}(\boldsymbol{x})=S'_{\boldsymbol{I}}(\boldsymbol{x})+1$ for $\boldsymbol{I}\ne\boldsymbol{I}'(\boldsymbol{x})$. For $\boldsymbol{I}\subset\boldsymbol{J}$, $\boldsymbol{i}=(i_1,\dots,i_N)\in \boldsymbol{I}$, and $n\in\{1,\dots,N\}$ let
$$
\begin{equation*}
P_n(\boldsymbol{I},\boldsymbol{i})=\{\boldsymbol{j}\in\boldsymbol{I} \colon j_m=i_m,\ m\in\{1,\dots,N\},\ m\ne n\},
\end{equation*}
\notag
$$
and let $P_n(\boldsymbol{J},\boldsymbol{i})=P_n(\boldsymbol{i})$. Definition 5.1. Let $n\in\{1,\dots,N\}$, $\boldsymbol{I}\subset\boldsymbol{J}$, and $\boldsymbol{i}\in\boldsymbol{I}$. Then we call a subset $\boldsymbol{L}_n=\boldsymbol{L}_n(\boldsymbol{i})$ of $\boldsymbol{I}$ a complete $n$-chain in $\boldsymbol{I}$ with vertex $\boldsymbol{i}$ if the following conditions are satisfied: Definition 5.2. Let $\boldsymbol{i}\in\boldsymbol{I}\subset\boldsymbol{J}$. Then $\varGamma\subset\boldsymbol{I}$ is called a complete group in $\boldsymbol{I}$ with vertex $\boldsymbol{i}$ if there exist complete $n$-chains $\boldsymbol{L}_n$ in $\boldsymbol{I}$, $n\in\{1,\dots,N\}$, with vertex $\boldsymbol{i}$ such that $\varGamma=\displaystyle\bigcup_{n=1}^N\boldsymbol{L}_n$. We partition the index set $\boldsymbol{J}$ into finitely many disjoint groups $\varGamma^s$, $s\in\{1,\dots,S\}$, defined recursively as follows. We introduce the natural lexigographic ordering in $\boldsymbol{J}$: for $\boldsymbol{j}\ne\boldsymbol{j}'$ in $\boldsymbol{J}$ we set $\boldsymbol{j}<\boldsymbol{j}'$ if there exists $n\in\{1,\dots,N\}$ such that $j_i=j'_i$ for all $i<n$, but $j_n<j'_n$. Consider the minimal $\boldsymbol{i}^1$ in $\boldsymbol{J}$. If there exists a complete group $\varGamma=\displaystyle\bigcup_{n=1}^N\boldsymbol{L}_n$ in $\boldsymbol{J}$ with vertex $\boldsymbol{i}^1$, then we set $\varGamma^1=\varGamma$. If there is no such $\varGamma$, then we set $\varGamma^1=P_n(\boldsymbol{J},\boldsymbol{i}^1)$, where $n\in\{1,\dots,N\}$ is the least index such that $\boldsymbol{L}_n$ is not defined. In this case we say that $\varGamma^1$ is an incomplete $n$-group. Once we have constructed $\varGamma^1,\dots,\varGamma^s$, we consider $\boldsymbol{J}^{s+1}=\boldsymbol{J}\setminus\bigl(\varGamma^1\cup\dots\cup \varGamma^s\bigr)$ and repeat the above construction for $\boldsymbol{J}^{s+1}$ in place of $\boldsymbol{J}$ by defining $\varGamma^{s+1}$ (which can be complete at step $s+1$ or incomplete). Let $S$ be the largest superscript such that $\boldsymbol{J}^S\ne\varnothing$. We fix the partition $\{\varGamma^s\}=\{\varGamma^s\}_{s=1}^S$ of $\boldsymbol{J}$. For each (complete or incomplete) group $\varGamma=\varGamma^s$, from (45)–(47) we obtain
$$
\begin{equation}
\begin{gathered} \, \alpha_\varGamma\leqslant A\delta^{N-1},\qquad c_0(g_\varGamma)=0,\qquad |\boldsymbol{c}^1(g_\varGamma)|\leqslant A\omega(\delta)\delta^N; \nonumber \\ |\nabla g_\varGamma(\boldsymbol{x})|\leqslant A\omega(\delta)S_{\varGamma}(\boldsymbol{x}),\qquad \|\nabla g_{\varGamma}\|\leqslant A\omega(\delta),\quad\text{and}\quad \|S_{\varGamma}\|\leqslant A. \end{gathered}
\end{equation}
\tag{49}
$$
The proof of the last estimates is quite simple; for instance, see [30], Lemmas 5.6 and 5.7. Lemma 5.5. For each complete group $\varGamma=\varGamma^s$ there exists a function $h_\varGamma\in A\omega(\delta)\mathcal I_1(G_\varGamma)\subset \mathcal A^1_{\mathcal L}(X)$ such that and for all $\boldsymbol{x}\in\mathbb R^N$ Proof. We follow mainly the proof of Lemma 2.7 in [31]. Let $\varGamma$ be a complete group in $\boldsymbol{J}$ with vertex $\boldsymbol{i}$ and complete $n$-chains $\boldsymbol{L}_n$, $n\in\{1,\dots,N\}$, and let $\boldsymbol{L}_n=\boldsymbol{L}_n^1\cup\boldsymbol{L}_n^2\cup \boldsymbol{L}_n^3$ (as in Definition 5.1). We begin by constructing functions $h_{\boldsymbol{L}_n}(\boldsymbol{x})\in A\mathcal I_1(G_{\boldsymbol{L}_n})$ such that
$$
\begin{equation}
c_0(h_{\boldsymbol{L}_n})=0\quad\text{and}\quad \boldsymbol{c}^1(h_{\boldsymbol{L}_n})= \delta^N(\boldsymbol{e}_n+\boldsymbol{u}_n),
\end{equation}
\tag{51}
$$
where $\boldsymbol{e}_n$ is the unit vector on the $Ox_n$-axis, $\boldsymbol{u}_n$ is a vector in $\mathbb C^N$ such that $|\boldsymbol{u}_n|<\varepsilon_N$, and
$$
\begin{equation}
|\nabla h_{\boldsymbol{L}_n}(\boldsymbol{x})|\leqslant AS_{\boldsymbol{L}_n}(\boldsymbol{x}).
\end{equation}
\tag{52}
$$
Here we choose $\varepsilon_N$ to be sufficiently small so that in looking for a suitable linear combination (with complex coefficients) of the functions $\{h_{\boldsymbol{L}_n}\}_{n=1}^N$ that yields the required function $h_\varGamma$ (see properties (49) and equalities (50)) we obtain a well-conditioned system of linear equations.
We construct the function $h_{\boldsymbol{L}_1}$; the other $\{h_{\boldsymbol{L}_n}\}$ can be constructed in a similar way. For each $\boldsymbol{j}\in\boldsymbol{L}_1$ we choose $h_{\boldsymbol{j}}\in 2\mathcal I_1(G_{\boldsymbol{j}})$ so that $c_0(h_{\boldsymbol{j}})=\alpha_{\boldsymbol{j}}= \alpha^1(G_{\boldsymbol{j}})$. Let $T_{\boldsymbol{j}}=\mathcal Lh_{\boldsymbol{j}}$ and thus $\alpha_{\boldsymbol{j}}=\langle T_{\boldsymbol{j}},1\rangle$. We fix $\boldsymbol{j}^1\in\boldsymbol{L}^1_1$, $\boldsymbol{j}^3\in\boldsymbol{L}^3_1$, and for $\theta=1$ and $\theta=3$ introduce the notation
$$
\begin{equation*}
\boldsymbol{a}^\theta=\boldsymbol{a}_{\boldsymbol{j}^\theta},\quad G^\theta=G_{\boldsymbol{j}^\theta},\quad \alpha^\theta=\alpha_{\boldsymbol{j}^\theta},\quad h^\theta=h_{\boldsymbol{j}^\theta},\quad\text{and}\quad T^\theta=\mathcal Lh^\theta.
\end{equation*}
\notag
$$
Set $M=|\boldsymbol{a}^1-\boldsymbol{a}^3|/\delta$. Let $\lambda^1\in(0,1)$ and $\lambda^3\in(0,1)$ be numbers such that $\lambda^1\alpha^1=\lambda^3\alpha^3=:\alpha$. Consider
$$
\begin{equation}
h^{13}(\boldsymbol{x})=h^{13} (\boldsymbol{j}^1,\boldsymbol{j}^3,\lambda^1,\lambda^3,\boldsymbol{x})= \frac{1}{M}(\lambda^1h^1(\boldsymbol{x})-\lambda^3h^3(\boldsymbol{x})).
\end{equation}
\tag{53}
$$
Then $c_0(h^{13})=0$, and by Lemma 5.2
$$
\begin{equation*}
\begin{aligned} \, M\boldsymbol{c}^1(h^{13})&=-\langle\lambda^1T^1(\boldsymbol{y})- \lambda^3T^3(\boldsymbol{y}),\boldsymbol{y}\rangle \\ &=-\lambda^1\langle T^1(\boldsymbol{y}),(\boldsymbol{y}- \boldsymbol{a}^1)\rangle-\lambda^1\langle T^1,\boldsymbol{a}^1\rangle \\ &\qquad+\lambda^3\langle T^3(\boldsymbol{y}),(\boldsymbol{y}- \boldsymbol{a}^3)\rangle+\lambda^3\langle T^3,\boldsymbol{a}^3\rangle \\ &=\alpha(\boldsymbol{a}^3-\boldsymbol{a}^1)+\boldsymbol{R}^{13}, \end{aligned}
\end{equation*}
\notag
$$
where $|\boldsymbol{R}^{13}|\leqslant A\delta\alpha$, as follows from (36) for $|\beta|=1$.
Therefore,
$$
\begin{equation}
\boldsymbol{c}^1(h^{13})=\delta\alpha(\boldsymbol{e}_1+ \boldsymbol{u}_1^{13}),\qquad |\boldsymbol{u}_1^{13}|=O\biggl(\frac{1}{M}\biggr).
\end{equation}
\tag{54}
$$
It is clear that
$$
\begin{equation}
|\nabla h^{13}(\boldsymbol{x})|\leqslant M^{-1}\bigl(\lambda_1|\nabla h^1(\boldsymbol{x})|+ \lambda_3|\nabla h^3(\boldsymbol{x})|\bigr).
\end{equation}
\tag{55}
$$
Moreover, for all $\boldsymbol{x}$ such that $|\boldsymbol{x}-\boldsymbol{a}^1|>p\delta$ and $|\boldsymbol{x}-\boldsymbol{a}^3|>p\delta$, in view of (38) we have
$$
\begin{equation*}
|\nabla h^{13}(\boldsymbol{x})|\leqslant \frac{\alpha}{M}|\nabla\Phi(\boldsymbol{x}-\boldsymbol{a}^1)- \nabla\Phi(\boldsymbol{x}-\boldsymbol{a}^3)|+ A\alpha\delta(|\boldsymbol{x}-\boldsymbol{a}^1|^{-N}+|\boldsymbol{x}- \boldsymbol{a}^3|^{-N}).
\end{equation*}
\notag
$$
Hence using the simple estimate
$$
\begin{equation*}
|\nabla\Phi(\boldsymbol{x}-\boldsymbol{a}^1)-\nabla\Phi(\boldsymbol{x}- \boldsymbol{a}^3)|\leqslant A_5|\boldsymbol{a}^1-\boldsymbol{a}^3| \bigl(|\boldsymbol{x}-\boldsymbol{a}^1|^{-N}+ |\boldsymbol{x}-\boldsymbol{a}^3|^{-N}\bigr)
\end{equation*}
\notag
$$
(which holds for $|\boldsymbol{x}-\boldsymbol{a}^1|>p\delta$ and $|\boldsymbol{x}-\boldsymbol{a}^3|>p\delta$) we obtain
$$
\begin{equation}
|\nabla h^{13}(\boldsymbol{x})|\leqslant A\delta\alpha(|\boldsymbol{x}-\boldsymbol{a}^1|^{-N}+ |\boldsymbol{x}-\boldsymbol{a}^3|^{-N}).
\end{equation}
\tag{56}
$$
Also note that when $|\boldsymbol{x}-\boldsymbol{a}^\theta|\leqslant p\delta$ (for $\theta=1$ or $\theta=3$), in view of (37) (and because $|\boldsymbol{a}^3-\boldsymbol{a}^1|=M\delta\geqslant q\delta\geqslant 3p\delta$) we have
$$
\begin{equation}
|\nabla h^{13}(\boldsymbol{x})|\leqslant \frac{\lambda^\theta}{p}+ A\,\frac{\alpha\delta}{|\boldsymbol{x}-\boldsymbol{a}^{4-\theta}|^N}\,.
\end{equation}
\tag{57}
$$
Next we sum the functions $h^{13}=h^{13}(\boldsymbol{j}^1,\boldsymbol{j}^3,\lambda^1,\lambda^3)$ we have constructed in a special way. It is easy to see that for each $\boldsymbol{j}\in\boldsymbol{L}_1^1\cup \boldsymbol{L}_1^3$ there exist numbers $\lambda(\boldsymbol{j},\kappa)$ (where $\kappa\in \{1,\dots,\kappa_{\boldsymbol{j}}\}$, $\kappa_{\boldsymbol{j}}\in\mathbb N$) with the following properties:
Set
$$
\begin{equation*}
h_{\boldsymbol{L}_1}(\boldsymbol{x})= \sum_{(\boldsymbol{j}^1,\kappa^1)\in\Psi^1}\frac{\delta} {|\boldsymbol{a}_{\boldsymbol{j}^3}-\boldsymbol{a}_{\boldsymbol{j}^1}|} \bigl(\lambda(\boldsymbol{j}^3,\kappa^3)h_{\boldsymbol{j}^3}- \lambda(\boldsymbol{j}^1,\kappa^1)h_{\boldsymbol{j}^1}\bigr),
\end{equation*}
\notag
$$
where $(\boldsymbol{j}^3,\kappa^3)$ corresponds to $(\boldsymbol{j}^1,\kappa^1)$ in the above sense. Each term of the above sum has the form (53) for $\lambda^\theta= \lambda(\boldsymbol{j}^\theta,\kappa^\theta)$. It is clear that $c_0(h_{\boldsymbol{L}_1})=0$, and from (54) we obtain
$$
\begin{equation}
\boldsymbol{c}^1(h_{\boldsymbol{L}_1})= \delta^N(\boldsymbol{e}_1+\boldsymbol{u}_1),\qquad |\boldsymbol{u}_1|=O\biggl(\frac{1}{M}\biggr).
\end{equation}
\tag{58}
$$
Estimate (52) (for $n=1$) follows from (56), (57), and condition (a) stated above.
It remains to select a parameter $q>3p$ in Definition 5.1 so that for $M\geqslant q$ we have $O(1/M)<\varepsilon_N$. Lemma 5.5 is proved. To prove Theorem 5.1 it remains to show that the function $\nabla f=\nabla\displaystyle\sum_{\boldsymbol{j}\in \boldsymbol{J}}f_{\boldsymbol{j}}$ can be approximated uniformly in $\mathbb R^N$ with an accuracy of $A\omega(\delta)$ by the function $\nabla F$, where
$$
\begin{equation*}
F=\sideset{}{'}\sum_s\,\sum_{\boldsymbol{j}\in \varGamma^s}f^*_{\boldsymbol{j}}+ \sideset{}{''}\sum_s\biggl(\,\sum_{\boldsymbol{j}\in \varGamma^s}f^*_{\boldsymbol{j}}+h_{\varGamma^s}\biggr)
\end{equation*}
\notag
$$
and the signs $\displaystyle\sideset{}{'}\sum_s$ and $\displaystyle\sideset{}{''}\sum_s$ indicate summation over all complete and incomplete groups, respectively. To prove the last assertion it is sufficient to verify that for each $\boldsymbol{x}\in\mathbb R^N$ we have
$$
\begin{equation*}
|\nabla(F(\boldsymbol{x})-f(\boldsymbol{x}))|\leqslant \sideset{}{'}\sum_s|\nabla g_{\varGamma^s}(\boldsymbol{x})|+ \sideset{}{''}\sum_s|\nabla(g_{\varGamma^s}(\boldsymbol{x})- h_{\varGamma^s}(\boldsymbol{x}))|\leqslant A\omega(\delta).
\end{equation*}
\notag
$$
After that it suffices to let $\delta$ approach zero. Now our situation is similar to [30], pp. 201–204; the reader can find there some simple details left out in the finial part of our proof. Notwithstanding, we present a detailed plan of the proof for the completeness of presentation and the harmonization of notation. First we find an estimate for $\displaystyle\sideset{}{'}\sum_s|\nabla g_{\varGamma^s}(\boldsymbol{x})|$. Fix $n\in\{1,\dots,N\}$ and recall that each $P_n(\boldsymbol{J},\boldsymbol{i})$ contains at most one incomplete group $\varGamma$ (that is, an $n$-incomplete chain $\boldsymbol{L}_n=\varGamma$). For $\boldsymbol{i}=(i_1,\dots,i_N)$ let $\boldsymbol{i}'_n=(i_1,\dots,i_{n-1},i_{n+1},\dots,i_N)$. For each $m\in\{0,1,\dots\}$ we let $\boldsymbol{S}_m$ denote the set of indices $s\in\{1,\dots,S\}$ such that $|(\boldsymbol{i}^s)'_n|\in[m,m+1)$. Hence in view of (49) and condition (3) in Definition 5.1, for $m>2p$ the estimate $S_{\boldsymbol{L}_n^{(s)}}(\boldsymbol{x}) \leqslant Am^{-N}$ holds for each $s\in\boldsymbol{S}_m$, while for $m\leqslant 2p$ we have the estimate $S_{\boldsymbol{L}_n^{(s)}}(\boldsymbol{x})\leqslant A$. Since the cardinality of $\boldsymbol{S}_m$ is at most $A(m+1)^{N-2}$, we can majorize the sum $\displaystyle\sideset{}{'}\sum_s|\nabla g_{\varGamma^s}(\boldsymbol{x})|$ under consideration by the series $A\omega(\delta)\displaystyle\sum_{m=1}^{+\infty}m^{-2}$, as required. An estimate for $\displaystyle\sideset{}{''}\sum_s|\nabla(g_{\varGamma^s}(\boldsymbol{x})- h_{\varGamma^s}(\boldsymbol{x}))|$ takes greater effort. For each complete group $\varGamma^s$ set $\chi^s=g_{\Gamma^s}-h_{\Gamma^s}$. Because of (49) and Lemma 5.5 we have
$$
\begin{equation}
|\nabla\chi^s(\boldsymbol{x})|\leqslant A\omega(\delta)S_{\varGamma^s}(\boldsymbol{x}),\quad c_0(\chi^s)=0,\quad\text{and}\quad \boldsymbol{c}^1(\chi^s)=0.
\end{equation}
\tag{59}
$$
It remains to show that
$$
\begin{equation*}
\sideset{}{''}\sum_s|\nabla\chi^s(\boldsymbol{x})|\leqslant A\omega(\delta)
\end{equation*}
\notag
$$
for each $\boldsymbol{x}\in\mathbb R^N$. Throughout what follows we fix $\boldsymbol{x}\in\mathbb R^N$; we assume without loss of generality that $|\boldsymbol{x}|<\delta$. All constructions that follow depend on this $\boldsymbol{x}$. For each complete group $\varGamma^s=\displaystyle\bigcup_{n=1}^N\boldsymbol{L}_n^{(s)}$ with vertex $\boldsymbol{i}^s$ we set $\boldsymbol{a}^s=\boldsymbol{a}_{\boldsymbol{i}^s}$,
$$
\begin{equation*}
M^s_n=\frac{1}{\delta}\operatorname{diam}(B^*_{\boldsymbol{L}_n^{(s)}}),\quad\text{and} \quad M^s=\max\bigl\{M^s_n \colon n\in\{1,\dots,N\}\bigr\}.
\end{equation*}
\notag
$$
Let $\theta=1/(N+2)$. We subdivide the set of complete groups into two classes. Class (1). It contains all complete groups $\varGamma^s$ such that $M^s\leqslant |\boldsymbol{i}^s|^{\theta}$. Clearly, this inequality is possible only for
$$
\begin{equation*}
|\boldsymbol{x}-\boldsymbol{a}^s|\geqslant (|\boldsymbol{i}^s|-1)\delta\geqslant ((M^s)^{N+2}-1)\delta>2pM^s\delta.
\end{equation*}
\notag
$$
Since $\chi^s\in A\omega(\delta) \mathcal I_1\bigl(B(\boldsymbol{a}^s,M^s\delta)\bigr)$ and (59) holds, in view of (39) we have the estimate
$$
\begin{equation*}
|\nabla\chi^s(\boldsymbol{x})|\leqslant A\omega(\delta)\,\frac{(M^s\delta)^{N+1}}{(|\boldsymbol{i}^s|\delta)^{N+1}} \leqslant A\omega(\delta)|\boldsymbol{i}^s|^{-N-\theta}.
\end{equation*}
\notag
$$
Each spherical shell $B(\boldsymbol{0},(m+1)\delta)\setminus \overline{B(\boldsymbol{0},m\delta)}$ (where $m>p$) contains at most $Am^{N-1}$ vertices of groups, so it is easy to see that
$$
\begin{equation*}
\sideset{}{^{(1)}}\sum_s|\nabla\chi^s(\boldsymbol{x})|\leqslant A\omega(\delta)\sum_{m>p}m^{-1-\theta}\leqslant A^2\omega(\delta),
\end{equation*}
\notag
$$
where the last sum is taken over all complete groups in class (1), so that it has the required estimate. Class (2). It contains all complete groups $\varGamma^s$ such that $M^s>|\boldsymbol{i}^s|^\theta$. In particular, for some $n\in\{1,\dots,N\}$ we have $M^s_n>|\boldsymbol{i}^s|^\theta$. For each group $\varGamma^s=\displaystyle\bigcup_{n=1}^N \boldsymbol{L}_n^{(s)}$ in this class (with vertex $\boldsymbol{i}^s$) we let $\boldsymbol{N}'s$ denote the set of indices $n\in\{1,\dots,N\}$ for which $M^s_n\leqslant|\boldsymbol{i}^s|^\theta$ and $\boldsymbol{N}''s$ denote the set of other indices in $\{1,\dots,N\}$. Thus, we always have $\boldsymbol{N}''s\ne\varnothing$. We start with the case when $\boldsymbol{N}'s\ne\varnothing$. For $n\in\boldsymbol{N}'s$ we have $|\boldsymbol{a}^s|>2p\delta$ and $M^s_n\leqslant |\boldsymbol{i}^s|^\theta$, so that $|\boldsymbol{x}-\boldsymbol{a}_{\boldsymbol{j}}|\geqslant 2^{-1}|\boldsymbol{x}-\boldsymbol{a}^s|$ for all $\boldsymbol{j}\in \boldsymbol{L}_n^{(s)}$. Therefore, from (48) and (49) we obtain
$$
\begin{equation*}
\sum_{n\in\boldsymbol{N}'s}S_{\boldsymbol{L}^{(s)}_n}(\boldsymbol{x})\leqslant \frac{A\omega(\delta)\delta^N}{|\boldsymbol{x}-\boldsymbol{a}^s|^N}\,,
\end{equation*}
\notag
$$
hence
$$
\begin{equation*}
|\nabla\chi^s(\boldsymbol{x})|\leqslant A\omega(\delta)\biggl(\frac{\delta^N}{|\boldsymbol{x}-\boldsymbol{a}^s|^N}+ \sum_{n\in\boldsymbol{N}''s}S_{\boldsymbol{L}^{(s)}_n}(\boldsymbol{x})\biggr).
\end{equation*}
\notag
$$
In the case when $\boldsymbol{N}'s=\varnothing$ the above estimate hods without the first term on the right. Note that the number of groups $\varGamma^s$ such that $|\boldsymbol{x}-\boldsymbol{a}^s|\leqslant 2p\delta$ is at most $A$, so the sum $\displaystyle\sum|\nabla\chi^s(\boldsymbol{x})|$ over such $s$ is at most $A\omega(\delta)$. Thus, it remains to establish the following result. Lemma 5.6. Fix $n\in\{1,\dots,N\}$, and let $\varSigma_n\subset\{1,\dots,S\}$ be the set of indices $s$ such that $\varGamma^s$ is a complete group in class (2) and, moreover, $|\boldsymbol{x}-\boldsymbol{a}^s|>2p\delta$ and $M^s_n>|\boldsymbol{i}^s|^\theta$. Then This statement coincides with Lemma 5.9 in [30] (where slightly different notation is adopted), where the full proof is presented. We only discuss the central idea of the proof. Fix $m\in\{0,1,\dots\}$ and $\boldsymbol{i}\in\boldsymbol{J}$ such that $|\boldsymbol{i}'_n|\in[m,m+1)$ (if there exist such $\boldsymbol{i}$). Let $\varSigma_n^m(\boldsymbol{i})$ denote the set of indices $s\in\varSigma_n$ such that $(\boldsymbol{i}^s)'_n=\boldsymbol{i}'_n$. Then for $m>2p$, in view of (48), condition (3) in Definition 5.1, and the inequality $M^s_n>|\boldsymbol{i}^s|^\theta\geqslant m^\theta$ we have the estimate
$$
\begin{equation*}
\sum_{s\in\varSigma_n^m(\boldsymbol{i})} \biggl(\frac{\delta^N}{|\boldsymbol{x}-\boldsymbol{a}^{s}|^N}+ S_{\boldsymbol{L}^{(s)}_n}(\boldsymbol{x})\biggr)\leqslant A\sum_{l=0}^{+\infty}\bigl(m^2+(m^\theta l)^2\bigr)^{-N/2}\leqslant Am^{-(N-1+\theta)},
\end{equation*}
\notag
$$
and for $m\leqslant 2p$ the penultimate sum has an estimate by a mere constant $A$. Since the number of distinct $\varSigma_n^m(\boldsymbol{i})$ is at most $A(m+1)^{N-2}$, the sum in Lemma 5.6 is majorized by the convergent series $A\displaystyle\sum_{m=1}^{+\infty}m^{-1-\theta}$, as required. It remains to sum with respect to $n$. Theorem 5.1 is proved. $\Box$ The capacity $\gamma^1_{\mathcal L}$ (respectively, $\alpha^1_{\mathcal L}$) in $\mathbb R^2$ is for all $\mathcal L$ comparable with the analytic capacity $\gamma$ (respectively, with the continuous analytic capacity $\alpha$) in $\mathbb C$; see [14]. A metric (more precisely, integro-geometric) description of $\gamma$ and $\alpha$ was obtained in [32] and [33]. We recall these results. Given a triple of points $z,w,\zeta\in\mathbb C$, let $R(z,w,\zeta)$ be the radius of the circle through these points (we set $R(z,w,\zeta)=+\infty$ if these are points on one straight line). The Menger curvature for $z$, $w$, and $\zeta$ is the quantity Let $\mu$ be a non-negative finite Borel measure on $\mathbb C$. Then the curvature of this measure is by definition the quantity
$$
\begin{equation*}
c^2(\mu)=\iiint\bigl(c(z,w,\zeta)\bigr)^2\,d\mu(z)\,d\mu(w)\,d\mu(\zeta).
\end{equation*}
\notag
$$
This concept was introduced by Mel’nikov [34], who considered a discrete version of the analytic capacity $\gamma$. For a bounded set $E\subset\mathbb C$ let
$$
\begin{equation*}
\begin{aligned} \, \mathcal M_c1(E)&=\bigl\{\mu\colon\operatorname{Supp}(\mu)\subset E,\ \mu(B(z,r))\leqslant r\ (\forall\,z\in\mathbb C,\ \forall\,r>0),\ c^2(\mu)\leqslant\mu(E)\bigr\}, \\ \kappa(E)&=\sup\biggl\{\int d\mu \colon \mu\in\mathcal M_c1(E)\biggr\}, \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\kappa_c(E)=\sup\biggl\{\int d\mu \colon \mu\in\mathcal M_c1(E),\ \forall\,z\in\mathbb C\, \ \frac{\mu(B(z,r))}{r}\to0\ (r\to0+)\biggr\}.
\end{equation*}
\notag
$$
By [32] and [33] there exists an absolute constant $A>1$ such that
$$
\begin{equation*}
A^{-1}\kappa(E)\leqslant\gamma(E)\leqslant A\kappa(E)\quad\text{and}\quad A^{-1}\kappa_c(E)\leqslant\alpha(E)\leqslant A\kappa_c(E)
\end{equation*}
\notag
$$
for all bounded sets $E$. In the recent paper [27], Corollaries 1.4–1.6, Tolsa obtained full integro-geometric descriptions of the capacities $\gamma^1_{\mathcal L}$ in all dimensions $N\geqslant2$. Let us state these results. Let $\mu$ be a positive Borel measure in $\mathbb R^N$. Consider the following characteristics of $\mu$:
$$
\begin{equation*}
\theta_{\mu}(\boldsymbol{x},r) = \frac{\mu(B(\boldsymbol{x},r))}{r^{N-1}}\,,\qquad \boldsymbol{x}\in\mathbb R^N,\quad r>0,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\beta_{2,\mu}(\boldsymbol{x},r) =\inf_{\{H\}}\biggl(\frac{1}{r^{N-1}} \int_{B(\boldsymbol{x},r)} \biggl(\frac{\operatorname{dist}(\boldsymbol{x},H)}{r}\biggr)^2\,d\mu (\boldsymbol{x})\biggr)^{1/2},
\end{equation*}
\notag
$$
where the infimum is taken over all hyperplanes $H$ in $\mathbb R^N$. Also consider the Jones–Wolff potential
$$
\begin{equation*}
U_\mu(\boldsymbol{x})=\sup_{r>0}\theta_\mu(\boldsymbol{x},r)+ \biggl(\int_0^{+\infty}\bigl(\beta_{2,\mu}(\boldsymbol{x},r)\bigr)^2 \theta_\mu(\boldsymbol{x},r)\,\frac{dr}{r}\biggr)^{1/2}.
\end{equation*}
\notag
$$
Some direct consequences are the countable subadditivity of all capacities $\gamma^1_{\mathcal L}$, their pairwise commensurability in any fixed direction $N\geqslant 2$, and two-sided estimates for the change of any capacity $\gamma^1_{\mathcal L}$ under bi-Lipschitz maps of $\mathbb R^N$. Also note that $\gamma^1_\Delta$ is proportional to the Calderón–Zygmund capacity (see [39], § 4), and the links between the Calderón–Zygmund capacities $\gamma_s$ and Hausdorff measures are well understood: for instance, see [39], §§ 5.2, 5.3, and 6, and the references in that paper. In a private communication Tolsa informed these authors on the following result, whose proof (using Theorem 5.2) follows the same scheme as the proof of Lemmas 4.4 and 4.5 in [14]. Under the assumptions of Theorem 5.2
$$
\begin{equation}
\alpha^1_{\mathcal L}(E)\asymp \sup\Bigl\{\mu(E) \colon \operatorname{Supp}(\mu)\subset E,\ U_\mu(\boldsymbol{x})\leqslant1,\ \lim_{r\to0}\theta_\mu(\boldsymbol{x},r)=0\ \forall\,\boldsymbol{x}\in E\Bigr\}.
\end{equation}
\tag{61}
$$
6. Description of the capacities $\gamma^0_{\mathcal L}$For $m=0$ Theorem 2.3 is stated in terms of the classical harmonic capacities from potential theory. This follows from the commensurability, established in [22], between the capacities $\gamma_{\mathcal L}=\gamma^0_{\mathcal L}$ for all $\mathcal L$, in the same dimension $N$. The capacities $\gamma_{\mathcal L}$ were introduced by Harvey and Polking [35] for the description of removable singularities in the classes of $L^\infty$-bounded $\mathcal L$-analytic functions. Namely, a compact set $K$ in $\mathbb R^N$ satisfies $\gamma_{\mathcal L}(K)=0$ if and only if for each neighbourhood $U$ of $K$ the condition $f\in L^\infty(U)\cap\mathcal A_{\mathcal L}(U\setminus K)$ implies that there exists $F\in\mathcal A_{\mathcal L}(U)$ such that $F=f$ on $U\setminus K$. Apart from $\gamma_{\mathcal L}$, we consider the capacities $\alpha_{\mathcal L}$, $\gamma_{\mathcal L,+}$, and $\alpha_{\mathcal L,+}$, so we recall the corresponding definitions. Let $K\subset{\mathbb R}^N$, $N\geqslant3$, be a compact set. Then by definition
$$
\begin{equation}
\gamma_{\mathcal L}(K)=\sup_T\bigl\{|\langle T,1\rangle| \colon \operatorname{Supp}(T)\subset K,\ \|T*\varPhi_{\mathcal L}\|_{L^\infty(\mathbb R^N)}\leqslant1\bigr\},
\end{equation}
\tag{62}
$$
where, as before, $\operatorname{Supp}(T)$ is the support of the distribution (function or measure) $T$, the supremum is taken over all distributions $T$ indicated, $*$ denotes convolution, and $\langle T,\varphi\rangle$ is the action of $T$ on the function $\varphi$ in $C^\infty(\mathbb R^N)$; for $K=\varnothing$ we set $\gamma_{\mathcal L}(K)=0$. The capacity $\alpha_{\mathcal L}(K)$ is defined by (62), where we assume in addition that $T*\varPhi_{\mathcal L}\in C(\mathbb R^N)$. The capacities $\gamma_{\mathcal L,+}$ and $\alpha_{\mathcal L,+}$ are defined similarly to $\gamma_{\mathcal L}$ and $\alpha_{\mathcal L}$, under the assumption that the distributions $T$ are non-negative Borel measures with support in $K$; in particular,
$$
\begin{equation}
\gamma_{\mathcal L,+}(K)=\sup_{\mu}\bigl\{\|\mu\| \colon \operatorname{Supp}(\mu)\subset K,\ \mu\geqslant0,\ \|\mu*\varPhi_{\mathcal L}\|_{L^\infty(\mathbb R^N)}\leqslant1\bigr\},
\end{equation}
\tag{63}
$$
where $\|\mu\|$ is the total mass of the measure $\mu$. In all cases above the capacity of a bounded (Borel) set is defined to be the supremum of the capacities of its compact subsets. In potential theory harmonic capacity is defined by (63), where $\mathcal L=\Delta$ is the Laplace operator. Namely, for $N\geqslant3$
$$
\begin{equation}
\varPhi_\Delta(\boldsymbol{x})= -\frac{1}{(N-2)\sigma_N|\boldsymbol{x}|^{N-2}}\,,
\end{equation}
\tag{64}
$$
where $\sigma_N$ is the area of the unit sphere in $\mathbb R^N$; thus, the capacity $\gamma_{\Delta,+}$ in (63) is precisely $(N-2)\sigma_N$ times greater than the (classical) harmonic capacity. In addition, for each bounded set $U$
$$
\begin{equation}
\alpha_{\Delta,+}(U)=\alpha_\Delta(U)=\gamma_{\Delta,+}(U)=\gamma_\Delta(U).
\end{equation}
\tag{65}
$$
The key results $\gamma_{\Delta,+}(U)=\gamma_\Delta(U)$ and $\alpha_{\Delta,+}(U)=\gamma_{\Delta,+}(U)$ were proved in [35], Theorem 3.1, and [36], Chap. III, Lemma 6, respectively (also see [37], Lemma XII, and [38], Chap. 1, § 3, Theorem 1.8). It is important to note that (65) is closely connected with the solvability of the Dirichlet problem and the maximum principle for harmonic functions. For arbitrary operators $\mathcal L$ with complex coefficients the appropriate machinery is not sufficiently well developed and the problem of the commensurability of capacities is much more complicated. By definition, for each operator $\mathcal L$ and each set $U$ we have
$$
\begin{equation}
\gamma_{\mathcal L}(U)\geqslant\alpha_{\mathcal L}(U),\ \gamma_{\mathcal L,+}(U)\geqslant\alpha_{\mathcal L,+}(U),\ \gamma_{\mathcal L}(U)\geqslant\gamma_{\mathcal L,+}(U),\ \text{and}\ \alpha_{\mathcal L}(U)\geqslant\alpha_{\mathcal L,+}(U).
\end{equation}
\tag{66}
$$
Because $\sup_{\boldsymbol{x}\ne0}|\varPhi_{\mathcal L}(\boldsymbol{x})|/ |\boldsymbol{x}|^{2-N}<+\infty$ for $N\geqslant3$, for each $\mathcal L$ it is obvious that there exists a constant $A(\mathcal L)\geqslant1$ such that for each bounded set $U\subset\mathbb R^N$ we have
$$
\begin{equation}
A(\mathcal L)\gamma_{{\mathcal L}}(U)\geqslant A(\mathcal L)\gamma_{{\mathcal L,+}}(U)\geqslant \gamma_{\Delta,+}(U).
\end{equation}
\tag{67}
$$
It is slightly more laborious to deduce that
$$
\begin{equation}
A(\mathcal L)\alpha_{\mathcal L,+}(U)\geqslant \alpha_{\Delta,+}(U).
\end{equation}
\tag{68}
$$
For the proof we can limit ourselves to convolutions $\mu*\varPhi_{\mathcal L}$ with non-negative measures $\mu$ such that
$$
\begin{equation*}
\operatorname{Supp}(\mu)\subset U,\quad \|\mu*\varPhi_{\Delta}\|_{L^\infty(\mathbb R^N)}\leqslant1,\quad\text{and}\quad \mu*\varPhi_{\Delta}\in C(\mathbb R^N).
\end{equation*}
\notag
$$
By Dini’s well-known theorem, uniformly in $\boldsymbol{x}\in\mathbb R^N$ we have
$$
\begin{equation*}
\lim_{\delta\to0}\int_{|\boldsymbol{y}-\boldsymbol{x}|<\delta} \frac{d\mu(\boldsymbol{y})}{|\boldsymbol{y}-\boldsymbol{x}|^{N-2}}=0.
\end{equation*}
\notag
$$
Since the quantity $|\varPhi_{\mathcal L}(\boldsymbol{x})|$ is majorized by $A(\mathcal L)|\boldsymbol{x}|^{2-N}$, we have $\|\mu*\varPhi_{\mathcal L}\|_{L^\infty(\mathbb R^N)}\leqslant A(\mathcal L)$, while it follows from Weierstrass’s uniform convergence theorem that $\mu*\varPhi_{\mathcal L}\in C(\mathbb R^N)$. Thus, to show that each capacity in (66) is commensurable (for fixed $N$) with harmonic capacity, by (65), (66), and (68) it is sufficient to prove the reverse inequality to (67), and the proof can be limited to compact sets. This was proved in Theorem 1 in [22]. Namely, the following result holds. Theorem 6.1. Let $N\geqslant3$. Then for each operator $\mathcal L$ there exists a constant $A=A(\mathcal L)\geqslant1$ such that for each compact set $K\subset\mathbb R^N$. Although the corresponding problem is longstanding, in fact, its in-depth study only started five years ago. In [20] explicit expressions were obtained for the fundamental solutions $\varPhi_{\mathcal L}$ in their dependence on $\mathcal L$; in particular, it was found that the capacities $\gamma_{\mathcal L,+}$ and $\gamma_{\Delta,+}$ are commensurable for $N=3$ and $N=4$ (see [20], Corollary 3, and also Theorem 3.1 above). That $\gamma_{\mathcal L,+}$ and $\gamma_{\Delta,+}$ are commensurable for any $N\geqslant3$ was shown in [21]. The proof uses an elementary result of Lemma 3 in [20], so that one can use the Fourier transform and energy approach to the problem of the commensurability of $\gamma_{\mathcal L,+}$ and $\gamma_{\Delta,+}$. However, the general problem of the commensurability of $\gamma_{\mathcal L}$ and $\gamma_{\Delta,+}$ is much more complicated. The proof of Theorem 6.1 also uses Vituskin’s scheme, methods of the theory of singular integrals, and inductive arguments similar to the ones Tolsa used in [32] to prove that analytic capacity is countably subadditive. It is also important to note that, in contrast to the odd Cauchy kernel, the kernels $\varPhi_{\mathcal L}$ are even and have non-zero means, so Tolsa’s arguments must be modified considerably. Also note that during the last decades a number of results on estimates and commensurability were obtained for various capacities, where, as a rule, the convolutions of distribution and measures with odd kernels were discussed (see the recent survey [39] by Volberg and Eiderman in this connection). The paper [40], in which a special case of Theorem 6.1 for Cantor-type sets was established, is of certain interest as concerns its methods. By treating sets with simple structure we can avoid most technical difficulties, preserving nonetheless the main ideas of the proof of Theorem 6.1. Let us look closer at the main ideas of the proof of Theorem 6.1. We begin with the commensurability of $\gamma_{\mathcal L,+}$ and $\gamma_{\Delta,+}$ for all $N\geqslant3$. Theorem 6.2. Let $N\geqslant3$. Then there exists a constant $A=A(\mathcal L)>1$ such that for each compact set $K\subset\mathbb R^N$. Proof. We present a short proof of Theorem 6.2 by following [21]. By (63) there exists a non-negative measure $\mu$ such that
We regularize $\mu$ in the standard way. Fix a function $\varphi_1\in C_0^\infty(B)$ (where $B$ is a unit ball in $\mathbb R^N$) such that $\varphi_1\geqslant0$ and For an arbitrary $\varepsilon>0$ let $\varphi_\varepsilon(\boldsymbol{x})= \varepsilon^{-N}\varphi_1(\boldsymbol{x}/\varepsilon)$, so that
$$
\begin{equation*}
\int_{|\boldsymbol{x}|\leqslant\varepsilon} \varphi_\varepsilon(\boldsymbol{x})\,d\boldsymbol{x}=1.
\end{equation*}
\notag
$$
Let $h_\varepsilon=\mu*\varphi_\varepsilon$. Then $h_\varepsilon\geqslant0$, $h_\varepsilon\in C_0^\infty(K_\varepsilon)$, where $K_\varepsilon$ is the closure of the $\varepsilon$-neighbourhood of $K$, and
$$
\begin{equation*}
\int_{\mathbb R^N}h_\varepsilon(\boldsymbol{x})\,d\boldsymbol{x}=\|\mu\|.
\end{equation*}
\notag
$$
Consider the following (in general, complex-valued) energy integral:
$$
\begin{equation}
\begin{aligned} \, \nonumber I_{h_\varepsilon,\mathcal L}&=-(N-2)\sigma_N \int_{K_\varepsilon}(h_\varepsilon*\Phi_{\mathcal L})(\boldsymbol{x}) \overline{h_\varepsilon(\boldsymbol{x})}\,d\boldsymbol{x} \\ &=-(N-2)\sigma_N\langle h_\varepsilon* \Phi_{\mathcal L},\overline{h_\varepsilon}\rangle, \end{aligned}
\end{equation}
\tag{69}
$$
where bars denote complex conjugation and corner brackets are used to denote the action of a distribution on a test function. Here we bear in mind that for $\mathcal L=\Delta$ we obtain the (positive) energy integral related to the kernel $1/|\boldsymbol{x}|^{N-2}$ (see [38], Chap. I, § 4).
Since $\Phi_{\mathcal L}$ is a locally integrable function in $\mathbb R^N$ and $h_\varepsilon\in C_0^\infty(\mathbb R^N)$, the integral in (69) is absolutely convergent. Since $h_\varepsilon*\Phi_{\mathcal L}= (\mu*\Phi_{\mathcal L})*\varphi_\varepsilon$ and $\|\mu*\Phi_{\mathcal L}\|_{L^\infty(\mathbb R^N)}\leqslant1$, we have $\|h_\varepsilon*\Phi_{\mathcal L}\|_{L^\infty(\mathbb R^N)}\leqslant1$, and therefore
$$
\begin{equation}
|I_{h_\varepsilon,\mathcal L}|\leqslant(N-2)\sigma_N\|\mu\|.
\end{equation}
\tag{70}
$$
Let $(\boldsymbol{y},\boldsymbol{x})=y_1x_1+\cdots+y_Nx_N$ for $\boldsymbol{y},\boldsymbol{x}\in\mathbb R^N$. Because $h_\varepsilon$ is a real function in $C_0^\infty(\mathbb R^N)$, its direct and inverse Fourier transforms
$$
\begin{equation*}
F[h_\varepsilon](\boldsymbol{x}) =\int_{\mathbb R^N} e^{-i(\boldsymbol{y},\boldsymbol{x})} h_\varepsilon(\boldsymbol{y})\,d\boldsymbol{y}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
F^{-1}[h_\varepsilon](\boldsymbol{x}) =\frac{1}{(2\pi)^N}\int_{\mathbb R^N} e^{i(\boldsymbol{y},\boldsymbol{x})} h_\varepsilon(\boldsymbol{y})\,d\boldsymbol{y}= \frac{1}{(2\pi)^N}\overline{F[h_\varepsilon](\boldsymbol{x})}
\end{equation*}
\notag
$$
belong to the Schwartz space $\mathcal S(\mathbb R^N)$ of functions rapidly decreasing at infinity. If $\psi\in\mathcal S(\mathbb R^N)$ and $\Psi\in\mathcal S'(\mathbb R^N)$, where $\mathcal S'(\mathbb R^N)$ is the space of tempered distribution in $\mathbb R^N$, then the Fourier transforms act on $\Psi$ by the formulae
$$
\begin{equation*}
\langle F[\Psi]|\psi\rangle=\langle \Psi|F[\psi]\rangle\quad\text{and}\quad \langle F^{-1}[\Psi]|\psi\rangle=\langle \Psi|F^{-1}[\psi]\rangle,
\end{equation*}
\notag
$$
and we have $\langle\Psi|\psi\rangle=\langle F[\Psi]|F^{-1}[\psi]\rangle$.
For $N\geqslant3$ the distribution $F[\Phi_{\mathcal L}]$ coincides with the function $-1/L$, which is locally integrable in $\mathbb R^N$, where $L=L(\boldsymbol{x})$ is the symbol of the operator $\mathcal L$. By the equality $F[h_{\varepsilon}*\Phi_{\mathcal L}]=-F[h_{\varepsilon}]/L$ we have
$$
\begin{equation}
\langle h_\varepsilon*\Phi_{\mathcal L},h_\varepsilon\rangle= -\biggl\langle\frac{F[h_{\varepsilon}]}{L}\,, F^{-1}[h_\varepsilon]\biggr\rangle=-\frac{1}{(2\pi)^N}\int_{\mathbb R^N} |F[h_\varepsilon(\boldsymbol{x})]|^2 \frac{1}{L(\boldsymbol{x})}\,d\boldsymbol{x}.
\end{equation}
\tag{71}
$$
We use the following result from [20], Lemma 3: there exist $\tau\in(0,1)$ and $\vartheta\in(-\pi,\pi]$ such that in $\mathbb R^N\setminus\{\boldsymbol{0}\}$
$$
\begin{equation}
\operatorname{Re}\bigl(e^{i\vartheta}L(\boldsymbol{x})\bigr)\geqslant \tau|L(\boldsymbol{x})|.
\end{equation}
\tag{72}
$$
For this $\vartheta$, from (69) and (71) we obtain
$$
\begin{equation}
I_{h_\varepsilon,\mathcal L}=e^{i\vartheta}\frac{(N-2)\sigma_N}{(2\pi)^N} \int_{\mathbb R^N}|F[h_\varepsilon(\boldsymbol{x})]|^2\, \frac{e^{-i\vartheta}}{L(\boldsymbol{x})}\,d\boldsymbol{x}.
\end{equation}
\tag{73}
$$
By (72), in $\mathbb R^N\setminus\{\boldsymbol{0}\}$ we have the following estimate (where $A_2=A_2(L)>0$):
$$
\begin{equation}
A_2\operatorname{Re}\frac{e^{-i\vartheta}}{L(\boldsymbol{x})}= A_2\frac{\operatorname{Re}(e^{i\vartheta}L(\boldsymbol{x}))} {|L(\boldsymbol{x})|^2}\geqslant \frac{A_2\tau}{|L(\boldsymbol{x})|}\geqslant\frac{1}{|\boldsymbol{x}|^2}\,.
\end{equation}
\tag{74}
$$
From (70), (73), and (74) we obtain where $I_{h_\varepsilon,\Delta}$ is the energy integral (69) for $\mathcal L=\Delta$ (and, correspondingly, $L(\boldsymbol{x})=|\boldsymbol{x}|^2$).Let $h_\varepsilon^0=h_\varepsilon/\|\mu\|$. Then $\displaystyle\int_{\mathbb R^N}h_\varepsilon^0 (\boldsymbol{x})\,d\boldsymbol{x}=1$, and relations (69) and (75) yield $I_{h_\varepsilon^0,\Delta}\leqslant A_3(L)/\|\mu\|$. Recall (for instance, see [38], Chap. II, § 1) one of the equivalent definitions of the harmonic capacity of a compact set $K_\varepsilon$: this is the quantity $1/\inf(I_{\mu^0,\Delta})$, where the infimum is taken over all non-negative measures $\mu^0$ such that $\operatorname{Supp}(\mu^0)\subset K_\varepsilon$ and $\|\mu^0\|=1$. Hence (taking the inequality $\|\mu\|\geqslant(1/2)\gamma_{\mathcal L,+}(K)$ into account)
$$
\begin{equation*}
A\gamma_{\Delta}(K_\varepsilon)\geqslant2\|\mu\|\geqslant \gamma_{\mathcal L,+}(K).
\end{equation*}
\notag
$$
It remains to let $\varepsilon$ tend to zero and use the relation $\lim_{\varepsilon\to0}\gamma_{\Delta}(K_\varepsilon)=\gamma_{\Delta}(K)$ (for instance, see [38], Chap. II, § 1.5, or [17], Proposition 3.1). $\Box$ Now we turn to the general case of Theorem 6.1. Proof of Theorem 6.1. For complex measures $\mu$ the situation is more complicated than for non-negative ones. Namely, in place of (70) we can directly obtain only the much weaker estimate
$$
\begin{equation}
|I_{h_\varepsilon,\mathcal L}|\leqslant (N-2)\sigma_N \int_{\mathbb R^N}|h_\varepsilon(\boldsymbol{x})|\,d\boldsymbol{x},
\end{equation}
\tag{76}
$$
which does not imply Theorem 6.1, of course, because it is possible that
This is not so grave for $\mathcal L=\Delta$. In fact, bearing in mind the same limiting procedure as in the proof of Theorem 6.2, we can assume that the compact set $K$ consists of a finite number of dyadic cubes. Then for a non-negative measure $\mu$ delivering the minimum energy in the class of measures with the same total mass, the function $\mu*|\boldsymbol{x}|^{2-N}$ is equal to a positive constant on the whole of $K$ (for instance, see [36], Chap. III, Theorem 3). Let $\nu$ be a complex measure on $K$ such that $I_{|\nu|,\Delta}<\infty$ and, in addition, $\langle\nu,1\rangle> 0$. We represent $\nu$ as a sum $\nu=\mu+\nu_1$, where $\mu$ is the non-negative measure mentioned above and $\langle\nu_1,1\rangle=0$ (so that $\langle\nu,1\rangle= \langle\mu,1\rangle$). Then
$$
\begin{equation*}
I_{\mu+\nu_1,\Delta}=I_{\mu,\Delta}+I_{\nu_1,\Delta}+ \int_{\mathbb R^N}\biggl(\mu*\frac{1}{|\boldsymbol{x}|^{N-2}}\biggr) (\boldsymbol{x})\,d(\nu_1(\boldsymbol{x})+\overline{\nu_1(\boldsymbol{x}}))= I_{\mu,\Delta}+I_{\nu_1,\Delta},
\end{equation*}
\notag
$$
where the energy $I_{\nu_1,\Delta}$ is defined by (69) and is non-negative because $L(\boldsymbol{x})=|\boldsymbol{x}|^2$ and in view of (69)–(71). Thus,
$$
\begin{equation}
\langle\nu,1\rangle=\langle\mu,1\rangle\quad\text{and}\quad I_{\nu,\Delta}\geqslant I_{\mu,\Delta},\qquad \mu\geqslant0.
\end{equation}
\tag{77}
$$
By contrast with the Laplace operator, there is no direct link between capacity and energy for arbitrary operators $\mathcal L$ with complex coefficients; it is not even known whether the weak maximum principle holds with a constant independent of the set. For this reason the proof of Theorem 6.1 has required novel approaches. First of all, note that the inequality $A\gamma_{\Delta,+}(K)\geqslant \gamma_{\mathcal L}(K)$ is trivial in the case when $K$ is a cube. Going over to compact sets with more complicated structure we use induction arguments similar to the ones in [32]. Their aim is to reduce the right-hand side of (76) by simplifying the function, which will be defined on a compact set containing $K$ but having a simpler structure than $K$. On the other hand the capacity $\gamma_{\Delta,+}$ of the enclosing set should not be significantly larger than that of $K$. This procedure takes the most transparent form for generations of Cantor sets [40]. We look at the general case of Theorem 6.1. Below $X$ can be, but not necessarily is, the original compact set $K$. Using Whitney’s famous construction ([3], Chap. 6, § 1) we obtain the following result ([22], Lemma 8). Recall that in Whitney’s construction dyadic cubes are said to be non-overlapping if their interiors do not intersect. Throughout what follows $s(D)$ is the side length of the cube $D$. Lemma 6.1. Let $X$ be a compact set consisting of a finite number of non-overlapping dyadic cubes $D_k$. Then for all sufficiently small $\lambda=\lambda(N)>0$ there exists a compact set $X'$ with the following properties: Assuming that the ratio $\gamma_{\Delta,+}(X)/(\operatorname{diam}(X))^{N-2}$ is sufficiently small (if the corresponding quantities are commensurate, then there is nothing to prove) we suppose (see [22], formula (4.6)) that the cubes $Q_j$ have diameters significantly less (at least with coefficient 2) than that of $X$. We see an alternative: either for all $j$ we have
$$
\begin{equation}
\frac{\gamma_{\mathcal L}(X)}{\gamma_{\Delta,+}(X)}\geqslant \frac{\gamma_{\mathcal L}(X\cap (17/16)Q_j)} {2\gamma_{\Delta,+}(X\cap (17/16)Q_j)}
\end{equation}
\tag{79}
$$
(case 1), or the reverse inequality holds for some $j$ (case 2). By the construction of $X$ and $Q_j$ it is obvious that the numerators and denominators of all fractions in (79) are non-zero. Consider case 1. The role of condition (79) consists in allowing us to transfer inequality (78) from $\gamma_{\Delta,+}(X)$ to $\gamma_{\mathcal L}(X)$. By (78) and (79) we have
$$
\begin{equation}
\sum_j\gamma_{\mathcal L}\biggl(X\cap \frac{17}{16}Q_j\biggr)\leqslant 2\frac{\gamma_{\mathcal L}(X)}{\gamma_{\Delta,+}(X)} \sum_j\gamma_{\Delta,+}\biggl(X\cap \frac{17}{16}Q_j\biggr)\leqslant 2A_4\gamma_{\mathcal L}(X).
\end{equation}
\tag{80}
$$
Estimate (80) is a key to the proof of the main lemma below ([22], Lemma 9). Lemma 6.2 (main lemma). Let $X$ be a compact set such that $\gamma_{\mathcal L}(X)>0$ and $f=(\mathcal Lf)*\Phi_{\mathcal L}$ be a function such that Also assume that there exists a compact set $X'$ equal to a finite union of disjoint dyadic cubes $Q_j$ and such that $X\subset X'$ and, moreover, Then for each $M\geqslant 2^N$ there exist $\alpha_M>0$, a compact set $\boldsymbol{X}$ equal to the union of a finite family of disjoint dyadic cubes, and a complex function $\nu\in C_0^\infty(\boldsymbol{X})$ such that
From Lemma 6.2 we can easily deduce the assertion of Theorem 6.1 for the compact set $X$. In fact, just as in (77), there exists a non-negative measure $\mu_{\boldsymbol{X}}$, $\operatorname{Supp}(\mu_{\boldsymbol{X}})\subset\boldsymbol{X}$, such that $\langle\mu_{\boldsymbol{X}},1\rangle=\langle\nu,1\rangle$ and $I_{\mu_{\boldsymbol{X}},\Delta}\leqslant I_{\nu,\Delta}$. Hence by property (c) (see the derivation of Theorem 6.2 from (75)) we have $\gamma_{\Delta,+}(\boldsymbol{X})\geqslant \gamma_{\mathcal L}(X)/(4\alpha_MM)$. In combination with (b) and condition (1) this yields
$$
\begin{equation*}
A_5\gamma_{\Delta,+}(X)\geqslant\gamma_{\Delta,+}(X')\geqslant \gamma_{\Delta,+}(\boldsymbol{X})-\frac{A_7}{M}\gamma_{\mathcal L}(X) \geqslant \frac{1}{M}\biggl(\frac{1}{4\alpha_M}-A_7\biggr) \gamma_{\mathcal L}(X).
\end{equation*}
\notag
$$
Using now property (a) we can select $M=M(L)$ so that $A\gamma_{\Delta,+}(X)\geqslant\gamma_{\mathcal L}(X)$. The proof of Lemma 6.2 is technical (see Lemmas 4, 5, 10, 11, and 12 in [22]); it uses Vitushkin’a approximation scheme and methods of geometric measure theory and the theory of singular integrals. We note briefly that condition (80), given that the structure of $\boldsymbol{X}$ is much simpler than that of $X$, enables one to deduce the estimate
$$
\begin{equation*}
\int_{\mathbb R^N}|\nu(\boldsymbol{x})|\,d\boldsymbol{x}\leqslant A\gamma_{\mathcal L}(X)
\end{equation*}
\notag
$$
(see assertion (i) of Lemma 11 in [22]). By contrast with Cantor sets (see [40]), here we cannot obtain a nice uniform estimate for $\nu*\Phi_{\mathcal L}$, but the combination of the inequalities $\gamma_{\Delta,+}(\boldsymbol{X})\leqslant \gamma_{\Delta,+}(X')+(A/M)\gamma_{\mathcal L}(X)$ and $I_{\nu,\Delta}\leqslant AM^{8/9}\gamma_{\mathcal L}(X)$ compensates for this adequately because $8/9<1$. The fact that we can replace $M$ by $M^{8/9}$ in the last bound is a result of two technical lemmas, Lemmas 5 and 12 in [22]. Next we look at case 2 (that is, when (79) fails for some $j$). Let $\gamma_{\mathcal L}(X)=C\gamma_{\Delta,+}(X)$, where $C$ is a positive constant of which we know nothing else. For the corresponding index $j$ case 2 yields $\gamma_{\mathcal L}(X\cap (17/16)Q_j)\geqslant 2C\gamma_{\Delta,+}(X\cap(17/16)Q_j)$, and we have $\operatorname{diam}(X\cap (17/16)Q_j)<(1/2)\operatorname{diam}(X)$. Setting $X:=X^{(1)}=X\cap(17/16)Q_j$ (where ${}^{(1)}$ denotes the step of iterations) we repeat the construction leading to Lemma 6.1 and consider the alternative related to (79). If the inequality corresponding to (79) holds for $X= X^{(1)}$, then using Lemma 6.2, our main lemma, we obtain $A\gamma_{\Delta,+}(X^{(1)})\geqslant \gamma_{\mathcal L}(X^{(1)})$. For the original compact set $K$ this yields the inequality
$$
\begin{equation*}
A\gamma_{\Delta,+}\biggl(K\cap\frac{17}{16}Q_j\biggr)\geqslant 2C\gamma_{\Delta,+}\biggl(K\cap\frac{17}{16}Q_j\biggr),
\end{equation*}
\notag
$$
and therefore $C\leqslant A/2$, so that $\gamma_{\mathcal L}(K)\leqslant(A/2)\gamma_{\Delta,+}(K)$, which provides an estimate required for the completion of the proof of Theorem 6.1 (with a twofold surplus). In the case when (79) fails we repeat the procedure for $X^{(k)}$, where $k=1,2,\dots$ . Then at some step we either arrive at case 1 directly (which completes the proof of Theorem 6.1, because at each successive step the estimate is at least twice better and $\gamma_{\mathcal L}(K)\leqslant(A/2^k)\gamma_{\Delta,+}(K)$), or we arrive at the situation where $Q_j\subset K$ – when we have nothing to prove because the estimate in Theorem 6.1 for an arbitrary $K$ follows immediately from the trivial inequality $A\gamma_{\Delta,+}(Q_j)\geqslant\gamma_{\mathcal L}(Q_j)$. This completes the proof of Theorem 6.1. $\Box$ Now consider the case of $\mathbb R^2$, which has some special features. If $\mathcal L$ is a strongly elliptic operator in $\mathbb R^2$, then $\lim_{|\boldsymbol{x}|\to\infty}\Phi_{\mathcal L}(\boldsymbol{x})=\infty$, so the capacities $\gamma_{\mathcal L}$, $\alpha_{\mathcal L}$, $\gamma_{\mathcal L,+}$, and $\alpha_{\mathcal L,+}$ are defined locally (for instance, see [19], § 1, formula (1.4)). In particular, this means that the norm $\|{\,\cdot\,}\|_{L^\infty(\mathbb R^2)}$ is replaced by $\|{\,\cdot\,}\|_{L^\infty(B)}$, where $B$ is a disc such that $K\subset(1/2)B$. For harmonic capacities in $\mathbb R^2$ equality (65) has a natural analogue ([19], Proposition 2.1), and the relevant capacities $\gamma_{\Delta,+}$ and $\gamma_{\mathcal L,+}$ are commensurable ([19], Proposition 2.3). This is a direct consequence of the structure of the fundamental solution. The main result of [19] is Theorem 1.1, which has the following meaning. Let $\mathcal L_2$ be an arbitrary strongly elliptic operator in $\mathbb R^2$. Then there exists $c_{33}\ne0$ such that $\mathcal L_3=\mathcal L_2+c_{33}\,\partial^2/\partial x_3^2$ is an elliptic operator in $\mathbb R^3$, and the local capacity $\gamma_{\mathcal L_2}(K)$ of $K$ relative to a disc $B$ of radius $R$ such that $K\subset(1/2)B$ is commensurable with $R^{-1}\gamma_{\mathcal L_3}(K')$, where $K'=K\times[-R,R]$ is the direct product in $\mathbb R^3$. By Proposition 2.1 and Theorem 1.1 in [19], Theorem 6.1 in $\mathbb R^3$ implies that the local capacities $\gamma_{\mathcal L_2}$ in $\mathbb R^2$ are commensurable with harmonic capacity. In fact, in all estimates we can limit ourselves to the case of the unit disc $B=B(\boldsymbol{0},1)$. Let $U\subset B(\boldsymbol{0},1/2)$; then
$$
\begin{equation}
\gamma_{\mathcal L}^0(U)=\sup_T\{|\langle T,1\rangle| \colon \operatorname{Supp}(T)\subset U,\ \|T*\varPhi_{{\mathcal L}}\|_{L^\infty(B(\boldsymbol{0},1))}\leqslant1\},
\end{equation}
\tag{81}
$$
and in the case of the capacity $\gamma_{\mathcal L,+}^0$ the distributions $T$ in (81) are non-negative measures with support in $U$, while for the capacities $\alpha_{\mathcal L}^0$ and $\alpha_{\mathcal L,+}^0$ in (81) one must additionally assume that $T*\varPhi_{\mathcal L}\in C(B(\boldsymbol{0},1))$. Recall that in $\mathbb R^2$ we have $\varPhi_{\Delta}=(2\pi)^{-1}\log|\boldsymbol{x}|$. Taking the sign of the logarithm into account, for $\gamma_{\Delta,+}$ we have an equivalent (non-local) definition (for instance, see [38], Ch. II, § 4):
$$
\begin{equation}
\gamma_{\Delta,+}(U)=\sup_{\mu}\{\|\mu\| \colon \operatorname{Supp}(\mu)\subset U,\ \mu\geqslant0,\ \mu*\varPhi_{\Delta}\geqslant-1\},
\end{equation}
\tag{82}
$$
and the capacity (82) is greater than the harmonic (Wiener) capacity with the precise coefficient $2\pi$. Thus, by Theorem 6.1 established above, and also by Proposition 2.1 and Theorem 1.1 in [19], the following result holds. Theorem 6.3. There exists a constant $A=A(\mathcal L)>1$ such that for each set $U\subset B(\boldsymbol{0},1/2)$ in $\mathbb R^2$, The case when $U\subset B(\boldsymbol{a},R/2)$ for an arbitrary disc $B=B(\boldsymbol{a},R)$ is considered using a translation and a dilation (see (1.4), (1.5), and Proposition 2.2 in [19]). Let $P(B)$ be a composition of a translation and a dilation taking $B$ to $B(\boldsymbol{0},1)$; then the capacity $\gamma_{\mathcal L}^{0,B}(U)$ of $U$ relative to $B$ satisfies
$$
\begin{equation}
\gamma_{\mathcal L}^{0,B}(U)=\gamma_{\mathcal L}^{0}(P(U)).
\end{equation}
\tag{83}
$$
Note some consequences of Theorems 6.1 and 6.3 above, Theorem 1.1 in [19], the results in [35], and well-known properties of harmonic capacity. 1. Each of the capacities $\gamma_{\mathcal L}$, $\gamma_{\mathcal L,+}$, $\alpha_{\mathcal L}$, and $\alpha_{\mathcal L,+}$ (as well as $\gamma_{\mathcal L}^0$, $\gamma_{\mathcal L,+}^0$, $\alpha_{\mathcal L}^0$, and $\alpha_{\mathcal L,+}^0$ in $\mathbb R^2$) is countably subadditive. Namely, if $U=\displaystyle\bigcup_jU_j$, then where $\gamma$ is any of these capacities.2. A compact set $K\subset\mathbb R^N$, $N\geqslant2$, is removable for solutions of the equation $\mathcal Lf=0$ (for $N=2$ only strongly elliptic equations are considered) in the class of bounded or the class of continuous functions if and only if $\gamma_{\Delta,+}(K)=0$. 3. Let $K\subset\mathbb R^2$ be a compact set. If for each $\varepsilon>0$ it can be covered by a finite family of open discs $B_j$ of radii $r_j<1$ such that
$$
\begin{equation*}
\displaystyle\sum_j\dfrac{1}{\log(1/r_j)}<\varepsilon,
\end{equation*}
\notag
$$
then $K$ is removable for solutions of strongly elliptic equations $\mathcal Lf=0$ in the classes of bounded and continuous functions for all $\mathcal L$. For all $\mathcal L$ and all $N\geqslant 2$ similar metric properties of removable sets can be established using the estimates for harmonic capacities from [36], Chap. 4. 7. Uniform approximation by $\mathcal L$-analytic functionsFor simpler notation we write here $\mathcal A_{\mathcal L}$ and $C_{\mathcal L}$ in place of $\mathcal A^0_{\mathcal L}$ and $C^0_{\mathcal L}$. Recall first of all that for each non-strongly elliptic operator $\mathcal L$ and each compact set $X$ in $\mathbb R^2$ we have $\mathcal A_{\mathcal L}(X)=C_{\mathcal L}(X)$ ([5], Theorem 1). Thus, in what follows $\mathcal L$ is an elliptic operator in $\mathbb R^N$ for $N\geqslant3$ or a strongly elliptic operator in $\mathbb R^2$. Let $N\geqslant2$, and let $X\subset\mathbb R^N$ be a non-empty compact set. We treat each function $f\in C(X)$ as extended outside $X$ (by the Urysohn–Brouwer theorem) as a continuous function with compact support, and let $\Omega_f$ be its modulus of continuity. A criterion for $f$ to belong to the space $\mathcal A_{\mathcal L}(X)$ was obtained for $N\geqslant3$ in [15] in terms of the capacities $\gamma_{\mathcal L}$, and for $N=2$ in [18] in terms of $\gamma^0_{\mathcal L}$ (up to a translation and a dilation). By Theorems 6.1 and 6.3, in these criteria $\gamma_{\mathcal L}$ and $\gamma^0_{\mathcal L}$ can be replaced by the classical harmonic capacities. For $N\geqslant3$ we have the following result. Theorem 7.1. Let $X$ be a compact subset of $\mathbb R^N$, $N\geqslant3$, $\mathcal L$ be an elliptic operator in $\mathbb R^N$ with symbol $L$, and let $f\in C(X)$. Then the following conditions are equivalent: Recall that the $L$-oscillation of a function $f$ over a ball $B$ was defined above in (2). The meaning of this quantity is easy to see in the special case (3) for $\mathcal L=\Delta$, when, bearing in mind the mean-value theorem for harmonic functions, $\mathcal O^\Delta_B(f)$ characterizes in a natural way the ‘measure of the deviation’ of the function $f$ in the ball $B$ ‘ from being harmonic’ (in fact, if $\Delta f=0$ in a neighbourhood of $\overline{B}$, then $\mathcal O^\Delta_B(f)=0$ by the mean-value theorem). In the general case (see Lemma 4.1), for $f\in C^\infty(\mathbb R^N)$ we have equality (19):
$$
\begin{equation*}
\int_B\psi_{\boldsymbol{a}}^r(\boldsymbol{x}) \mathcal Lf(\boldsymbol{x})\,d\boldsymbol{x}=\mathcal O^L_B(f),
\end{equation*}
\notag
$$
where $\psi_{\boldsymbol{a}}^r(\boldsymbol{x})= (r^2-|\boldsymbol{x}-\boldsymbol{a}|^2)/(2N|B|)$ in $B=B(\boldsymbol{a},r)$ and $\psi_{\boldsymbol{a}}^r(\boldsymbol{x})=0$ outside $B(\boldsymbol{a},r)$. Since the right-hand side of (2) contains no derivatives of $f$, by (19), for $f\in C(\mathbb R^N)$ it is natural to interpret the $L$-oscillation $\mathcal O^L_B(f)$ as the action $\langle\mathcal Lf,\psi_{\boldsymbol{a}}^r\rangle$ of the distribution $\mathcal Lf$ on the test function $\psi_{\boldsymbol{a}}^r$. We return to the statement of Theorem 7.1. The implication (c) $\Rightarrow$ (b) is trivial and the implication $f\in\mathcal A_{\mathcal L}(X)\Rightarrow{\rm (c)}$ is standard. The difficult part of Theorem 7.1 is the implication ${\rm (b)}\Rightarrow f\in\mathcal A_{\mathcal L}(X)$, that is, showing that condition (b) is sufficient. We begin with some comments to the implication $f\in\mathcal A_{\mathcal L}(X)\Rightarrow{\rm(c)}$. It is proved in the same way as its special case for $\mathcal L=\Delta$ and $N=3$ (see [10], Lemma 2.5). Namely, by (2) and (19), for $f\in C(\mathbb R^N)$ the products $\psi_{\boldsymbol{a}}^r\mathcal Lf$ are well defined as distributions. Just as in Lemma 4.4, we introduce the Vitushkin localization operator $V_{\psi_{\boldsymbol{a}}^r}(f)= (\psi_{\boldsymbol{a}}^r\mathcal Lf)*\varPhi_{\mathcal L}$. Similarly to Lemma 2.4 in [10] we prove the following standard result. Lemma 7.1. The function $(\psi_{\boldsymbol{a}}^r {\mathcal L}f)*\varPhi_{\mathcal L}$ has the following properties: Note that this lemma is significantly simpler than Lemmas 4.4 and 5.1, which claim similar results. Since $f\in\mathcal A_{\mathcal L}(X)$, for each $\varepsilon>0$ there exists a function $F\in C(\mathbb R^N)$ that is $\mathcal L$-analytic in a neighbourhood of $X$ and such that $\|f-F\|_{L^\infty(\mathbb R^N)}\leqslant\varepsilon$ (the Urysohn–Brouwer extension theorem can be used here if necessary). Applying Lemma 7.1 to the function $(\psi_{\boldsymbol{a}}^r\mathcal L(f-F))*\varPhi_{\mathcal L}$ we can assume that
$$
\begin{equation*}
\|(\psi_{\boldsymbol{a}}^r\mathcal LF)* \varPhi_{\mathcal L}\|_{L^\infty(\mathbb R^N)}\leqslant (A_1+1)\Omega_f(r)r^{2-N}.
\end{equation*}
\notag
$$
Hence for
$$
\begin{equation}
\langle T,1\rangle=\langle\psi_{\boldsymbol{a}}^r\mathcal LF,1\rangle= \langle\mathcal LF,\psi_{\boldsymbol{a}}^r\rangle=\mathcal O^L_B(F),
\end{equation}
\tag{85}
$$
by the definition (62) of capacity, for each $\lambda>1$ we obtain
$$
\begin{equation*}
|\mathcal O^L_{B(\boldsymbol{a},r)}(F)|\leqslant (A_1+1)\Omega_f(r)r^{2-N}\gamma_{\mathcal L} \bigl(B(\boldsymbol{a},\lambda r)\setminus X\bigr).
\end{equation*}
\notag
$$
It remains to use the estimate $\|f-F\|_{L^\infty(\mathbb R^N)}\leqslant \varepsilon$, where $\varepsilon>0$ can be arbitrary, and the continuity of $\mathcal O^L_{B(\boldsymbol{a},\lambda r)}(f)$ as a function of $\lambda$. As a result, since $\gamma_{\mathcal L}$ and $\gamma_{\Delta,+}$ are commensurable, we have $f\in\mathcal A_{\mathcal L}(X)\Rightarrow{\rm(c)}$. Now consider the implication ${\rm(b)}\Rightarrow f\in\mathcal A_{\mathcal L}(X)$. Note that the case $m=0$ is much more complicated than $m\in(0,1)\cup(1,2)$, because singular integral operators with Calderón–Zygmund kernels are unbounded in $C(\mathbb R^N)$. The case $m=0$ is also more technically involved than $m=1$ because in each localization of the original function we must equate the Laurent coefficients of order one higher. Theorem 7.1 reduces to the following main lemma (see [15], Theorem 1). Lemma 7.2. If there exist $k\geqslant 1$ and a function $\epsilon$, $\epsilon(r)\to0+$ as $r\to 0+$, such that for each ball $B=B(\boldsymbol{a},r)$ and each function $\varphi\in C_0^\infty(\mathbb R^N)$ such that $\operatorname{Supp}(\varphi)\subset B$ the estimate holds, then $f\in\mathcal A_{\mathcal L}(X)$. Of course, condition (84) is easier to verify than (86), because it does not involve arbitrary test functions $\varphi$. In fact (for instance, see (85)), it is the special case of (86) for the special test function $\psi_{\boldsymbol{a}}^r$. However, (86) is more convenient to deduce that $f\in\mathcal A_{\mathcal L}(X)$. For $k=1$ and $\epsilon=A\Omega_f$ condition (86), similarly to (84), is necessary in order that $f\in\mathcal A_{\mathcal L}(X)$ (see [15], Lemma 5). The scheme of transition from (84) to (86) was in fact proposed by Paramonov [31] for holomorphic functions. It is based on the construction of special partitions of the unity using repeated convolutions (see the proof of Lemma 5.4 above). In [15], § 7, a modification of this scheme was used for second-order operators. The proof of Lemma 7.2 uses the scheme of separation of singularities and approximation of functions by portions (Vitushkin’s scheme) proposed by Vitushkin in [8], which was developed further and adapted to general elliptic equations by many authors. In accordance with Vitushkin’s scheme, the original function $f$ is represented, by means of an appropriate partition of the unity, as a finite sum of functions with localized singularities (so-called localizations), and the problem consists in equating an appropriate number of coefficients of the Laurent series of these localizations with respect to partial derivatives of the fundamental solution. The general scheme of partitions of unity was proposed in [41], Lemma 3.1. Namely, let $\{Q_j\}$ be a finite family of disjoint dyadic cubes with edges $s_j$. Then for each $\lambda>0$ there exist functions $\varphi_j\in C_0^\infty(\mathbb R^N)$ such that $\operatorname{Supp}(\varphi_j)\subset(1+\lambda)Q_j$, $0\leqslant \varphi_j(\boldsymbol{x})\leqslant1$ for all $\boldsymbol{x}$, $\displaystyle\sum_j\varphi_j(\boldsymbol{x})=1$ on the set $\displaystyle\bigcup_jQ_j$, and for each $j$
$$
\begin{equation*}
\|\nabla^2\varphi_j\|_{L^\infty(\mathbb R^N)}\leqslant A(\lambda,N)s_j^{-2}.
\end{equation*}
\notag
$$
Note that $\lambda=1/32$ is taken in [15] (this small value of $\lambda$ is required for intricate geometric constructions, in particular, in the use of so-called compatible pairs of cubes), and the set $\{Q_j\}$ depends quite non-trivially on the distribution of the capacity over the complement of $X$. The functions $\varphi=\varphi_j$ are used in the construction of the localizations (32), and estimate (86) is only required for such $\varphi$ (that is, for functions with standard estimates for derivatives); furthermore, $r$ and $s_j$ are commensurate. The properties of localizations are standard and well known (for instance, see [8], Chap. 2, § 3, Lemma 1, [42], Lemma 14.10, [5], Lemma 1.2, and also Lemma 7.1 presented above). Namely,
Localizations $(\varphi_j\mathcal Lf)*\varPhi_{\mathcal L}$ can be expanded in Laurent series (34), where $\boldsymbol{a}$ is the centre of $Q_j$, $T=\varphi_j\mathcal Lf$, and $c_0=\langle\varphi_j\mathcal Lf,1\rangle= \langle\mathcal Lf,\varphi_j\rangle$ is the leading Laurent coefficient. By the definition of capacity, condition (86) expresses just the possibility to equate the coefficients $c_0$ of all localizations using functions $F_j$ such that $\|F_j\|_{L^\infty(\mathbb R^N)}\leqslant A_1\epsilon_f(s_j)$ and $\operatorname{Supp}(\mathcal LF_j)\subset A_1Q_j\setminus X$. In addition, we have the asymptotic expressions $(\varphi_j\mathcal Lf)*\varPhi_{\mathcal L}-F_j=O(|\boldsymbol{x}|^{1-N})$ as $\boldsymbol{x}\to\infty$. However, these relations are not sufficient in order that a direct estimate for $\displaystyle\sum_j|(\varphi_j\mathcal Lf)*\varPhi_{\mathcal L}-F_j|$ yields the inclusion $f\in\mathcal A_{\mathcal L}(X)$. Were we reproducing formally Vitushkin’s scheme from [8], Chap. 2, then the relation $(\varphi_j\mathcal Lf)* \varPhi_{\mathcal L}-F_j=O(|\boldsymbol{x}|^{-N-1})$ as $\boldsymbol{x}\to\infty$ would suffice. Note that for uniform approximations by holomorphic functions for $N=2$ (see [8]), and for approximations in the $C^1$-norm (for gradients), after equating the leading Laurent coefficients we a priori have the asymptotic behaviour $O(|\boldsymbol{x}|^{-N})$ at infinity, rather than $O(|\boldsymbol{x}|^{1-N})$ (see § 5, inequality (38)), which is technically easier for approximation. In these cases an effective method for grouping indices was proposed by Paramonov; we described it in § 5. The following theorem on the approximation of function by portions ([5], Theorem 2) turned out to be very useful for the proof of Lemma 7.2. Theorem 7.2. If for some localization $(\varphi_j\mathcal Lf)*\varPhi_{\mathcal L}$ there exists a function $F_j$ such that The proof of this theorem takes account of the fact that second partial derivatives of $\varPhi_{\mathcal L}$ are nice Calderón–Zygmund kernels with zero means, so the expressions $\displaystyle\sum_j((\varphi_j\mathcal Lf)*\varPhi_{\mathcal L}-F_j)$ have much better $L^2$-estimates in $\mathbb R^N$ than $\displaystyle\sum_j|(\varphi_j\mathcal Lf)* \varPhi_{\mathcal L}- F_j|$. Using partitions of unity on various scales and summing by means of convex combinations, from $L^2$-estimates one can go over to uniform ones. From Theorem 7.2 on the approximation of a function by portions ([5], Theorem 2) it follows that to prove Lemma 7.2 it is sufficient (and necessary) to equate in each localization, along the leading Laurent coefficient $c_0$, also the $N$ coefficients of the first partial derivatives of the fundamental solution, with appropriate estimates. It is a remarkable fact that we can do this as an implicit consequence of estimate (86). The problem of equating these coefficients causes the most trouble in the proof of Lemma 7.2. Certain special Lipschitz graphs are constructed in the proof; their idea was put forward in [43], where $\mathcal L$ was the square of the Cauchy–Riemann operator in $\mathbb R^2$. As this operator has a bounded fundamental solution, one needs no capacities and the construction is fully geometric. We present a model argument by following [43]; the reasoning in the proof of Lemma 7.2 is a considerable development of this argument, and we do not consider it here in full generality because of its complexity. In the space $\mathbb R^2$ of the variables $x_1$ and $x_2$ let $z=x_1+ix_2$ and $\overline{z}=x_1-ix_2$, let $\overline\partial=(\partial/\partial x_1+i\partial/\partial x_2)/2$ be the Cauchy–Riemann operator, $\overline\partial^2$ be its square, and $\varPhi=\pi^{-1}\overline{z}\big/z$ be the standard fundamental solution for $\overline\partial^2$. Let $X\subset\mathbb R^2$ be a non-empty compact set and $g\not\equiv0$ be a localization of a function $f$ such that $f\in C(\mathbb R^2)$ and $\overline\partial^2f=0$ in $X^\circ$. Without loss of generality we assume that $\operatorname{Supp}(\overline\partial^2g)\subset (1/2)Q_0\setminus X^\circ$, where $Q_0=[0,1]\times[0,1]$, and the problem consists in estimating and equating the Laurent coefficient of $g$ for $\partial\varPhi/\partial x_2$. Let $\mathcal X=Q_0\setminus X^\circ$. We cover $\mathcal X$ by finite families of closed dyadic squares
$$
\begin{equation*}
Q=Q^{m_1,m_2}_k=[m_1\,2^{-k},(m_1+1)\,2^{-k}]\times [m_2\,2^{-k},(m_2+1)\,2^{-k}],
\end{equation*}
\notag
$$
where $k$, $m_1$, and $m_2$ are integers. We construct recursively a non-increasing sequence of covers $\mathcal Q(k)$, $k=1,2,3,\dots$, were $\mathcal Q(1)$ is the set of four dyadic squares with side length $2^{-1}$ which form $Q_0$. Each square in $\mathcal Q(k+1)$ lies in (and maybe coincides with) a suitable square in $\mathcal Q(k)$. Each cover contains squares of two types: white squares, partitioned at the next step, and red squares, preserved in what follows (the four starting squares of the construction are white). To specify the local stopping times the following concept is central. Two squares $Q^{m_1,m_2}_k$ and $Q^{n_1,n_2}_k$ (of the same size, with side length $s=2^{-k}$) are said to be compatible (a compatible pair) if
$$
\begin{equation*}
|m_1-n_1|\leqslant1\quad\text{and}\quad 2\leqslant|m_2-n_2|\leqslant3.
\end{equation*}
\notag
$$
We call the set of squares $\{Q^{m_1,m_2}_k\}$, where $k$ and $m_1$ are fixed, a vertical row (in particular, two compatible squares must lie in the same or adjacent vertical rows). The step of recursion: constructing $\mathcal Q(k)$. We partition each white square $Q$ in $\mathcal Q(k- 1)$ into four squares with side $s(Q)/2$. Of the resulting squares, we include into $\mathcal Q(k)$ only the ones intersecting $\mathcal X$ (that is, not lying fully in $X^\circ$); we call them squares of the $k$th generation; the cover $\mathcal Q(k)$ consists of the squares of the $k$th generation and all the red squares of the previous generations. We define the colours of squares of the $k$th generation as follows. If their set contains a compatible pair $(Q_1,Q_2)$, then we say that $Q_1$ and $Q_2$ are red, and so will also be each square in $\mathcal Q(k)$ lying in the same vertical row as $Q_1$ or $Q_2$; then we repeat the search for compatible pairs among the squares whose colour we have not yet determined (defining red squares in the same way as above). After we have exhausted all compatible pairs, we call the remaining squares of the $k$th generation white; this completes the construction of the cover $\mathcal Q(k)$. Each cover $\mathcal Q(k)$ has the following properties ([43], Lemma 2.1), which follow from the constructions and are easy to verify using induction.
Now we can defined the global stopping time. Fix $k_0$, the smallest index $k$ such that where $\Omega_g$ is the modulus of continuity of $g$ and on the right-hand side we see the sum of the quantities $s(Q)$ over all (fixed) red squares $Q$. Since the compact set $\mathcal X$ contains a non-empty open set, such an index $k_0$ certainly exists.Fixing $k_0$ we select one square in each vertical row of the cover $\mathcal Q(k_0)$. Let $\varGamma$ be the polygonal line through the centres of these squares and $x_2=\Psi(x_1)$ be the equation of $\varGamma$. From properties $(1)$ and $(4)$ of $\mathcal{Q}(k_0)$ it is clear that this curve $\varGamma$ has the following properties: $|\Psi'(x_1)|<7$ and for all $Q$ in $\mathcal Q(k_0)$ we have $\operatorname{dist}(Q,\varGamma)\leqslant(5/2)s(Q)$. The construction is finished. Now using the (finite) set of cubes $\{Q_j\}$ making up the cover $\mathcal Q(k_0)$ we construct the partition of unity $\{\varphi_j\}$ from Lemma 3.1 in [41], decompose $g$ into a sum of repeated localizations: and equate the leading Laurent coefficients of all localizations. Namely, let $\boldsymbol{x}_j$ be any point in the square $(11/10)Q_j$ which lies in the complement of $X$, and let
$$
\begin{equation}
g_0(\boldsymbol{x})=g(\boldsymbol{x})- \sum_j\langle\varphi_j\overline\partial^2g,1\rangle \varPhi(\boldsymbol{x}-\boldsymbol{x}_j).
\end{equation}
\tag{88}
$$
Then $\langle\overline\partial^2g_0,1\rangle=0$, and the modulus of the coefficient of $\partial\varPhi/\partial x_2$ in the expansion of $g_0$ is estimated from above by the right-hand side of (87). The main ideas behind equating of this coefficient are as follows. Let $Q'=Q^{n_1,n_2}_k$ and $Q''=Q^{m_1,m_2}_k$ be a compatible pair of red squares, where we assume that $m_2>n_2$ (in other words, $Q''$ lies higher than $Q'$ relative to the $x_2$-axis), and let
$$
\begin{equation*}
\boldsymbol{x}'=(x_1',x_2')\in \frac{11}{10}Q'\quad\text{and}\quad \boldsymbol{x}''=(x_1'',x_2'')\in \frac{11}{10}Q''
\end{equation*}
\notag
$$
be the corresponding pair of points in the complement of $X$. We look at a function of the form
$$
\begin{equation}
r(\boldsymbol{x})=\sum_t\lambda_t(\varPhi(\boldsymbol{x}-\boldsymbol{x}_t'')- \varPhi(\boldsymbol{x}-\boldsymbol{x}_t')),\qquad 0\leqslant\lambda_t\leqslant1,
\end{equation}
\tag{89}
$$
where the sum is taken over all compatible pairs of red squares in $\mathcal Q(k_0)$. If all coefficients $\lambda_t$ are equal to 1, then, as follows from the definition of a compatible pair, the modulus of the Laurent coefficient at $\partial\varPhi/\partial x_2$ in the expansion of $r$ has at least the same order of magnitude as the right-hand side of (87). The main problem is that the uniform norm of $r$ can increase unboundedly with the number of compatible pairs. Here it is crucial that $\varGamma$ is a Lipschitz graph whose Lipschitz constant is under control and the function $r$ in (89) behaves as a singular integral with antisymmetric Calderón–Zygmund kernel of order $|\boldsymbol{x}|^{-1}$. Using methods of the theory of singular integrals we can achieve that, at the same time,
$$
\begin{equation*}
\|r\|_{\mathbb R^2}\leqslant A\quad\text{and}\quad \sum_t\lambda_ts(Q_t')>\frac{1}{100}\sum_{\text{red}}s(Q),
\end{equation*}
\notag
$$
where $Q_t'$ is some cube in the compatible pair $(Q_t',Q_t'')$ and the sum on the right in the second inequality is just as in (87). In fact, this is a discrete analogue of Lemma 4.2 in [44]. The uniform norm of $g_0$ in (88) can also increase without limit with the number of compatible pairs. Here the situation is even more complicated than in the case of $r$, and apart from methods of the theory of singular integrals we must now vary the cover $\mathcal Q(k_0)$. Of course, the need to use capacities and, in particular, the uneven distribution of capacity over the complement of $X$ makes the construction of equating the Laurent coefficients at $\partial\varPhi_{\mathcal L}/\partial x_n$ in Lemma 7.2 $n=1,2,\dots,N$, considerably more profound than in [43] (see [15], §§ 4 and 5 and Lemma 16 on a cumulative estimate for coefficients). The notion of a compatible pair and the definition of local stopping times become more involved. However, the key idea remains the same: a Lipschitz graph is plotted and the coefficient to be estimated is in fact a Carleson measure with respect to such a graph, while the construction itself is a corona-type construction. In total, we construct $2N$ functions in $\mathbb R^N$, two functions in each orthogonal direction, $N$ of which are analogues of $g_0$ in (88) and the other $N$ are analogues of $r$ in (89). The function equating the $N$ coefficients of first-order partial derivatives of the fundamental solution in the expansions of localizations is constructed as a linear combination of these functions ([15], Lemmas 12 and 13). To complete the proof of Lemma 7.2 (see [15], § 6) the technique of singular integrals on Lipschitz graphs is employed, where first-order derivatives of the fundamental solution $\varPhi_{\mathcal L}$ are used as (odd) Calderón–Zygmund kernels. The transition from $L^2$- to uniform estimates is in fact performed as in Theorem 2 in [5]. Now we look at the case $N=2$. Taking account of the commensurability of the capacities (81) and (82) and of equality (83), the criterion in [18] for solutions of strongly elliptic equations leads to the following result. Theorem 7.3. Let $X$ be a compact set in $\mathbb R^2$ and $\mathcal L$ be a strongly elliptic operators with symbol $L$, and let $f\in C(X)$. Then the following conditions are equivalent: The general scheme of the proof of Theorem 7.3 is the same as for Theorem 7.1, but there are technical features reflecting the non-standard behaviour of capacities in $\mathbb R^2$ by comparison with $\mathbb R^N$ for $N\geqslant3$, due to the unboundedness of fundamental solutions at infinity. 8. Approximation criteria for classes of functions. Sets of removable singularitiesWe present criteria, established earlier in [7], [25], [14], [16], [15], and [18], for the $C^m$-$\mathcal L$-approximability of classes of functions in the case $m\in[0,2)$. They are standard consequences of Theorem 2.3 (for instance, see details in [30], Theorem 6.1). In view of the above metric descriptions of the capacities $\alpha^m_{\mathcal L}$ these criteria can now be said to be of metric nature, that is, they can be reformulated in metric (or integro-metric, for $m=1$) terms. Corollary 8.1. For all operators $\mathcal L$ in $\mathbb R^N$ (except when $N=2$ and $m=0$) and all compact sets $X\subset\mathbb R^N$ the following conditions are equivalent: The following result, linking the concept of a set of $C^m$-removable singularities ($ \operatorname{\textit{Lip}} ^m$-removable singularities) for $\mathcal L$-analytic functions with the $C^m$-$\mathcal L$-capacity (respectively, $ \operatorname{\textit{Lip}} ^m$-$\mathcal L$-capacity) introduced above is well known. We mentioned it already in § 6, in the context of the uniform capacities $\alpha_{\mathcal L}$ and $\gamma_{\mathcal L}$. Theorem 8.1. A compact set $K$ in $\mathbb R^N$ satisfies the condition $\alpha^m_{\mathcal L}(K)=0$ (the condition $\gamma^m_{\mathcal L}(K)=0$) if and only if $K$ is a set of $C^m$-removable singularities (respectively, a set of $ \operatorname{\textit{Lip}} ^m$-removable singularities) for $\mathcal L$-analytic functions. Recall that a compact set $K\subset\mathbb R^N$ is a set of $C^m$-removable singularities for $\mathcal L$-analytic functions if the following holds: let $U$ be a neighbourhood of $K$, and let $f\in C^m(U)\cap {\mathcal A}_{\mathcal L}(U\setminus K)$; then $f\in {\mathcal A}_{\mathcal L}(U)$. Accordingly, a compact set $K\subset\mathbb R^N$ is a set of $ \operatorname{\textit{Lip}} ^m$-removable singularities for $\mathcal L$-analytic functions if for each function $f\in \operatorname{\textit{Lip}} ^m(U)\cap\mathcal A_{\mathcal L}(U\setminus K)$ there exists $F\in\mathcal A_{\mathcal L}(U)$ such that $F=f$ on $U\setminus K$. The proof of Theorem 8.1 is standard: for harmonic functions it was presented in [1] (Theorem 1.12). In other words, the quantities $\alpha^m_{\mathcal L}(E)$ and $\gamma^m_{\mathcal L}(E)$ are ‘measures of $\mathcal L$-analytic non-removability’ of the set $E$ for functions in the classes $C^m$ and $ \operatorname{\textit{Lip}} ^m$, respectively. Since all capacities under consideration here can be described in metric or integro-geometric terms, sets of $C^m$-removable singularities for $\mathcal L$-analytic functions also have this property. It is interesting that for fixed $N$ and $m>0$ the capacities $\gamma^m_{\mathcal L}$ (respectively, $\alpha^m_{\mathcal L}$) corresponding to different operators $\mathcal L$ are commensurable; moreover, for $m=0$ and fixed $N$ even all $\gamma^m_{\mathcal L}$ and $\alpha^m_{\mathcal L}$ are commensurable, so that the sets of $C^0$- and $L^\infty$-removable singularities for $\mathcal L$-analytic functions are the same. 9. $C^m$-approximation by $\mathcal L$-analytic polynomialsWe go over to the problem of the approximation of functions by $\mathcal L$-analytic polynomials. This problems has much poorer been investigated than the problem of the approximation by $\mathcal L$-analytic functions with singularities outside the compact set of approximation. However, there are several classical fundamental results on this problem which underlie the further studies to a considerable extent. The first result is the Walsh–Lebesgue criterion of the uniform approximation of functions by harmonic polynomials on compact sets in $\mathbb R^2$ [45] (1929; also see [46]). By this criterion, each continuous function on a compact set $X\subset\mathbb R^2$ that is harmonic in $X^\circ$ can be approximated by a sequence of harmonic polynomials uniformly on $X$ if and only if $\partial X=\partial\widehat{X}$, where $\widehat{X}$ is the topological hull of $X$ (which is by definition equal to the union of $X$ with all bounded connected components of the set $\mathbb R^2\setminus X$). Note also that for compact sets $X$ in $ \mathbb R^2$ we have $\widehat{X}= \{z\in\mathbb R^2 \colon |p(z)|\leqslant\max_{w\in X}|p(w)|$ for each polynomial $p$ of the complex variable$\}$, so that $\widehat{X}$ is also called the polynomial hull or polynomially convex hull of $X$. Compact sets $X$ such that $\partial X=\partial\widehat{X}$ are called Carathéodory compact sets. This class of sets arises in a natural way in many problems of the theory of approximations by analytic functions. The second result we must mention is the classical Mergelyan theorem of 1952 on the uniform approximation of functions by polynomials of the complex variables [47]: if $X\subset\mathbb R^2$ is a compact set, then all functions in $C(X)$ that are holomorphic on $X^\circ$ can be approximated uniformly on $X$ by polynomials of the complex variable to an arbitrary accuracy if and only if $X=\widehat{X}$, that is, the set $\mathbb R^2\setminus X$ is connected. In the general case the problem of $C^m$-approximation of functions by $\mathcal L$-analytic polynomials is also stated for classes of functions (when it is required to find necessary and sufficient conditions on a compact set $X$ ensuring that each function in the class $C^m_{\mathcal L}(X)$ can be approximated in the $C^m$-norm on $X$ by a sequence of $\mathcal L$-analytic polynomials). For a formal statement we introduce an appropriate function space. Recall that an $\mathcal L$-analytic polynomial is a (complex-valued) polynomial $P$ of $N$ real variables $x_1,\dots,x_n$ that satisfies the equation $\mathcal LP=0$ everywhere in $\mathbb R^N$. We denote the space of $\mathcal L$-analytic polynomials by $\mathcal P_{\mathcal L}$. Given a compact set $X\subset\mathbb R^N$, $N\geqslant2$, we define the space $\mathcal P^m_{\mathcal L}(X)$ to consist of all functions $f\in BC(\mathbb R^N)$ such that there exists a sequence of functions $\{g_n\}_{n=1}^{\infty}\subset BC^m(\mathbb R^N)$ with the following properties:
Note that in place of the condition that $g_n$ coincides with an $\mathcal L$-analytic polynomial in a neighbourhood of $X$ we can use the (equivaent) condition that the restriction to $X$ of the function $g_n$ and all of its partial derivatives of order up to $[m]$ inclusive (more precisely, the restriction to $X$ of the $[m]$-jet of $g_n$) coincides with the corresponding jet of some $\mathcal L$-analytic polynomial. Using the theory of Whitney spaces $C^m_{\rm jet}(X)$ and, in particular, Whitney’s extension theorem the space $\mathcal P^m_{\mathcal L}(X)$ can be defined ‘intrinsically’, from within $X$ (see [2], [3], Chap 6, and also [4], § 2). Throughout this section the superscript $m=0$ is omitted from the notation for all spaces: in place of $\mathcal P^0_{\mathcal L}(X)$ we write $\mathcal P_{\mathcal L}(X)$, in place of $C^0_{\mathcal L}(X)$ we write $C_{\mathcal L}(X)$, and so on. In this section we are interested in the following problem: let $m\geqslant0$, and let $\mathcal L$ be an operator in $\mathbb R^N$ of the form under consideration; describe compact sets $X$ in $\mathbb R^N$ such that Recall that $C^m_{\mathcal L}(X)= BC^m(\mathbb R^N)\cap \mathcal A_{\mathcal L}(X^\circ)$ and the following embeddings are clear:
$$
\begin{equation*}
\mathcal P^m_{\mathcal L}(X)\subset \mathcal A^m_{\mathcal L}(X)\subset C^m_{\mathcal L}(X).
\end{equation*}
\notag
$$
We consider the problem of the coincidence of the spaces $\mathcal P^m_{\mathcal L}(X)$ and $C^m_{\mathcal L}(X)$ for $N=2$ first. The multidimensional case is much poorer understood, and we discuss it briefly at the end of the section. The following theorem combines the criteria for the spaces $\mathcal P^m_{\mathcal L}(X)$ and $C^m_{\mathcal L}(X)$ to coincide in the case of plane compact sets and $m\geqslant1$. Theorem 9.1. Let $X$ be a compact subset of $\mathbb R^2$, and let $\mathcal L$ be an elliptic differential operator of the second order with constant complex coefficients. Then the following holds. Note that in this theorem the second condition in assertion (2) and the condition $\overline{X^\circ}=X$ in assertion (3) are just the conditions ensuring in Corollary 8.1 and Theorem 2.2, respectively, that the classes $\mathcal A^m_{\mathcal L}(X)$ and $C^m_{\mathcal L}(X)$ coincide for $m$ in question. Also note that for $N=2$ and $m=1$ the condition $X=\widehat{X}$ implies that condition (b) in Corollary 8.1 holds. In fact, the capacity $\alpha^1_{\mathcal L}$ in $\mathbb R^2$ has the property $\alpha^1_{\mathcal L}(D)\asymp\mathop{\mathrm{diam}}(D)$ for each domain $D\subset\mathbb R^2$ (see [14], § 5). In all the three cases the approximation criteria in Theorem 9.1 are deduced from the corresponding criteria for the equality of the spaces $\mathcal A^m_{\mathcal L}(X)$ and $C^m_{\mathcal L}(X)$, using the classical Runge method of the ‘motion of singularities’, whose application to the problem of the approximation by $\mathcal L$-analytic functions was justified, for example, in [4] (also see [48], § 3.10, for the case of classical uniform approximation). For completeness we show why the connected complement of the compact set $X$ is necessary in order that we could approximate each function $f$ in the class $C^1(\mathbb R^2)\cap\mathcal A_{\mathcal L}(X^\circ)$ by a sequence of polynomials $\{P_k\}$, $P_k\in\mathcal P_{\mathcal L}$ in the following sense: $P_k\to f$ and $\nabla P_k\to \nabla f$ uniformly on $X$ as $k\to+\infty$. Note that this is weaker than the condition $f\in\mathcal P^m_{\mathcal L}(K)$ for all $m\geqslant1$. Suppose that $\mathbb R^2\setminus X$ is disconnected and the origin lies in some bounded connected component of this set. Consider a function $f$ equal to the fundamental solution $\varPhi_{\mathcal L}(\boldsymbol{x})$ for $\mathcal L$ in an annulus $\{\rho<|\boldsymbol{x}|<R\}\subset\mathbb R^2$ containing $X$, and extended to a function in the class $C^1(\mathbb R^2)$ in an arbitrary way. Note that for $N=2$ the operator $\mathcal L$ can be factored into a product of two operators of the first order: $\mathcal L= c_{11}(\partial/\partial x_1-\lambda_1\partial/\partial x_2) (\partial/\partial x_1-\lambda_2\partial/\partial x_2)$, where $\lambda_{1,2}$ are the zeros of the characteristic polynomial of the operator $\mathcal L$. Furthermore, polynomials in the class $\mathcal P_{\mathcal L}$ have the form $Q_1(z_1)+Q_2(z_2)$ in the case when $\lambda_1\ne\lambda_2$ and the form $z_1Q_1(z_2)+Q_2(z_2)$ for $\lambda_1=\lambda_2$, where $Q_{1,2}$ are some polynomials in the complex variable with complex coefficients and $z_1$ and $z_2$ are certain special linear combinations of $x_1$ and $x_2$ with (complex) coefficients depending on $\lambda_{1,2}$. Here we do not need the explicit form of these combination (for instance, it can be found in [24], § 2), but the following holds: for $\lambda_1\ne\lambda_2$ we have
$$
\begin{equation*}
\biggl(\frac{\partial}{\partial x_1}- \frac{\lambda_s\partial}{\partial x_2}\biggr)z_s=1 \quad\text{and}\quad \biggl(\frac{\partial}{\partial x_1}- \frac{\lambda_s\partial}{\partial x_2}\biggr)z_{3-s}=0,\qquad s=1,2,
\end{equation*}
\notag
$$
while for $\lambda_1=\lambda_2=\lambda$ we have similar equalities with $\lambda_1$ replaced by $\lambda$ and $\lambda_2$ by $-\lambda$. Taking these equalities into account, it follows from the convergence $\nabla P_k\to\nabla f$ and the explicit form of the fundamental solution $\varPhi_{\mathcal L}$ for $N=2$ (for instance, see [24], Proposition 2.2) that the function $1/z_s$ where $s=1$ or $s=2$, can be approximated by polynomials in $z_s$ uniformly on $X$. However, this is impossible by the maximum principle for holomorphic functions. Details of the corresponding argument can be found in the proof of Theorem 3.2 in [24]. Now let $m<1$ (and $N=2$). Then the condition $X=\widehat{X}$ implies that $\mathcal A^m_{\mathcal L}(X)=C^m_{\mathcal L}(X)$. From this, in turn, using Runge’s method we can deduce equality (90) for all $\mathcal L$ and $m$ under consideration. But for $m<1$ it is no longer necessary that the complement be connected, and the full criteria for polynomial approximation in this case are not yet known for most operators $\mathcal L$. The above Walsh–Lebesgue criterion for the uniform approximation of functions by harmonic polynomials remains the same for all $m\in[0,1/2)$ (see Corollary 1.2 in [49]), and the following approximability criterion holds (recall that we defined the hull $\widehat{X}$ of the compact set $X$ before). Theorem 9.2. Let $X$ be a compact set in $\mathbb R^2$, and let $m\in[0,1/2)$. For the equality $\mathcal P^m_{\Delta}(X)=C^m_{\Delta}(X)$ to hold it is necessary and sufficient that $X$ be a Carathéodory compact set, that is, $\partial X=\partial\widehat{X}$. For $m\in[1/2,1)$ the situation is considerably more complex. We present a recent result in this direction. Let $D$ be a simple Carathéodory domain in $\mathbb R^2$, that is, a non-empty bounded domain in $\mathbb R^2$ such that the set $\Omega=\mathbb R^2\setminus\overline{D}$ is connected and $\partial D=\partial\Omega$ (such a domain $D$ must be simply connected: see below). In such a domain the Poisson operator $P_D$ if defined in a natural way: to a function $\varphi\in C(\partial D)$ it assigns a function $f\in C_{\Delta}(\overline{D})$ such that $f\big|_{\partial D}=\varphi$ (recall that by Lebesgue’s theorem of 1907 the classical Dirichlet problem for harmonic functions is solvable in each bounded simply connected domain $\mathbb R^2$, for any continuous boundary data; see below). The concept of a simple Carathéodory domain can also be introduced in the spaces $\mathbb R^N$ of dimension $N\geqslant3$, but we defer this to what follows. Recall that for $m\in(0,1)$ and a compact set $X\subset\mathbb R^N$, $N\geqslant2$, the space $C^m(X)$ can be defined as the closure of the subspace $C^\infty(\mathbb R^N)\big|_X$ in $ \operatorname{\textit{Lip}} ^m(X)$, while $ \operatorname{\textit{Lip}} ^m(X)$ consists of all functions $f\in C(X)$ such that $\|h\|'_{mX}<+\infty$, where the norm is defined by $\|f\|_{mX}=\max\{\|f\|_X,\|f\|'_{mX}\}$ (in all (semi)norms that follow the subscript $X$ means that in their definitions upper bounded are taken over the points in $X$). It is easy to show that if $D$ is a simple Carathéodory domain and $P_D$ acts continuously from $C^m(\partial D)$ to $C^m(\overline{D})\cap\mathcal A_\Delta(D)$ (in which case it is natural to say that $P_D$ is a $C^m$-continuous operator), then $\mathcal P_\Delta^m(\overline{D})=C_\Delta^m(\overline{D})$ and $\mathcal P^m_\Delta(\partial D)=C^m(\partial D)$. Thus there arises a natural question of the values of $m$ for which the Poisson operator $P_D$ is $C^m$-continuous. The following result, which we state here in the particular case of dimension $N=2$, was established in [50]. Proposition 9.1. (1) Let $D$ be a simple Carathéodotry domain in $\mathbb R^2$. Then there exists $m_D\in[0,1]$ such that the Poisson operator $P_D$ is $C^m$-continuous for all $m\in(0,m_D)$ and is not so for $m\in(m_D,1)$. (2) Let $D$ be a Jordan domain in $\mathbb R^2$ with piecewise smooth boundary, and let $\beta\in(0,1]$ be a number such that the smaller exterior angle $S_a$, $a\in\partial D$, between the two distinct rays with vertex $a$ that are tangent to $\partial D$ is equal to $\pi\beta$. Then $m_D=1/(2-\beta)$. In the investigations of the problem of uniform polynomial $\mathcal L$-analytic approximation a considerable progress has been achieves in the case of $\mathcal L=\overline\partial{}^2$ (the square of the Cauchy–Riemann operator $\overline\partial=(\partial/\partial x_1+i\partial/\partial x_2)/2$). In this case the problem concerns the approximation of functions by bianalytic polynomials, that is, ones of the form $\overline{z}p_1(z)+p_0(z)$, where $p_{0,1}$ are polynomials of the complex variable $z$, and $\overline{z}$ is the complex conjugate variable. Theorem 2.2 in [51] provides the following criterion for bianalytic polynomial approximation on Carathéodory compact sets in $\mathbb R^2$. Theorem 9.3. Let $X\subset\mathbb R^2$ be a Carathéodory compact set. Then $\mathcal P_{\overline\partial^2}(X)=C_{\overline\partial^2}(X)$ if and only if no bounded connected component of the set $\mathbb R^2\setminus X$ is a Nevanlinna domain. The property of being a Nevanlinna domain is a special analytic feature of a bounded simply connected domain in the complex plane, whose formal definition is presented in [52], Definition 1, and [51], Definition 2.1. Speaking informally, a bounded simply connected domain $\Omega$ is a Nevanlinna domain $\Omega$ if the function $\overline{z}$ coincides with a ratio of two bounded holomorphic functions in this domain almost everywhere on $\partial\Omega$ in the sense of a conformal mapping. The precise definition is as follows. Let $\mathbb D$ be the unit disc in $\mathbb C$, that is, $\mathbb D=\{|z|<1\}$, and let $\mathbb T=\partial\mathbb D$ be the unit circle. Recall that the notation $H^\infty(U)$, where $U$ is an open set in $\mathbb C$, is used for the space of bounded holomorphic functions in $U$. Also recall that by Fatou’s classical theorem, for any function $f$ in the class $H^\infty=H^\infty(\mathbb D)$, for almost all points $\zeta\in\mathbb T$ the function $f$ has finite limit angular (boundary) values $f(\zeta)$ at $\zeta$. Definition 9.1. A bounded simply connected domain $\Omega$ in $\mathbb C$ is called a Nevanlinna function if there exist functions $u,v\in H^\infty(\Omega)$ (where $v\not\equiv 0$) such that $\overline{w}=u(w)/v(w)$ on $\partial\Omega$ almost everywhere in the sense of conformal mappings. This means that for almost all points $\zeta\in\mathbb T$ the following equality of angular boundary values holds: where $\varphi$ is a conformal mapping of $\mathbb D$ onto $\Omega$. The definition of a Nevanlinna domain is consistent in the sense of its independent on the choice of $\varphi$. By the Luzin–Privalov boundary uniqueness theorem the ratio $u/v$ is uniquely defined in $\Omega$ (provided that $\Omega$ is a Nevanlinna domain). This function $S=u/v$ is called the Schwarz function of $\Omega$. Note that if the Nevanlinna domain $\Omega$ is a Jordan domain with rectifiable boundary, then the equality $\overline{w}=u(w)/v(w)$ can be treated directly as the equality of angular limit values almost everywhere on $\partial\Omega$. It is rather simple to present examples of both Nevanlinna domains (an arbitrary disc in the plane) and domains that do not have this property (any domain bounded by a closed Jordan polygonal line or any domain bounded by an ellipse other than a circle). An interesting example of a Nevanlinna domain is a so-called Neumann oval, the domain bounded by the image of an ellipse with centre at the origin under the map $z\mapsto1/z$. As in the previous examples, this property of the Neumann oval can be verified by direct calculations. An effective description of Nevanlinna domains $\Omega$ in terms of conformal mappings $\varphi$ of the disc $\mathbb D$ on $\Omega$ can be given in terms of pseudocontinuations of bounded holomorphic functions. Recall (see [53], Definition 2) that a function $f\in H^\infty(\mathbb D)$ admits a pseudocontinuation of Nevanlinna type if there exist functions $u_1,v_1\in H^\infty(\mathbb D_e)$, where $\mathbb D_e=\mathbb C\setminus\overline{\mathbb D}$, $v_1\not\equiv0$, such that for almost all $\zeta\in\mathbb T$ the equality $f(\zeta)=u_1(\zeta)/v_1(\zeta)$ holds in the sense of angular limit values, where $f(\zeta)$ is the limit value of $f$ from inside $\mathbb D$, while $u_1(\zeta)$ and $v_1(\zeta)$ are the limit values of $u_1$ and $v_1$ from inside $\mathbb D_e$. The next result is easy to verify (in particular, see [51], Proposition 3.1, where the connection between the notion of a Nevanlinna domain and the property of pseudocontinuation was established). Proposition 9.2. Assume that a function $\varphi$ mapping the disc $\mathbb D$ conformally onto a bounded simply connected domain $\Omega$ admits a pseudocontinuation $u_1/v_1$ of Nevanlinna type. Then $\Omega$ is a Nevanlinna domain. Moreover, let $Z'=\{z'_j\}_{j\in J}$ be the set of poles of $S_1(z)=u_1(z)/v_1(z)$ in $\mathbb D_e\cup\{\infty\}$ (the set $Z'$ and thus the index set $J$ are non-empty and at most countable). Set $Z=\{z_j=1/\overline{z'_j}\}_{j\in J}$ (where $1/\infty=0$) and $\Sigma=\{w_j=\varphi(z_j)\}_{j\in J}$. Then the Schwarz function $S$ of $\Omega$ can be defined by Moreover, $S$ has a pole of the same order at $w_j$ as the pole of $S_1$ at $z'_j$, $j\in J$. Conversely, if $\Omega$ is a Nevanlinna domain with Schwarz functions $S$ which has a (non-empty) set of poles $\Sigma$ in $\Omega$, and $\varphi$ is a conformal mapping of $\mathbb D$ onto $\Omega$, then $\varphi$ admits a pseudocontinuation $S_1$ of Nevanlinna type defined by The poles $Z'$ of $S_1$ are related to the poles $\Sigma$ of $S$ in the same way as above.Remark 9.1. Let $z_1=0,z_2,\dots,z_N$, $N\in\mathbb N$, be arbitrary different points in $\mathbb D$, $p_1,\dots,p_N$ be positive integers, and let $J=\{1,\dots, N\}$. We present an algorithm of the construction of a Nevanlinna domain $\Omega$ whose Schwarz function has poles $z_n$ of orders $p_n$, $n\in\{1,\dots,N\}$, and no other poles in $\Omega$. Using the notation of Proposition 9.2 for $z'_1=\infty$ ($z_1=0$) and $z'_n=1/\overline{z_n}$ ($n\in\{2,\dots,N\}$), for a sufficiently small $\varepsilon>0$ we look at the rational function which defines both $\varphi$ in $\mathbb D$ and $S_1$ in $\mathbb D_e$. Then $w_n=\varphi(z_n)=z_n$ for $n\in\{1,\dots,N\}$. Furthermore, to describe Nevanlinna domains in terms of conformal mappings we need the concept of a model space. Recall it. Let $\varTheta$ be an inner function (that is, a function in $H^\infty$ such that $|\varTheta(\xi)|=1$ for almost all $\xi\in\mathbb T$), and let $K_\varTheta\subset H^2$ (where $H^2$ is the standard Hardy Hilbert space in $\mathbb D$) be defined by $K_\varTheta=H^2\ominus\varTheta H^2$. Recall that by Beurling’s classical theorem all subspaces $K_\varTheta\subset H^2$ and only them are invariant subspaces of the backward shift operator $f\mapsto(f(z)-f(0))/z$ in $H^2$. Spaces $K_\varTheta$ are called model subspaces. This term was proposed by N. K. Nikolski in view of the prominent role these spaces play in the context of the Nagy–Foias functional model. The following result was established in [53], Theorem 1: Let $\varTheta$ be an inner function. Then each bounded univalent function in the space $K_\varTheta$ maps $\mathbb D$ conformally onto a Nevanlinna domain. The converse is also true: if a univalent function $\varphi\in H^\infty$ maps $\mathbb D$ conformally onto a Nevanlinna domain, then there exists an inner function $\varTheta$ such that $\varphi\in K_\varTheta$. The following question is natural: what are inner functions $\varTheta$ such that the spaces $K_\varTheta$ contain univalent functions? The answer is due to Belov and Fedorovskiy [54] and has a rather simple statement: A space $K_\varTheta$ contains bounded univalent functions if and only if one of the following two conditions is satisfied: $\varTheta$ has a zero in $\mathbb D$, or $\varTheta$ is a singular inner function such that the singular measure $\mu$ on $\mathbb T$ defining it satisfies $\mu(E)>0$ for some Beurling–Carleson measure $E\subset\mathbb T$. Recall that $E\subset\mathbb T$ is a Beurling–Carleson set (a Carleson set, or a set with finite entropy) if $\displaystyle\int\log\operatorname{dist}(\zeta,E)\,dm_1(\zeta)>-\infty$, where $m_1(\,\cdot\,)$ is the Lebesgue measure on $\mathbb T$. For details of the connection between the Nevanlinna property of a domain with pseudocontinuation and questions relating to univalent functions in model spaces the reader can consult [55] and [56]. The properties of Nevanlinna domains were extensively investigated during the last two decades; for instance, see [53]–[55] and [57]–[61]. It was shown that the Nevanlinna domains form quite a rich class, in spite of the fact that this property is defined in terms of a rather stringent condition on the regularity of the boundary of the domain. In the above papers the authors used essentially the following construction. Let $B$ be an infinite Blaschke product, that is,
$$
\begin{equation*}
B(z)=\prod_{n=1}^\infty\frac{\overline{a_n}}{|a_n|}\, \frac{z-a_n}{\overline{a_n}z-1}\,,
\end{equation*}
\notag
$$
where $(a_n)_{n=1}^\infty$ is a sequence of points such that $|a_n|<1$, $n\in\mathbb N$, and Also assume that $B$ satisfies the Carleson condition
$$
\begin{equation*}
\inf_{n\in\mathbb{N}}\prod_{\substack{k=1\\k\ne n}}^\infty \biggl|\frac{a_n-a_k}{1-a_n\overline{a_k}}\biggr|>0.
\end{equation*}
\notag
$$
In this case it is known that the system of functions $\{\psi_n\}_{n=1}^{+\infty}$, where
$$
\begin{equation*}
\psi_n(z)=\frac{\sqrt{1-|a_n|^2}}{1-\overline{a_n}z}\,,
\end{equation*}
\notag
$$
forms a Riesz basis in $K_B$. Using this observation we can construct Nevanlinna domains with various properties as the images of $\mathbb D$ under bounded univalent functions of the form for various, appropriately selected sequences of zeros $\{a_n\}_{n=1}^{+\infty}$ of the corresponding Blaschke product $B$ and appropriate coefficients $\{c_n\}_{n=1}^{+\infty}$. In [57] the first example of a Jordan Nevanlinna domain with nowhere analytic boundary was presented. After that, in [53] and [58] examples of Jordan Nevanlinna domains with $C^1$-boundaries that are not $C^{1+\delta}$-smooth for $\delta\in(0,1)$ were constructed. These domains have the form $\varphi(\mathbb D)$, where $\varphi$ is an appropriate functions of the form (91) which is univalent in $\mathbb D$. In the construction of such examples we must ‘keep the balance’ in providing a sufficiently rapid decrease of the coefficients $\{c_n\}_{n=1}^{+\infty}$ (to ensure that the mapping is univalent), which cannot be too rapid (to ensure the required regularity of the boundary of the image of the disc under this mapping). Now, [58] contains examples of Jordan Nevanlinna domains with ‘almost non-rectifiable boundaries’. This means that the corresponding function (91) is univalent in the disc $\mathbb D$, but $\varphi'\notin H^p$ for all $p>1$ (here $H^p$ is a classical Hardy space in the unit disc). Next, in [59] an example of a Nevanlinna domain with non-rectifiable boundary was presented. This construction was also made in the general framework of the above techniques, but involved a special construction of a so-called ‘Nevanlinna needle’ (a Nevanlinna domain having the shape of a disc with a very long and narrow spike similar to a ‘needle’ sticking from a boundary point; its length and width are described in terms of independent parameters). After that, in [61] an example of a Nevanlinna domain with fractal boundary was constructed whose Hausdorff dimension is $\log_23$. As in the previous examples, it has the form $\varphi(\mathbb D)$ for a suitable function $\varphi$ of the form (91). Finally, it was shown in [55] that the Hausdorff dimension of the boundary of a Nevanlinna domain (and even the dimension of the part of the boundary attainable from inside) can be equal to any number from $1$ to $2$ inclusive. All these examples were also constructed in the framework of the general scheme described above, where a modified construction of a Nevanlinna needle played a considerable part (enabling one to ‘grow’ needles sequentially from points close to the tip of the original needle). An essential part of all these constructions was delicate estimates for sums of the form (91) near points on $\mathbb T$ mapped to the ‘tip’ of a needle. For technical details, see [55], § 3. For compact sets $X$ that are not Carathéodory compact sets the question of the equality of the spaces $\mathcal P_{\overline\partial^2}(X)$ and $C_{\overline\partial^2}(X)$ has only been answered from some special classes of compact sets. In this connection we can mention Theorem 4 in [62], Theorem 4.3 in [51], and Theorems 3 and 4 in [63], whose precise statements we leave out. Although a definitive criterion of the uniform $\mathcal L$-analytic polynomial approximation for general operators $\mathcal L$ is not known, there is a reductive criterion for approximability, established in [62] (in the bianalytic case) and [64] (where the proof was adapted to general second-order elliptic operators $\mathcal L$ in $\mathbb R^2$). For a formulation we need a further function space. For a pair of compact sets $X$ and $Y$ in $\mathbb R^2$ such that $X\subseteq Y$, let $\mathcal A_{\mathcal L}(X,Y)$ be the space of functions $f$ than can be approximated, uniformly on $X$, by a sequence of $\mathcal L$-analytic functions in neighbourhoods of $Y$ (as usual, an own neighbourhood for each function). Clearly, $\mathcal A_{\mathcal L}(X,X)=\mathcal A_{\mathcal L}(X)$. As the results below show, an important special case of such spaces is $\mathcal A_{\mathcal L}(\partial G,\overline{G})$ for a fixed simply connected domain $G$ in $\mathbb R^2$: approximations are considered on the boundary of this domain and the singularities of approximating functions lie outside its closure. Theorem 9.4. Let $X$ be a compact set in $\mathbb R^2$ and $\mathcal L$ be an operator of the form considered above. Then $\mathcal P^0_{\mathcal L}(X)=C^0_{\mathcal L}(X)$ if and only if for each connected component $G$ of the set $(\widehat{X})^\circ$ that satisfies the condition $G\cap(\mathbb C\setminus X)\ne \varnothing$ the equality holds, where $X_G=\overline{G}\cap X$. This result allows us to reduce the problem of the equality of the spaces $\mathcal P_{\mathcal L}(X)$ and $C_{\mathcal L}(X)$ for a compact set $X$ to the approximation of functions by $\mathcal L$-analytic functions with singularities localized in a special way on compact subsets of $X$ whose structure can be simpler. For example, for Carathéodory spaces we obtain the following criterion of approximation. Corollary 9.1. Let $X\subset\mathbb R^2$ be a Carathéodory compact set. Then $\mathcal P_{\mathcal L}(X)=C_{\mathcal L}(X)$ if and only if $\mathcal A_{\mathcal L}(\partial G,\overline{G})=C(\partial G)$ for each bounded connected component $G$ of the complement $\mathbb R^2\setminus X$. In this statement we deal with domains $G$ such that $\partial G=\partial G_\infty$, where $G_\infty$ is the unbounded connected component of the set $\mathbb R^2\setminus\overline{G}$. They are called Carathéodory domains, and it is easy to verify that a Carathéodory domain is simply connected and has the property $G=(\overline{G})^\circ$. Thus, to solve the problem of the approximation of functions by $\mathcal L$-analytic polynomials on Carathéodory compact sets we must solve the problem of the description of Catathéodory domains $G$ such that $\mathcal A_{\mathcal L}(\partial G,\overline{G})=C(\partial G)$. In the case of operators with equal characteristic roots this problem is quite easily reduced to the case considered above, when $\mathcal L$ is the Bitsadze operator. For operators $\mathcal L$ in $\mathbb R^2$ with distinct characteristic roots the problem of the description of Carathéodory domains $G$ such that $\mathcal A_{\mathcal L}(\partial G,\overline{G})=C(\partial G)$ was solved using the concept of an $\mathcal L$-special domain (a special analytic feature of Carathéodory domains), which can be regarded as an analogue of the concept of a Nevanlinna domain that arises in the bianalytic case. For an accurate definition of an $\mathcal L$-special domain for operators $\mathcal L$ under consideration we require one more technical notion. Let $G$ be a Carathéodory domain in $\mathbb R^2$ and $\varphi$ be a conformal mapping of the unit disc $\mathbb D$ onto $G$. We say that a holomorphic function $f$ in $G$ belongs to the space $AC(G)$ if the compositon $f\circ\varphi$ extends to a continuous function in $\overline{\mathbb D}$ that is absolutely continuous on $\mathbb T$. If $f\in AC(G)$ and $\zeta\in\partial G$ is an arbitrary attainable boundary point of $G$, then for any path $\varGamma$ in $G\cup\{\zeta\}$ with endpoint $\zeta$ the limit of $f$ along $\varGamma$ exists and has the same value $f(\zeta)$, which is called the boundary value of $f$ at $\zeta$. For second-order operators $\mathcal L$ in $\mathbb R^2$ with distinct characteristic roots $\lambda_1$ and $\lambda_2$ the concept of an $\mathcal L$-special domain was introduced in [64]–[66]. For such an operator $\mathcal L$ let $T_j$, $j=1,2$, be the transformations of $\mathbb R^2$ taking a point $(x_1,x_2)$ to $(x_1,x_2/\lambda_{3-j})$. Definition 9.2. A Carathéodory domain $G$ is said to be $\mathcal L$-special (for $\mathcal L$ under consideration) if there exist functions $F_1$ and $F_2$ in the classes $AC(T_1G)$ and $AC(T_2G)$, respectively, such that $F_1(T_1\zeta)=F_2(T_2\zeta)$ for each attainable boundary point $\zeta$ of $G$. For each operator $\mathcal L$ under consideration (with distinct characteristic roots) we can easily present an example of an $\mathcal L$-special domain. This is a domain bounded by a specially selected ellipse with parameters expressed by explicit formulae in terms of $\lambda_1$ and $\lambda_2$. Using examples of Nevanlinna domains we can also present a number of examples of domains that are not $\mathcal L$-special for any operator $\mathcal L$. Unfortunately, so far we know of no other explicit examples of $\mathcal L$-special domains. Neither do we know of any results concerning the properties of conformal or univalent harmonic mappings of a disc onto such domains. The properties of $\mathcal L$-domains is an open problem which is yet to be solved. Using the concept of an $\mathcal L$-special domain we can state the following result. Theorem 9.5. Let $G$ be a Carathéodory domain in $\mathbb R^2$, and let $\mathcal L$ be an operator with distinct characteristic roots. Then the following results hold. Originally, this result was obtained in [65], and the recent paper [67] contains a new proof of it, which is significantly simpler and is based on some recent results in [63] on the properties of measures orthogonal to rational functions on Carathéodory compact sets (also see [68]). Corollary 9.2. Let $X\subset\mathbb R^2$ be a Carathéodory compact set and the operator $\mathcal L$ be strongly elliptic. Then $\mathcal P_{\mathcal L}(X)=C_{\mathcal L}(X)$. Thus, the sufficient condition for approximation $\partial X=\partial\widehat{X}$ which arises in the Walsh–Lebesgue theorem for approximation by harmonic polynomials also holds for a general strongly elliptic operator $\mathcal L$. It is quite a deep and interesting question, whether this condition is necessary for $\mathcal L$-analytic polynomial approximation in the case of a general second-order strongly elliptic operator in $\mathbb R^2$. Let us repeat a conjecture (stated originally in [24]) which we think to be fairly plausible. Conjecture 9.1. Let $\mathcal L$ be a strongly elliptic operator and $X$ be a compact set in $\mathbb R^2$. Then $\mathcal P_{\mathcal L}(X)=C_{\mathcal L}(X)$ if and only if $X$ is a Carathéodory compact set, that is, $\partial X=\partial\widehat{X}$. This could formally also be conjectured for $\mathcal L$ that is not strongly elliptic. However, the conjecture fails for such operators. In fact, given such an operator $\mathcal L$, we can find a nowhere dense compact set $X$ (the union of a specially selected ellipse and its centre) that is not a Carathéodory compact set but satisfies $C(X)=\mathcal P_{\mathcal L}(X)$. Note that a critically important ingredient in the proof of the Walsh–Lebesgue criterion for harmonic functions is Lebesgue’s classical result [46] of 1907 that the harmonic Dirichlet problem is solvable for any continuous boundary data in an arbitrary bounded simply connected domain in $\mathbb R^2$. In other words, it says that each bounded simply connected plane domain is regular for the Dirichlet problem under consideration. For general second-order strongly elliptic operators $\mathcal L$ with constant complex coefficients the description of domains regular for the corresponding $\mathcal L$-analytic Dirichlet problem is an open question. There is a conjecture that each bounded simply connected domain must be regular for any such $\mathcal L$, but so far regularity has only been established for domains with quite nice boundaries. For example, each Jordan domain with piecewise $C^1$-smooth boundary is regular for the $\mathcal L$-analytic Dirichlet problem, for each strongly elliptic operator $\mathcal L$ (see [69]), but even for Jordan domains with rectifiable boundaries the corresponding result is unknown. In the conclusion of this section we discuss quite briefly the case $N\geqslant3$. Then, just as for $N=2$, we can use Runge’s method of the motion of singularities and conclude that the complement being connected in combination with the equality of the spaces $\mathcal A^m_{\mathcal L}(X)$ and $C^m_{\mathcal L}(X)$ (for $m$ and $\mathcal L$ in question) is sufficient for equality (90) to hold for $X$ under consideration. However, no full criterion for approximation is known. In the harmonic case (when $\mathcal L=\Delta$), for $m\geqslant1$ these sufficient conditions are also necessary; see [49], Theorems 1.7 and 1.8. For $N\geqslant3$ and $m\in(0,1)$ there also is a connection between problems of $C^m$-approximation of functions by harmonic polynomials and the property of $C^m$-continuity of the Poisson operator $P_D$, which is well defined for simple Carathéodory domains. However, the concept of a simple Carathéodory domain $D$ must be modified in $\mathbb R^N$, $N\geqslant3$: the assumptions that $\Omega=\mathbb R^N\setminus\overline{D}$ is connected and $\partial D=\partial\Omega$ must be complemented by the requirement that both $D$ and $\Omega$ are regular for the harmonic Dirichlet problem (Lebesgue’s theorem mentioned above does not hold in dimension $N\geqslant3$, and there are examples of bounded simply connected domain that are not regular in the above sense). Note that the first part of Proposition 9.1 also holds for $N\geqslant3$. Moreover, for domains with sufficiently regular boundaries in $\mathbb R^N$ for $N\geqslant3$, [50] contains explicit expressions for $m_D$ (we do not present neither the relevant conditions of regularity, which are stated in terms of the outer cone condition, nor the explicit expressions for $m_D$ because they are slightly cumbersome). As in dimension 2, it is easy to show that for a simple Carathéodory domain $D\subset\mathbb R^N$, if the Poisson operator $P_D$ is $C^m$-continuous, then the relations $\mathcal P^m_\Delta(\overline{D})=C^m_\Delta(\overline{D})$ and $\mathcal P^m_\Delta(\partial D)=C^m(\partial D)$ are equivalent. As shown above, $\mathcal P^m_\Delta(\overline{D})=C^m_\Delta(\overline{D})$ if and only if for $\mathcal M_*^{N-2+m}$-almost all points $\boldsymbol{a}\in\partial D$ we have
$$
\begin{equation}
\limsup_{r\to0}\frac{\mathcal M^{N-2+m}(B(\boldsymbol{a},r)\setminus \overline{D})}{r^{N-2+m}}>0.
\end{equation}
\tag{92}
$$
Thus, for $m\in(0,m_D)$ condition (92) is necessary and sufficient for the equality $\mathcal P^m_\Delta(\partial D)=C^m(\partial D)$ to hold. Finally, it should be observed that the equalities $\mathcal P^m_\Delta(\overline{D})=C^m_\Delta(\overline{D})$ and $\mathcal P^m_\Delta(\partial D)=C^m(\partial D)$ can also hold simultaneously in the case when $P_D$ is not $C^m$-continuous (see [49], Proposition 3.5). The authors are grateful to the referees for their valuable comments, which helped us to improve the presentation. Part of this work was carried out when the third-named author was visiting the Sirius Mathematical Center in the framework of the Program of Small Research Groups. K. Yu. Fedorovskiy is sincerely grateful to the Sirius Mathematical Center for their hospitality and perfect research ambiance.
Citation in format AMSBIB
\Bibitem{MazParFed24} |
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