| Izvestiya: Mathematics |
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M. Arsenovića, J. Gajićb, M. Mateljevića a Department of Mathematics, University of Belgrade, Belgrade, Serbia b Faculty of Natural Sciences and Mathematics, University of Banja Luka, Banja Luka, Bosnia and Herzegovina
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  § 1. Introduction and notationThe first line of research that motivates our investigations is concerned with $(\alpha, \beta)$-harmonic functions, that is, functions $u$ satisfying equation $L_{\alpha, \beta} u = 0$, where $L_{\alpha, \beta} = \Delta + \alpha L_1 + \beta L_2 + Q_{\alpha, \beta}$. Here $\Delta$ is the Laplace–Beltrami operator on a domain $\Omega$, $L_1$ and $L_2$ are linear differential operators of first order and $Q_{\alpha, \beta}$ is a zero order operator. An early paper touching on this topic is [1], where $\Omega$ is the Siegel upper half plane $\mathbf{U^{n+1}}$. There $(\alpha, \beta)$-harmonic functions were used as a tool for obtaining results on $H^p$ theory on the Heisenberg group $\partial \mathbf{U^{n+1}} = \mathbf{H^n}$. The case, where $\Omega$ is the unit ball in $\mathbb C^n$ was considered in [2], where series expansion of $(\alpha, \beta)$-harmonic functions was derived and $H^p$ theory results for such functions were obtained. A detailed investigation of series expansion of $(\alpha, \beta)$-harmonic functions in the unit disc can be found in [3]. In [4], the authors obtained Schwarz–Pick type inequality for $(\alpha, \beta)$-harmonic functions in the unit disc. Let us mention that special one parameter cases also appeared, see for example [5]–[11]. In the cited papers the operators $L_{\alpha, \beta}$ are uniformly elliptic, but degenerate near the boundary of the domain: the lower order terms $\alpha L_1$, $\beta L_2$ and $Q_{\alpha, \beta}$ are not dominated by the principal part unless $\alpha = \beta = 0$. A related class of operators, in dimension two, was extensively studied by I. N. Vekua in connection with the theory of generalized analytic functions, a standard reference is [12]. Operators of the form $Lu = \Delta u + A(z)u_z + B(z) u_{\overline z}$ were studied there, note that the term of zero order here vanishes. These operators are defined in more general domains than the unit disc, the coefficients $A(z)$ and $B(z)$ are more general then those from the above references. Estimates in Sobolev norm are obtained for solutions in terms of boundary values and inhomogeneous part. The operators that are investigated by Vekua include, as a very special case, the following operators acting on functions defined in the unit disc:
$$
\begin{equation*}
L^{0*}_\alpha u:=u_{\overline{z}z} + \alpha z (1-|z|^2)^{-1} u_z, \qquad L^{0}_\alpha u:=u_{\overline{z}z} + \alpha \overline{z} (1-|z|^2)^{-1} u_{\overline{z}},
\end{equation*}
\notag
$$
a function $u$ is said to be $\alpha$-harmonic if $L^{0*}_\alpha u=0$. This is one parameter version of so-called $(\alpha, \beta)$-harmonic functions, namely the case $(0, \alpha)$, see Definition 1 below. The second line of research relevant for us is the study of $n$-harmonic functions, that is functions defined on the unit polydisc $\mathbb D^n \subset \mathbb C^n$ which are harmonic in each of the $n$ variables $z_1, \dots, z_n$. Such functions appeared already in $1939$ (for $n = 2$) in the context of harmonic analysis, see [13]. In fact, the main results in that paper were on summability, in the sense of Cesàro and Abel, of multiple Fourier series. More details on $n$-harmonic functions can be found in Chapt. XVII of [14] and in [15]. A probabilistic approach to some aspects of $H^p$ theory of $n$-harmonic functions is given in [16], a more recent paper on $n$-harmonic functions is [17]. In this paper, we merge these two avenues of investigation. In particular, we show that a significant part of the theory of separately $(\alpha, \beta)$-harmonic functions can be developed without reference to Fourier analysis. An exception is the treatment of series expansion of separately $(\alpha, \beta)$-harmonic functions in § 2.1, which heavily relies on Fourier series methods. By $(\alpha, \beta)$-harmonic function we mean a function $u$ in the unit polydisc which satisfy a system of second order elliptic PDEs: $L_{\alpha_j, \beta_j} u = 0$, $1 \leqslant j \leqslant n$, see Definition 1. Each of the operators $L_{\alpha_j, \beta_j}$ acts on the $z_j = x_j + i y_j$ variable and annihilates the same functions as the Laplacian if $\alpha_j = \beta_j = 0$. Our results extend one dimensional results from [11], [3] and [10] on a single operator $L_{\alpha, \beta}$ to the multidimensional case of a system of operators, these results are also generalizations of results on $n$-harmonic functions. For this system of PDEs, like in the special $n$-harmonic case, a particular phenomenon appears: boundary value data specified on a thin portion of the boundary (the so-called distinguished boundary) uniquelly determine solution of the system. The paper is organized as follows. In this section we establish notation used throughout the paper. In § 2 we define separately $(\alpha, \beta)$-harmonic functions in the unit polydisc and derive series expansion for such functions, see Theorem 2. In the same section $(\alpha, \beta)$-Poisson integral of measures and functions on $\mathbb T^n$ is introduced and one dimensional results are reviewed as needed. In § 3 we prove Proposition 4 on existence and uniqueness of the solution of the Dirichlet problem with continuous boundary data and as a consequence, we obtain Proposition 5, which is an estimate of a family of hypergeometric functions. A deeper result on continuity at a point $\zeta \in \mathbb T^n$ is given in Theorem 9. Next $H^p$ spaces of separately $(\alpha, \beta)$-harmonic functions are defined, integral representations are obtained and convergence in norm and in the weak$^\ast$ sense is discussed. We should mention an unexpected role of the above mentioned estimate of a family of hypergeometric functions. This estimate is a key in proving Theorem 8, which in turn is vital in proving results in § 3 on integral representations. The final § 4 contains results on relevant maximal functions, the main one is Theorem 16. It is used to derive Fatou type theorems, for example Theorem 17. We should point out that Corollary 2 is new even in dimension one. Since we are dealing with measures and $L^1$ functions on $\mathbb T^n$ our results are on the so-called restricted limits, we plan to treat unrestricted limits in another publication. For a discussion of restricted and non-restricted limits as well as approach regions, in a more general setting of symmetric spaces, see [18]. We use standard notation and terminology. The boundary of the unit disc $\mathbb D = \{ z \in \mathbb C \colon | z | < 1 \}$ is $\mathbb T = \{ \zeta \in \mathbb C \colon | \zeta | = 1 \}$. Given $\zeta = e^{i\varphi}$ in $\mathbb T$ we denote the Stolz region at $\zeta$ of aperture $0 < A< +\infty$ by
$$
\begin{equation*}
S_A (e^{i\varphi}) = \{ r e^{i (\varphi - \theta)} \colon | \theta | \leqslant A(1-r) \}.
\end{equation*}
\notag
$$
Non-tangential limit at a boundary point is denoted by NT-$\lim$. The unit polydisc $\mathbb D^n$ in $\mathbb C^n$ is the Cartesian product of $n$ copies of $\mathbb D$, and the unit torus $\mathbb T^n$ in $\mathbb C^n$ is the Cartesian product of $n$ copies of $\mathbb T$. If $n>1$, $\mathbb T^n$ is only a small part of the boundary of $\mathbb D^n$ and the unit torus is usually called the distinguished boundary of $\mathbb D^n$. We set $\mathbf{1} = (1, \dots, 1)$ and $\mathbf{0} = (0, \dots, 0)$. For $z = (z_1, \dots, z_n) \in \mathbb C^n$ we set $ \overline z = (\overline{z_1}, \dots, \overline{z_n})$. For $z, w \in \mathbb C^n$ we set $z \cdot w = (z_1 w_1, \dots, z_n w_n)$. We also write $z = (z', z_n)$, where $z = (z_1, \dots, z_n)$ is in $\mathbb C^n$ and $z' = (z_1, \dots, z_{n-1})$ is in $\mathbb C^{n-1}$. Euclidean balls in $\mathbb C^n$ are denoted by $B(z, r)$. We will use the normalized Haar measure $ m_n$ on $\mathbb T^n$. So, we have $m_n(\mathbb T^n)\,{=}\,1$. The space of complex Borel measures on $\mathbb T^n$ is denoted by $\mathcal M (\mathbb T^n)$, it is the dual space of $C(\mathbb T^n)$. The pairing of duality is given by
$$
\begin{equation*}
\langle \varphi, d\mu \rangle = \int_{\mathbb T^n} \varphi \, d\mu, \qquad \varphi \in C(\mathbb T^n), \quad \mu \in \mathcal M (\mathbb T^n).
\end{equation*}
\notag
$$
The total variation of $\mu$ is denoted by $\| \mu \|$, it is equal to $|\mu| (\mathbb T^n)$. For $z$ in $\mathbb D^n$ and a function $u \colon \mathbb D^n \to \mathbb C$ we define $u_z \colon \mathbb T^n \to \mathbb C$ by $u_z (\zeta) = u(z \cdot \zeta)$. If $z = (r, r, \dots, r)$, where $0 \leqslant r < 1$ we write $u_r$ instead of $u_z$. In particular, if $\mathbf{r} = (r_1, \dots, r_n) \in [0, 1)^n$, then $u_{\mathbf{r}}(\zeta) = u(r_1 \zeta_1, \dots, r_n\zeta_n)$. Also, for $\zeta \in \mathbb T^n$ we define a function $\zeta \cdot u$ on $\mathbb D^n$ by $\zeta \cdot u(z) = u(\zeta \cdot z)$. Clearly, this defines an action of the group $\mathbb T^n$ on the set of all functions on $\mathbb D^n$. Given a real number $x$, We set $x^+ = \max (x, 0)$ and $x^- = \max (-x, 0)$. For $m = (m_1, \dots, m_n)$ in $\mathbb Z^n$, we also define $m^+ = (m_1^+, \dots, m_n^+) \in \mathbb Z^n_+$, and similarly, $m^- = (m_1^-, \dots, m_n^-) \in \mathbb Z^n_+$. $| m | = | m_1 | + \dots + | m_n |$ is the order of a multiindex $m \in \mathbb Z^n$. Also, $m! = m_1! \cdots m_n!$ for $m \in \mathbb Z^n_+$. For $p = (p_1, \dots, p_n)$ in $\mathbb Z_+^n$ and $z \in \mathbb C^n$ we set $z^p = z_1^{p_1} \cdots z_n^{p_n}$, $\overline z^p = \overline z_1^{p_1} \cdots \overline z_n^{p_n}$. If $\zeta = (\zeta_1, \dots, \zeta_n) $ is in $\mathbb C^n$ and $\zeta_j \ne 0$ for all $1 \leqslant j \leqslant n$, then $\zeta^m = \zeta_1^{m_1} \cdots \zeta_n^{m_n}$ is defined for all $m \in \mathbb Z^n$. The order of a multiindex $(p, q)$ of length $2n$ is $| (p, q) | = | p | + | q |$. We set
$$
\begin{equation*}
\partial^{(p, q)} = \frac{\partial^{p_1}}{\partial z_1^{p_1}} \cdots \frac{\partial^{p_n}}{\partial z_n^{p_n}} \, \frac{\partial^{q_1}}{\partial \overline z_1^{q_1}} \cdots \frac{\partial^{q_n}}{\partial \overline z_n^{q_n}}, \qquad \partial^p = \partial^{(p, 0)}, \quad {\overline\partial}^q = \partial^{(0, q)}.
\end{equation*}
\notag
$$
Here $\partial/\partial z_j = 2^{-1}(\partial/\partial x_j - i \,\partial/\partial y_j)$ and $\partial/\partial \overline z_j = 2^{-1}(\partial/\partial x_j + i \, \partial/\partial y_j)$ are standard first order linear differential operators. The space $C^\infty (\mathbb D^n)$ is a locally convex complete metrizable topological vector space, it is endowed with the following family of seminorms:
$$
\begin{equation*}
\| u \|_{p, q, K} = \max_{z \in K} | \partial^{(p, q)} u(z) |, \qquad K \subset \mathbb D^n \quad \text{is compact}, \quad p, q \in \mathbb Z^n_+.
\end{equation*}
\notag
$$
For complex numbers $ a, b$ and $c$ such that $c \ne 0, -1, -2, \dots$, the hypergeometric function $F(a, b; c; z)$ is defined by
$$
\begin{equation}
F(a, b; c; z) = \sum_{k=0}^\infty \frac{(a)_k (b)_k}{(c)_k} \, \frac{z^k}{k!}, \qquad | z | < 1,
\end{equation}
\tag{1.1}
$$
where $(a)_k $ is the Pochhammer symbol:
$$
\begin{equation*}
(a)_0 = 1 \quad \text{and} \quad (a)_k = a(a+1) \cdots (a+k-1)\quad \text{for } \ k = 1, 2, \dots\, .
\end{equation*}
\notag
$$
If $\operatorname{Re} (a+ b - c) < 0$, then the series in (1.1) converges absolutely for $| z | \leqslant 1$. More details about the hypergeometric functions can be found in [19].
§ 2. Separately $(\alpha, \beta)$-harmonic functions in $\mathbb D^n$ Definition 1. Let $\alpha = (\alpha_1, \dots, \alpha_n)$ and $\beta = (\beta_1, \dots, \beta_n)$ are in $\mathbb C^n$. A function $ u $ in $C^2( \mathbb D^n)$ is separately $ (\alpha, \beta)$-harmonic if $L_{\alpha_j, \beta_j} u=0$ for $1\leqslant j\leqslant n$, where Note that $L_{\alpha_j, \beta_j}$ is a second order linear partial differential operator which acts only with respect to $z_j = x_j + iy_j$ variable, hence the term “separately”. The first order coefficients $\alpha_j z_j$ and $\beta_j \overline{z_j}$ are not dominated by $1 - | z_j |^2$, unless $\alpha_j = \beta_j\,{=}\, 0$, thus although the operator $L_{\alpha_j, \beta_j}$ is uniformly elliptic when considered as an operator acting on functions of a single varable in $\mathbb D$, it is degenerate. If $n > 1$ the operators $L_{\alpha_j, \beta_j}$ are not elliptic as operators acting on functions defined in $\mathbb D^n$. If $\alpha_j = \beta_j = 0$, then $L_{\alpha_j, \beta_j} = 4^{-1} (1 - | z_j |^2) \Delta_j$, where $\Delta_j$ is the Laplace operator with respect to $x_j$ and $y_j$ variables. Thus $\operatorname{sh}_{0, 0}(\mathbb D^n)$ is the space of $n$-harmonic functions. Even in this special case we do not get an algebra, because the square of a harmonic function is not a harmonic function in general. Proof. Any separately $(\alpha, \beta)$-harmonic function $u$ is a solution of a linear elliptic partial differential equation $Lu = 0$, where $L = L_{\alpha_1, \beta_1} + \dots + L_{\alpha_n, \beta_n}$ has $C^\infty$ coefficients. Therefore, by elliptic regularity, $u$ is in $C^\infty (\mathbb D^n)$. Lemma 1 is proved. Since
$$
\begin{equation*}
\partial_z[u(\zeta z)] = \zeta (\partial_z u) (\zeta z), \quad \partial_{\overline z} [u(\zeta z)] = \overline\zeta (\partial_{\overline z} u)(\zeta z), \qquad \zeta \in \mathbb C, \quad u \in C^1(\mathbb D),
\end{equation*}
\notag
$$
we get
$$
\begin{equation}
L_{\alpha, \beta}[u(\zeta z)] = (1 - |z|^2) |\zeta|^2 \,\frac{\partial^2 u}{\partial z \, \partial \overline z} (\zeta z) + \alpha \zeta z (\partial_z u)(\zeta z) + \beta \overline {\zeta z} (\partial_{\overline z} u)(\zeta z) - \alpha\beta u(\zeta z).
\end{equation}
\tag{2.1}
$$
A consequence of this formula is the following rotational invariance of $L_{\alpha, \beta}$, which appeared in [3]: $L_{\alpha, \beta}(\zeta \cdot u) = \zeta \cdot (L_{\alpha, \beta} u)$ if $| \zeta | = 1$. This rotational invariance extends immediately to any dimension $n \geqslant 1$:
$$
\begin{equation}
L_{\alpha_j, \beta_j}(\zeta \cdot u) = \zeta \cdot (L_{\alpha_j, \beta_j} u), \qquad \zeta \in \mathbb T^n, \quad 1 \leqslant j \leqslant n.
\end{equation}
\tag{2.2}
$$
2.1. Series expansion of separately $(\alpha, \beta)$-harmonic functionsOur goal is to extend the following theorem to the $n$-dimensional setting. Theorem 1 (see Theorems 5.1 and 5.3 in [3]). Let $u$ be $(\alpha, \beta)$-harmonic in $\mathbb D$. Then We say $(p, q) \in \mathbb Z^n_+ \times \mathbb Z^n_+$ is pure if $p \cdot q = (p_1 q_1, \dots, p_n q_n) = \mathbf{0}$. We denote the set of all pure multiindexes $(p, q)$ by $H$. We will use without comment the natural bijection $\mathbb Z^n \ni m \mapsto (m^+, m^-) \in H$ to change the index of summation. Let $z^{(m)}$ denote $z^{m^+} {\overline z}^{m^-}$, where $m \in \mathbb Z^n$ and $z \in \mathbb C^n$. Let us assume, for a moment, that $n=1$. Then for a pure $(p, q)$, which in this case means $p=0$ or $q=0$, we set
$$
\begin{equation*}
F_{p,q} (z) = F(p-\beta, q - \alpha; p + q + 1; z), \qquad | z | < 1.
\end{equation*}
\notag
$$
With this notation at hand we can write (2.3) as
$$
\begin{equation}
u(z) = \sum_{(p,q) \in H} \frac{ \partial^{(p, q)}u}{p!\, q!} (0) \, F_{p,q}(| z |^2) z^p \overline z^q, \qquad | z | < 1.
\end{equation}
\tag{2.4}
$$
Returning to the general case $n \geqslant 1$, for $(p, q) \in H$ we define
$$
\begin{equation}
\mathcal F_{p, q} (z) = \prod_{j=1}^n F_{p_j, q_j} ( | z_j |^2), \qquad z \in \mathbb D^n.
\end{equation}
\tag{2.5}
$$
Theorem 2. Let $u$ be separately $(\alpha, \beta)$-harmonic in $\mathbb D^n$. Then and the convergence is in the topology of the space $C^\infty (\mathbb D^n)$. We prove this theorem closely following methods used by Klintborg and Olofsson ([3], $n=1$). We need some preliminary concepts and results. Definition 2. A continuous function $u$ on $\mathbb D^n_0 = \{ z \in \mathbb D^n \colon \prod_j z_j \ne 0 \}$ is said to be $m$-homogeneous, where $m \in \mathbb Z^n$, if This means $u(e^{i\theta_1} z_1, \dots, e^{i\theta_n} z_n) = e^{im_1 \theta_1} \cdots e^{im_n\theta_n} u(z)$ for all $z \in \mathbb D^n_0$ and $\theta_j \in \mathbb R$. Let us note, see [3], that for every $(p, q) \in H$ the function $\mathcal F_{p,q}(z) z^p \overline z^q$ is separately $(\alpha, \beta)$-harmonic in $\mathbb D^n$ and $(p-q)$-homogeneous. Definition 3. For $u \in C(\mathbb D^n)$ and $m \in \mathbb Z^n$, the $m$th homogeneous component $u_m$ of $u$ is defined by Note that $u_m$ is $m$-homogeneous by a simple change of variables. If $u \in C^k(\mathbb D^n)$, then, by differentiation under the integral sign, $u_m \in C^k(\mathbb D^n)$ for $0 \leqslant k \leqslant \infty$. Proposition 1. Let $u_m(z)$ be the $m$th homogeneous component of $u \in C^\infty (\mathbb D^n)$ for $m \in \mathbb Z^n$. Then Proof. We fix a compact set $K = r \overline{\mathbb D}^n$ and $(\gamma, \delta) \in \mathbb Z_+^n \times \mathbb Z^n_+$. For a given $m \in \mathbb Z^n$ we set $l_j = N$ if $m_j \ne 0$ and $l_j = 0$ otherwise. This gives us $l \in \mathbb Z_+^n$ with $| l | \leqslant Nn$. We can differentiate under the integral sign and perform partial integration to obtain
$$
\begin{equation*}
\begin{aligned} \, \partial_z^{(\gamma, \delta)} u_m(z) &= \int_{-\pi}^\pi \dots \int_{-\pi}^\pi \partial_z^{(\gamma, \delta)} u(e^{i\theta_1}z_1, \dots, e^{i\theta_n}z_n) \prod_{j=1}^n e^{-i m_j\theta_j} \, \frac{d\theta_1 \cdots d\theta_n}{ (2\pi)^n} \\ &= \frac{(2\pi)^{-n}}{(im_1)^{l_1} \cdots (im_n)^{l_n}} \int_{-\pi}^\pi \dots \int_{-\pi}^\pi \partial^l_\theta \, \partial_z^{(\gamma, \delta)} u (e^{i\theta_1}z_1, \dots, e^{i\theta_n}z_n) \\ &\qquad\times \prod_{j=1}^n e^{-i m_j\theta_j} \, d\theta_1 \cdots d\theta_n, \qquad z \in \mathbb D^n. \end{aligned}
\end{equation*}
\notag
$$
This implies, for $z \in K$,
$$
\begin{equation}
| \partial_z^{(\gamma, \delta)} u_m(z) | \leqslant \max_{| (s, t) | \leqslant | \gamma | + | \delta | + Nn} \|\partial^{(s, t)} u \|_{C(K)} \prod_{j=1}^n \frac{1}{m_j^N}, \qquad N \in \mathbb N,
\end{equation}
\tag{2.10}
$$
in the above product, $1/m_j^N$ is interpreted as $1$ if $m_j = 0$. Since the multiple series $\sum_{m \in \mathbb Z^n_+} m_1^{-2}\cdots m_n^{-2}$ converges, this ensures the convergence in the $C^\infty (\mathbb D^n)$ topology of the multiple series in (2.9). The sum converges, for a fixed $z \in \mathbb D^n$ to $u(z)$. Indeed, $u_m(z)$ is the $m$th Fourier coefficient of the function $u_z \in C^\infty (\mathbb T^n)$ and therefore $u_z(\zeta) = \sum_{m \in \mathbb Z^n} u_m(z) \zeta^m$. Taking $\zeta = \mathbf{1}$ we obtain (2.9). Proposition 1 is proved. Lemma 2 (see Proposition 4.2 for $n=1$ in [3]). Let $u \in \operatorname{sh}_{\alpha, \beta} (\mathbb D^n)$. Then for every $m \in \mathbb Z^n$ its homogeneous part $u_m$ is also separately $(\alpha, \beta)$-harmonic. Proof. Using (2.2) we obtain, for every $1 \leqslant j \leqslant n$,
$$
\begin{equation*}
L_{\alpha_j, \beta_j} u_m(z) = L_{\alpha_j, \beta_j} \int_{\mathbb T^n} \overline\zeta^m (\zeta \cdot u)(z) \, dm_n(\zeta) = \int_{\mathbb T^n} \overline\zeta^m L_{\alpha_j, \beta_j} (\zeta \cdot u)(z) \, dm_n(\zeta) = 0.
\end{equation*}
\notag
$$
Lemma 2 is proved. Proposition 2 (see Theorem 2.4 for $n=1$ in [3]). Assume that $u \in \operatorname{sh}_{\alpha, \beta} (\mathbb D^n)$ is $m$-homogeneous for some $m \in \mathbb Z^n$. Then, for some constant $c$, Conversely, every function $u(z)$ of the above form is $m$-homogeneous and separately $(\alpha, \beta)$-harmonic. Proof. We proceed by induction on $n$. The case $n=1$ is Theorem 2.4 in [3]. Now let $n > 1$ and assume that the proposition holds for $n-1$. Let us fix $z' \in \mathbb D^{n-1}$. The function $U_{z'} (z_n) = u(z', z_n)$ is $(\alpha_n, \beta_n)$-harmonic. Since the theorem holds for $n=1$ we have $u(z) = v(z') F_{m_n^+, m_n^-} (| z_n |^2) z_n^{(m_n)}$. Clearly, $v$ is $(\alpha', \beta')$-harmonic in $\mathbb D^{n-1}$ and $m' = (m_1, \dots, m_{n-1})$-homogeneous. Therefore, the inductive hypothesis provides us representation (2.11). The converse follows from the fact that each of the functions $F_{m_j^+, m_j^-} (| z_j |^2) z_j^{(m_j)}$ is $(\alpha_j, \beta_j)$-harmonic in $\mathbb D$. Proposition 2 is proved. Proof of Theorem 2. The function $u$ is in $C^\infty (\mathbb D^n)$ by Lemma 1, hence Proposition 1 gives us the expansion $u(z) = \sum_{m \in \mathbb Z^n} u_m(z)$ of $u$ into its $m$th homogeneous components $u_m$, where the convergence is in the topology of $C^\infty (\mathbb D^n)$. By Lemma 2 each $u_m$ is separately $(\alpha, \beta)$-harmonic. Next, Proposition 2 gives representation of each $u_m$ in the form given in (2.11). Therefore,
$$
\begin{equation}
u(z) = \sum_{m \in \mathbb Z^n} c_m \prod_{j=1}^n F_{m_j^+, m_j^-} (| z_j |^2) {z_j}^{(m_j)} = \sum_{(t,s) \in H} c_{t, s} \mathcal F_{t, s} (z) z^t \overline z^s, \qquad z \in \mathbb D^n.
\end{equation}
\tag{2.12}
$$
It remains to show that $c_{p, q} = \partial^{(p, q)}u (0)/(p!\, q!)$ for each pure multiindex $(p, q) $ in $\mathbb Z^n_+ \times \mathbb Z^n_+$. Fix $(p, q)$ in $H$, set $k = | p | + | q |$ and consider the Taylor expansion of $u$ at zero:
$$
\begin{equation}
u(z) = \sum_{j=0}^k \sum_{| t | + | s | = j} \frac{\partial^{(t, s)}u(0)}{t! \, s!} z^t \overline z^s + O(\| z \|^{k+1}).
\end{equation}
\tag{2.13}
$$
Using (2.13) and orthogonality relations for the standard basis for $L^2(\mathbb T^n)$, we obtain
$$
\begin{equation}
I(r) = \int_{\mathbb T^n} u_r(\zeta) \overline\zeta^p \zeta^q \, dm_n(\zeta) = \frac{\partial^{(p, q)}u(0)}{p! \, q!}r^k + O(r^{k+1}).
\end{equation}
\tag{2.14}
$$
On the other hand, $F(a, b; c; 0) = 1$ and hence $\mathcal F_{t, s} (0) = 1$ for all $(s, t)$ in $H$. Setting $z = r\zeta$ in (2.12) and using the same orthogonality relations we obtain This combined with (2.14) gives us the desired formula for $c_{p,q}$. Theorem 2 is proved. 2.2. $(\alpha, \beta)$-Poisson integrals Theorem 3 (see Theorem 1.4 and Theorem 6.4 in [3]). Let $\alpha, \beta \in \mathbb C$. Then the function We also define, for complex $\alpha, \beta \notin \mathbb Z^- = \{-1, -2, \dots \}$ and $\operatorname{Re} \alpha + \operatorname{Re} \beta > -1$,
$$
\begin{equation}
c_{\alpha, \beta}=\frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{\Gamma(\alpha + \beta + 1)},
\end{equation}
\tag{2.17}
$$
$$
\begin{equation}
v_{\alpha, \beta}(z) = c_{\alpha, \beta} u_{\alpha, \beta}(z), \qquad z \in \mathbb D.
\end{equation}
\tag{2.18}
$$
For $|\zeta| = 1$ the function $f(z) = u_{\alpha, \beta}(\zeta z)$ is $(\alpha, \beta)$-harmonic in $\mathbb D$, by rotational invariance (2.2) and Theorem 3. On the other hand, taking $\zeta = r$ in (2.1) we obtain
$$
\begin{equation}
L_{\alpha, \beta}[u(rz)] = (L_{\alpha, \beta} u)(rz) - (1-r^2) \, \frac{\partial^2 u}{\partial z \, \partial \overline z} (rz), \qquad 0 < r < 1,
\end{equation}
\tag{2.19}
$$
which shows that for $(\alpha, \beta)$-harmonic function $u$ the function $z \mapsto u(rz)$ need not be $(\alpha, \beta)$-harmonic. From this point on we assume that $\alpha, \beta \in \mathbb C^n$ satisfy the conditions
$$
\begin{equation}
\operatorname{Re} \alpha_j + \operatorname{Re} \beta_j > -1 \quad \text{and} \quad \alpha_j, \beta_j \notin \mathbb Z^- \quad \text{for all} \quad 1 \leqslant j \leqslant n.
\end{equation}
\tag{2.20}
$$
Definition 4. We define $(\alpha, \beta)$-Poisson kernel by If $ \operatorname{Re}\alpha_j=\alpha_j=\beta_j>-1/2$ for all $ j=1,\dots, n, $, then $ P_{\alpha, \beta}$ is real and positive. This case will be important in § 4. On the other hand, the case $n=1$, $\alpha = 0$ and $\beta > -1$ appeared in [11]. Setting
$$
\begin{equation}
V_{\alpha, \beta}(z) = \prod_{j=1}^n v_{\alpha_j, \beta_j}(z_j), \qquad z \in \mathbb D^n,
\end{equation}
\tag{2.22}
$$
we can write, for $z \in \mathbb D^n$ and $\zeta \in \mathbb T^n$,
$$
\begin{equation}
P_{\alpha, \beta}(z, \zeta) = \prod_{j=1}^n P_{\alpha_j, \beta_j}(z_j, \zeta_j) = \prod_{j=1}^n v_{\alpha_j, \beta_j}(z_j \overline \zeta_j) = V_{\alpha, \beta}(z \cdot \overline\zeta).
\end{equation}
\tag{2.23}
$$
By Theorem 3 and (2.2) we see that $u_{\alpha_j, \beta_j}(\overline \zeta_j z_j)$ is $(\alpha_j, \beta_j)$-harmonic for $1 \leqslant j \leqslant n$. Thus the following lemma immediately follows. Lemma 3. The $(\alpha, \beta)$-Poisson kernel $P_{\alpha, \beta}(z, \zeta)$ is separately $(\alpha, \beta)$-harmonic with respect to $z \in \mathbb D^n$ for each fixed $\zeta \in \mathbb T^n$. Definition 5. We define $(\alpha, \beta)$-Poisson integral of $\mu \in \mathcal M(\mathbb T^n)$ by the formula For $f \in L^1(\mathbb T^n)$ we define $P_{\alpha, \beta}[f] = P_{\alpha, \beta}[f\, dm_n]$, that is We use the same notation $P_{\alpha, \beta}$ for the operator and its integral kernel, this abuse of notation is quite harmless. Let $\mu \in \mathcal M(\mathbb T^n)$ and $\mathbf{r} = (r_1, \dots, r_n) \in [0, 1)^n$. For $u = P_{\alpha, \beta}[d\mu]$ we have
$$
\begin{equation}
u(r_1 e^{i t_1}, \dots, r_n e^{i t_n}) = \int_{\mathbb T^n} \prod_{j=1}^n v_{\alpha_j, \beta_j} (r_j e^{i (t_j - \zeta_j)}) \, d\mu (\zeta)
\end{equation}
\tag{2.26}
$$
and therefore
$$
\begin{equation}
(P_{\alpha, \beta}[d\mu])_{\mathbf{r}} = (V_{\alpha, \beta})_{\mathbf{r}} \ast d\mu,
\end{equation}
\tag{2.27}
$$
where the convolution is taken with respect to the group structure of $\mathbb T^n$. Taking $d\mu = f \, dm_n$ in (2.27) we obtain
$$
\begin{equation}
(P_{\alpha, \beta}[f])_{\mathbf{r}} = (V_{\alpha, \beta})_{\mathbf{r}} \ast f, \qquad f \in L^1(\mathbb T^n).
\end{equation}
\tag{2.28}
$$
From (2.15)–(2.18) and (2.22) using Fubini’s theorem we obtain
$$
\begin{equation}
\| (V_{\alpha, \beta})_{\mathbf{r}} \|_{L^1(\mathbb T^n)} \leqslant K(\alpha, \beta), \qquad \mathbf{r} \in [0, 1)^n,
\end{equation}
\tag{2.29}
$$
where
$$
\begin{equation*}
\begin{aligned} \, K(\alpha, \beta) &= \exp\biggl\{ \frac{\pi}{2} \sum_{j=1}^n |{\operatorname{Im} \alpha_j - \operatorname{Im} \beta_j}|\biggr\} \\ &\qquad\times\prod_{j=1}^n \biggl| \frac{\Gamma(\alpha_j + 1)\Gamma(\beta_j + 1)}{\Gamma(\alpha_j + \beta_j + 1)} \biggr| \frac{\Gamma(\operatorname{Re} \alpha_j + \operatorname{Re} \beta_j +1)}{\Gamma^2 ((\operatorname{Re} \alpha_j + \operatorname{Re} \beta_j)/2+1)}. \end{aligned}
\end{equation*}
\notag
$$
Inequality (2.29) can also be written as
$$
\begin{equation}
\int_{\mathbb{T}^n}| P_{\alpha,\beta}(z, \zeta) | \, dm_n(\zeta) \leqslant K(\alpha, \beta), \qquad z \in \mathbb D^n.
\end{equation}
\tag{2.30}
$$
An immediate consequence of Lemma 3 is the next proposition. Proposition 3. For any complex Borel measure $\mu$ on $\mathbb T^n$ its $(\alpha, \beta)$-Poisson integral $u = P_{\alpha, \beta} [d\mu]$ is separately $(\alpha, \beta)$-harmonic function on $\mathbb D^n$. § 3. $H^p$ theory for $\operatorname{sh}_{\alpha, \beta} (\mathbb D^n)$ functions3.1. Dirichlet problem for the system $L_{\alpha_j, \beta_j} u = 0$Let $ \varphi \in C(\mathbb T)$. Consider the following Dirichlet problem: find a function $u$ in $C(\overline{\mathbb D})$ such that $L_{\alpha, \beta} u = 0$ in $\mathbb D$ and $u = \varphi$ on $\mathbb T$. Theorem 4 (see Theorem 7.1 in [3]). The above Dirichlet problem has a unique solution $u \in C(\overline{\mathbb D})$, it is given by We generalize this result to separately $(\alpha, \beta)$-harmonic functions with continuous boundary data on $\mathbb T^n$, see Proposition 4 below. We begin with the following definition. Definition 6. We define $\operatorname{sh}_{\alpha, \beta}^\infty (\mathbb D^n) = \operatorname{sh}_{\alpha, \beta}(\mathbb D^n) \cap L^\infty (\mathbb D^n)$ and The space $C\mathrm{sh}_{\alpha, \beta}(\mathbb D^n)$ can be identified with a subspace of $\operatorname{sh}_{\alpha, \beta}(\mathbb D^n)$ consisting of functions which have continuous extension to $\overline{\mathbb D^n}$. Clearly, $C\mathrm{sh}_{\alpha, \beta}(\mathbb D^n) \subset \operatorname{sh}_{\alpha, \beta}^\infty (\mathbb D^n)$ and both spaces are Banach spaces with respect to the supremum norm. Using the conditions $\operatorname{Re} \alpha_j + \operatorname{Re} \beta_j > -1$ and Proposition 2 we deduce that, for each $(p, q)$ in $H$, the function
$$
\begin{equation}
\varphi_{p,q}(z) = \mathcal F_{p,q} (z) z^p \overline z^q, \qquad z \in \overline{\mathbb D}^n,
\end{equation}
\tag{3.2}
$$
belongs to the space $C\mathrm{sh}_{\alpha, \beta}(\mathbb D^n)$. Theorem 5. If $\varphi \in L^\infty(\mathbb T^n)$, then $P_{\alpha,\beta}[\varphi] \in \operatorname{sh}_{\alpha, \beta}^\infty (\mathbb D^n) $. Moreover, In particular, $P_{\alpha, \beta}$ is a bounded linear operator from $L^\infty(\mathbb T^n)$ to $\operatorname{sh}_{\alpha, \beta}^\infty (\mathbb D^n)$. Proof. By Proposition 3 the function $P_{\alpha, \beta}[\varphi]$ is in $\operatorname{sh}_{\alpha, \beta}(\mathbb D^n)$. Now estimate (3.3) follows from (2.30) and (2.25). Theorem 5 is proved. Lemma 4. Assume $u \in C\mathrm{sh}_{\alpha, \beta}(\mathbb D^n)$, $1 \leqslant k < n$, and $\zeta_j \in \mathbb T$, $k < j \leqslant n$. Set $\alpha' = (\alpha_1, \dots, \alpha_k)$, $\beta' = (\beta_1, \dots, \beta_k)$. Then the function belongs to the space $C\mathrm{sh}_{\alpha', \beta'} (\mathbb D^k)$. Proof. Choose a sequence of points $w_l = (w_{k+1}^l, \dots, w_n^l)$ in $\mathbb D^{n-k}$ which converges to $\zeta = (\zeta_{k+1}, \dots, \zeta_n)$ and set
$$
\begin{equation*}
u_l(z_1, \dots, z_k) = u(z_1, \dots, z_k, w_{k+1}^l, \dots, w_n^l), \qquad (z_1, \dots, z_k) \in \overline{\mathbb D^k}.
\end{equation*}
\notag
$$
Then $u_l$ uniformly converges to $u_{\zeta_{k+1}, \dots, \zeta_n}$ on $\overline{\mathbb D^k}$. Clearly, $u_{\zeta_{k+1}, \dots, \zeta_n}$ is continuous on $\overline{\mathbb D^k}$. Since $L_{\alpha_j, \beta_j} u_l = 0$ on $\mathbb D^k$, the mentioned uniform convergence gives us that $L_{\alpha_j, \beta_j}u_{\zeta_{k+1}, \dots, \zeta_n} = 0$ in the weak sense, $1 \leqslant j \leqslant k$. Hence $L u_{\zeta_{k+1}, \dots, \zeta_n} = 0$ in the weak sense as well, where $L = L_{\alpha_1, \beta_1} + \dots + L_{\alpha_k, \beta_k}$. Since $L$ is elliptic, elliptic regularity gives us $u_{\zeta_{k+1}, \dots, \zeta_n} \in C^\infty (\mathbb D^k)$ and therefore $L_{\alpha_j, \beta_j}u_{\zeta_{k+1}, \dots, \zeta_n} = 0$ in the classical sense for $1 \leqslant j \leqslant k$. This implies that $u_{\zeta_{k+1}, \dots, \zeta_n} \in C\mathrm{sh}_{\alpha', \beta'} (\mathbb D^k)$. Lemma 4 is proved. Of course, any other $n-k$ variables belonging to $\mathbb T$ could have been fixed, with analogous conclusion. Let us note that the above lemma holds for any choice of parameters $\alpha$, $\beta$ in $\mathbb C^n$. Theorem 6. If $u \in C\mathrm{sh}_{\alpha, \beta}(\mathbb D^n)$, then $u (z)= P_{\alpha, \beta}[u|_{\mathbb T^n}] (z)$ for all $z \in \mathbb D^n$. Proof. Let $z = (z', z_n) \in \mathbb D^n$. We set $g_{z'}(\lambda)= u(z', \lambda)$, $| \lambda | \leqslant 1$. Clearly, $g_{z'}$ is continuous on $\overline{\mathbb D}$ and $(\alpha_n,\beta_n)$-harmonic in $ \mathbb D$. Then, by Theorem 4,
$$
\begin{equation*}
u(z', z_n)=g_{z'}(z_n) = \int_{\mathbb T} P_{\alpha_n, \beta_n}(z_n, \zeta_n) u(z', \zeta_n) \, dm_1 (\zeta_n);
\end{equation*}
\notag
$$
$u_{\zeta_n}(z_1, \dots, z_{n-1}) = u(z_1, \dots, z_{n-1}, \zeta_n)$ is $((\alpha_1, \dots, \alpha_{n-1}), (\beta_1, \dots, \beta_{n-1}))$-harmonic for every $\zeta_n$ in $\mathbb T$ by Lemma 4. Hence we analogously obtain
$$
\begin{equation*}
\begin{aligned} \, u(z) &= \int_{\mathbb T} P_{\alpha_n, \beta_n}(z_n, \zeta_n) \biggl[ \int_{\mathbb T} P_{\alpha_{n-1}, \beta_{n-1}}(z_{n-1}, \zeta_{n-1}) \\ &\qquad\times u(z_1, \dots, z_{n-2},\zeta_{n-1}, \zeta_n) \, dm_1(\zeta_{n-1}) \biggr] \, dm_1(\zeta_n). \end{aligned}
\end{equation*}
\notag
$$
Clearly, we can iterate this process, using Lemma 4 at each step and obtain $n$-times iterated integral, which is equal to $ P_{\alpha, \beta}[u|_{\mathbb T^n}] (z)$ by Fubini’s theorem. Theorem 6 is proved. Theorem 7. If $\psi \in C({\mathbb T^n})$, then $P_{\alpha, \beta}[\psi]$ has continuous extension $u$ on $\overline{\mathbb D^n}$, where $u \in C\mathrm{sh}_{\alpha, \beta}(\mathbb D^n)$ and $u |_{\mathbb T^n} = \psi$. Proof. The set $Y$ of complex valued functions on $\mathbb D^n$ which extend continuously to $\mathbb D^n$ is a closed vector subspace of the Banach space $X= CB(\mathbb D^n)$ of continuous bounded functions on $\mathbb D^n$, with supremum norm. Let $A$ be the set of all finite sums of the functions of the form
$$
\begin{equation}
\psi (\zeta) = \psi_1 (\zeta_1) \psi_2 (\zeta_2) \dotsb \psi_n(\zeta_n), \qquad \zeta = (\zeta_1, \dots, \zeta_n) \in \mathbb T^n,
\end{equation}
\tag{3.4}
$$
where $\psi_j$ is a continuous complex valued function on $\mathbb T$ for $1 \leqslant j \leqslant n$. It is easy to verify that $A$ is an algebra of continuous functions on a compact set $\mathbb T^n$ which is closed under conjugation, separates points on $\mathbb T^n$ and contains constants. Hence, by the Stone–Weierstrass theorem, it is dense in $C(\mathbb T^n)$. By (2.23) we have, for $\psi$ as in (3.4),
$$
\begin{equation*}
P_{\alpha, \beta}[\psi] (z_1, \dots, z_n) = \prod_{j=1}^n (P_{\alpha_j, \beta_j}[\psi_j])(z_j).
\end{equation*}
\notag
$$
Using Theorem 4, this shows that $P_{\alpha, \beta}$ maps $A$ into $Y$ and that restriction of $P_{\alpha, \beta}[\psi]$ to $\mathbb T^n$ is equal to $\psi$, for $\psi \in A$. Moreover, $P_{\alpha, \beta} \colon C(\mathbb T^n) \to CB(\mathbb D^n)$ is continuous by Theorem 5. Since $A$ is dense in $C(\mathbb T^n)$ and $Y$ is closed in $X$, it follows that $P_{\alpha, \beta}$ maps $C(\mathbb T^n)$ into $Y$, which completes the proof. Theorem 7 is proved. Theorems 7 and 6 imply that $C(\mathbb T^n)$ and $C\mathrm{sh}_{\alpha, \beta} (\mathbb D^n)$ are isomorphic as Banach spaces, the operator $P_{\alpha, \beta} \colon C(\mathbb T^n) \to C\mathrm{sh}_{\alpha, \beta} (\mathbb D^n)$ provides an isomorphism (not necessarily isometric). Another consequence of Theorems 7 and 6 is the following result on the Dirichlet problem for our system of PDEs. Proposition 4. For every $\varphi \in C(\mathbb T^n)$ the Dirichlet problem has a unique solution $u$ in the space $C\mathrm{sh}_{\alpha, \beta} (\mathbb D^n)$ and $u(z) = P_{\alpha, \beta}[\varphi](z)$ for $z \in \mathbb D^n$. We can apply Proposition 4 to the function $\varphi_{p, q} (z) = \mathcal F_{p,q} (z) z^p \overline z^q$, which is in the space $C\mathrm{sh}_{\alpha, \beta} (\mathbb D^n)$. Since $\varphi_{p,q} (\zeta) = \mathcal F_{p,q}(\mathbf{1}) \zeta^p \overline \zeta^q$ for $\zeta \in \mathbb T^n$ we get
$$
\begin{equation}
P_{\alpha, \beta}[\zeta^p \overline \zeta^q] (z) = \frac{\mathcal F_{p,q} (z)}{\mathcal F_{p,q}(\mathbf{1})} z^p \overline z^q, \qquad z \in \mathbb D^n.
\end{equation}
\tag{3.5}
$$
Combining (3.5) with Theorem 5 and the obvious equality $\| \zeta^p \overline \zeta^q \|_{L^\infty (\mathbb T^n)} = 1$, we deduce the following result, which is, in the particular case $n = 1$, an estimate of a family of hypergeometric functions. We have already noted that $u(rz)$ is not necessarily separately $(\alpha, \beta)$-harmonic if $u$ is separately $(\alpha, \beta)$-harmonic, see (2.19). This leads to difficulties in applying standard arguments on integral representations which involve the Banach–Alaouglu theorem. The next theorem allows us to surmount these difficulties, see the proofs of Theorems 13 and 14. Proof. We recall locally uniform expansion of $u$ from Theorem 2,
$$
\begin{equation}
u(z) = \sum_{(p,q) \in H} c_{p, q} \mathcal F_{p,q} (z) z^p \overline z^q, \qquad z \in \mathbb D^n, \quad c_{p, q} = \frac{ \partial^{(p, q)}u}{p!\, q!} (0).
\end{equation}
\tag{3.8}
$$
Therefore, by (3.5),
$$
\begin{equation}
\begin{aligned} \, \lim_{\mathbf{r} \to 1} P_{\alpha, \beta}[u_{\mathbf{r}}] (z) & = \lim_{\mathbf{r} \to 1} \int_{\mathbb T^n} P_{\alpha, \beta} (z, \zeta) u ( \mathbf{r} \cdot \zeta) \, dm_n(\zeta) \nonumber \\ &= \lim_{\mathbf{r} \to 1} \int_{\mathbb T^n} P_{\alpha, \beta} (z, \zeta) \sum_{(p, q) \in H} c_{p, q} \mathbf{r}^p \mathbf{r}^q \mathcal F_{p, q} (\mathbf{r}) \zeta^p \overline \zeta^q \, dm_n(\zeta) \nonumber \\ &= \lim_{\mathbf{r} \to 1} \sum_{(p, q) \in H} c_{p, q} \mathbf{r}^p \mathbf{r}^q \mathcal F_{p, q} (\mathbf{r}) P_{\alpha, \beta}[\zeta^p \overline \zeta^q](z) \nonumber \\ &= \lim_{\mathbf{r} \to 1} \sum_{(p, q) \in H} c_{p, q} \mathbf{r}^p \mathbf{r}^q \frac{\mathcal F_{p,q} (\mathbf{r})}{\mathcal F_{p,q}(\mathbf{1})} \mathcal F_{p,q} (z) z^p \overline z^q. \end{aligned}
\end{equation}
\tag{3.9}
$$
Next let us estimate each term $A(p, q, \mathbf{r}, z)$ in the above series using (2.10) and (3.6). We have
$$
\begin{equation*}
| A(p, q, \mathbf{r}, z) | \leqslant C_z \frac{\mathcal F_{p, q} (\mathbf{r})}{\mathcal F_{p, q} (\mathbf{1})} \mathbf{r}^p \mathbf{r}^q \prod_{j=1}^n p_j^{-2} q_j^{-2} \leqslant C_z K(\alpha, \beta) \prod_{j=1}^n p_j^{-2} q_j^{-2},
\end{equation*}
\notag
$$
where we follow the earlier convention of interpreting $0^{-2}$ as $1$. Since the multiple series $\sum_H \prod_{j=1}^n p_j^{-2} q_j^{-2}$ converges, and the terms do not depend on $\mathbf{r}$, the interchange of limit and summation in (3.9) is legitimate. This completes the proof of Theorem 8. Before we apply this theorem to representation theorems in the next subsection, we use it to derive the following corollary. Corollary 1. Assume $u \in \operatorname{sh}_{\alpha, \beta} (\mathbb D^n)$ has continuous extension to $\mathbb D^n \cup \mathbb T^n$. Then $u$ has continuous extension to $\overline{\mathbb D^n}$, that is, $u$ is in the space $C\mathrm{sh}_{\alpha, \beta}(\mathbb D^n)$. Proof. We assume $u$ is extended by continuity to $\mathbb D^n \cup \mathbb T^n$. Set $\varphi (\zeta) = u(\zeta)$ for $\zeta \in \mathbb T^n$. The function $(r, \zeta) \to u(r\zeta)$ is continuous on the compact set $[0, 1] \times \mathbb T^n$ and therefore $u_r$ converges uniformly to $\varphi$ over $\mathbb T^n$ as $r \to 1^-$. Thus, by Theorem 8, we have, for a fixed $z \in \mathbb D^n$,
$$
\begin{equation*}
\begin{aligned} \, u(z) &= \lim_{r \to 1}P_{\alpha, \beta}[u_r](z) = \lim_{r \to 1} \int_{\mathbb T^n} P_{\alpha, \beta}(z, \zeta) u_r(\zeta) \, dm_n(\zeta) \\ &= \int_{\mathbb T^n} P_{\alpha, \beta}(z, \zeta) \varphi(\zeta) \, dm_n(\zeta), \end{aligned}
\end{equation*}
\notag
$$
which gives us $u = P_{\alpha \beta}[\varphi]$, where $\varphi \in C(\mathbb T^n)$. In view of Theorem 7, this suffices to conclude $u \in C\mathrm{sh}_{\alpha, \beta}(\mathbb D^n)$. Corollary 1 is proved. Clearly, the assumptions can be relaxed considerably, for example, instead of continuity one could assume convergence in the sense of distributions of $u_r(\zeta)$ to $\varphi \in C(\mathbb T^n)$ as $r \to 1$. The next theorem is a local version of Theorem 7. Theorem 9. Assume $\varphi \colon \mathbb T^n \to \mathbb C$ is continuous at $\zeta^0 = (e^{i\theta^0_1}, \dots, e^{i\theta^0_n}) $ in $ \mathbb T^n$ and belongs to $L^\infty(\mathbb T^n)$. Then $u = P_{\alpha, \beta} [\varphi]$ has continuous extension to $\mathbb D^n \cup \{ \zeta^0 \}$. Proof. Let us consider an auxiliary function $\varphi (\zeta) - \varphi (\zeta^0)$. An application of Theorem 7 to the constant function $\psi (\zeta) = \varphi (\zeta^0)$ shows that we can assume, without loss of generality, that $\varphi (\zeta^0) = 0$. For $z = (r_1 e^{i \theta_1}, \dots, r_n e^{i \theta_n})$ in $\mathbb D^n$ we have, see (2.25).
$$
\begin{equation*}
\begin{aligned} \, | u(z) | &\leqslant \int_{\mathbb T^n} | \varphi (\zeta) | \prod_{j = 1}^n | P_{\alpha_j, \beta_j}(r_j e^{i \theta_j}, \zeta_j) | \, dm_n(\zeta) \\ &= (2\pi)^{-n} \int_{-\pi}^\pi \dots \int_{-\pi}^\pi | \varphi (e^{it_1}, \dots, e^{it_n}) | \prod_{j=1}^n | v_{\alpha_j, \beta_j} (r_j e^{i(\theta_j - t_j)}) | \, dt_1 \cdots dt_n. \end{aligned}
\end{equation*}
\notag
$$
For $0 < \delta < \pi$ and $\theta = (\theta_1, \dots, \theta_n)$ in $[-\pi, \pi]^n$ we write
$$
\begin{equation*}
M_\infty (\varphi, \theta, \delta) = \sup_{| \theta_j' - \theta_j | < \delta} | \varphi (e^{i\theta_1'}, \dots, e^{i \theta_n'}) |.
\end{equation*}
\notag
$$
The complement of $[-\delta, \delta]^n$ in $[-\pi, \pi]^n$ is contained in the union of the sets
$$
\begin{equation*}
S_j = \{ \theta \in [-\pi, \pi ]^n \colon | \theta_j | \geqslant \delta \}, \qquad 1 \leqslant j \leqslant n.
\end{equation*}
\notag
$$
Thus we have $| u(z) | \leqslant I_0(z, \delta) + I_1(z, \delta) + I_2(z, \delta) + \dots + I_n(z, \delta)$, where
$$
\begin{equation*}
\begin{aligned} \, I_0 (z, \delta) &= \frac{1}{(2\pi)^n} \int_{[-\delta, \delta]^n} | \varphi (e^{i (\theta_1 - t_1)}, \dots, e^{i (\theta_n - t_n)}) | \prod_{j=1}^n | v_{\alpha_j, \beta_j}(r_j e^{i t_j}) | \, dt_1\cdots dt_n \\ &\leqslant \frac{ M_\infty (\varphi, \theta, \delta)}{(2\pi)^n} \int_{[-\delta, \delta]^n} \prod_{j=1}^n | v_{\alpha_j, \beta_j}(r_j e^{i t_j}) | \, dt_1 \cdots dt_n \leqslant K(\alpha, \beta) M_\infty (\varphi, \theta, \delta) \end{aligned}
\end{equation*}
\notag
$$
and, for $1 \leqslant j \leqslant n$,
$$
\begin{equation*}
\begin{aligned} \, I_j(z, \delta) &= \frac{1}{(2\pi)^n} \int_{S_j} | \varphi (e^{i (\theta_1 - t_1)}, \dots, e^{i (\theta_n - t_n)}) | \prod_{k=1}^n | v_{\alpha_k, \beta_k} (r_k e^{i t_k}) | \, dt_1 \cdots dt_n \\ &\leqslant \frac{\| \varphi \|_\infty}{(2\pi)^n} \prod_{k \ne j} \int_{-\pi}^\pi | v_{\alpha_k, \beta_k} (r_k e^{i t_k}) | \, dt_k \cdot \int_{\delta < | t_j | \leqslant \pi} | v_{\alpha_j, \beta_j}(r_j e^{i t_j}) | \, dt_j \\ &\leqslant \frac{K_j(\alpha, \beta)}{2\pi} \| \varphi \|_\infty \sup_{\delta < | t_j | \leqslant \pi} | v_{\alpha_j, \beta_j} (r_j e^{i t_j}) | \\ &\leqslant C_j(\delta) K_j(\alpha, \beta)(1 - r_j^2)^{\operatorname{Re} \alpha_j + \operatorname{Re} \beta_j + 1} \| \varphi \|_\infty. \end{aligned}
\end{equation*}
\notag
$$
Hence, setting $C_{\alpha, \beta} (\delta) = \max_j C_j(\delta) K_j(\alpha, \beta)$, we have
$$
\begin{equation}
| u(z) | \leqslant K(\alpha, \beta) M_\infty (\varphi, \theta, \delta) + C_{\alpha, \beta} (\delta) \| \varphi \|_\infty \sum_{j=1}^n (1 - r_j^2)^{\operatorname{Re} \alpha_j + \operatorname{Re} \beta_j + 1}.
\end{equation}
\tag{3.10}
$$
Given $\varepsilon > 0$ we can choose $\delta > 0$ such that $|\theta_j - \theta_j^0| < 2\delta$, $1 \leqslant j \leqslant n$, implies $|\varphi (\zeta)| < \varepsilon$, where $\zeta = (e^{i\theta_1}, \dots, e^{i\theta_n})$. Therefore, if $|\theta_j - \theta_j^0| < \delta$, $1 \leqslant j \leqslant n$, then $ M_\infty (\varphi, \theta, \delta) \leqslant \varepsilon$. Now the result follows from (3.10), the conditions $\operatorname{Re} \alpha_j + \operatorname{Re} \beta_j > -1$ and since $z = (r_1 e^{i \theta_1}, \dots, r_n e^{i \theta_n})$ converges to $\zeta^0$ if and only if $r_j \to 1$ for all $1 \leqslant j \leqslant n$ and $\theta_j \to \theta_j^0$ for all $1 \leqslant j \leqslant n$. Theorem 9 is proved. 3.2. The $\operatorname{sh}^p_{\alpha, \beta} (\mathbb D^n)$ spacesDefinition 7. Let $1 \leqslant p < \infty$. We define the space $\operatorname{sh}_{\alpha, \beta}^p({\mathbb D^n})$ as the set of all $u \in \operatorname{sh}_{\alpha, \beta} (\mathbb D^n)$ such that It is easy to verify that the spaces $\operatorname{sh}_{\alpha, \beta}^p (\mathbb D^n)$ are Banach spaces with respect to the above norm. Proposition 6. If $\mu \in \mathcal M (\mathbb T^n)$, then $u = P_{\alpha,\beta}[d\mu]$ belongs to $\operatorname{sh}_{\alpha, \beta}^1 (\mathbb D^n)$ and $\| u \|_{\alpha, \beta;1} \leqslant K(\alpha, \beta) \| \mu \|$. Proof. By Proposition 3, $u$ is separately $(\alpha, \beta)$-harmonic and by (2.27) and (2.29) we have
$$
\begin{equation*}
\| u_{\mathbf{r}} \|_1 = \| (V_{\alpha, \beta})_{\mathbf{r}} \ast d \mu \|_1 \leqslant \| (V_{\alpha, \beta})_{\mathbf{r}} \|_1 \| \mu \| \leqslant K(\alpha, \beta) \| \mu \|, \qquad \mathbf{r} \in [0, 1)^n.
\end{equation*}
\notag
$$
Proposition 6 is proved. We proved that $P_{\alpha, \beta}$ is a bounded linear operator from $\mathcal M (\mathbb T^n)$ to $\operatorname{sh}_{\alpha, \beta}^1 (\mathbb D^n)$, with norm bounded by $K(\alpha, \beta)$. The following lemma will be useful in arguments involving duality. Lemma 5. Assume that $\mu$ and $\nu$ are complex Borel measures on $\mathbb T^n$, $u = P_{\alpha, \beta}[d\mu]$, $v = P_{\beta, \alpha}[d\nu]$ and let $\mathbf{r} \in [0, 1)^n$. Then $\langle u_{\mathbf{r}}, d\nu \rangle = \langle v_{\mathbf{r}}, d\mu \rangle$, or explicitly Proof. Since $P_{\alpha, \beta}(\mathbf{r} \cdot \zeta, \xi) = P_{\beta,\alpha}(\mathbf{r} \cdot \xi, \zeta)$ we have
$$
\begin{equation*}
\begin{aligned} \, &\int_{\mathbb T^n} (P_{\alpha, \beta}[d\mu]) (\mathbf{r} \cdot \zeta)\, d\nu(\zeta) = \int_{\mathbb T^n} \biggl( \int_{\mathbb T^n} P_{\alpha, \beta} (\mathbf{r} \cdot \zeta, \xi) \, d\mu (\xi) \biggr) \, d\nu(\zeta) \\ &\qquad= \int_{\mathbb T^n} \biggl( \int_{\mathbb T^n} P_{\beta, \alpha} (\mathbf{r} \cdot \xi, \zeta) \, d\nu (\zeta)\biggr) \, d\mu (\xi) = \int_{\mathbb T^n} (P_{\beta, \alpha}[d\nu]) (\mathbf{r} \cdot \xi) \, d\mu (\xi). \end{aligned}
\end{equation*}
\notag
$$
Lemma 5 is proved. It was proved in [17] that if $ f (z) = P[d\mu](z)$, where $\mu \in \mathcal M (\mathbb T^n)$, then $ f_r \, dm \to d \mu $ weak$^\ast$ as $r \to 1$, where $P$ is the Poisson kernel for $n$-harmonic functions (the case $\alpha_j = \beta_j = 0$). The next theorem generalizes this result to separately $(\alpha, \beta)$-harmonic functions, at the same time allowing for more general limits. Theorem 10. Let $u = P_{\alpha,\beta}[d\mu]$, where $\mu \in \mathcal M (\mathbb T^n)$. Then $u_{\mathbf{r}} \, dm_n$ converges weak$^\ast$ to $d\mu$ as $\mathbf{r} = (r_1, \dots, r_n) \to (1, \dots, 1) = \mathbf{1}$ in $[0, 1)^n$. Proof. We fix $\varphi \in C(\mathbb T^n)$ and set $v = P_{\beta, \alpha}[\varphi]$. Using Lemma 5 with $d\nu = \varphi \, dm_n$ we obtain
$$
\begin{equation*}
\int_{\mathbb T^n} u(\mathbf{r} \cdot \zeta) \varphi (\zeta) \, dm_n(\zeta) = \int_{\mathbb T^n} v(\mathbf{r} \cdot \xi)\, d\mu (\xi).
\end{equation*}
\notag
$$
However, by Theorem 7, $v(\mathbf{r} \cdot \xi)$ converges uniformly on $\mathbb T^n$ to $\varphi (\xi)$ as $\mathbf{r}$ converges to $\mathbf 1$. Hence, $\lim_{\mathbf{r} \to \mathbf{1}} \int_{\mathbb T^n} u(\mathbf{r} \cdot \zeta) \varphi (\zeta) \, dm_n(\zeta) = \int_{\mathbb T^n} \varphi (\xi) \, d\mu (\xi)$. Theorem 10 is proved. Theorem 11. If $\psi \in L^p(\mathbb T^n)$ for some $1 \leqslant p \leqslant \infty$, then $P_{\alpha, \beta}[\psi]$ is in the space $\operatorname{sh}_{\alpha, \beta}^p (\mathbb D^n)$ and For $1 \leqslant p < \infty$, Proof. In proving (3.13) we argue as in Proposition 6 and use Young’s inequality for convolutions. Indeed, $u = P_{\alpha, \beta}[\psi]$ is separately $(\alpha, \beta)$-harmonic by Proposition 3 and by (2.28) and (2.29) we have
Now we assume $1 \leqslant p < \infty$. Let $\psi \in L^p(\mathbb T^n)$ and let $\varepsilon > 0$. Since $C(\mathbb T^n) $ is dense in $L^p(\mathbb T^n)$ we can choose $\varphi \in C(\mathbb T^n)$ such that $\|\varphi-\psi \|_p < \varepsilon $. Then, by (3.13),
$$
\begin{equation*}
\begin{aligned} \, \| (P_{\alpha, \beta}[\psi])_{\mathbf{r}} - \psi\|_p &\leqslant \| (P_{\alpha, \beta}[\psi - \varphi])_{\mathbf{r}} \|_p + \| (P_{\alpha, \beta}[\varphi])_{\mathbf{r}} - \varphi \|_p + \| \varphi-\psi \|_p \\ &\leqslant [K(\alpha, \beta) + 1] \varepsilon + \| (P_{\alpha, \beta}[\varphi])_{\mathbf{r}} - \varphi \|_p \\ &\leqslant [K(\alpha, \beta) + 1] \varepsilon + \| (P_{\alpha, \beta}[\varphi])_{\mathbf{r}} - \varphi \|_{C(\mathbb T^n)}. \end{aligned}
\end{equation*}
\notag
$$
Since, by Theorem 7, $\lim_{\mathbf{r} \to 1} \| (P_{\alpha, \beta}[\varphi])_{\mathbf{r}} - \varphi \|_{C(\mathbb T^n)} = 0$, we obtain from the above estimates $\limsup_{\mathbf{r} \to 1} \| (P_{\alpha, \beta}[\psi])_{\mathbf{r}} - \psi\|_p \leqslant [K(\alpha, \beta) + 1] \varepsilon$, which gives us (3.14). Theorem 11 is proved. Theorem 12. Let $u = P_{\alpha,\beta}[f]$, where $f \in L^\infty (\mathbb T^n)$. Then $u_{\mathbf{r}}$ converges weak$^\ast$ to $f$ as $\mathbf{r} = (r_1, \dots, r_n) \to (1, \dots, 1) = \mathbf{1}$ in $[0, 1)^n$. Proof. We fix $\varphi \in L^1(\mathbb T^n)$ and set $v = P_{\beta, \alpha}[\varphi]$. Using Lemma 5 with $d\nu = \varphi \, dm_n$ and $d\mu = f\, dm_n$ we obtain
$$
\begin{equation*}
\int_{\mathbb T^n} u(\mathbf{r} \cdot \zeta) \varphi (\zeta) \, dm_n(\zeta) = \int_{\mathbb T^n} v(\mathbf{r} \cdot \xi) f(\xi) \, dm_n (\xi).
\end{equation*}
\notag
$$
However, by Theorem 11, $v(\mathbf{r} \cdot \xi)$ converges in the $L^1(\mathbb T^n)$ norm to $\varphi (\xi)$ as $\mathbf{r}$ tends to $\mathbf 1$. Hence, $\lim_{\mathbf{r} \to \mathbf{1}} \int_{\mathbb T^n} u(\mathbf{r} \cdot \zeta) \varphi (\zeta) \, dm_n(\zeta) = \int_{\mathbb T^n} f(\xi) \varphi (\xi) \, dm_n (\xi)$. Theorem 12 is proved. Next we turn to integral representation theorems for functions in $\operatorname{sh}_{\alpha, \beta}^p(\mathbb D^n)$ spaces. Theorem 13. Assume $u$ is separately $(\alpha, \beta)$-harmonic function on $\mathbb D^n$ such that Then there is a unique $ \mu \in \mathcal{M}(\mathbb{T}^n)$ such that $u = P_{\alpha, \beta}[d\mu]$. Proof. Since a sequence $u((1-1/k)\zeta) \, dm_n(\zeta)$ of complex Borel measures on $\mathbb T^n$ is bounded in $\mathcal M(\mathbb T^n) = C(\mathbb T^n)^\ast$ by (3.15), the Banach–Alaoglu theorem and separability of $C(\mathbb T^n)$ give us a weak$^\ast$ convergent subsequence. This gives us a sequence $r_l$ in $[0, 1)$ such that $\lim_{l \to \infty} r_l = 1$ and a complex Borel measure $\mu$ on $\mathbb T^n$ such that
$$
\begin{equation}
\lim_{l \to \infty} \int_{\mathbb T^n} g(\zeta) u (r_l \zeta) \, dm_n(\zeta) = \int_{\mathbb T^n} g(\zeta) \, d\mu(\zeta) \quad \text{for all } \ g \in C(\mathbb T^n).
\end{equation}
\tag{3.16}
$$
We apply (3.16) to $g_z(\zeta) = P_{\alpha, \beta} (z, \zeta)$, where $z$ in $\mathbb D^n$ is fixed. This is legitimate because $g_z$ is continuous on $\mathbb T^n$. By Theorem 8 we have
$$
\begin{equation*}
\begin{aligned} \, P_{\alpha, \beta}[d\mu] (z) & = \int_{\mathbb T^n} P_{\alpha, \beta} (z, \zeta) \, d\mu (\zeta) = \lim_{l \to \infty} \int_{\mathbb T^n} P_{\alpha, \beta} (z, \zeta) u (r_l \zeta) \, dm_n(\zeta) \\ & = \lim_{l \to \infty} P_{\alpha, \beta}[u_{r_l}](z) = u(z). \end{aligned}
\end{equation*}
\notag
$$
Uniqueness follows from Theorem 10. Theorem 13 is proved. Theorem 13 and Proposition 6 imply that the following conditions are equivalent for a given separately $(\alpha, \beta)$-harmonic function $u$ on $\mathbb D^n$:
$$
\begin{equation*}
\begin{gathered} \, \sup_{0 \leqslant r <1} \int_{\mathbb T^n} | u(r \zeta) | \, dm_n(\zeta) < \infty, \qquad \sup_{\mathbf{r} \in [0,1)^n} \int_{\mathbb T^n}| u_{\mathbf{r}} (\zeta) | \, dm_n (\zeta) < \infty, \\ u = P_{\alpha, \beta}[d\mu], \quad \text{where } \ \mu \in \mathcal M (\mathbb T^n). \end{gathered}
\end{equation*}
\notag
$$
For $n$-harmonic functions equivalence of the last two conditions appeared in [20]. Theorem 14. Let $ 1 < p\leqslant\infty $ and assume $u \in \operatorname{sh}_{\alpha, \beta} (\mathbb D^n)$ satisfies Then there exists a unique function $ \psi \in L^p ({\mathbb T^n}) $ such that $u = P_{\alpha, \beta}[\psi]$. Proof. Since $p > 1$, $L^p (\mathbb T^n)$ is the dual of $ L^q ({\mathbb T^n})$, where $1/p + 1/q = 1$. As in the proof of Theorem 13, using this duality and the Banach–Alaoglu theorem, we obtain a sequence $r_k$ in $[0, 1)$ converging to $1$ and a function $\psi$ in $L^p(\mathbb T^n)$ such that
$$
\begin{equation}
\lim_{k \to \infty} \int_{\mathbb T^n} \varphi(\zeta)u(r_k \zeta) \, dm_n(\zeta) =\int_{\mathbb T^n} \varphi(\zeta)\psi(\zeta) \, dm_n(\zeta) \quad \text{for all }\ \varphi \in L^q (\mathbb T^n).
\end{equation}
\tag{3.18}
$$
We apply (3.18) to $\varphi _z (\zeta) = P(z, \zeta)$, where $z$ in $\mathbb D^n$ is fixed. Since $\varphi_z \in C(\mathbb T^n) \subset L^q(\mathbb T^n)$ this application is legitimate. By Theorem 8 we have
$$
\begin{equation*}
\begin{aligned} \, P_{\alpha, \beta}[\psi] (z) &= \int_{\mathbb T^n} P_{\alpha, \beta} (z, \zeta) \psi(\zeta) \, dm_n (\zeta) = \lim_{k \to \infty} \int_{\mathbb T^n} P_{\alpha, \beta} (z, \zeta) u (r_k \zeta) \, dm_n(\zeta) \\ & = \lim_{k \to \infty} P_{\alpha, \beta}[u_{r_k}](z) = u(z). \end{aligned}
\end{equation*}
\notag
$$
Uniqueness follows from Theorems 12 and 11. Theorem 14 is proved. We have the following chain of continuous embeddings of Banach spaces
$$
\begin{equation}
C\mathrm{sh}_{\alpha, \beta} (\mathbb D^n) \subset \operatorname{sh}_{\alpha, \beta}^\infty (\mathbb D^n) \subset \operatorname{sh}_{\alpha, \beta}^q (\mathbb D^n) \subset \operatorname{sh}_{\alpha, \beta}^p (\mathbb D^n) \subset \operatorname{sh}_{\alpha, \beta}^1 (\mathbb D^n),
\end{equation}
\tag{3.19}
$$
where $1 < p < q < \infty$. The above results show that these spaces are isomorphic, as Banach spaces, respectively to the spaces
$$
\begin{equation}
C(\mathbb T^n) \subset L^\infty (\mathbb T^n) \subset L^q (\mathbb T^n) \subset L^p (\mathbb T^n) \subset \mathcal M (\mathbb T^n),
\end{equation}
\tag{3.20}
$$
an isomorphism is provided by the operator $P_{\alpha, \beta}$. Our next aim is to generalize Lemma 4 to the case $u \in \operatorname{sh}^p_{\alpha, \beta} (\mathbb D^n)$. To do so, we observe that most of the above results extend to the vector valued case, with values in a complex Banach space $X$. In fact, Definition 1 extends to the case $u \in C^2(\mathbb D^n, X)$, Definition 5 to the case of $X$-valued measures $\mu$ on $\mathbb T^n$ and functions $f$ in $L^1 (\mathbb T^n, X)$ and Definitions 6 and 7 have obvious generalizations which yield spaces $\operatorname{sh}^p_{\alpha, \beta} (\mathbb D^n, X)$ for $1 \leqslant p \leqslant \infty$ and $C\mathrm{sh}_{\alpha, \beta} (\mathbb D^n, X)$. Further, Proposition 3 extends to the case of $X$-valued measures and Theorem 4 extends to functions $\varphi \in C(\mathbb T, X)$. We need the following $L^1$-variant of Theorem 4. Proposition 7. Assume $\varphi \in L^1(\mathbb T, X)$, and let $u = P_{\alpha, \beta}[\varphi]$. Then A proof can be based on estimates on the kernel $P_{\alpha, \beta}$, use of maximal function, and the density of $C(\mathbb T, X)$ in $L^1(\mathbb T, X)$. In [2] one can find the scalar valued case discussed in the more general situation of the unit ball in $\mathbb C^n$. In addition, the following results hold in the context of $X$-valued functions, with essentially the same proofs: Theorems 5 and 9, Proposition 6 and the first part of Theorem 11. The following theorem is an extension of Lemma 4. Theorem 15. Assume $\varphi \colon \mathbb T^n \to \mathbb C$ is in $L^p (\mathbb T^n)$, where $1 \leqslant p \leqslant +\infty$ and let $u = P_{\alpha, \beta}[\varphi]$. Then: $1^\circ$. For each $z_n$ in $\mathbb D$ the function $v_{z_n} (z') = u(z', z_n)$, $z' \in \mathbb D^{n-1}$, belongs to the space $X = \operatorname{sh}_{\alpha', \beta'}^p (\mathbb D^{n-1})$, where $\alpha' = (\alpha_1, \dots, \alpha_{n-1})$ and $\beta' = (\beta_1, \dots, \beta_{n-1})$. $2^\circ$. There is a set $N \subset \mathbb T$ of measure zero such that for every $\zeta_n$ in $\mathbb T \setminus N$ there exists a function $w_{\zeta_n}$ in $\operatorname{sh}_{\alpha', \beta'}^p (\mathbb D^{n-1})$ such that where the limit is both in $\operatorname{sh}_{\alpha', \beta'}^p (\mathbb D^{n-1})$-norm and locally uniform over $z' \in \mathbb D^{n-1}$. Proof. There is a set $E \subset \mathbb T$ of measure zero such that for $\zeta_n \in \mathbb T \setminus E$ the function $\varphi_{\zeta_n} \colon \mathbb T^{n-1} \to \mathbb C$ defined by $\varphi_{\zeta_n}(\zeta') = \varphi (\zeta', \zeta_n)$ belongs to $L^p (\mathbb T^{n-1})$. We define $\Phi (\zeta_n) = P_{\alpha', \beta'}[\varphi_{\zeta_n}]$ for $\zeta_n \in \mathbb T \setminus E$. The mapping $\Phi \colon \mathbb T \to X$ is defined almost everywhere and so
$$
\begin{equation*}
\begin{aligned} \, \| \Phi \|_{L^1(\mathbb T, X)}^p &= \biggl( \int_{\mathbb T} \| \Phi (\zeta_n) \|_X \, dm_1 (\zeta_n) \biggr)^p \leqslant \int_{\mathbb T} \| P_{\alpha', \beta'}[\varphi_{\zeta_n}] \|_{\alpha', \beta';p}^p \, dm_1(\zeta_n) \\ &\leqslant K^p (\alpha', \beta') \int_{\mathbb T} \| \varphi_{\zeta_n} \|^p_{L^p(\mathbb T^{n-1})} \, dm_1 (\zeta_n) = K^p (\alpha', \beta') \| \varphi \|_{L^p(\mathbb T^n)}^p. \end{aligned}
\end{equation*}
\notag
$$
We proved that $\Phi \in L^1(\mathbb T, \operatorname{sh}_{\alpha', \beta'}^p (\mathbb D^{n-1}))$, in fact $\| \Phi \|_1 \leqslant K (\alpha', \beta') \| \varphi \|_p$. We have
$$
\begin{equation}
\Phi (\zeta_n) (z') = \int_{\mathbb T^{n-1}} V_{\alpha', \beta'} (z' \cdot \overline{\zeta'}) \varphi (\zeta', \zeta_n) \, dm_{n-1} (\zeta'), \qquad z' \in \mathbb D^{n-1}, \quad \zeta_n \in \mathbb T \setminus E.
\end{equation}
\tag{3.23}
$$
Let us set $U(z_n) = P_{\alpha_n, \beta_n}[\Phi] (z_n)$, $| z_n | < 1$. Then, for each $z_n \in \mathbb D$, $U(z_n)$ is in $\operatorname{sh}_{\alpha', \beta'}^p (\mathbb D^{n-1})$. Since evaluation functionals are continuous on the space $X$, we use (3.23) to obtain, for $z = (z', z_n) \in \mathbb D^n$,
$$
\begin{equation*}
\begin{aligned} \, U(z_n)(z') & = \biggl( \int_{\mathbb T} V_{\alpha_n, \beta_n} (z_n, \overline{\zeta_n}) \Phi (\zeta_n) \, dm_1(\zeta_n) \biggr) (z') \\ &= \int_{\mathbb T} V_{\alpha_n, \beta_n} (z_n, \overline{\zeta_n}) \Phi (\zeta_n) (z') \, dm_1(\zeta_n) \\ &= \int_{\mathbb T} V_{\alpha_n, \beta_n} (z_n, \overline{\zeta_n}) \int_{\mathbb T^{n-1}} V_{\alpha', \beta'} (z' \cdot \overline{\zeta'}) \varphi (\zeta', \zeta_n) \, dm_{n-1} (\zeta') \, dm_1(\zeta_n) \\ & = u(z', z_n). \end{aligned}
\end{equation*}
\notag
$$
Hence $v_{z_n} = U(z_n)$ and assertion $1^\circ$ is proved.
Assertion $2^\circ$ follows from Proposition 7, indeed one can take $w_{\zeta_n} = \Psi (\zeta_n)$ almost everywhere. Locally uniform convergence follows from interior elliptic estimates and boundedness in the $\operatorname{sh}_{\alpha', \beta'}^p (\mathbb D^{n-1})$-norm. Theorem 15 is proved. Of course, in assertion $1^\circ$ any variable $z_k$ could have been selected. The special case of $\alpha = \beta = \mathbf{0}$ appeared in [17] as Theorem 8. For the case of analytic functions, see [20] and [21]. § 4. Maximal functions and Fatou type theoremsIn § 3 we studied convergence in $L^p(\mathbb T^n)$ norm, or in weak$^\ast$ topology, of $u_{\mathbf{r}}$ for $u$ in the appropriate $\operatorname{sh}^p_{\alpha, \beta} (\mathbb D^n)$ space. In this section we study almost everywhere convergence of $u_{\mathbf{r}}$ for a separately $(\alpha, \beta)$-harmonic function $u$. The first result of this nature was related to Cesàro summability of double Fourier series in [13]. In that paper authors noted that their results hold for Poisson kernels as well as for Fejer kernels, thus obtaining a result for $2$-harmonic functions. Our approach is based on the presentation given in [15], where $n$-harmonic functions are considered, see Theorem 2.3.1 from [15]. An important tool is a special maximal function $M_q$, it is used to obtain estimates of relevant restricted non-tangential maximal functions which lead to Fatou type theorems. Since the proof of crucial Theorem 16 is rather involved, we split this section into subsections. 4.1. A maximal function $M_q$For $\gamma$ in $\mathbb Z_+^n$ we define a $\gamma$ box as a subset $Q$ of $\mathbb T^n$ of the following form:
$$
\begin{equation*}
Q = I_1 \times \cdots \times I_n,\quad \text{where } \ I_j = \{ e^{i\theta} \colon a_j \leqslant \theta < b_j \}
\end{equation*}
\notag
$$
is a semi-open arc on $\mathbb T$ of length $0 < b_j -a_j \leqslant 2\pi$ and these lengths have the same ratio as $(2^{\gamma_1}, \dots, 2^{\gamma_n})$. The center of such a box is denoted by $c(Q) = ( e^{ic_1}, \dots, e^{ic_n})$, where $c_j = (a_j + b_j)/2$. We denote the family of all $\gamma$ boxes by $\mathcal Q_\gamma$. Next, we define the $\gamma$-maximal function of a complex Borel measure $\mu$ on $\mathbb T^n$ by
$$
\begin{equation}
M_\gamma \mu (\zeta) = \sup_{Q \in \mathcal Q_\gamma, \, c(Q) = \zeta} \frac{|\mu| (Q)}{m_n(Q)}, \qquad \mu \in \mathcal M(\mathbb T^n), \quad \zeta \in \mathbb T^n.
\end{equation}
\tag{4.1}
$$
As in the standard situation of $\mathbb R^n$ with balls instead of $\gamma$ boxes, the following basic estimate holds (see pp. 26, 27 in [15]):
$$
\begin{equation}
m_n (\{ \zeta \in \mathbb T^n \colon M_\gamma \mu (\zeta) > \lambda \}) \leqslant 3^n \frac{ \| \mu \| }{\lambda}, \qquad \mu \in \mathcal M (\mathbb T^n), \quad \lambda > 0.
\end{equation}
\tag{4.2}
$$
The fact that the constant $3^n$ is independent of $\gamma \in \mathbb Z_+^n$ is of crucial importance. Next, for $0 < q < 1$, we introduce the maximal function $M_q$ by
$$
\begin{equation}
M_q \mu (\zeta) = \sum_{\gamma \in \mathbb Z_+^n} q^{| \gamma |} M_\gamma \mu (\zeta), \qquad \mu \in \mathcal M(\mathbb T^n), \quad \zeta \in \mathbb T^n.
\end{equation}
\tag{4.3}
$$
Lemma 6 (see p. 26 for $q=1/2$ in [15]). Let $0 < q < 1$ and let $\mu \in \mathcal M (\mathbb T^n)$. Then Proof. We set, for $\lambda > 0$ and $\gamma \in \mathbb Z_+^n$,
$$
\begin{equation*}
E(\lambda) = \{ \zeta \in \mathbb T^n \colon M_q \mu (\zeta) > \lambda \}, \qquad E_\gamma(\lambda) = \{ \zeta \in \mathbb T^n \colon M_\gamma \mu (\zeta) > \lambda \}.
\end{equation*}
\notag
$$
Note that $ \sum_{k=0}^\infty q^{k/2} = 1/(1-\sqrt{q})=\eta (q)$ and $\sum_{\gamma \in \mathbb Z_+^n} q^{|\gamma|/2}=\eta(q)^n$ for $0 < q < 1$. Hence, taking into account (4.3), we see that
$$
\begin{equation*}
E(\lambda) \subset \bigcup_{\gamma \in \mathbb Z_+^n} E_\gamma(\eta(q)^{-n} q^{- | \gamma |/2} \lambda).
\end{equation*}
\notag
$$
Therefore, using (4.2), we obtain
$$
\begin{equation*}
m_n(E(\lambda)) \leqslant \sum_{\gamma \in \mathbb Z_+^n} 3^n \frac{\| \mu \|}{ \eta(q)^{-n} q^{- | \gamma |/2} \lambda} = 3^n \eta(q)^{2n} \| \mu \| \lambda^{-1},
\end{equation*}
\notag
$$
which is the desired estimate (4.4). Lemma 6 is proved. 4.2. An estimate of $P_{\alpha, \beta}[d\mu]$ by $P_\mathbf{t} [d\mu]$For real $t > -1$ and $z$ in $\mathbb D$ we set
$$
\begin{equation}
u_t(z) = u_{t/2, t/2} (z) = \frac{(1 - | z |^2)^{t+1}}{| 1 - z |^{t+2}},
\end{equation}
\tag{4.5}
$$
$$
\begin{equation}
v_t(z) = v_{t/2, t/2} (z) = \frac{\Gamma^2(t/2 + 1)}{\Gamma(t+1)} \frac{(1 - | z |^2)^{t+1}}{| 1 - z |^{t+2}}.
\end{equation}
\tag{4.6}
$$
An elementary inequality $| 1 - z | \geqslant \pi^{-1} | \theta |$, which is valid for $0 \leqslant r < 1$ and $-\pi \leqslant \theta \leqslant \pi$, implies the estimate
$$
\begin{equation}
0 < u_t (re^{i\theta}) \leqslant 2^{t+1} \pi^{t+2} \frac{(1 - r)^{t+1}}{| \theta |^{t+2}}, \qquad 0 \leqslant r < 1, \quad -\pi \leqslant \theta \leqslant \pi.
\end{equation}
\tag{4.7}
$$
The kernel $v_t(z)$ appeared in [11], where the operator $L_{t/2, t/2}$ was analysed and a Fatou type theorem was obtained in the one dimensional setting of $\mathbb D$. For $\mathbf{t} = (t_1, \dots, t_n) \in (-1, +\infty)^n$ we set
$$
\begin{equation}
U_\mathbf{t} (z) = \prod_{j=1}^n u_{t_j} (z_j) , \quad V_\mathbf{t} (z) = \prod_{j=1}^n v_{t_j} (z_j), \qquad z \in \mathbb D^n.
\end{equation}
\tag{4.8}
$$
Now we have the positive kernels $K_\mathbf{t} (z, \zeta) = V_\mathbf{t}(z \cdot \overline\zeta)$ and $K'_\mathbf{t} (z, \zeta) = U_\mathbf{t}(z \cdot \overline\zeta)$, where $z \in \mathbb D^n$, $\zeta \in \mathbb T^n$. Clearly, $K_\mathbf{t} = P_{\alpha, \beta}$ with $\alpha = \beta =\mathbf{t}/2$. The next lemma follows from (2.24), (2.21), (4.8) and (4.5). It allows us to pass from complex valued kernels $P_{\alpha, \beta}$ to positive ones in various estimates, see (4.14) below. Lemma 7. Let $\mu \in \mathcal M (\mathbb T^n)$ and let $z \in \mathbb T^n$. Then Here $\mathbf{t} = (t_1, \dots, t_n)$, where $t_j = \operatorname{Re} \alpha_j + \operatorname{Re} \beta_j$ for $1 \leqslant j \leqslant n$ and 4.3. Estimates within Stolz cone with fixed $\mathbf{r}$We define the full Stolz cone at $\zeta \in \mathbb T^n$ of aperture $0 < A< \infty$ by the restricted Stolz cone $S_{A, B}(\zeta)$ for $0 < A< \infty$, $1 \leqslant B < \infty$ at $\zeta \in \mathbb T^n$ is defined by
$$
\begin{equation}
S_{A, B}(\zeta) = \biggl\{ (r_1e^{i\theta_1}, \dots, r_n e^{i\theta_n}) \in S_A(\zeta) \colon \max_{j,k} \frac{1-r_j}{1-r_k} \leqslant B \biggr\}.
\end{equation}
\tag{4.11}
$$
It is an elementary fact that there is a constant $\delta_A > 0$ such that
$$
\begin{equation}
\delta_A | 1 - re^{i\varphi} | \leqslant \inf_{| \theta | \leqslant A(1-r)} | 1 - re^{i(\varphi - \theta)} | \leqslant | 1 - re^{i\varphi} | , \qquad 0 < A< \infty.
\end{equation}
\tag{4.12}
$$
Let us set
$$
\begin{equation}
c(\mathbf{t}) = \sum_{j=1}^n (t_j + 2), \qquad \mathbf{t} \in (-1, +\infty)^n.
\end{equation}
\tag{4.13}
$$
Proposition 8. Let $\eta = ( e^{i\psi_1}, \dots, e^{i\psi_n} )$ in $\mathbb T^n$, $\mathbf{r} \in [0, 1)^n$, $\mathbf{t} \in (-1, +\infty)^n$ and $0 < A< \infty$. Then Proof. Using (4.12) we obtain
$$
\begin{equation}
\begin{aligned} \, \sup_{z \in S_A(\eta),\, | z_j | = r_j} U_\mathbf{t} (z \cdot \overline\zeta) &= \sup_{z \in S_A(\eta),\, | z_j | = r_j} \prod_{j=1}^n \frac{ (1 - r_j^2)^{t_j + 1} }{ | 1 - z_j \overline{\zeta_j} |^{t_j + 2} } \nonumber \\ &= \sup_{| \theta_j | \leqslant A(1 - r_j)} \prod_{j=1}^n \frac{ (1 - r_j^2)^{t_j + 1} }{ | 1 - r_j e^{i [ (\psi_j - \theta_j) -\varphi_j ]} |^{t_j + 2} } \nonumber \\ &\leqslant \delta_A^{-c(\mathbf{t})} \prod_{j=1}^n \frac{ (1 - r_j^2)^{t_j + 1} }{ | 1 - r_j e^{i (\psi_j -\varphi_j) } |^{t_j + 2} } \nonumber \\ &= \delta_A^{-c(\mathbf{t})} U_\mathbf{t} ( \mathbf{r} \cdot \eta \cdot \overline\zeta), \end{aligned}
\end{equation}
\tag{4.15}
$$
where $z = (r_1 e^{i ( \psi_1 - \theta_1)}, \dots, r_n e^{i ( \psi_n - \theta_n)} ) $ is in $ S_A(\eta)$ and $\zeta = ( e^{i\varphi_1}, \dots, e^{i\varphi_n})$ is in $\mathbb T^n$. This and Lemma 7 give us the desired estimate. Proposition 8 is proved. Note that in (4.14) both $\eta \in \mathbb T^n$ and $\mathbf{r} \in [0, 1)^n$ are fixed. In order to estimate the integral on the right-hand side of (4.14) we need a special partition of $\mathbb T^n$ adapted to given $\mathbf{r}$. 4.4. Partitions $\mathcal P_{\mathbf{r}}$ of $\mathbb T^n$ and estimates of $|\mu| (Q)$, $Q \in \mathcal P_{\mathbf{r}}$For a given $0 \leqslant \rho < 1$ we denote by $\kappa = \kappa (\rho)$ the unique $\kappa \in [1, 2)$ such that
$$
\begin{equation}
\frac{\pi}{(1-\rho) \kappa (\rho)} = 2^p,\quad \text{where } \ p \in \mathbb N_0.
\end{equation}
\tag{4.16}
$$
This $p = p(\rho)$ is uniquely determined by $\rho$. We set
$$
\begin{equation}
x_0 = 0, \qquad x_j = x_j(\rho) = 2^{j-1} (1-\rho)\kappa (\rho) \quad \text{for } \ 1 \leqslant j \leqslant p+1.
\end{equation}
\tag{4.17}
$$
Clearly, $0 = x_0 < x_1 < \dots < x_{p+1} = \pi$ and $2x_j = x_{j+1}$ for $1 \leqslant j \leqslant p$. We set, for $j = 0, \dots, p$, $J_j^+ = J_j^+ (\rho) = [x_j, x_{j+1})$ and $J_j^- = J_j^- (\rho) = [-x_{j+1}, -x_j)$. These $2(p+1)$ semi-open intervals form a partition $\widetilde{\mathcal P}_\rho$ of $[-\pi, \pi)$ and the corresponding $2(p+ 1)$ semi-open arcs $I_j^+ = I_j^+(\rho)$ and $I_j^- = I_j^-(\rho)$ on $\mathbb T$ form a partition $\mathcal P_\rho$ of $\mathbb T$. Note that $m_1(I_j^{\pm}) = 2^{j-2} (1-\rho)\kappa (\rho)/ \pi$ for $1 \leqslant j \leqslant p$ and $m_1(I_1^{\pm}) = m_1(I_0^{\pm})$. Let us fix $1 \leqslant B < +\infty$ and choose the smallest integer $b \geqslant 0$ such that $B \leqslant 2^b$. Assume that $\mathbf{r} \in [0, 1)^n$ satisfies the condition $\max_j (1 - r_j) \leqslant B \min_j (1 - r_j)$. We can assume, without loss of generality, that $r_1 \leqslant r_2 \leqslant \dots \leqslant r_n$. The partitions $\mathcal P_{r_k}$, $1 \leqslant k \leqslant n$, of $\mathbb T$ induce a partition of $\mathbb T^n$. Namely, we choose $\varepsilon$ in $\{+, - \}^n$, $j$ in $\{0, \dots, p(r_1)\} \times \dots \times \{0, \dots, p(r_n)\} $ and set $Q(\varepsilon, j) = \prod_{l=1}^n I_{j_l}^{\varepsilon_l}$. These $[2(p(r_1)+ 1)] \times \dots \times [2(p(r_n)+1)]$ boxes form a partition $\mathcal P_{\mathbf{r}}$ of $\mathbb T^n$. For each choice of $\varepsilon$ and $j$ there is an decreasing sequence $\iota = (\iota_l)_{l=1}^n$ of integers such that $\iota_1 \leqslant b$, $\iota_n = 0$ and $Q(\varepsilon, j)$ is a $(j_1 + \iota_1, \dots, j_n+\iota_n)$ box. The sequence $\iota$ depends only on $j$ and not on $\varepsilon$. Every box $Q(\varepsilon, j)$ is contained in a box $R(j)$ consisting of all $(e^{i\theta_1}, \dots, e^{i\theta_n})$ such that $-x_{j_k+1}(r_k) \leqslant \theta_k < x_{j_k+1}(r_k)$ for all $1 \leqslant k \leqslant n$. The box $R(j)$ is, being homothetic to $Q(\varepsilon, j)$, also a $j + \iota$ box, it is clearly centered at $\mathbf 1$ and $m_n(R(j)) = 4^n m_n(Q(\varepsilon, j))$. Hence we obtain
$$
\begin{equation}
|\mu| (Q(\varepsilon, j)) \leqslant |\mu| (R(j)) \leqslant m_n(R(j)) (M_{j + \iota} \mu)(\mathbf{1}) = 4^n m_n(Q(\varepsilon, j) ) (M_{j + \iota} \mu)(\mathbf{1}).
\end{equation}
\tag{4.18}
$$
It easily follows from (4.17) and (4.16) that
$$
\begin{equation}
m_n(Q(\varepsilon, j)) = 4^{-n} 2^{j_1 + \dots + j_n} \prod_{k=1}^n \frac{(1 - r_k) \kappa (r_k)}{\pi} = 4^{-n} \prod_{k=1}^n 2^{j_k - p(r_k)},
\end{equation}
\tag{4.19}
$$
which, combined with the previous inequality, gives the following proposition. Proposition 9. Let $\mu$ be a complex Borel measure on $\mathbb T^n$, let $1 \leqslant B < \infty$, assume $\mathbf{r} \in [0, 1)^n$ satisfies the condition $\max_j (1-r_j) \leqslant B \min_j (1-r_j)$ and let $\mathcal P_{\mathbf{r}}$ be the partition of $\mathbb T^n$ associated with $\mathbf{r}$. Then, for every $\varepsilon \in \{ -1, +1 \}^n$ and $j = (j_1, \dots, j_n)$ such that $0 \leqslant j_k \leqslant p(r_k)$ for all $1 \leqslant k \leqslant n$, 4.5. Estimate of $P_{\alpha, \beta}[d\mu]$ over $S_{A, B}(\eta)$Let us fix $0 \leqslant \rho < 1$ and consider points $x_0, \dots, x_{p+1}$ from (4.17) which generate partition $\widetilde{\mathcal P}_\rho$ of $[0, \pi)$. We apply (4.7) to $\theta = x_1, \dots, x_p$ and obtain, using (4.17) and $\kappa (\rho) \geqslant 1$, the following inequality for $1 \leqslant j \leqslant p = p(\rho)$:
$$
\begin{equation*}
u_t(\rho e^{i x_j}) \leqslant \frac{2^{t+1} \pi^{t+2}}{1-\rho} \biggl( \frac{1 - \rho}{ 2^{j-1} (1-\rho) \kappa (\rho)}\biggr)^{t+2} \leqslant \frac{2^{2t+3} \pi^{t+2}}{1-\rho} 2^{-(t+2)j}.
\end{equation*}
\notag
$$
It is valid also for $j = 0$ by definition (4.5) of $u_t$, so we obtained
$$
\begin{equation}
u_t(\rho e^{i x_j}) \leqslant \frac{2^{2t+3} \pi^{t+2}}{1-\rho} 2^{-(t+2)j}, \qquad 0 \leqslant j \leqslant p = p(\rho).
\end{equation}
\tag{4.21}
$$
Note that $u_t(\rho e^{i\theta})$ is even in $\theta \in (-\pi, \pi)$ and decreasing in $\theta \in (0, \pi)$ for a fixed $0 \leqslant \rho < 1$ (see [11]). Thus (4.21) gives, for $\mathbf{t}$ in $(-1, +\infty)^n$ and $\mathbf{r}$ in $[0, 1)^n$:
$$
\begin{equation}
U_\mathbf{t} (\mathbf{r} \cdot \zeta) \leqslant C_\mathbf{t}\prod_{k=1}^n 2^{-(t_k + 2)j_k} \prod_{k=1}^n \frac{1}{1-r_k}, \qquad \zeta \in Q(\varepsilon, j).
\end{equation}
\tag{4.22}
$$
Now we assume that $\mathbf{r}$ satisfies the condition $\max_j (1-r_j) \leqslant B \min_j (1-r_j)$, where $B \geqslant 1$ and estimate the integral appearing in Proposition 8, taking $\eta = \mathbf{1}$ for convenience of notation. Using (4.22) and Proposition 9 we obtain
$$
\begin{equation}
\begin{aligned} \, \int_{\mathbb T^n} K_\mathbf{t} (\mathbf{r} \cdot \mathbf{1}, \zeta) \, d| \mu| (\zeta) &= k_\mathbf{t} \int_{\mathbb T^n} U_\mathbf{t} (\mathbf{r} \cdot \zeta) \, d| \mu| (\zeta) = k_\mathbf{t} \sum_{Q \in \mathcal P_{\mathbf{r}}} \int_Q U_\mathbf{t} (\mathbf{r} \cdot \zeta) \, d| \mu| (\zeta) \nonumber \\ &\leqslant k_\mathbf{t} C_\mathbf{t} \prod_{k=1}^n \frac{1}{1-r_k} \sum_{(\varepsilon, j)} |\mu| (Q(\varepsilon, j)) \prod_{k=1}^n 2^{-(t_k + 2)j_k} \nonumber \\ &\leqslant 2^n k_\mathbf{t} C_\mathbf{t} \sum_j \prod_{k=1}^n 2^{-(t_k + 2)j_k} 2^{j_k} \frac{\kappa(r_k)}{\pi} (M_{j + \iota} \mu)(\mathbf{1}) \nonumber \\ &\leqslant \frac{2^{2n}}{\pi^n} k_\mathbf{t} C_\mathbf{t} \sum_j \prod_{k=1}^n 2^{-(t_k + 1)j_k} (M_{j + \iota} \mu) (\mathbf{1}) \nonumber \\ &\leqslant \frac{2^{2n}}{\pi^n} k_\mathbf{t} C_\mathbf{t} \sum_j q^{| j |} (M_{j + \iota} \mu) (\mathbf{1}) \nonumber \\ &\leqslant \frac{2^{2n}}{\pi^n} \frac{ k_\mathbf{t} C_\mathbf{t}}{q^{(n-1)b}} \sum_j q^{| j + \iota |} (M_{j + \iota} \mu) (\mathbf{1}), \end{aligned}
\end{equation}
\tag{4.23}
$$
where
$$
\begin{equation*}
\quad q = q(\mathbf{t}) = 2^{- \min_{1 \leqslant k \leqslant n} (t_k + 1)} < 1.
\end{equation*}
\notag
$$
For each $\gamma \in \mathbb Z_+^n$ there are at most $(b+1)^{n-1}$ representations $\gamma = j + \iota$. Hence we obtained
$$
\begin{equation}
\begin{aligned} \, \int_{\mathbb T^n} K_\mathbf{t} (\mathbf{r} \cdot \mathbf{1}, \zeta) \, d| \mu| (\zeta) & \leqslant \frac{2^{2n}}{\pi^n} \, \frac{ k_\mathbf{t} C_\mathbf{t}}{q^{(n-1)b}} (b+1)^{n-1} \sum_{\gamma \in \mathbb Z_+^n} q^{| \gamma |} (M_\gamma \mu) (\mathbf{1}) \nonumber \\ & = C(\mathbf{t}, B) (M_q \mu) (\mathbf{1}). \end{aligned}
\end{equation}
\tag{4.24}
$$
Note that the right-hand side does not depend on $\mathbf{r}$. Now we use (4.14) to get
$$
\begin{equation}
\sup_{\mathbf{r} \cdot \xi \in S_{A, B}(\mathbf{1})} | P_{\alpha,\beta}[d\mu](\mathbf{r} \cdot \xi)| \leqslant \frac{ C( \mathbf{t}, B )}{\delta_A^{c ( \mathbf{t} ) } }(M_q \mu)(\mathbf{1}).
\end{equation}
\tag{4.25}
$$
Since (4.25) holds at any point $\eta \in \mathbb T^n$ we proved the following proposition. Proposition 10. Let $0 < A< +\infty$ and $1 \leqslant B < +\infty$. Set $q = 2^{- m(\alpha, \beta)} < 1$, where $m(\alpha, \beta) = \min_{1 \leqslant k \leqslant n} (\operatorname{Re} \alpha_k + \operatorname{Re} \beta_k + 1)$, $t_j = \operatorname{Re} \alpha_j + \operatorname{Re} \beta_j$ for $1 \leqslant j \leqslant n$ and $c(\mathbf{t}) = \sum (t_j + 2)$. Then there are constants $\delta_A > 0$ and $C_{\alpha, \beta, B} < \infty$ such that 4.6. Restricted non-tangential maximal functionIn view of Proposition 10 it is natural to introduce a restricted non-tangential maximal function of aperture $0 < A< +\infty$ and the restriction $1 \leqslant B < \infty$ of a function $u \in C(\mathbb D^n)$ by $(\widetilde M_{A, B}^{\mathrm{NT}} u) (\eta) = \sup_{z \in S_{A, B} (\eta)} | u(z) | $, $\eta \in \mathbb T^n$, and of a complex Borel measure $\mu$ on $\mathbb T^n$ by
$$
\begin{equation}
(M_{A, B}^{\mathrm{NT}} \mu) (\eta) = \sup_{z \in S_{A, B} (\eta)} | P_{\alpha,\beta}[d\mu](z)| , \qquad \eta \in \mathbb T^n.
\end{equation}
\tag{4.27}
$$
Of course, $M_{A, B}^{\mathrm{NT}} \mu = \widetilde M_{A, B}^{\mathrm{NT}}( P_{\alpha, \beta}[d\mu])$. If $f \in L^1(\mathbb T^n)$ we write simply $M_{A, B}^{\mathrm{NT}} f$ instead of $M_{A, B}^{\mathrm{NT}} f \, dm_n$. Combining Proposition 10 and Lemma 6 we obtain the following weak $(1, 1)$-type estimate for the restricted non-tangential maximal function $M_{A, B}^{\mathrm{NT}}$. Theorem 16. For every $0 < A< +\infty$ and $1 \leqslant B < \infty$, there is a constant $C = C_{\alpha, \beta, A, B}$ such that 4.7. Applications to Fatou type theoremsNow we turn to Fatou type theorems, Theorem 16 will play a key role. Definition 8. We write $\mathrm{NT}_{A, B}\text{-}\lim_{z \to \eta} u(z) = l$ if for every $\varepsilon > 0$ there is a $\delta > 0$ such that $| u(z) - l | < \varepsilon$ whenever $| z - \eta | < \delta$ and $z \in S_{A, B}(\eta)$. Here $u$ is a function on $\mathbb D^n$, $0 < A< \infty$, $1 \leqslant B < \infty$ and $\eta$ is in $\mathbb T^n$. If $\mathrm{NT}_{A, B}\text{-}\lim_{z \to \eta} u(z) = l$ for every $0 < A< \infty$ and $1 \leqslant B < \infty$ we write $\mathrm{RNT}\text{-}\lim_{z \to \eta} u(z) = l$ (the restricted non-tangential limit). If $u$ is real valued we can analogously consider $\mathrm{NT}_{A, B}\text{-}\limsup_{z \to \eta} u(z)$. Proof. We fix $0 < A<\infty$ and $1 \leqslant B < \infty$. Let us define for $\delta > 0$, $g \in L^1(\mathbb T^n)$ and $\eta \in \mathbb T^n$ and note the following properties of $\Omega^\delta (g, \eta)$.
$1^\circ$. $\Omega^\delta (g, \eta)$ is increasing in $\delta > 0$ and $\Omega^\delta (g, \eta) \leqslant 2 (M_{A, B}^{\mathrm{NT}} g) (\eta)$. $2^\circ$. $\lim_{\delta \to 0} \Omega^\delta (g, \eta) = 0$ if and only if $\mathrm{NT}_{A, B}\text{-}\lim_{z \to \eta} P_{\alpha, \beta}[g](z)$ exists. $3^\circ$. $\lim_{\delta \to 0} \Omega^\delta (g, \eta) = \lim_{\delta \to 0} \Omega^\delta (g - \varphi, \eta)$ for every $\varphi \in C(\mathbb T^n)$. The first two properties are obvious, the third one follows from Theorem 7. Set $E_\varepsilon = \{ \zeta \in \mathbb T^n \colon \lim_{\delta \to 0} \Omega^\delta (f, \zeta) > \varepsilon \}$ for $\varepsilon > 0$. By properties $1^\circ$ and $3^\circ$ we have, for any $\varphi \in C(\mathbb T^n)$,
$$
\begin{equation*}
E_\varepsilon = \Bigl\{ \zeta \in \mathbb T^n \colon \lim_{\delta \to 0} \Omega^\delta (f - \varphi, \zeta) > \varepsilon \Bigr\} \subset \biggl\{ \zeta \in \mathbb T^n \colon [M_{A, B}^{\mathrm{NT}} (f - \varphi)] (\zeta) > \frac{\varepsilon}2 \biggr\}.
\end{equation*}
\notag
$$
Using Theorem 16 we deduce
$$
\begin{equation*}
m_n \biggl( \biggl\{ \zeta \in \mathbb T^n \colon [M_{A, B}^{\mathrm{NT}} (f - \varphi)] (\zeta) > \frac{\varepsilon}2 \biggr\} \biggr) \leqslant \frac{2C_{\alpha, \beta, A, B}}{\varepsilon} \| f - \varphi \|_{L^1 (\mathbb T^n)}.
\end{equation*}
\notag
$$
Since $C (\mathbb T^n)$ is dense in $L^1 (\mathbb T^n)$, the set $ E_\varepsilon $ is contained in a set of arbitrarily small measure and therefore $E_\varepsilon$ is a null set. Since $\varepsilon > 0$ is arbitrary it follows that $\lim_{\delta \to 0} \Omega^\delta (f, \eta) = 0$ for almost every $\eta \in \mathbb T^n$. Therefore, using property $2^\circ$, $\mathrm{NT}_{A, B}\text{-}\lim_{z \to \eta} P_{\alpha, \beta}[f](z) = F(\eta)$ exists for almost every $\eta \in \mathbb T^n$. Now we have $F(\eta) = f(\eta)$ by Theorem 11, formula (3.14). Theorem 17 is proved. A local version of the above theorem for $n$-harmonic functions can be found in Chap. XVII, Theorem 4.13 of [14]. It is not possible to replace $\mathrm{RNT}$ limits in the above theorem by non-restricted $\text{NT}$ limits, that is, by limits over full Stoltz cones $S_A(\eta)$, see [22] and [13]. A special case of the following lemma appeared in [15], p. 29, Lemma 3, where $n$-harmonic functions were considered. Lemma 8. If a complex Borel measure $ \mu $ on $ \mathcal M (\mathbb T^n)$ vanishes in an open set $ V\subset \mathbb{T}^n$ and $u = P_{\alpha, \beta}[d\mu]$, then Proof. We fix an aperture $0 < A< \infty$ and $1 \leqslant B < \infty$. By Lemma 6 $M_q \mu$ is finite almost everywhere on $\mathbb T^n$ and in particular on $V$. Hence it suffices to prove that $\mathrm{NT}_{A, B}\text{-}\lim_{z \to \eta} u(z) = 0$ whenever $\eta$ is a point in $V$ such that $(M_q \mu)(\eta) < +\infty$. We assume, for convenience of notation, that $\eta = \mathbf{1}$. Since $\mathbf{1}$ is in $V$, there is a positive integer $N$ such that $I_N^n \subset V$, where $I_N = \{ e^{i\theta} \colon | \theta | \leqslant \pi 2^{-N} \}$. Now assume that $\mathbf{r}$ in $[0, 1)^n$ satisfies $\max_j (1-r_j) \leqslant B \min_j (1-r_j)$. Using (4.14) and (4.23) we obtain
$$
\begin{equation}
\begin{aligned} \, &\sup_{\xi \colon \mathbf{r} \cdot \xi \in S_{A, B} (\mathbf{1})} | P_{\alpha,\beta}[d\mu](\mathbf{r} \cdot \xi) | \leqslant \frac{k_{\alpha, \beta}}{k_\mathbf{t} \delta_A^{c(\mathbf{t})}} \int_{\mathbb T^n} K_\mathbf{t} (\mathbf{r}, \zeta) \, d| \mu| (\zeta) \nonumber \\ &\qquad \leqslant C_\mathbf{t} \frac{k_{\alpha, \beta}}{\delta_A^{c(\mathbf{t})} } \prod_{k=1}^n \frac{1}{1-r_k} \sum_{(\varepsilon, j)} |\mu| (Q(\varepsilon, j)) \prod_{k=1}^n \frac{1}{2^{(t_k + 2)j_k}} \nonumber \\ &\qquad \leqslant C_\mathbf{t} 2^n \frac{k_{\alpha, \beta}}{\delta_A^{c(\mathbf{t})} } \prod_{k=1}^n \frac{1}{1-r_k} \sum_{j \in I(\mathbf{r})} |\mu| (R(j)) \prod_{k=1}^n \frac{1}{2^{(t_k + 2)j_k}}, \end{aligned}
\end{equation}
\tag{4.32}
$$
where $I(\mathbf{r}) = \{0, \dots, p(r_1)\} \times \dots \times \{0, \dots, p(r_n)\}$. We see, using (4.16) and (4.17), that $|\mu| (R(j)) = 0$ whenever $j_k < p(r_k) - N$ for all $1 \leqslant k \leqslant n$. Next, for $\mathbf{r} \in [0, 1)^n$ we set $Z( \mathbf{r} ) = \{ (j_1, \dots, j_n) \in \mathbb Z_+^n \colon j_k < p(r_k) - N, k = 1, \dots, n \} $. Therefore, arguing as in derivation of (4.25), we deduce from (4.32),
$$
\begin{equation}
\begin{aligned} \, \varphi (\mathbf{r}) & = \sup_{\xi \colon \mathbf{r} \cdot \xi \in S_{A, B} (\mathbf{1})} | P_{\alpha,\beta}[d\mu](\mathbf{r} \cdot \xi) | \nonumber \\ &\leqslant C(\alpha, \beta, A) \prod_{k=1}^n \frac{1}{1-r_k} \sum_{j \in \mathbb Z_+^n \setminus Z( \mathbf{r} )} |\mu| (R(j)) \prod_{k=1}^n \frac{1}{2^{(t_k + 2)j_k}} \nonumber \\ &\leqslant C(\alpha, \beta, A) B^{n-1} \sum_{j \in \mathbb Z_+^n \setminus Z( \mathbf{r} )} q^{| j |} (M_{j + \iota} \mu) (\mathbf{1}) \nonumber \\ &\leqslant C(\alpha, \beta, A) \biggl( \frac{ B }{ q^{b+1} } \biggr)^{n-1} \sum_{j \in \mathbb Z_+^n \setminus Z( \mathbf{r} )} q^{| j + \beta|} (M_{j + \iota} \mu) (\mathbf{1}). \end{aligned}
\end{equation}
\tag{4.33}
$$
By our assumption $(M_q \mu) (\mathbf{1}) = \sum_{\gamma \in Z_+^n} q^{| \gamma |} (M_\gamma \mu) (\mathbf{1}) < +\infty$. Since $p(\rho) $ tends to $ +\infty$ as $\rho \uparrow 1$ we see that for every finite $F \subset \mathbb Z_+^n$ there is an $\delta > 0$ such that $F \subset Z ( \mathbf{r} )$ whenever $1-r_k < \delta$ for all $1 \leqslant k \leqslant n$. Thus the sum appearing in (4.33) can be seen as a remainder of a convergent sum, it tends to zero as $\mathbf{r} \to \mathbf{1}$. Hence $\lim_{\mathbf{r} \to 1} \varphi (\mathbf{r}) = 0$, which is equivalent to $\mathrm{NT}_{A, B}\text{-}\lim_{z \to \mathbf{1}} u(z) = 0$. Lemma 8 is proved. Convergence to $0$ at every point $\eta$ in $V$ fails even in the special case of $n$-harmonic functions (that is, $\alpha = \beta = 0$). See p. 25 in [15], for an example. This failure of localization is specifically multidimensional phenomenon: if $n=1$ there are pointwise results of this nature, for example, Theorem 6.2 from [11]. Proof. We fix $0 < A< \infty$ and $1 \leqslant B < \infty$ and set for $\nu \in \mathcal M(\mathbb T^n)$ and $\delta > 0$
$$
\begin{equation}
\Lambda_{A, B} (\nu, \delta) = \{ \zeta \in \mathbb T^n \colon \mathrm{NT}_{A, B}\text{-}\limsup_{z \to \zeta} | P_{\alpha,\beta}[d\nu](z) | > \delta \},
\end{equation}
\tag{4.34}
$$
$$
\begin{equation}
\Phi_{A, B} (\nu, \delta) = \{ \zeta \in \mathbb T^n \colon (M_{A, B}^{\mathrm{NT}} \nu) (\zeta) > \delta \}.
\end{equation}
\tag{4.35}
$$
Clearly, $\Lambda_{A, B} (\nu, \delta) \subset \Psi_{A, B} (\nu, \delta)$. It suffices to prove that $\Lambda_{A, B} (\mu, \delta)$ is a null set for every $\delta > 0$. Let us fix $\delta > 0$. By Lemma 8 the sets $\Lambda_{A, B} (\nu, \delta)$ and $\Lambda_{A, B} (\nu - \sigma, \delta)$ are equal up to a set of measure zero whenever $\sigma \in \mathcal M(\mathbb T^n)$ is supported on a compact set $K$ of Lebesgue measure $0$. Hence, for such $\sigma$ we have by Theorem 16
$$
\begin{equation}
m_n( \Lambda_{A, B} (\mu, \delta)) = m_n( \Lambda_{A, B} (\mu - \sigma, \delta)) \leqslant m_n( \Psi_{A, B} (\mu - \sigma, \delta)) \leqslant C_{\alpha, \beta, A, B} \frac{ \| \mu - \sigma \|}{\delta}.
\end{equation}
\tag{4.36}
$$
By regularity properties of complex Borel measures, for a given $\varepsilon > 0$ there is a compact set $K \subset \mathbb T^n$ such that $m_n (K) = 0$ and $\| \mu - \mu_K \| < \varepsilon$, where $\mu_K (E) = \mu (E \cap K)$. Hence, taking $\sigma = \mu_K$ in (4.36) we see that $\Lambda_{A, B} (\mu, \delta)$ is contained in a set of arbitrarily small measure, hence $\Lambda_{A, B} (\mu, \delta)$ is a null set. Theorem 18 is proved. Combining Theorems 17 and 18 we obtain the following corollary. In the special case of $n$-harmonic functions, thus result appeared in [23]. However, for $(\alpha, \beta)$-harmonic functions it is new even in dimension $1$, although in [11] one parameter case was studied. Corollary 2. Let $\mu \in \mathcal M (\mathbb T^n)$ and $u = P_{\alpha, \beta}[d\mu]$. Then where $d\mu = f \, dm_n + d\sigma$ is Lebesgue–Radon–Nikodým decomposition of $d\mu$ into absolutely continuous part $f\, dm_n$ and singular part $d\sigma$. Proposition 11. Let $u \in \operatorname{sh}_{\alpha, \beta} (\mathbb D^n)$ and assume $M^+ u (\zeta) = \sup_{0 \leqslant r < 1} | u(r\zeta) |$ is in $L^1(\mathbb T^n)$. Then $u = P_{\alpha, \beta} [f]$ for some $f \in L^1 (\mathbb T^n)$. Proof. Clearly, $u \in \operatorname{sh}^1_{\alpha, \beta} (\mathbb D^n)$ and therefore $u = P_{\alpha, \beta} [d\mu]$ for some $\mu$ in $\mathcal M (\mathbb T^n)$. We have Lebesgue–Radon–Nikodým decomposition $d \mu = f \, dm_n + d \sigma$ and by Corollary 2 we have $\lim_{r \to 1} u(r\zeta) = f(\zeta)$ for almost every $\zeta \in \mathbb T^n$. Since $M^+ u$ is integrable, the Dominated Convergence Theorem implies that convergence is also in the $L^1 (\mathbb T^n)$ norm. Since $u_r \, dm_n \to d\mu$ in the weak$^\ast$ sense, this implies $d\mu = f \, dm_n$. Proposition 11 is proved.
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