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Sbornik: Mathematics, 2024, Volume 215, Issue 11, Pages 1468–1498
DOI: https://doi.org/10.4213/sm10067e (Mi sm10067) 		 
This article is cited in 1 scientific paper (total in 1 paper)

Kolmogorov widths of a Sobolev class with constraints on derivatives in different metrics

A. A. Vasil'evaab

a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
b Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
English version PDF (677 kB) HTML full-text Citations (1) Russian version article
References:
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DOI: https://doi.org/10.4213/sm10067e
Abstract: We obtain order estimates for the Kolmogorov widths of periodic Sobolev classes defined by constraints on the $L_{p_j}$-norm of the $r_j$th derivative with respect to the $j$th variable for $1\le j\le d$.
Bibliography: 31 titles.
Keywords: Kolmogorov width, anisotropic Sobolev class.
Received: 18.01.2024 and 28.06.2024
Published: 04.02.2025
Bibliographic databases:
Document Type: Article
MSC: 41A46
Language: English
Original paper language: Russian

§ 1. Introduction

We study the Kolmogorov widths of the periodic Sobolev class on the $d$-dimensional torus $\mathbb{T}^d$ defined by the constraints

$$ \begin{equation} \biggl\| \frac{\partial^{r_j}}{\partial x^{r_j}_j}f\biggr\|_{L_{p_j}(\mathbb{T}^d)} \leqslant 1, \qquad j=1,\dots,d, \end{equation} \tag{1.1} $$
on the partial derivatives. This class of functions is an example of the intersection of several Sobolev classes defined by constraints on one of the partial derivatives (see [1]–[3]). Functional classes of this kind on $\mathbb R^d$ were studied in [4], § 6. Sufficient conditions for an embedding of such a class in the Lorentz space were obtained; the particular case $p_1=\dots=p_d$ was considered earlier in [5]. Anisotropic Sobolev classes on a domain $\Omega \subset \mathbb R^d$ defined by constraints on $\biggl\| \dfrac{\partial^{r_j}}{\partial x^{r_j}_j}f\biggr\|_{L_{p_j}(\Omega)}$ ($j=1,\dots,d$) and the norm of the function $f$ in the space $L_{p_0}$ with a special weight were studied in [6], where it was assumed that
$$ \begin{equation} 1-\sum_{k=1}^d \frac{1}{p_kr_k}>0\quad\text{and} \quad q\geqslant \max_{1\leqslant k\leqslant d} p_k. \end{equation} \tag{1.2} $$
Embedding theorems in the weighted space $L_q$ were obtained, and some (not order-sharp in general) estimates for the Kolmogorov widths were put forward. Here we consider nonweighted periodic classes, where condition (1.2) is not assumed, and the width estimates we obtain are order sharp.

The class of functions with constraints of the form (1.1) is an example of an anisotropic Sobolev class. Anisotropic function classes of another type (defined by constraints on derivatives in a mixed norm) were studied in [1] and [7]–[11].

Definition 1. Let $X$ be a normed linear space, and let $M\subset X$ and $n\in \mathbb Z_+$. Then the Kolmogorov $n$-width of $M$ in $X$ is defined by

$$ \begin{equation*} d_n(M,X)= \mathop{\smash\inf\vphantom\sup} _{L\in \mathcal L_n(X)} \sup_{x\in M} \mathop{\smash\inf\vphantom\sup} _{y\in L} \|x-y\|; \end{equation*} \notag $$
here $\mathcal L_n(X)$ is the class of all subspaces in $X$ of dimension $\leqslant n$.

Widths of finite-dimensional balls were estimated in [12]–[17]. For a historical account of this problem, see also [18]–[20].

The papers [2], [3] and [21]–[24] were concerned with the problem of estimates for widths of a periodic Sobolev class on $\mathbb{T}^d:=[0,2\pi]^d$ defined by a constraint on the $L_p$-norm of one or several (fractional) partial derivatives, and also for widths of the intersection of some periodic Sobolev classes on $\mathbb{T}^1$; see also [25] and [20]. The result of [3] on widths of an intersection of one-dimensional Sobolev classes was extended in [26] to the case of ‘small smoothness’ for $q>2$, with the exception of some ‘limiting’ cases.

Recall the definition of the Weyl derivative of a periodic function (see, for example, [18], Ch. 2, § 2). For $d\in \mathbb N$, $d\geqslant 2$, let $\mathbb{T}^d=[0,2\pi]^d$. We denote the space of distributions on $\mathbb{T}^d$ by $\mathcal S'(\mathbb{T}^d)$ (the corresponding space of test functions consists of infinitely smooth periodic functions). To each distribution $f\in \mathcal S'(\mathbb{T}^d)$ there corresponds its Fourier series expansion $f=\sum_{\overline{k}\in \mathbb Z^d} c_{\overline{k}}(f) e^{i(\overline{k},\cdot)}$ with convergence in the topology of $\mathcal S'(\mathbb{T}^d)$; here and in what follows $(\cdot,\cdot)$ is the standard inner product on $\mathbb R^d$. We set

$$ \begin{equation*} \mathring{\mathbb Z}^d=\{(k_1,k_2,\dots,k_d)\in \mathbb Z^d\colon k_1k_2\dotsb k_d\ne 0\} \end{equation*} \notag $$
and
$$ \begin{equation*} \mathring{\mathcal S}'(\mathbb{T}^d)=\biggl\{ f\in \mathcal S'(\mathbb{T}^d)\colon f=\sum_{\overline{k}\in \mathring{\mathbb Z}^d} c_{\overline{k}}(f) e^{i(\overline{k}, \cdot)}\biggr\}. \end{equation*} \notag $$

Let $r_j>0$, $1\leqslant j\leqslant d$. The Weyl derivative of order $r_j$ with respect to $x_j$ of a distribution $f\in \mathring{\mathcal S}'(\mathbb{T}^d)$ is defined by

$$ \begin{equation*} \partial_j^{r_j}f:=\frac{\partial^{r_j}f}{\partial x_j^{r_j}} :=\sum_{\overline{k}\in \mathring{\mathbb Z}^d} c_{\overline{k}}(f) (ik_j)^{r_j} e^{i(\overline{k},\cdot)}, \end{equation*} \notag $$
where $(ik_j)^{r_j}=|k_j|^{r_j}e^{\operatorname{sgn}k_j\cdot i\pi r_j/2}$.

Let $1<q<\infty$, $1<p_j<\infty$, $r_j>0$, $1\leqslant j\leqslant d$, $\overline{p}=(p_1,\dots, p_d)$ and $\overline{r}=(r_1,\dots,r_d)$. We set

$$ \begin{equation*} W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d) =\{f\in \mathring{\mathcal S}'(\mathbb{T}^d)\colon \|\partial_j^{r_j}f\|_{L_{p_j}(\mathbb{T}^d)}\leqslant 1, \, 1\leqslant j\leqslant d\}. \end{equation*} \notag $$

In the present paper we consider the order estimate problem for the Kolmogorov width of the class $W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d)$ in the space $L_q(\mathbb{T}^d)$.

Given $\overline{a}=(a_1,\dots,a_d) \in \mathbb R^d$, the harmonic mean of $a_1,\dots,a_d$ is defined by

$$ \begin{equation*} \langle \overline{a}\rangle=\frac{d}{1/a_1+\dots+1/a_d}. \end{equation*} \notag $$
For $\overline{a}=(a_1,\dots,a_d)\in \mathbb R^d$ and $\overline{b}=(b_1,\dots,b_d)\in \mathbb R^d$ we set $\overline{a}\circ\overline{b}=(a_1b_1,\dots,a_db_d)$.

From Theorem 1 in [1] it follows that the set $W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d)$ is bounded in $L_q(\mathbb{T}^d)$ if and only if

$$ \begin{equation*} \frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ\overline{r}\rangle} \geqslant 0. \end{equation*} \notag $$
The above embedding is compact if this inequality is strict (see Theorem 5 in [1]).

Let us formulate some theorems on estimates for widths.

First we consider the case involving some additional constraints on the parameters. Then estimates of widths are explicitly found (except in the case where certain ‘limiting’ relations hold between the parameters).

Let us introduce some notation for order equalities and inequalities. Let $X$ and $Y$ be sets, and let $f_1,f_2\colon X\times Y\to \mathbb{R}_+$. We write $f_1(x,y)\underset{y}{\lesssim} f_2(x,y)$ (or $f_2(x,y)\underset{y}{\gtrsim} f_1(x,y)$ if for each $y\in Y$ there exists $c(y)>0$ such that $f_1(x,y)\leqslant c(y)f_2(x,y)$ for all $x\in X$; $f_1(x,y)\underset{y}{\asymp} f_2(x,y)$ if $f_1(x,y) \underset{y}{\lesssim} f_2(x,y)$ and $f_2(x,y)\underset{y}{\lesssim} f_1(x,y)$.

Theorem 1. Let $d\in \mathbb N$, $d\geqslant 2$, $1<q<\infty$, $1<p_j<\infty$ and $r_j>0$, $j=1,\dots,d$, and let $\frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ\overline{r}\rangle} > 0$. Assume that

$$ \begin{equation} \sum_{i=1}^d \frac{1}{r_i}\biggl(\frac{1}{p_i}-\frac{1}{p_j}\biggr)<1, \qquad j=1,\dots,d. \end{equation} \tag{1.3} $$

1. Let $p_j\geqslant q$, $j=1,\dots,d$. Then

$$ \begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{r}, \overline{p}, q, d}{\asymp} n^{-\langle \overline{r}\rangle /d}. \end{equation*} \notag $$

2. Let $1<q\leqslant 2$.

3. Let $2<q<\infty$. Assume that there exists $i\in \{1,\dots,d\}$ such that $p_i<q$. Also set $\theta_1=\frac{\langle \overline{r}\rangle}{d}$, $\theta_2=\frac{\langle \overline{r}\rangle}{d}+\frac 12-\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ\overline{r}\rangle}$ and $\theta_3=\frac q2\bigl( \frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ \overline{r}\rangle}\bigr)$.

Remark 1. We will also show that in the case where $p_j\geqslant q$ for $1\leqslant j\leqslant d$ the assertion of the theorem also holds without condition (1.3).

Now we consider the general case where (1.3) is not assumed to hold.

In what follows we set $\max \varnothing:=-\infty$.

Theorem 2. Let $d\in \mathbb N$, $d\geqslant 2$, $1<q<\infty$, $1<p_j<\infty$ and $r_j>0$, $j=1, \dots,d$, and let $\frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ\overline{r}\rangle} > 0$.

1. Let $1<q\leqslant 2$. Set

$$ \begin{equation} \begin{gathered} \, I_0=\{j\in 1,\dots,d\colon p_j\geqslant q\}, \qquad J_0=\{j\in 1,\dots,d\colon p_j\leqslant q\}, \\ I_0'=\{j\in 1,\dots,d\colon p_j> q\}, \qquad J_0'=\{j\in 1,\dots,d\colon p_j< q\}; \end{gathered} \end{equation} \tag{1.4} $$
and let $\lambda_{i,j}\in [0,1]$ be defined by
$$ \begin{equation} \frac 1q=\frac{1-\lambda_{i,j}}{p_i}+\frac{\lambda_{i,j}}{p_j}, \qquad i\in I_0', \quad j\in J_0'. \end{equation} \tag{1.5} $$
For $\alpha_1,\dots,\alpha_d\in \mathbb R$ set
$$ \begin{equation*} \begin{gathered} \, h_1(\alpha_1,\dots,\alpha_d)=\max_{j\in I_0} r_j\alpha_j, \\ h_2(\alpha_1,\dots,\alpha_d)=\max_{j\in J_0}\biggl(r_j\alpha_j-\frac1{p_j}+\frac1q\biggr), \\ h_3(\alpha_1,\dots,\alpha_d)=\max_{i\in I_0',j\in J_0'} ((1-\lambda_{i,j})r_i\alpha_i+\lambda_{i,j}r_j\alpha_j) \end{gathered} \end{equation*} \notag $$
and
$$ \begin{equation} h(\alpha_1,\dots,\alpha_d)=\max_{1\leqslant j\leqslant 3} h_j(\alpha_1,\dots,\alpha_d). \end{equation} \tag{1.6} $$
Assume that the function $h$ has a unique minimum point $(\widehat\alpha_1,\dots,\widehat\alpha_d)$ on the set
$$ \begin{equation} D=\{(\alpha_1,\dots,\alpha_d)\in \mathbb R^{d}\colon \alpha_1\geqslant 0,\dots,\alpha_d\geqslant 0,\, \alpha_1+\dots+\alpha_d=1\}. \end{equation} \tag{1.7} $$
Then
$$ \begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{r},\overline{p},q,d}{\asymp} n^{-h(\widehat \alpha_1,\dots,\widehat \alpha_d)}. \end{equation*} \notag $$

2. Let $2<q<\infty$. Set

$$ \begin{equation} \begin{gathered} \, I=\{j\in 1,\dots,d\colon p_j\geqslant q\}, \qquad J=\{j\in 1,\dots,d\colon 2\leqslant p_j\leqslant q\}, \\ K=\{j\in 1,\dots,d\colon p_j\leqslant 2\}, \\ I'=\{j\in 1,\dots,d\colon p_j> q\}, \qquad J'=\{j\in 1,\dots,d\colon 2< p_j< q\}, \\ K'=\{j\in 1,\dots,d\colon p_j< 2\}; \end{gathered} \end{equation} \tag{1.8} $$
and let $\lambda_{i,j}\in [0,1]$ and $\mu_{i,j}\in [0,1]$ be defined by
$$ \begin{equation} \begin{gathered} \, \frac 1q=\frac{1-\lambda_{i,j}}{p_i}+\frac{\lambda_{i,j}}{p_j}, \qquad i\in I', \quad j\in J'\cup K, \\ \frac 12=\frac{1-\mu_{i,j}}{p_i}+\frac{\mu_{i,j}}{p_j}, \qquad i\in I\cup J', \quad j\in K'. \end{gathered} \end{equation} \tag{1.9} $$
For $\alpha_1,\dots,\alpha_d\in \mathbb R$ and $s\in \mathbb R$ set
$$ \begin{equation*} \begin{gathered} \, \widetilde h_1(\alpha_1,\dots,\alpha_d,s)=\max_{j\in I} r_j\alpha_j, \\ \widetilde h_2(\alpha_1,\dots,\alpha_d,s)=\max_{j\in J} \biggl(r_j\alpha_j -\frac 12\cdot \frac{1/p_j-1/q}{1/2-1/q}(s-1)\biggr), \\ \widetilde h_3(\alpha_1,\dots,\alpha_d,s)=\max_{j\in K}\biggl(r_j\alpha_j-\frac{s}{p_j}+\frac12\biggr), \\ \widetilde h_4(\alpha_1,\dots,\alpha_d,s)=\max_{i\in I',j\in J'\cup K} ((1-\lambda_{i,j})r_i\alpha_i+\lambda_{i,j}r_j\alpha_j), \\ \widetilde h_5(\alpha_1,\dots,\alpha_d,s)=\max_{i\in I\cup J',j\in K'} \biggl((1-\mu_{i,j})r_i\alpha_i+\mu_{i,j}r_j\alpha_j-\frac s2+\frac12\biggr) \end{gathered} \end{equation*} \notag $$
and
$$ \begin{equation*} \widetilde h(\alpha_1,\dots,\alpha_d,s)=\max_{1\leqslant j\leqslant 5} \widetilde h_j(\alpha_1,\dots, \alpha_d,s). \end{equation*} \notag $$
Assume that the function $\widetilde h$ has a unique minimum point $(\widehat \alpha_1,\dots,\widehat \alpha_d,\widehat s)$ on the set
$$ \begin{equation*} \begin{aligned} \, \widetilde D &=\biggl\{(\alpha_1,\dots,\alpha_d,s)\in \mathbb R^{d+1}\colon 1\leqslant s \leqslant \frac q2, \\ &\qquad\qquad \alpha_1\geqslant 0,\dots,\alpha_d\geqslant 0, \, \alpha_1+\dots+\alpha_d=s\biggr\}. \end{aligned} \end{equation*} \notag $$
Then
$$ \begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{r},\overline{p},q,d}{\asymp} n^{-\widetilde h(\widehat \alpha_1,\dots,\widehat \alpha_d,\widehat s)}. \end{equation*} \notag $$

Now consider the case where condition (1.3) is not satisfied.

Theorem 3. Let $d\in \mathbb N$, $d\geqslant 2$, $r_k>0$ and $1<p_k\leqslant q<\infty$ for $k=1,\dots,d$, and assume that there exists $j\in \{1,\dots,d\}$ such that

$$ \begin{equation*} \sum_{i=1}^d \frac{1}{r_i}\biggl(\frac{1}{p_i}-\frac{1}{p_j}\biggr)\geqslant 1. \end{equation*} \notag $$
Then $W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d)$ is not compactly embedded in $L_q(\mathbb{T}^d)$.

For $d=2$, if condition (1.3) is not met, then the orders of widths can be calculated explicitly. Assume without loss of generality that $p_1>p_2$. Then $r_2\leqslant \frac{1}{p_2} -\frac{1}{p_1}$. By Remark 1 and Theorem 3 it suffices to consider the case where $p_2<q<p_1$.

Theorem 4. Let $1<p_2<q<p_1<\infty$, $r_1>0$ and $r_2>0$. Assume that

$$ \begin{equation} r_2\leqslant \frac{1}{p_2} -\frac{1}{p_1}. \end{equation} \tag{1.10} $$
Then the following hold.

1. Let $1<q\leqslant 2$. Let $\lambda\in (0,1)$ be defined by $\frac 1q=\frac{1-\lambda}{p_1}+ \frac{\lambda}{p_2}$. Assume that $\frac{\langle \overline{r}\rangle}{2} \ne \lambda r_2$. Then

$$ \begin{equation} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^2),L_q(\mathbb{T}^2)) \underset{\overline{r},\overline{p},q}{\asymp} n^{-\min \{\langle \overline{r}\rangle/2, \lambda r_2\}}. \end{equation} \tag{1.11} $$

2. Let $2<q<\infty$. Let $\lambda\in (0,1)$ be defined by $\frac 1q= \frac{1-\lambda}{p_1}+\frac{\lambda}{p_2}$, and let $\widehat s$ be defined by $\widehat s\bigl(1-\frac{r_2(1-2/q)}{1/p_2-1/p_1}\bigr)=1$.

§ 2. Preliminaries

Let $\overline{m}=(m_1,\dots,m_d)\in \mathbb N^d$. Set

$$ \begin{equation} \begin{gathered} \, m=m_1+\dots+m_d, \\ \square_{\overline{m}}=\{k\in \mathbb Z^d\colon 2^{m_j-1}\leqslant |k_j|<2^{m_j}, \, 1\leqslant j\leqslant d\} \notag \end{gathered} \end{equation} \tag{2.1} $$
and $\mathcal T_{\overline{m}}=\operatorname{span} \{e^{i(\overline{k},\, \cdot)}\}_{\overline{k}\in \square_{\overline{m}}}$. Given $f(\cdot)=\sum_{\overline{k}\in \mathring{\mathbb Z}^d} c_{\overline{k}}(f)e^{i(\overline{k},\, \cdot)}$, we define
$$ \begin{equation*} \delta_{\overline{m}} f(\cdot)=\sum_{\overline{k}\in \square_{\overline{m}}} c_{\overline{k}}(f)e^{i(\overline{k},\, \cdot)}. \end{equation*} \notag $$

For $f\in \mathring{\mathcal S}'(\mathbb{T}^d)$ set

$$ \begin{equation*} Pf(t)=\biggl(\sum_{\overline{m}\in \mathbb N^d}|\delta_{\overline{m}}f(t)|^2\biggr)^{1/2}. \end{equation*} \notag $$

Theorem 5 (Littlewood–Paley theorem; see [27], § 1.5.2; [18], Ch. 2, § 2.3, Theorem 15; [7], Ch. III, § 15.2; and [8]). Let $1<q<\infty$. Then $f\in L_q(\mathbb{T}^d)$ if and only if $Pf\in L_q(\mathbb{T}^d)$; in addition,

$$ \begin{equation*} \|f\|_{L_q(\mathbb{T}^d)} \underset{q,d}{\asymp} \|Pf\|_{L_q(\mathbb{T}^d)}. \end{equation*} \notag $$

Theorem 6 (see [20], Theorem 3.3.1, and [18], Ch. 2, § 2.3, Theorem 18 for $r_j\geqslant 0$). Let $1<p_j<\infty$ and $r_j\in \mathbb R$. Then for $f\in \mathcal T_{\overline{m}}$,

$$ \begin{equation*} \|\partial_j^{r_j} f\|_{L_{p_j}(\mathbb{T}^d)} \underset{\overline{p},\overline{r},d}{\asymp} 2^{m_jr_j} \|x\|_{L_{p_j}(\mathbb{T}^d)}. \end{equation*} \notag $$

For arbitrary $r_j\in \mathbb R$, this result is a consequence of Marcinkiewicz’s multiplier theorem (see [27], § 1.5.3, and [7], Ch. III, § 15.3).

Theorem 7 (see [3], Theorem B, and [28], vol. 2, Ch. X, Theorem 7.5). There exists an isomorphism $A\colon \mathcal T_{\overline{m}} \to \mathbb R^{2^m}$ such that for all $q\in (1,\infty)$ and $f\in \mathcal T_{\overline{m}}$,

$$ \begin{equation*} \|f\|_{L_q(\mathbb{T}^d)} \underset{q,d}{\asymp} 2^{-m/q} \|Ax\|_{l_q^{2^m}}. \end{equation*} \notag $$

Given $N\in \mathbb N$, $1\leqslant q\leqslant \infty$, $(x_i)_{i=1}^N\in \mathbb{R}^N$, set

$$ \begin{equation*} \|(x_i)_{i=1}^N\|_{l_q^N}=\biggl(\sum_{i=1}^N |x_i|^q\biggr)^{1/q} \quad\text{for } q<\infty \end{equation*} \notag $$
and
$$ \begin{equation*} \|(x_i)_{i=1}^N\|_{l_q^N}=\max_{1\leqslant i\leqslant N}|x_i| \quad\text{for } q=\infty. \end{equation*} \notag $$
We denote the space $\mathbb{R}^N$ equipped with this norm by $l_q^N$, and $B_q^N$ is the unit ball in $l_q^N$.

The widths $d_n(B_p^N,l_q^N)$ were estimated by Pietsch, Stesin, Kashin, Gluskin and Garnaev (see [12]–[17]). In what follows we present results on width estimates in a number of cases considered below.

Theorem 8 ([16]). Let $1\leqslant p\leqslant q<\infty$ and $0\leqslant n\leqslant N/2$.

1. Let $1\leqslant q\leqslant 2$. Then $d_n(B_p^N,l_q^N) \asymp 1$.

2. Let $2<q<\infty$ and $\omega_{pq}=\min \bigl\{1,\frac{1/p-1/q}{1/2-1/q}\bigr\}$. Then

$$ \begin{equation*} d_n(B_p^N,l_q^N) \underset{q}{\asymp} \min \{1,n^{-1/2}N^{1/q}\} ^{\omega_{pq}}. \end{equation*} \notag $$

Theorem 9 (see [12] and [13]). Let $1\leqslant q\leqslant p\leqslant \infty$ and $0\leqslant n\leqslant N$. Then

$$ \begin{equation*} d_n(B_p^N,l_q^N)=(N-n)^{1/q-1/p}. \end{equation*} \notag $$

Order estimates for Kolmogorov widths of intersections of $N$-dimensional balls were obtained in [2] for $N=2n$ and in [29] for $N\geqslant 2n$. In [29] an order estimate was written down explicitly under the additional assumption that none of these balls contains another. In [26], Proposition 1, an estimate for the intersection of a finite number of balls was given in a general case. Let us formulate this result.

Theorem 10 (see [26], Proposition 1). Let $A$ be a nonempty finite set, let $1\leqslant p_\alpha\leqslant \infty$ and $\nu_\alpha>0$, $\alpha \in A$, let

$$ \begin{equation} M_0=\bigcap_{\alpha \in A} \nu_\alpha B_{p_\alpha}^N, \end{equation} \tag{2.2} $$
$N\geqslant 2n$, and let the numbers $\lambda_{\alpha,\beta}$ and $\widetilde \lambda_{\alpha,\beta}$ be defined by
$$ \begin{equation} \frac{1}{q}=\frac{1-\lambda_{\alpha,\beta}}{p_\alpha}+ \frac{\lambda_{\alpha,\beta}}{p_\beta} \quad\textit{if } p_\alpha > q\textit{ and } p_\beta < q \end{equation} \tag{2.3} $$
and
$$ \begin{equation} \frac{1}{2}=\frac{1-\widetilde\lambda_{\alpha,\beta}}{p_\alpha}+ \frac{\widetilde\lambda_{\alpha,\beta}}{p_\beta} \quad\textit{if } p_\alpha > 2\textit{ and } p_\beta < 2. \end{equation} \tag{2.4} $$
Then for $q\leqslant 2$
$$ \begin{equation} d_n(M_0,l_q^N) \asymp \min \Bigl\{ \min_{\alpha \in A}d_n(\nu_\alpha B_{p_\alpha}^N, l_q^N),\min_{p_\alpha>q,p_\beta< q} \nu_\alpha ^{1-\lambda_{\alpha,\beta}}\nu_\beta^{\lambda_{\alpha,\beta}}\Bigr\}, \end{equation} \tag{2.5} $$
and for $q>2$
$$ \begin{equation} \begin{aligned} \, d_n(M_0,l_q^N) &\underset{q}{\asymp} \min \Bigl\{ \min_{\alpha \in A}d_n(\nu_\alpha B_{p_\alpha}^N,l_q^N),\min_{p_\alpha>q,p_\beta< q} \nu_\alpha ^{1-\lambda_{\alpha,\beta}}\nu_\beta^{\lambda_{\alpha,\beta}}, \nonumber \\ &\qquad\qquad \min_{p_\alpha> 2,p_\beta< 2} \nu_\alpha ^{1-\widetilde\lambda_{\alpha,\beta}}\nu_\beta^{\widetilde\lambda_{\alpha,\beta}}d_n(B_2^N, l_q^N)\Bigr\}. \end{aligned} \end{equation} \tag{2.6} $$

The original formulation of Proposition 1 in [26] involved nonstrict inequalities in place of the strict ones $p_\alpha> q$, $p_\beta< q$, $p_\alpha> 2$ and $p_\beta< 2$ in (2.5) and (2.6); the resulting estimate remains the same.

For $k\in \{1,\dots,N\}$ let the sets $V_k\subset \mathbb R^N$ be defined by

$$ \begin{equation*} V_k=\operatorname{conv}\{(\varepsilon_1 \widehat{x}_{\sigma(1)},\dots, \varepsilon_N \widehat{x}_{\sigma(N)})\colon \varepsilon_j=\pm 1,\,1\leqslant j\leqslant N,\, \sigma \in S_N\}, \end{equation*} \notag $$
where $\widehat{x}_j=1$ for $1\leqslant j\leqslant k$, $\widehat{x}_j=0$ for $k+1\leqslant j\leqslant N$ and $S_N$ is the permutation group on $N$ elements. Note that $V_1 =B_1^N$ and $V_N=B_\infty^N$.

For $2\leqslant q<\infty$ lower estimates for $d_n(V_k,l_q^N)$ were obtained by Gluskin [15].

Theorem 11 ([15]). Let $2\leqslant q<\infty$ and $1\leqslant k\leqslant N$. Then

$$ \begin{equation} d_n(V_k,l_q^N) \underset{q}{\gtrsim} \begin{cases} k^{1/q} & \textit{for }n\leqslant \min \biggl\{N^{2/q}k^{1 -2/q},\dfrac N2\biggr\}, \\ k^{1/2}n^{-1/2}N^{1/q} & \textit{for }N^{2/q}k^{1 -2/q} \leqslant n\leqslant \dfrac N2. \end{cases} \end{equation} \tag{2.7} $$

The following result was obtained by Gluskin [30] (with a constant depending on $q$ in the order inequality) and by Malykhin and Ryutin [31] (with a constant independent of $q$). It was noted in [30], p. 39, that the equality $d_n(V_k,l_1^N)=\min\{k,N-n\}$ is due to Galeev.

Theorem 12 (see [30] and [31]). Let $1\leqslant q\leqslant 2$, $n\leqslant N/2$. Then

$$ \begin{equation} d_n(V_k,l_q^N) \gtrsim k^{1/q}. \end{equation} \tag{2.8} $$

§ 3. On estimates for widths of intersections of finite-dimensional balls

In this section we refine the width estimates from Theorem 10.

Assume first that $1\leqslant q\leqslant 2$. From (2.5) and Theorems 8 and 9 it follows that, for $n\leqslant N/2$,

$$ \begin{equation} d_n(M_0,l_q^N) \asymp \min \Bigl\{\min_{p_\alpha \geqslant q} \nu_\alpha N^{1/q-1/p_\alpha},\,\min_{p_\alpha \leqslant q} \nu_\alpha,\, \min_{p_\alpha> q,p_\beta< q} \nu_\alpha ^{1-\lambda_{\alpha,\beta}}\nu_\beta^{\lambda_{\alpha,\beta}}\Bigr\}. \end{equation} \tag{3.1} $$

Lemma 1. For $1\leqslant q\leqslant 2$ let $p_\alpha\ne q$ for each $\alpha\in A$, let $n \leqslant N/2$, and let the set $M_0$ be defined by (2.2).

1. Let $p_{\alpha_*} < q$ and $\nu_{\alpha_*} \leqslant \nu_\beta$ for each $\beta \in A$. Then

$$ \begin{equation*} d_n(M_0,l_q^N) \asymp \nu_{\alpha_*}. \end{equation*} \notag $$

2. Let $p_{\alpha_*} > q$ and $\nu_{\alpha_*} N^{1/p_\beta -1/p_{\alpha_*}}\leqslant \nu_\beta$ for each $\beta \in A$. Then

$$ \begin{equation*} d_n(M_0,l_q^N) \asymp \nu_{\alpha_*} N^{1/q-1/p_{\alpha_*}}. \end{equation*} \notag $$

3. Let $p_{\alpha_*} > q$ and $p_{\beta_*}< q$, and let

$$ \begin{equation} \begin{gathered} \, \nu_{\alpha_*} ^{1-\lambda_{\alpha_*,\beta_*}}\nu_{\beta_*}^{\lambda_{\alpha_*,\beta_*}} \leqslant \nu_{\alpha_*} ^{1-\lambda_{\alpha_*,\gamma}}\nu_\gamma^{\lambda_{\alpha_*,\gamma}}, \quad \gamma \in A, \quad\textit{for } p_\gamma < q, \\ \nu_{\alpha_*} ^{1-\lambda_{\alpha_*,\beta_*}}\nu_{\beta_*}^{\lambda_{\alpha_*,\beta_*}} \leqslant \nu_\gamma ^{1-\lambda_{\gamma,\beta_*}}\nu_{\beta_*}^{\lambda_{\gamma,\beta_*}}, \quad \gamma \in A, \quad\textit{for } p_\gamma > q, \end{gathered} \end{equation} \tag{3.2} $$
$$ \begin{equation} \nu_{\alpha_*} \leqslant \nu_{\beta_*}\quad\textit{and} \quad \nu_{\alpha_*}\geqslant \nu_{\beta_*} N^{1/p_{\alpha_*}-1/p_{\beta_*}}. \end{equation} \tag{3.3} $$
Then
$$ \begin{equation*} d_n(M_0,l_q^N) \asymp \nu_{\alpha_*} ^{1-\lambda_{\alpha_*,\beta_*}}\nu_{\beta_*}^{\lambda_{\alpha_*,\beta_*}}. \end{equation*} \notag $$

Proof. In view of (3.1) it suffices to prove the lower estimates for the widths $d_n(M_0,l_q^N)$.

In part 1 we use the embedding $\nu_{\alpha_*} B^N_1 \subset M_0$ and Theorem 8, and in part 2, the embedding $\nu_{\alpha_*} N^{-1/p_{\alpha_*}} B^N_{\infty} \subset M_0$ and Theorem 9.

In part 3 let the number $l$ be defined by ${\nu_{\alpha_*}}/{\nu_{\beta_*}}= l^{1/p_{\alpha_*}-1/p_{\beta_*}}$; we also set $k=\lceil l\rceil$. From (3.3) it follows that $1\leqslant l\leqslant N$, and so $1\leqslant k\leqslant N$. We claim that $\nu_{\alpha_*} ^{1-\lambda_{\alpha_*,\beta_*}}\nu_{\beta_*}^{\lambda_{\alpha_*,\beta_*}} k^{-1/q}V_k \subset 2M_0$. Now the lower estimate for $d_n(M_0,l_q^N)$ is secured by (2.8). It suffices to verify that for each $\gamma\in A$

$$ \begin{equation} \nu_{\alpha_*} ^{1-\lambda_{\alpha_*,\beta_*}}\nu_{\beta_*}^{\lambda_{\alpha_*,\beta_*}} l^{1/p_\gamma-1/q}\leqslant \nu_\gamma, \end{equation} \tag{3.4} $$
that is, $\nu_{\alpha_*} l^{1/p_\gamma-1/p_{\alpha_*}} \leqslant \nu_\gamma$ (see (2.3) and the definition of $l$). The last inequality follows from (3.2) by the same argument as in [29], p. 6.

This proves Lemma 1.

Now consider the case $q>2$. From (2.6) and Theorems 8, 9, for $N^{2/q}\leqslant n \leqslant {N}/{2}$ we have

$$ \begin{equation} \begin{aligned} \, \notag d_n(M_0,l_q^N) &\underset{q}{\asymp} \min \Bigl\{ \min_{p_\alpha \geqslant q}\nu_\alpha N^{1/q-1/p_\alpha}, \min_{2\leqslant p_\alpha \leqslant q}\nu_\alpha (n^{-1/2}N^{1/q})^{\frac{1/p_\alpha-1/q}{1/2-1/q}}, \\ \notag &\qquad\qquad \min_{p_\alpha \leqslant 2} \nu_\alpha n^{-1/2}N^{1/q},\min_{p_\alpha> q,p_\beta< q} \nu_\alpha ^{1-\lambda_{\alpha,\beta}}\nu_\beta^{\lambda_{\alpha,\beta}}, \\ &\qquad\qquad \min_{p_\alpha> 2,p_\beta< 2} \nu_\alpha ^{1-\widetilde\lambda_{\alpha,\beta}}\nu_\beta^{\widetilde\lambda_{\alpha,\beta}}n^{-1/2}N^{1/q} \Bigr\}. \end{aligned} \end{equation} \tag{3.5} $$

Lemma 2. Let $2< q<\infty$ and $p_\alpha\notin \{2,q\}$ for each $\alpha\in A$, let $N^{2/q}\leqslant n \leqslant N/2$, and let the set $M_0$ be defined by (2.2).

1. Let $p_{\alpha_*} < 2$ and $\nu_{\alpha_*} \leqslant \nu_\beta$ for each $\beta \in A$. Then

$$ \begin{equation*} d_n(M_0,l_q^N) \underset{q}{\asymp} \nu_{\alpha_*}n^{-1/2}N^{1/q}. \end{equation*} \notag $$

2. Let $p_{\alpha_*} > q$ and $\nu_{\alpha_*} N^{1/p_\beta -1/p_{\alpha_*}}\leqslant \nu_\beta$ for each $\beta \in A$. Then

$$ \begin{equation*} d_n(M_0,l_q^N) \underset{q}{\asymp} \nu_{\alpha_*} N^{1/q-1/p_{\alpha_*}}. \end{equation*} \notag $$

3. Let $2<p_{\alpha_*}<q$ and

$$ \begin{equation} \nu_{\alpha_*}(n^{1/2}N^{-1/q})^{\frac{1/p_\beta-1/p_{\alpha_*}}{1/2-1/q}} \leqslant \nu_\beta \end{equation} \tag{3.6} $$
for each $\beta \in A$. Then
$$ \begin{equation} d_n(M_0,l_q^N) \underset{q}{\asymp} \nu_{\alpha_*} (n^{-1/2}N^{1/q})^{\frac{1/p_{\alpha_*}-1/q}{1/2-1/q}}. \end{equation} \tag{3.7} $$

4. Let $p_{\alpha_*} > q$ and $p_{\beta_*}< q$, let (3.2) hold, and let

$$ \begin{equation} \nu_{\alpha_*} \leqslant \nu_{\beta_*}(n^{1/2}N^{-1/q})^{\frac{1/p_{\alpha_*}-1/p_{\beta_*}}{1/2-1/q}}\quad\textit{and} \quad \nu_{\alpha_*}\geqslant \nu_{\beta_*} N^{1/p_{\alpha_*}-1/p_{\beta_*}}. \end{equation} \tag{3.8} $$
Then
$$ \begin{equation*} d_n(M_0,l_q^N) \underset{q}{\asymp} \nu_{\alpha_*} ^{1-\lambda_{\alpha_*,\beta_*}}\nu_{\beta_*}^{\lambda_{\alpha_*,\beta_*}}. \end{equation*} \notag $$

5. Let $p_{\alpha_*} > 2$ and $p_{\beta_*}< 2$, and let

$$ \begin{equation} \begin{gathered} \, \nu_{\alpha_*} ^{1-\widetilde\lambda_{\alpha_*,\beta_*}}\nu_{\beta_*}^{\widetilde\lambda_{\alpha_*,\beta_*}} \leqslant \nu_{\alpha_*} ^{1-\widetilde\lambda_{\alpha_*,\gamma}}\nu_\gamma^{\widetilde\lambda_{\alpha_*,\gamma}}, \quad \gamma \in A,\quad\textit{for } p_\gamma < 2, \\ \nu_{\alpha_*} ^{1-\widetilde\lambda_{\alpha_*,\beta_*}}\nu_{\beta_*}^{\widetilde\lambda_{\alpha_*,\beta_*}} \leqslant \nu_\gamma ^{1-\widetilde\lambda_{\gamma,\beta_*}}\nu_{\beta_*}^{\widetilde\lambda_{\gamma,\beta_*}}, \quad \gamma \in A,\quad\textit{for } p_\gamma > 2, \end{gathered} \end{equation} \tag{3.9} $$
$$ \begin{equation} \nu_{\alpha_*} \leqslant \nu_{\beta_*}\quad\textit{and} \quad \nu_{\alpha_*}\geqslant \nu_{\beta_*} (n^{1/2}N^{-1/q})^{\frac{1/p_{\alpha_*}-1/p_{\beta_*}}{1/2-1/q}}. \end{equation} \tag{3.10} $$
Then
$$ \begin{equation*} d_n(M_0,l_q^N) \underset{q}{\asymp} \nu_{\alpha_*} ^{1-\widetilde\lambda_{\alpha_*,\beta_*}}\nu_{\beta_*}^{\widetilde \lambda_{\alpha_*,\beta_*}}n^{-1/2}N^{1/q}. \end{equation*} \notag $$

Proof. By (3.5) it suffices to prove the lower estimate.

In part 1 we use the embedding $\nu_{\alpha_*}B_1^N \subset M_0$ and Theorem 8, and in part 2 we employ the embedding $\nu_{\alpha_*}N^{-1/p_{\alpha_*}}B_\infty^N \subset M_0$ and Theorem 9.

In part 3 we set

$$ \begin{equation*} l=(n^{1/2}N^{-1/q})^{\frac{1}{1/2-1/q}}, \qquad k=\lceil l\rceil. \end{equation*} \notag $$
In this case we have $1\leqslant l\leqslant N$, $1\leqslant k\leqslant N$ and $n \leqslant N^{2/q} k^{1-2/q}$. We claim that $\nu_{\alpha_*}k^{-1/p_{\alpha_*}}V_k \,{\subset}\, 2M_0$. As a result, estimate (3.7) follows from (2.7). The claim follows from the inequality $\nu_{\alpha_*} l^{1/p_\beta-1/p_{\alpha_*}} \leqslant \nu_\beta$ for $\beta\in A$, which is in turn secured by (3.6).

In parts 4 and 5 we define the number $l$ by

$$ \begin{equation} \frac{\nu_{\alpha_*}}{\nu_{\beta_*}}=l^{1/p_{\alpha_*}-1/p_{\beta_*}}. \end{equation} \tag{3.11} $$

In part 4 we set $k=\lceil l\rceil$. From (3.8) we obtain

$$ \begin{equation*} (n^{1/2}N^{-1/q})^{\frac{1}{1/2-1/q}}\leqslant l\leqslant N. \end{equation*} \notag $$
Hence $1\!\leqslant\! k\leqslant\! N$ and $n \!\leqslant\! N^{2/q} k^{1-2/q}$. Now we show that $\nu_{\alpha_*}^{1-\lambda_{\alpha_*,\beta_*}} \nu_{\beta_*}^{\lambda_{\alpha_*,\beta_*}} k^{-1/q}V_k\! \subset 2M_0$ and use (2.7). To do this, it suffices to verify that (3.4) holds for each $\gamma\in A$; in view of (2.3) and (3.11) this is equivalent to the inequality $\nu_{\alpha_*} l^{1/p_\gamma-1/p_{\alpha_*}} \leqslant \nu_\gamma$. It follows from (3.2) by the same argument as in [29], pp. 11–12.

In part 5 we set $k=\lfloor l\rfloor$. From (3.10) we obtain

$$ \begin{equation*} 1\leqslant l\leqslant (n^{1/2}N^{-1/q})^{\frac{1}{1/2-1/q}}. \end{equation*} \notag $$
Hence $1\!\leqslant\! k\!\leqslant\! N$ and $n \!\geqslant\! N^{2/q} k^{1-2/q}$. Now we show that $\nu_{\alpha_*}^{1-\widetilde\lambda_{\alpha_*,\beta_*}} \nu_{\beta_*}^{\widetilde\lambda_{\alpha_*,\beta_*}} k^{-1/2}V_k \!\subset\! 2M_0$ and use (2.7). To do this it suffices to verify that for each $\gamma\in A$,
$$ \begin{equation*} \nu_{\alpha_*} ^{1-\widetilde\lambda_{\alpha_*,\beta_*}}\nu_{\beta_*}^{\widetilde\lambda_{\alpha_*,\beta_*}} l^{1/p_\gamma-1/2}\leqslant \nu_\gamma; \end{equation*} \notag $$
in view of (2.4) and (3.11) this is equivalent to the inequality $\nu_{\alpha_*} l^{1/p_\gamma-1/p_{\alpha_*}} \leqslant \nu_\gamma$. It is secured by (3.9), where we proceed as in [29], pp. 12–13.

This proves Lemma 2.

§ 4. Proof of Theorem 2

The following two results will be used to reduce the problem of estimates for widths of $W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d)$ to estimates for widths of finite-dimensional balls. Recall that for $\overline{m}\in \mathbb N^d$ the number $m$ was defined in (2.1).

Lemma 3. Let $n\in \mathbb Z_+$. Then

$$ \begin{equation} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{p},q,\overline{r},d}{\gtrsim} d_n\biggl(\bigcap_{j=1}^d 2^{-m_jr_j-m/q+m/p_j} B^{2^m}_{p_j},l_q^{2^m}\biggr), \qquad \overline{m}\in \mathbb N^d. \end{equation} \tag{4.1} $$

Proof. Proceeding as in [3], Theorem 1, and using Theorems 5–7 we have the order inequalities
$$ \begin{equation*} \begin{aligned} \, d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) &\underset{\overline{p},q,\overline{r},d}{\gtrsim} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d)\cap \mathcal T_{\overline{m}}, L_q(\mathbb{T}^d)\cap \mathcal T_{\overline{m}}) \\ &\underset{\overline{p},q,\overline{r},d}{\gtrsim} d_n \biggl(\bigcap_{j=1}^d 2^{-m_jr_j-m/q+m/p_j} B^{2^m}_{p_j},l_q^{2^m}\biggr). \end{aligned} \end{equation*} \notag $$

The following result is a consequence of Theorems 5–7.

Lemma 4. Let $k\in \mathbb Z_+$. Then

$$ \begin{equation} d_k(\delta_{\overline{m}}W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d), L_q(\mathbb{T}^d)) \underset{\overline{p},q,\overline{r},d}{\lesssim} d_k \biggl(\bigcap_{j=1}^d 2^{-m_jr_j-m/q+m/p_j} B^{2^m}_{p_j},l_q^{2^m}\biggr), \qquad \overline{m}\in \mathbb N^d. \end{equation} \tag{4.2} $$

In particular, for any function $f\in W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d)$,

$$ \begin{equation} \|\delta_{\overline{m}}f\|_{L_q(\mathbb{T}^d)} \underset{\overline{p},q,\overline{r},d}{\lesssim} d_0 \biggl(\bigcap_{j=1}^d 2^{-m_jr_j-m/q+m/p_j} B^{2^m}_{p_j},l_q^{2^m}\biggr)=:C_{\overline{m}}. \end{equation} \tag{4.3} $$
Using Theorems 8–10 and recalling (1.4) and (1.5), for $q\leqslant 2$ we have
$$ \begin{equation*} C_{\overline{m}}\lesssim \min \Bigl\{ \min_{j\in I_0} 2^{-m_jr_j}, \,\min_{j\in J_0} 2^{-r_jm_j-m/q+m/p_j}, \,\min_{i\in I_0',j\in J_0'} 2^{-(1-\lambda_{i,j})r_im_i-\lambda_{i,j}r_jm_j}\Bigr\}. \end{equation*} \notag $$
For $2<q<\infty$, recalling the notation (1.8) and (1.9) we obtain
$$ \begin{equation*} \begin{aligned} \, C_{\overline{m}} &\underset{q}{\lesssim} \min \Bigl\{ \min_{j\in I} 2^{-m_jr_j}, \,\min_{j\in J\cup K} 2^{-r_jm_j-m/q+m/p_j}, \\ &\qquad\qquad \min_{i\in I',j\in J'\cup K} 2^{-(1-\lambda_{i,j})r_im_i-\lambda_{i,j}r_jm_j}, \\ &\qquad\qquad \min_{i\in I\cup J',j\in K'} 2^{-(1-\mu_{i,j})r_im_i-\mu_{i,j}r_jm_j -m/q+m/2}\Bigr\} \\ &=\min \Bigl\{ \min_{j\in I} 2^{-m_jr_j},\,\min_{j\in J\cup K} 2^{-r_jm_j-m/q+m/p_j}, \\ &\qquad\qquad \min_{i\in I',j\in J'\cup K} 2^{-(1-\lambda_{i,j})r_im_i-\lambda_{i,j}r_jm_j}\Bigr\}; \end{aligned} \end{equation*} \notag $$
the last equality holds because for $i\in I\cup J'$ and $j\in K'$ we have
$$ \begin{equation} \begin{aligned} \, \notag &(1-\mu_{i,j})r_im_i+\mu_{i,j}r_jm_j+\frac mq-\frac m2 \\ &\qquad \leqslant\max \biggl\{(1-\lambda_{i,j})r_im_i+\lambda_{i,j}r_jm_j ,r_jm_j+\frac mq-\frac{m}{p_j} \biggr\} \quad\text{if } p_i> q \end{aligned} \end{equation} \tag{4.4} $$
and
$$ \begin{equation} \begin{aligned} \, \notag &(1-\mu_{i,j})r_im_i+\mu_{i,j}r_jm_j+\frac mq-\frac m2 \\ &\qquad \leqslant\max \biggl\{ r_im_i+\frac mq-\frac{m}{p_i}, r_jm_j+\frac mq-\frac{m}{p_j}\biggr\} \quad \text{if } 2< p_i\leqslant q. \end{aligned} \end{equation} \tag{4.5} $$

Thus, for $q\leqslant 2$ and $q>2$ alike we have

$$ \begin{equation} C_{\overline{m}} \underset{\overline{p},q,\overline{r},d}{\lesssim} 2^{-\varphi(m_1,\dots,m_d)}, \end{equation} \tag{4.6} $$
where
$$ \begin{equation} \begin{aligned} \, \notag \varphi(t_1,\dots,t_d) &=\max \biggl\{\max_{p_j\geqslant q} t_jr_j,\max_{p_j\leqslant q} \biggl(t_jr_j+\frac tq-\frac{t}{p_j}\biggr), \\ &\qquad\qquad\max_{p_i> q,p_j< q} \bigl((1-\lambda_{i,j})r_it_i+\lambda_{i,j}r_jt_j\bigr)\biggr\}, \qquad t=t_1+\dots+t_d. \end{aligned} \end{equation} \tag{4.7} $$

Let the function $h$ and the set $D$ be defined by (1.6) and (1.7) (for $q\leqslant 2$ and $q>2$ alike).

Lemma 5. Let $(\alpha_1^*,\dots,\alpha_d^*)$ be a minimum point of the function $h$ on the set $D$, and let $h(\alpha_1^*,\dots,\alpha_d^*)>0$. Let the numbers $C_{\overline{m}}$ be defined by (4.3). Then for each $N\in \mathbb N$,

$$ \begin{equation} \sum_{m\geqslant N} \|\delta_{\overline{m}}f\|_{L_q(\mathbb{T}^d)} \underset{\overline{p},q,\overline{r},d}{\lesssim} \sum_{m\geqslant N} C_{\overline{m}} \underset{\overline{p},q,\overline{r},d}{\lesssim} 2^{-N\cdot h(\alpha_1^*,\dots, \alpha_d^*)} N^{d-1}. \end{equation} \tag{4.8} $$
If $h$ has a unique minimum point on $D$, then
$$ \begin{equation} \sum_{m\geqslant N} \|\delta_{\overline{m}}f\|_{L_q(\mathbb{T}^d)} \underset{\overline{p},q,\overline{r},d}{\lesssim} \sum_{m\geqslant N} C_{\overline{m}} \underset{\overline{p},q,\overline{r},d}{\lesssim} 2^{-N\cdot h(\alpha_1^*,\dots, \alpha_d^*)}. \end{equation} \tag{4.9} $$

Proof. The left-hand order inequalities in (4.8) and (4.9) are secured by (4.3).

Let us prove the right-hand inequalities in (4.8) and (4.9). From (4.6) we obtain

$$ \begin{equation*} \sum_{m\geqslant N} C_{\overline{m}} \underset{\overline{p},q,\overline{r},d}{\lesssim} \sum_{m\geqslant N} 2^{-\varphi(\overline{m})} \underset{\overline{p},q,\overline{r},d}{\lesssim} \int_{t\geqslant N,t_1, \dots,t_d\geqslant 0} 2^{-\varphi(t_1,\dots,t_d)}\, dt_1\dotsb dt_d=:\Sigma, \end{equation*} \notag $$
where $t=t_1+\dots+t_d$. We also set $\alpha_j={t_j}/{t}$, $1\leqslant j\leqslant d$. Comparing (4.7) and the function $h$, we have $\varphi(t_1,\dots,t_d)=t \cdot h(\alpha_1,\dots,\alpha_d)$, $(\alpha_1,\dots,\alpha_d)\in D$. Setting $E_t=\{(t_1,\dots, t_{d-1})\colon t_1+\dots+t_{d-1}\leqslant t,\, t_j\geqslant 0,\, 1\leqslant j\leqslant d-1\}$ we find that
$$ \begin{equation*} \begin{aligned} \, \Sigma &=\int_N^\infty \int_{E_t}2^{-\varphi(t_1,\dots,t_{d-1}, t-t_1-\dots-t_{d-1})}\, dt_1\dotsb dt_{d-1}\, dt \\ &\leqslant \int_N^\infty 2^{-t\cdot h(\alpha_1^*,\dots,\alpha_d^*)} t^{d-1}\, dt \underset{\overline{p},q,\overline{r},d}{\lesssim} 2^{-N\cdot h(\alpha_1^*,\dots, \alpha_d^*)}N^{d-1}. \end{aligned} \end{equation*} \notag $$

If $h$ has a unique minimum point on $D$, then $(N\alpha_1^*,\dots,N\alpha_d^*)$ is a unique minimum point of the function $\varphi$ on $G_N:=\{(t_1,\dots,t_d)\colon t_1+\cdots+t_d\geqslant N$, $t_j\geqslant 0$, $1\leqslant j\leqslant d\}$. By (4.7) there exists $b=b(\overline{p},q,\overline{r},d)> 0$ such that

$$ \begin{equation*} \varphi(t_1,\dots,t_d)\geqslant \varphi(N\alpha_1^*,\dots,N\alpha_d^*)+b\sum_{j=1}^d |t_j-N\alpha_j^*|, \qquad (t_1, \dots,t_d)\in G_N. \end{equation*} \notag $$
Hence
$$ \begin{equation*} \Sigma \underset{\overline{p},q,\overline{r},d}{\lesssim} 2^{-\varphi(N\alpha_1^*, \dots,N\alpha_d^*)}=2^{-N\cdot h(\alpha_1^*,\dots,\alpha_d^*)}. \end{equation*} \notag $$

This proves Lemma 5.

Let $q>2$, and let the function $\widetilde h$ and the set $\widetilde D$ be defined as in part 2 of Theorem 2. Set

$$ \begin{equation*} \widetilde D_{q/2}=\biggl\{(\alpha_1,\dots,\alpha_d,s)\in \widetilde D\colon s=\frac q2\biggr\}. \end{equation*} \notag $$

Lemma 6. Let $q>2$. Then for each $(\alpha_1,\dots,\alpha_d,q/2)\in \widetilde D_{q/2}$,

$$ \begin{equation} \widetilde h\biggl(\alpha_1,\dots,\alpha_d,\frac q2\biggr) =\frac q2 \cdot h\biggl(\frac{2\alpha_1}q,\dots, \frac{2\alpha_d}q\biggr). \end{equation} \tag{4.10} $$
In particular,
$$ \begin{equation} \min_{\widetilde D_{q/2}}\widetilde h=\frac q2 \min_D h. \end{equation} \tag{4.11} $$

Proof. Let us prove (4.10). We have
$$ \begin{equation*} \begin{aligned} \, \widetilde h\biggl(\alpha_1,\dots,\alpha_d,\frac q2\biggr) &=\max \biggl\{ \max_{p_j\geqslant q} r_j\alpha_j,\, \max_{p_j\leqslant q} \biggl(r_j\alpha_j+\frac12-\frac{q}{2p_j}\biggr), \\ &\qquad\qquad \max_{p_i> q,p_j< q} ((1-\lambda_{i,j})r_i\alpha_i+\lambda_{i,j}r_j\alpha_j), \\ &\qquad\qquad \max_{p_i> 2,p_j< 2} \biggl((1-\mu_{i,j})r_i\alpha_i +\mu_{i,j}r_j\alpha_j+\frac12-\frac q4\biggr)\biggr\} \\ &=\max \biggl\{ \max_{p_j\geqslant q} r_j\alpha_j, \,\max_{p_j\leqslant q} \biggl(r_j\alpha_j+\frac 12- \frac{q}{2p_j}\biggr), \\ &\qquad\qquad \max_{p_i> q,p_j< q} \bigl((1-\lambda_{i,j})r_i\alpha_i +\lambda_{i,j}r_j\alpha_j\bigr)\biggr\} \\ &=\frac q2 \cdot h\biggl(\frac{2\alpha_1}q,\dots,\frac{2\alpha_d}q\biggr); \end{aligned} \end{equation*} \notag $$
the second equality is secured by (4.4) and (4.5).

The lemma is proved.

Lemma 7. Assume that $\min_D h>0$, and let $k_{\overline{m}}\in \mathbb Z_+$, $\overline{m}\in \mathbb N^d$, be such that $\sum_{\overline{m}\in \mathbb N^d}k_{\overline{m}}\leqslant Cn$ for some constant $C\in \mathbb N$. Then

$$ \begin{equation} d_{Cn} (W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{p},q,\overline{r},d}{\lesssim} \sum_{\overline{m}\in \mathbb N^d} d_{k_{\overline{m}}} \biggl(\bigcap_{j=1}^d 2^{-m_jr_j-m/q+m/p_j} B^{2^m}_{p_j}, l_q^{2^m}\biggr). \end{equation} \tag{4.12} $$

Proof. We have $\min_{D}h>0$, and so it follows from (4.8) that the partial sums $S_Nf:=\sum_{m\leqslant N} \delta_{\overline{m}}f$ form a Cauchy sequence in $L_q(\mathbb{T}^d)$. On the other hand $S_Nf\underset{N\to \infty}{\to} f$ in $\mathcal S'(\mathbb{T}^d)$. Hence $f\in L_q(\mathbb{T}^d)$ and $S_Nf\underset{N\to \infty}{\to} f$ in $L_q(\mathbb{T}^d)$, which implies that
$$ \begin{equation*} f=\sum_{N\in \mathbb N} \, \sum_{\overline{m}\in \mathbb N^d\colon m=N} \delta_{\overline{m}}f \end{equation*} \notag $$
(the series is convergent in $L_q(\mathbb{T}^d)$). It remains to invoke Lemma 4.

This proves Lemma 7.

Proof of Theorem 2. First we prove the upper estimate under the assumption that $\min_{D}h>0$. Then we prove the lower estimate, and obtain in passing that if $\min_{D}h\leqslant 0$, then $W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d)$ is not compact in $L_q(\mathbb{T}^d)$. Hence, under the assumptions of Theorem 2 the inequality $\min_{D}h>0$ holds automatically, since the embedding is compact because
$$ \begin{equation*} \frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ\overline{r}\rangle}>0. \end{equation*} \notag $$

Thus, we assume that $\max_{D}h>0$.

We set $q_*=\min\{q,2\}$. Let $\overline{m}_*=(m_1^*,\dots,m_d^*)\in \mathbb R_+^d$, $2^{m_*}\in [n,n^{q_*/2}]$ and $\varepsilon>0$ ($\overline{m}_*$ and $\varepsilon$ will be chosen below depending on $\overline{p}$, $q$, $\overline{r}$ and $d$). We also set

$$ \begin{equation*} |\overline{m}-\overline{m}_*| :=\sum_{j=1}^d |m_j-m_j^*|. \end{equation*} \notag $$
Setting
$$ \begin{equation} k_{\overline{m}}= \begin{cases} 0 &\text{for }2^m > n^{q_*/2}, \\ \min \{\lfloor n\cdot 2^{-\varepsilon|\overline{m}-\overline{m}_*|}\rfloor,2^m\} & \text{for } 2^m \leqslant n^{q_*/2}, \end{cases} \end{equation} \tag{4.13} $$
we have
$$ \begin{equation*} \sum_{\overline{m}\in \mathbb N^d} k_{\overline{m}} \leqslant \sum_{\overline{m}\in \mathbb N^d} n\cdot 2^{-\varepsilon|\overline{m}-\overline{m}_*|} \underset{\varepsilon,d}{\lesssim} n. \end{equation*} \notag $$

Next we use Lemma 7 and estimate the right-hand side of (4.12) from above as follows:

$$ \begin{equation*} \begin{aligned} \, & \sum_{\overline{m}\in \mathbb N^d} d_{k_{\overline{m}}}\biggl(\bigcap_{j=1}^d 2^{-m_jr_j-m/q+m/p_j} B^{2^m}_{p_j},l_q^{2^m}\biggr) \\ &\qquad \leqslant \sum_{2^m > n^{q_*/2}} d_0\biggl(\bigcap_{j=1}^d 2^{-m_jr_j-m/q+m/p_j} B^{2^m}_{p_j},l_q^{2^m}\biggr) \\ &\qquad\qquad +\sum_{ n\cdot 2^{-\varepsilon|\overline{m}-\overline{m}_*|}\leqslant 2^m \leqslant n^{q_*/2}} d_{k_{\overline{m}}}\biggl(\bigcap_{j=1}^d 2^{-m_jr_j-m/q+m/p_j} B^{2^m}_{p_j}, l_q^{2^m}\biggr). \end{aligned} \end{equation*} \notag $$

Consider the case $q>2$ (the case $q\leqslant 2$ is simpler and can be analyzed similarly). Let

$$ \begin{equation*} S_1=\sum_{2^m > n^{q/2}} d_0\biggl(\bigcap_{j=1}^d 2^{-m_jr_j-m/q+m/p_j} B^{2^m}_{p_j},l_q^{2^m}\biggr) \end{equation*} \notag $$
and
$$ \begin{equation*} S_{2,\varepsilon}=\sum_{n\cdot 2^{-\varepsilon|\overline{m}-\overline{m}_*|}\leqslant 2^m \leqslant n^{q/2}} d_{k_{\overline{m}}}\biggl(\bigcap_{j=1}^d 2^{-m_jr_j-m/q+m/p_j} B^{2^m}_{p_j}, l_q^{2^m}\biggr). \end{equation*} \notag $$
We claim that
$$ \begin{equation} S_1 \underset{\overline{p},q,\overline{r},d}{\lesssim} n^{-\widetilde h(\widehat \alpha_1,\dots,\widehat \alpha_d,\widehat s)}. \end{equation} \tag{4.14} $$
Setting $c_*\!=\!\min_{\widetilde D_{q/2}}\widetilde h$ and $\log x\! :=\!\log_2 x$, and using (4.3) and (4.8) for $N\!=\!\lfloor\frac q2 \log n\rfloor$, we have
$$ \begin{equation*} S_1 \underset{\overline{p},q,\overline{r},d}{\lesssim} 2^{-\frac{q \log n}{2}\min_D h}(\log n)^{d-1} \stackrel{(4.11)}{=} n^{- c_*}(\log n)^{d-1}. \end{equation*} \notag $$
If $c_*> \widetilde h(\widehat \alpha_1,\dots,\widehat\alpha_d,\widehat s)$, then
$$ \begin{equation*} n^{- c_*}(\log n)^{d-1} \underset{\overline{p},q,\overline{r},d}{\lesssim} n^{-\widetilde h(\widehat \alpha_1,\dots,\widehat \alpha_d,\widehat s)}, \end{equation*} \notag $$
which gives (4.14). If $c_*=\widetilde h(\widehat \alpha_1, \dots,\widehat \alpha_d,\widehat s)$, then the function $\widetilde h$ has a unique minimum point on $\widetilde D_{q/2}$; by Lemma 6, the function $h$ has a unique minimum point on $D$. Applying (4.9), we arrive at (4.14) again.

Let us now estimate $S_{2,\varepsilon}$. First consider the case $\varepsilon=0$. We have

$$ \begin{equation*} S_{2,0}=\sum_{n\leqslant 2^m\leqslant n^{q/2}} d_n\biggl(\bigcap_{j=1}^d 2^{-m_jr_j-m/q+m/p_j} B^{2^m}_{p_j},l_q^{2^m}\biggr). \end{equation*} \notag $$
Using Theorems 8–10 this gives
$$ \begin{equation*} S_{2,0}\underset{q}{\lesssim} \sum_{n\leqslant 2^m\leqslant n^{q/2}} 2^{-\psi_n(\overline{m}, m)}, \end{equation*} \notag $$
where
$$ \begin{equation} \begin{aligned} \, \notag &\psi_n(t_1,\dots,t_d,t) \\ &\qquad=\max \biggl\{ \max_{j\in I} r_jt_j, \,\max_{j\in J} \biggl( r_jt_j -\frac 12\cdot \frac{1/p_j-1/q}{1/2-1/q}t+\frac 12\cdot \frac{1/p_j-1/q}{1/2-1/q} \log n\biggr), \notag \\ &\qquad\qquad\qquad \max_{j\in K} \biggl(r_jt_j-\frac{t}{p_j}+\frac 12 \log n\biggr), \, \max_{i\in I',j\in J'\cup K}((1-\lambda_{i,j})r_it_i+\lambda_{i,j}r_jt_j), \notag \\ &\qquad\qquad\qquad \max_{i\in I\cup J',j\in K'} \biggl((1-\mu_{i,j})r_it_i+\mu_{i,j}r_jt_j -\frac t2+\frac 12\log n\biggr)\biggr\}. \end{aligned} \end{equation} \tag{4.15} $$
Setting
$$ \begin{equation*} G_n=\biggl\{(t_1,\dots,t_d)\in \mathbb R^d_+\colon \log n \leqslant t_1+\dots+t_d \leqslant \frac q2 \log n\biggr\}, \end{equation*} \notag $$
we obtain
$$ \begin{equation*} \sum_{n\leqslant 2^m\leqslant n^{q/2}} 2^{-\psi_n(\overline{m},m)} \underset{\overline{p}, \overline{r},q,d}{\lesssim} \int_{G_n} 2^{-\psi_n(t_1,\dots,t_d,t_1+\dots+t_d)}\, dt_1\dotsb dt_d. \end{equation*} \notag $$
Note that
$$ \begin{equation} \psi_n(t_1,\dots,t_d,t)=\widetilde h \biggl(\frac{t_1}{\log n},\dots,\frac{t_d}{\log n}, \frac{t}{\log n}\biggr)\cdot\log n. \end{equation} \tag{4.16} $$
The function $\widetilde h$ has a unique minimum point on the set $\widetilde D$, and so the function $f_n(t_1, \dots,t_d):=\psi_n(t_1,\dots,t_d,t_1+\dots+t_d)$ also has a unique minimum point on $G_n$, and this point has the form $(\widehat \alpha_1,\dots,\widehat \alpha_d)\log n$. In addition, if $\widehat s=\widehat \alpha_1+\dots+\widehat \alpha_d>1$, then $(\widehat \alpha_1,\dots,\widehat \alpha_d)\log n$ is the unique minimum point of the function $f_n$ on the set
$$ \begin{equation*} \widehat G_n=\biggl\{(t_1,\dots,t_d)\in \mathbb R^d_+\colon t_1+\dots+t_d \leqslant \frac q2 \log n\biggr\}. \end{equation*} \notag $$

We set

$$ \begin{equation} \overline{m}_*=(m_1^*,\dots,m_d^*)=(\widehat \alpha_1,\dots,\widehat \alpha_d)\log n. \end{equation} \tag{4.17} $$
By (4.15) and (4.16) there exists $c_{\overline{p},q,\overline{r},d} > 0$ such that
$$ \begin{equation} \begin{gathered} \, \psi_n(t_1,\dots,t_d,t_1+\dots+t_d) \geqslant \widetilde h(\widehat \alpha_1,\dots,\widehat \alpha_d,\widehat s)\log n+ c_{\overline{p},q,\overline{r},d} \sum_{j=1}^d |t_j-m^*_j|, \\ (t_1,\dots,t_d)\in \begin{cases} G_n, & \widehat s=1, \\ \widehat G_n, & \widehat s > 1. \end{cases} \end{gathered} \end{equation} \tag{4.18} $$
Hence
$$ \begin{equation*} \int_{G_n} 2^{-\psi_n(t_1,\dots,t_d,t_1+\dots+t_d)}\, dt_1\dotsb dt_d \underset{\overline{p},\overline{r},q,d}{\lesssim} n^{-\widetilde h(\widehat \alpha_1,\dots,\widehat \alpha_d,\widehat s)}, \end{equation*} \notag $$
that is,
$$ \begin{equation*} S_{2,0}\underset{\overline{p},\overline{r},q,d}{\lesssim} n^{-\widetilde h(\widehat \alpha_1, \dots,\widehat \alpha_d,\widehat s)}. \end{equation*} \notag $$

Let us now estimate $S_{2,\varepsilon}$ for small $\varepsilon >0$. We set

$$ \begin{equation*} G_{n,\varepsilon}= \biggl\{(t_1,\dots,t_d)\in \mathbb R^d_+\colon \log n-\varepsilon \sum_{j=1}^d |t_j-m_j^*| \leqslant t_1+\dots+t_d\leqslant \frac q2 \log n\biggr\}. \end{equation*} \notag $$
From (4.18), for sufficiently small $\varepsilon>0$ we obtain
$$ \begin{equation} \begin{gathered} \, \psi_n(t_1,\dots,t_d,t_1+\dots+t_d) \geqslant \widetilde h(\widehat \alpha_1,\dots,\widehat \alpha_d,\widehat s)\log n+ \frac{c_{\overline{p},q,\overline{r},d}}{2} \sum_{j=1}^d |t_j-m^*_j|, \\ (t_1,\dots,t_d) \in G_{n,\varepsilon} \end{gathered} \end{equation} \tag{4.19} $$
(the greatest possible $\varepsilon$ for which (4.19) holds depends on $\overline{p}$, $q$, $\overline{r}$ and $d$).

Let us use Theorems 8–10 and (4.13). We have $2^{m_*}\in [n,n^{q/2}]$, and so for sufficiently small $\varepsilon>0$ and for $n\cdot 2^{-\varepsilon|\overline{m}-\overline{m}_*|}\leqslant 2^m \leqslant n^{q/2}$ we have $k_n \asymp {n\cdot 2^{-\varepsilon|\overline{m}-\overline{m}_*|}}$. Hence for some $b=b(\overline{p},\overline{r},q,d)>0$,

$$ \begin{equation*} \begin{aligned} \, S_{2,\varepsilon} &\underset{q}{\lesssim} \sum_{n\cdot 2^{-\varepsilon|\overline{m}-\overline{m}_*|}\leqslant 2^m \leqslant n^{q/2}} 2^{-\psi_n(\overline{m}, m)}\cdot 2^{\varepsilon b |\overline{m}-\overline{m}_*|} \\ &\!\!\!\!\underset{\overline{p}, \overline{r},q,d}{\lesssim} \int_{G_{n,\varepsilon}} 2^{-\psi_n(t_1,\dots,t_d,t_1+\dots+t_d)+\varepsilon b\sum_{j=1}^d |t_j-m^*_j|}\, dt_1\dotsb dt_d \\ &\!\!\!\stackrel{(4.19)}{\leqslant} \int_{G_{n,\varepsilon}} 2^{-\widetilde h(\widehat \alpha_1,\dots,\widehat \alpha_d,\widehat s)\log n-(c_{\overline{p},q,\overline{r},d}/2-b\varepsilon) \sum_{j=1}^d |t_j-m^*_j|}\, dt_1\dotsb dt_d \\ &\!\!\!\!\underset{\overline{p},\overline{r},q,d}{\lesssim} n^{-\widetilde h(\widehat \alpha_1,\dots,\widehat \alpha_d,\widehat s)} \end{aligned} \end{equation*} \notag $$
for small $\varepsilon >0$. Now from (4.14) we have the required upper estimate for the width.

We prove the lower estimate. Again, we consider the more involved case ${q>2}$. Let $\overline{m} \in \mathbb N^d$, $2n\leqslant 2^m \leqslant n^{q/2}$, and let $\alpha_j=m_j/\log n$ for $1\leqslant j\leqslant d$ and ${s=\alpha_1+\dots+\alpha_d}$. From Lemma 3 and Theorems 8–10 we obtain

$$ \begin{equation} \begin{aligned} \, \notag d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) &\underset{\overline{p},\overline{r},q,d}{\gtrsim} d_n \biggl(\bigcap_{i=1}^d 2^{-m_ir_i-m/q+m/p_i}B^{2^m}_{p_i},l_q^{2^m}\biggr) \\ &\ \ \, \underset{q}{\asymp} 2^{-\psi_n(\overline{m},m)} \stackrel{(4.16)}{=} n^{-\widetilde h( \alpha_1,\dots, \alpha_d,s)}. \end{aligned} \end{equation} \tag{4.20} $$
Let $\overline{m}_*\in G_n$ be defined by (4.17), and let $\overline{m}\in G_n$ be a nearest point to $\overline{m}_*$ (with respect to the Euclidean norm) with positive integer coordinates such that $m\geqslant \log(2n)$. Then
$$ \begin{equation} n^{-\widetilde h( \alpha_1,\dots,\alpha_d,s)} \underset{\overline{p},\overline{r},q,d}{\asymp} n^{\widetilde h(\widehat \alpha_1,\dots,\widehat\alpha_d,\widehat s)}, \end{equation} \tag{4.21} $$
from which the required lower estimate for the width follows.

If $\min_D h\leqslant 0$, then $\min_{\widetilde D} \widetilde h\leqslant\min_{\widetilde D_{q/2}} \widetilde h\leqslant 0$ (see Lemma 6), so that by (4.20) and (4.21) we have

$$ \begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{p},\overline{r},q,d}{\gtrsim} 1, \end{equation*} \notag $$
that is, $W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d)$ is not compact in $L_q(\mathbb{T}^d)$.

Remark 2. We have shown that if $\min_D h\leqslant 0$, then

$$ \begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{p},\overline{r},q,d}{\gtrsim} 1. \end{equation*} \notag $$
From (4.20) and (4.21) it also follows that the same estimate holds for $q>2$ if $\min_{\widetilde D} \widetilde h\leqslant 0$.

§ 5. Proof of Theorem 1

By Theorem 2 it suffices to find a minimum point of the function $h$ on $D$ for ${q\leqslant 2}$ and of the function $\widetilde h$ on $\widetilde D$ for $q>2$.

Consider the more involved case $q>2$ (the argument for $q\leqslant 2$ is similar and uses Lemma 1).

First we prove the theorem under the additional assumption that $p_i\notin \{2,q\}$, $1\leqslant i\leqslant d$. We have $I=I'$, $J=J'$ and $K=K'$.

As above, let the numbers $\lambda_{i,j}$ and $\mu_{i,j}$ be defined by (1.9).

Lemma 8. Let $p_i\notin \{2,q\}$ and $\alpha_i\geqslant 0$ $(1\leqslant i\leqslant d)$, let $\alpha_1+\dots+\alpha_d=s$, and assume that $1\leqslant s\leqslant q/2$.

1. Let $j\in I$. Then $\widetilde h(\alpha_1,\dots,\alpha_d,s)=r_j\alpha_j$ if and only if $\alpha_jr_j-\alpha_ir_i\geqslant 0$, $1\leqslant i\leqslant d$.

2. Let $j\in J$. Then

$$ \begin{equation*} \widetilde h(\alpha_1,\dots,\alpha_d,s)=r_j\alpha_j- \frac 12\cdot \frac{1/p_j-1/q}{1/2-1/q}(s-1) \end{equation*} \notag $$
if and only if
$$ \begin{equation*} \alpha_jr_j-\alpha_ir_i\geqslant \frac 12\cdot \frac{1/p_j-1/p_i}{1/2-1/q}(s-1), \qquad 1\leqslant i\leqslant d. \end{equation*} \notag $$

3. Let $j\in K$. Then

$$ \begin{equation*} \widetilde h(\alpha_1,\dots,\alpha_d,s)=r_j\alpha_j- \frac{s}{p_j}+\frac 12 \end{equation*} \notag $$
if and only if
$$ \begin{equation*} \alpha_jr_j-\alpha_ir_i\geqslant \frac{s}{p_j}-\frac{s}{p_i}, \qquad 1\leqslant i\leqslant d. \end{equation*} \notag $$

4. Let $i\in I$ and $j\in J\cup K$. Then

$$ \begin{equation*} \widetilde h(\alpha_1,\dots,\alpha_d,s)=(1-\lambda_{i,j})r_i\alpha_i+\lambda_{i,j} r_j\alpha_j \end{equation*} \notag $$
if and only if
$$ \begin{equation*} \begin{gathered} \, \alpha_ir_i-\alpha_jr_j\leqslant 0, \qquad \alpha_ir_i-\alpha_jr_j\geqslant \frac 12\cdot \frac{1/p_i-1/p_j}{1/2-1/q}(s-1), \\ \frac{\alpha_ir_i-\alpha_jr_j}{1/p_i-1/p_j} \geqslant \frac{\alpha_ir_i- \alpha_kr_k}{1/p_i-1/p_k}, \qquad k\in J\cup K, \end{gathered} \end{equation*} \notag $$
and
$$ \begin{equation*} \frac{\alpha_ir_i-\alpha_jr_j}{1/p_i-1/p_j} \leqslant \frac{\alpha_kr_k- \alpha_jr_j}{1/p_k-1/p_j}, \qquad k\in I. \end{equation*} \notag $$

5. Let $i\in I\cup J$ and $j\in K$. Then

$$ \begin{equation*} \widetilde h(\alpha_1,\dots,\alpha_d,s)=(1-\mu_{i,j})r_i\alpha_i+\mu_{i,j} r_j\alpha_j -\frac s2+\frac 12 \end{equation*} \notag $$
if and only if
$$ \begin{equation*} \begin{gathered} \, \alpha_ir_i-\alpha_jr_j\leqslant \frac 12\cdot \frac{1/p_i-1/p_j}{1/2-1/q}(s-1), \qquad \alpha_ir_i-\alpha_jr_j\geqslant \frac{s}{p_i}-\frac{s}{p_j}, \\ \frac{\alpha_ir_i-\alpha_jr_j}{1/p_i-1/p_j} \geqslant \frac{\alpha_ir_i- \alpha_kr_k}{1/p_i-1/p_k}, \qquad k\in K, \end{gathered} \end{equation*} \notag $$
and
$$ \begin{equation*} \frac{\alpha_ir_i-\alpha_jr_j}{1/p_i-1/p_j} \leqslant \frac{\alpha_kr_k- \alpha_jr_j}{1/p_k-1/p_j}, \qquad k\in I\cup J. \end{equation*} \notag $$

Proof. Necessity. Part 1 holds because we have $\alpha_jr_j\geqslant \alpha_ir_i$ for $i\in I$ and $r_j\alpha_j \geqslant (1- \lambda_{j,i})r_j\alpha_j+ \lambda_{j,i}r_i\alpha_i$ for $i\in J\cup K$. For part 2 we invoke the inequalities
$$ \begin{equation*} \begin{gathered} \, r_j\alpha_j-\frac 12\cdot \frac{1/p_j-1/q}{1/2-1/q}(s-1) \geqslant r_i\alpha_i-\frac 12\cdot \frac{1/p_i-1/q}{1/2-1/q}(s-1), \qquad i\in J, \\ r_j\alpha_j-\frac 12\cdot \frac{1/p_j-1/q}{1/2-1/q}(s-1) \geqslant (1-\lambda_{i,j})r_i\alpha_i+\lambda_{i,j}r_j\alpha_j, \qquad i\in I, \end{gathered} \end{equation*} \notag $$
and
$$ \begin{equation*} r_j\alpha_j-\frac 12\cdot \frac{1/p_j-1/q}{1/2-1/q}(s-1) \geqslant (1-\mu_{j,i})r_j\alpha_j+ \mu_{j,i}r_i\alpha_i -\frac s2+\frac 12, \qquad i\in K. \end{equation*} \notag $$
In part 3 we use the inequalities
$$ \begin{equation*} \begin{gathered} \, r_j\alpha_j+\frac 12 -\frac{s}{p_j} \geqslant r_i\alpha_i+\frac 12 -\frac{s}{p_i}, \qquad i\in K, \\ r_j\alpha_j+\frac 12 -\frac{s}{p_j} \geqslant (1-\mu_{i,j})\alpha_ir_i+\mu_{i,j} \alpha_jr_j+\frac 12 -\frac s2, \qquad i\in I\cup J, \end{gathered} \end{equation*} \notag $$
and in part 4 we employ the inequalities
$$ \begin{equation*} \begin{gathered} \, (1-\lambda_{i,j})r_i\alpha_i+\lambda_{i,j} r_j\alpha_j\geqslant r_i\alpha_i, \\ (1-\lambda_{i,j})r_i\alpha_i+\lambda_{i,j} r_j\alpha_j\geqslant \alpha_jr_j-\frac 12\cdot \frac{1/p_j-1/q}{1/2-1/q}(s-1) , \qquad j\in J, \\ (1-\lambda_{i,j})r_i\alpha_i+\lambda_{i,j} r_j\alpha_j\geqslant (1-\mu_{i,j})r_i\alpha_i+\mu_{i,j} r_j\alpha_j-\frac 12(s-1), \qquad j\in K, \\ (1-\lambda_{i,j})r_i\alpha_i+\lambda_{i,j} r_j\alpha_j\geqslant (1-\lambda_{i,k})r_i\alpha_i+ \lambda_{i,k} r_k\alpha_k, \qquad k\in J\cup K, \end{gathered} \end{equation*} \notag $$
and
$$ \begin{equation*} (1-\lambda_{i,j})r_i\alpha_i+\lambda_{i,j} r_j\alpha_j\geqslant (1-\lambda_{k,j})r_k\alpha_k+ \lambda_{k,j} r_j\alpha_j, \qquad k\in I. \end{equation*} \notag $$
The analysis of part 5 is similar to that of part 4.

Sufficiency. Consider part 1 (the other assertions of the theorem are dealt with similarly). Let $\alpha_1^*+\dots+\alpha_d^*=s^*\in [1,q/2]$, $\alpha_j^*\geqslant 0$ ($1\leqslant j\leqslant d$), and let $r_j\alpha_j^*\geqslant r_i\alpha_i^*$ for all $i=1,\dots,d$, but $\widetilde h(\alpha_1^*,\dots, \alpha_d^*,s^*) > r_j\alpha_j^*$. There exist $c>0$ and an open subset $U$ of the set

$$ \begin{equation*} \biggl\{(\alpha_1,\dots,\alpha_d,s)\colon \alpha_1+\dots+\alpha_d=s,\, \alpha_i\geqslant 0,\, 1\leqslant s\leqslant \frac q2,\, r_j\alpha_j\geqslant r_i\alpha_i,\,1\leqslant i\leqslant d\biggr\} \end{equation*} \notag $$
such that for each $(\alpha_1,\dots, \alpha_d,s)\in U$,
$$ \begin{equation} \widetilde h(\alpha_1,\dots,\alpha_d,s)-r_j\alpha_j \geqslant c. \end{equation} \tag{5.1} $$

For sufficiently large $n\in \mathbb N$ there exists $(\alpha_1,\dots,\alpha_d,s)\in U$ such that $m_k:=\alpha_k\log n \in \mathbb N$ ($1\leqslant k\leqslant d$) and $m\geqslant \log(2n)$ (recall that $m:=m_1+\dots+m_d$). Hence

$$ \begin{equation*} 2^{-r_jm_j-m/q+m/p_j} \cdot 2^{m(1/p_i-1/p_j)} \leqslant 2^{-r_im_i-m/q+m/p_i}, \qquad 1\leqslant i\leqslant d. \end{equation*} \notag $$
By part 2 of Lemma 2,
$$ \begin{equation*} d_n\biggl(\bigcap_{i=1}^d 2^{-r_im_i-m/q+m/p_i}B_{p_i}^{2^m},l_q^{2^m}\biggr) \underset{q}{\asymp} 2^{-m_jr_j}=n^{-r_j\alpha_j}. \end{equation*} \notag $$
On the other hand, by (4.20)
$$ \begin{equation*} d_n\biggl(\bigcap_{i=1}^d 2^{-r_im_i-m/q+m/p_i}B_{p_i}^{2^m},l_q^{2^m}\biggr) \underset{q}{\asymp} n^{-\widetilde h(\alpha_1, \dots,\alpha_d,s)}\stackrel{(5.1)}{\leqslant} n^{-r_j\alpha_j-c}, \end{equation*} \notag $$
which is a contradiction.

Lemma 8 is proved.

Proof of Theorem 1. First we verify the theorem for $p_i\notin\{2,q\}$, $1\leqslant i\leqslant d$.

Consider the points

$$ \begin{equation} \xi_k=(\alpha_1^k,\dots,\alpha_d^k,s^k), \qquad 1\leqslant k\leqslant 4, \end{equation} \tag{5.2} $$
where $s^1=s^2=1$, $s^3=s^4=q/2$,
$$ \begin{equation} \alpha^1_j=\frac{1/r_j}{\sum_{i=1}^d 1/r_i}\quad\text{and} \quad \alpha^2_j=\frac{1-\sum_{i=1}^d\frac{1}{r_i}(1/p_i-1/p_j)}{r_j\sum_{i=1}^d 1/r_i}, \qquad 1\leqslant j\leqslant d, \end{equation} \tag{5.3} $$
and
$$ \begin{equation} \alpha^3_j=\frac{q}{2}\alpha_j^2\quad\text{and} \quad \alpha^4_j=\frac{q}{2}\alpha_1^j, \qquad 1\leqslant j\leqslant d. \end{equation} \tag{5.4} $$
Note that by (1.3) we have $\alpha^k_j>0$ for $1\leqslant k\leqslant 4$ and $1\leqslant j\leqslant d$.

We claim that the minimum of the function $\widetilde h$ on $\widetilde D$ can only be attained at $\xi_1$, $\xi_2$ or $\xi_3$. We will also evaluate $\widetilde h$ at these points.

We need the following notation. Let $\widehat l_{m,t}$, $1\leqslant m,t\leqslant 4$, $m\ne t$, be the line segments between $\xi_m$ and $\xi_t$. For $1\leqslant k\leqslant d$ we define line segments $l_k$, $\widetilde l_k$ and $\widehat l_k$ as follows: $l_k$ is defined by

$$ \begin{equation} \begin{gathered} \, r_1\alpha_1=\dots=r_{k-1}\alpha_{k-1}= r_{k+1}\alpha_{k+1}=\dots=r_d\alpha_d, \\ \alpha_1+\dots+\alpha_d=s=1, \\ r_k\alpha_k- r_j\alpha_j \leqslant 0, \qquad j\ne k, \quad\alpha_i\geqslant 0, \quad 1\leqslant i\leqslant d, \end{gathered} \end{equation} \tag{5.5} $$
$\widehat l_k$ is defined by
$$ \begin{equation} \begin{gathered} \, \begin{aligned} \, r_1\alpha_1-\frac{1}{p_1} &=\dots=r_{k-1}\alpha_{k-1}-\frac{1}{p_{k-1}} \\ &= r_{k+1}\alpha_{k+1}-\frac{1}{p_{k+1}}=\dots=r_d\alpha_d-\frac{1}{p_d}, \end{aligned} \\ \alpha_1+\dots+\alpha_d=s=1, \\ r_k\alpha_k-r_j\alpha_j\leqslant \frac{1}{p_k}-\frac{1}{p_j}, \qquad j\ne k, \quad\alpha_i\geqslant 0, \quad 1\leqslant i\leqslant d, \end{gathered} \end{equation} \tag{5.6} $$
and $\widetilde l_k$ is defined by
$$ \begin{equation} \begin{gathered} \, \begin{aligned} \, r_1\alpha_1-\frac{q}{2p_1}&=\dots=r_{k-1}\alpha_{k-1}-\frac{q}{2p_{k-1}} \\ &= r_{k+1}\alpha_{k+1}-\frac{q}{2p_{k+1}}=\dots=r_d\alpha_d-\frac{q}{2p_d}, \end{aligned} \\ \alpha_1+\dots+\alpha_d=s=\frac q2, \\ r_k\alpha_k-r_j\alpha_j\leqslant \frac{q}{2p_k}-\frac{q}{2p_j}, \qquad j\ne k, \quad \alpha_i\geqslant 0, \quad 1\leqslant i\leqslant d. \end{gathered} \end{equation} \tag{5.7} $$

Note that $\xi_1\in l_k$, $\xi_2\in \widehat l_k$ and $\xi_3\in \widetilde l_k$ are endpoints of the corresponding segments; in addition, the systems of equalities and inequalities (5.5)–(5.7) have the same matrix. Hence the segments $l_k$, $\widetilde l_k$ and $\widehat l_k$ have the form

$$ \begin{equation} \begin{gathered} \, l_k=\xi_1+tv_k, \qquad 0\leqslant t\leqslant \tau_k, \\ \widehat l_k=\xi_2+tv_k, \qquad 0\leqslant t\leqslant \widehat\tau_k, \\ \widetilde l_k=\xi_3+tv_k, \qquad 0\leqslant t\leqslant \widetilde \tau_k, \end{gathered} \end{equation} \tag{5.8} $$
where $\tau_k$, $\widehat \tau_k$ and $\widetilde \tau_k$ are some positive numbers.

We set $\xi_{1,k}=\xi_1+\tau_kv_k$ (this is the second endpoint of $l_k$). Now $\xi_{1,k}$ is given by

$$ \begin{equation} \begin{gathered} \, \alpha_k=0, \qquad r_1\alpha_1=\dots=r_{k-1}\alpha_{k-1}=r_{k+1}\alpha_{k+1}=\dots=r_d\alpha_d, \\ \alpha_1+\dots+\alpha_d=s=1. \end{gathered} \end{equation} \tag{5.9} $$
Setting
$$ \begin{equation} \psi_j(\alpha_1,\dots,\alpha_d,s)=r_j\alpha_j, \qquad 1\leqslant j\leqslant d, \end{equation} \tag{5.10} $$
we have
$$ \begin{equation} \begin{gathered} \, \psi_j(\xi_1) \stackrel{(5.3)}{=} \frac{1}{\sum_{i=1}^d 1/r_i}, \quad \psi_j(\xi_4) \stackrel{(5.4)}{=} \frac{q}{2\sum_{i=1}^d 1/r_i}, \qquad 1\leqslant j\leqslant d, \\ \psi_j(\xi_{1,k}) \stackrel{(5.9)}{=} \frac{1}{\sum_{i\ne k}1/r_i}, \qquad j\ne k. \end{gathered} \end{equation} \tag{5.11} $$

The set $\widetilde D$ is partitioned into polytopes on which $\widetilde h$ is an affine function. For such a polytope we now find the set of its vertices with positive $\alpha_j$; we will also indicate the set of edges outgoing of these vertices.

Let $V$ be such a polytope.

1. Let $V=\{(\alpha_1,\dots,\alpha_d,s)\in \widetilde D\colon \widetilde h(\alpha_1,\dots, \alpha_d,s)=r_j\alpha_j\}$, where $j\in I$. We use part 1 of Lemma 8. At vertices of $V$ with positive coordinates we have $r_1\alpha_1=\dots= r_d\alpha_d$, where $\alpha_1+\dots+\alpha_d=s=1$ or $\alpha_1+\dots+\alpha_d=s=q/2$. These equalities define the points $\xi_1$ and $\xi_4$. The edges going out of $\xi_1$ are given by

$$ \begin{equation*} \biggl\{(\alpha_1,\dots,\alpha_d,s)\colon r_1\alpha_1=\dots=r_d\alpha_d,\, \alpha_1+\dots+\alpha_d=s\in \biggl[1,\frac q2\biggr]\biggr\} \end{equation*} \notag $$
or by (5.5) for $k\ne j$ (see part 1 of Lemma 8); these are $\widehat l_{1,4}$ and $l_k$, $k\ne j$. From (5.10) and (5.11) we obtain
$$ \begin{equation} \widetilde h(\xi_1)< \widetilde h(\xi_4)\quad\text{and} \quad \widetilde h(\xi_1)< \widetilde h(\xi_{1,k}), \quad k\ne j. \end{equation} \tag{5.12} $$
Thus, the function $\widetilde h$ attains its minimum on $V$ only at $\xi_1$, and
$$ \begin{equation} \min_V \widetilde h=\widetilde h(\xi_1)= \psi_j(\xi_1)\stackrel{(5.11)}{=}\frac{\langle \overline{r}\rangle}{d}. \end{equation} \tag{5.13} $$

2. Let

$$ \begin{equation} V=\biggl\{(\alpha_1,\dots,\alpha_d,s)\in \widetilde D\colon \widetilde h(\alpha_1,\dots,\alpha_d, s)=r_j\alpha_j-\frac 12\cdot \frac{1/p_j-1/q}{1/2-1/q}(s-1)\biggr\}, \end{equation} \tag{5.14} $$
where $j\in J$. By part 2 of Lemma 8 the vertices of $V$ with positive coordinates satisfy
$$ \begin{equation*} r_1\alpha_1-\frac 12\cdot \frac{1/p_1-1/q}{1/2-1/q}(s-1)=\dots=r_d\alpha_d-\frac 12\cdot \frac{1/p_d-1/q}{1/2-1/q}(s-1) \end{equation*} \notag $$
and
$$ \begin{equation*} \alpha_1+\dots+\alpha_d=s, \qquad s=1 \text{ or } s=\frac q2. \end{equation*} \notag $$

For $s=1$ we have the point $\xi_1$, and for $s=q/2$ we have $\xi_3$.

An edge going out of $\xi_1$ is either $l_k$ ($k=1,\dots,d$, $k\ne j$; see (5.5) and part 2 of Lemma 8), or the line segment $\widehat l_{1,3}$ between $\xi_1$ and $\xi_3$. By (5.14), for ${s=1}$ we have $\widetilde h(\alpha_1,\dots, \alpha_d,1)=r_j\alpha_j$, and so from (5.10) and (5.11) we find that ${\widetilde h(\xi_1)<\widetilde h(\xi_{1,k})}$, $k\ne j$. Hence if $\widetilde h(\xi_1)\leqslant \widetilde h(\xi_3)$, then $\xi_1$ is a minimum point of $\widetilde h$ on $V$ and ${\widetilde h (\xi_1)={\langle \overline{r}\rangle}/{d}}$.

An edge going out of the point $\xi_3$ is either $\widehat l_{1,3}$, or the segment $\widetilde l_k$ ($k=1,\dots,d$, $k\ne j$; see (5.7)). Let $\xi_{3,k}\ne \xi_3$ be an endpoint of $\widetilde l_k$. From (5.8), (5.10), (5.11) and (5.14) we obtain

$$ \begin{equation} \widetilde h(\xi_3) < \widetilde h(\xi_{3,k}). \end{equation} \tag{5.15} $$
Hence if $\widetilde h(\xi_1)\geqslant \widetilde h(\xi_3)$, then $\xi_3$ is a minimum point of $\widetilde h$ on $V$; in addition,
$$ \begin{equation*} \widetilde h(\xi_3)=\frac q2 \biggl( \frac{\langle \overline{r}\rangle}{d}+\frac 1q -\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ \overline{r}\rangle}\biggr). \end{equation*} \notag $$

Thus, we have

$$ \begin{equation} \min_V \widetilde h=\min \bigl\{\widetilde h(\xi_1),\widetilde h(\xi_3)\bigr\}= \min \biggl \{\frac{\langle \overline{r}\rangle}{d},\frac q2 \biggl( \frac{\langle \overline{r}\rangle}{d}+\frac 1q -\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ \overline{r}\rangle}\biggr)\biggr\}; \end{equation} \tag{5.16} $$
and if, in addition,
$$ \begin{equation*} \frac{\langle \overline{r}\rangle}{d} \ne \frac q2 \biggl( \frac{\langle \overline{r}\rangle}{d}+\frac 1q -\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ \overline{r}\rangle}\biggr), \end{equation*} \notag $$
then the minimum on $V$ is attained at a unique point.

3. Let

$$ \begin{equation} V=\biggl\{(\alpha_1,\dots,\alpha_d,s)\in \widetilde D\colon \widetilde h(\alpha_1,\dots,\alpha_d, s)=r_j\alpha_j+\frac 12-\frac{s}{p_j}\biggr\}, \end{equation} \tag{5.17} $$
where $j\in K$. By part 3 of Lemma 8 the vertices with positive coordinates of the polytope $V$ are given by
$$ \begin{equation*} r_1\alpha_1-\frac{s}{p_1}=\dots=r_d\alpha_d-\frac{s}{p_d} \end{equation*} \notag $$
and
$$ \begin{equation*} \alpha_1+\dots+\alpha_d=s, \qquad s=1 \text{ or } s=\frac q2. \end{equation*} \notag $$

For $s=1$ we have the point $\xi_2$, and for $s=q/2$ we have $\xi_3$.

An edge going out of $\xi_3$ is either $\widetilde l_k$ ($k\ne j$; see (5.7) and part 3 of Lemma 8), or the segment $\widehat l_{2,3}$. An edge going out of $\xi_2$ is either $\widehat l_{2,3}$, or a segment $\widehat l_k$ ($k\ne j$; see (5.6)).

Let $\xi_{2,k}\ne \xi_2$ be an endpoint of the edge $\widehat l_k$ and $\xi_{3,k}\ne \xi_3$ be an endpoint of $\widetilde l_k$. From (5.8), (5.10), (5.11) and (5.17) we obtain $\widetilde h(\xi_2)< \widetilde h(\xi_{2,k})$ and $\widetilde h(\xi_3)< \widetilde h(\xi_{3,k})$.

Thus,

$$ \begin{equation} \begin{aligned} \, \notag \min_V\widetilde h &=\min\bigl\{\widetilde h(\xi_2),\widetilde h(\xi_3)\bigr\} \\ &=\min \biggl\{ \frac{\langle \overline{r}\rangle}{d}+\frac 12 -\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ \overline{r}\rangle},\frac q2 \biggl( \frac{\langle \overline{r}\rangle}{d}+\frac 1q -\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ \overline{r}\rangle}\biggr)\biggr\}; \end{aligned} \end{equation} \tag{5.18} $$
and if, in addition,
$$ \begin{equation*} \frac{\langle \overline{r}\rangle}{d}+\frac 12 -\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ \overline{r}\rangle}\ne \frac q2 \biggl( \frac{\langle \overline{r}\rangle}{d}+\frac 1q -\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ \overline{r}\rangle}\biggr), \end{equation*} \notag $$
then the function $\widetilde h$ attains its minimum on $V$ at a unique point.

4. Let

$$ \begin{equation} V=\bigl\{(\alpha_1,\dots,\alpha_d,s)\in \widetilde D\colon \widetilde h(\alpha_1, \dots,\alpha_d,s)=(1-\lambda_{i,j})r_i\alpha_i+\lambda_{i,j}r_j\alpha_j\bigr\}, \end{equation} \tag{5.19} $$
where $i\in I$ and $j\in J\cup K$. By part 4 of Lemma 8 the vertices with positive coordinates of $V$ are given by the equalities
$$ \begin{equation} \frac{\alpha_ir_i-\alpha_jr_j}{1/p_i-1/p_j}=\frac{\alpha_ir_i-\alpha_kr_k}{1/p_i-1/p_k}, \qquad k\in J\cup K, \end{equation} \tag{5.20} $$
$$ \begin{equation} \frac{\alpha_ir_i-\alpha_jr_j}{1/p_i-1/p_j}=\frac{\alpha_kr_k-\alpha_jr_j}{1/p_k-1/p_j}, \qquad k\in I, \end{equation} \tag{5.21} $$
$$ \begin{equation} \nonumber \alpha_1+\dots+\alpha_d=s, \quad\text{where }s=1 \text{ or } s=\frac q2, \end{equation} \notag $$
and
$$ \begin{equation*} \alpha_ir_i-\alpha_jr_j=0 \text{ or } \alpha_ir_i-\alpha_jr_j=\frac 12\cdot \frac{1/p_i-1/p_j}{1/2-1/q}(s-1). \end{equation*} \notag $$

In the case where $\alpha_ir_i-\alpha_jr_j=0$ we have

$$ \begin{equation*} \alpha_1r_1=\dots=\alpha_dr_d\quad\text{and} \quad \alpha_1+\dots+\alpha_d=s, \quad s=1 \text{ or } s=\frac q2. \end{equation*} \notag $$

For $s=1$ we have the point $\xi_1$, and for $s=q/2$, the point $\xi_4$.

Let

$$ \begin{equation*} \alpha_ir_i-\alpha_jr_j=\frac 12\cdot \frac{1/p_i-1/p_j}{1/2-1/q}(s-1). \end{equation*} \notag $$
For $s=1$ this is the equality $\alpha_ir_i-\alpha_jr_j=0$, and so we obtain the point $\xi_1$ again. For $s=q/2$ we have $\alpha_ir_i-\alpha_jr_j=\frac q2 (1/p_i-1/p_j)$; now from (5.20) and (5.21) we obtain
$$ \begin{equation*} \alpha_1-\frac{q}{2p_1}=\dots=\alpha_d-\frac{q}{2p_d}\quad\text{and}\quad \alpha_1+\dots+\alpha_d=s=\frac q2. \end{equation*} \notag $$
These equalities define the vertex $\xi_3$.

An edge going out of $\xi_1$ is either $l_k$ ($k\ne i$, $j$), $\widehat l_{1,3}$, or $\widehat l_{1,4}$. On the edges $l_k$ and $\widehat l_{1,4}$ we have $r_i\alpha_i=r_j\alpha_j$, and the function $\widetilde h$ coincides with $\alpha_ir_i$. Now from (5.10) and (5.11) we obtain $\widetilde h(\xi_1)< \widetilde h(\xi_4)$ and $\widetilde h(\xi_1)< \widetilde h(\xi_{1,k})$, $k\ne i,j$. Hence if $\widetilde h(\xi_1)\leqslant \widetilde h(\xi_3)$, then $\min_{V} \widetilde h=\widetilde h(\xi_1)={\langle \overline{r} \rangle}/{d}$.

An edge going out of $\xi_3$ is either $\widetilde l_k$ ($k\ne i$, $j$), $\widehat l_{1,3}$, or $\widehat l_{3,4}$. On the edges $\widetilde l_k$ we have $r_i\alpha_i-{q}/(2p_i)=r_j\alpha_j-q/(2p_j)$, and so on $\widetilde l_k$ the function $\widetilde h$ is

$$ \begin{equation*} (1-\lambda_{i,j})r_i\alpha_i+\lambda_{i,j}r_j\alpha_j=r_i\alpha_i+\frac 12 -\frac{q}{2p_i}. \end{equation*} \notag $$
Hence by (5.8), (5.10) and (5.11) $\widetilde h(\xi_3)<\widetilde h(\xi_{3,k})$ for $k\ne i,j$. So if $\widetilde h(\xi_3)\leqslant \widetilde h(\xi_1)$, then by the inequality $\widetilde h(\xi_1) < \widetilde h(\xi_4)$ we have
$$ \begin{equation*} \min_V \widetilde h=\widetilde h(\xi_3)=\frac q2 \biggl( \frac{\langle \overline{r}\rangle}{d}+\frac 1q -\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ \overline{r}\rangle}\biggr). \end{equation*} \notag $$

As a result,

$$ \begin{equation} \min_V \widetilde h=\min \{\widetilde h(\xi_1),\widetilde h(\xi_3)\} = \min \biggl \{\frac{\langle \overline{r}\rangle}{d},\frac q2 \biggl( \frac{\langle \overline{r}\rangle}{d}+\frac 1q -\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ \overline{r}\rangle}\biggr)\biggr\}; \end{equation} \tag{5.22} $$
in addition, if
$$ \begin{equation*} \frac{\langle \overline{r}\rangle}{d} \ne \frac q2 \biggl( \frac{\langle \overline{r}\rangle}{d}+\frac 1q -\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ \overline{r}\rangle}\biggr), \end{equation*} \notag $$
then the minimum on $V$ is attained at a unique point.

5. A similar argument shows that if

$$ \begin{equation*} V=\biggl\{(\alpha_1,\dots,\alpha_d,s)\in \widetilde D\colon \widetilde h(\alpha_1,\dots,\alpha_d, s)=(1-\mu_{i,j})r_i\alpha_i+\mu_{i,j}r_j\alpha_j+\frac 12 -\frac s2\biggr\}, \end{equation*} \notag $$
where $i\in I\cup J$ and $j\in K$, then $\xi_1$, $\xi_2$ and $\xi_3$ are vertices of $V$ with positive coordinates, and
$$ \begin{equation} \min_V \widetilde h=\min \{h(\xi_1),h(\xi_2),h(\xi_3)\}=\min \{\theta_1,\theta_2,\theta_3\} \end{equation} \tag{5.23} $$
(recall the notation in the statement of Theorem 1); by the assumptions of the theorem there exists $j_*\in \{1,2,3\}$ such that $\theta_{j_*}=\min_{j\ne j_*} \theta_j$, and therefore the minimum in (5.23) is attained at a unique point.

So the set $\widetilde D$ falls into closed polytopes $V^{(k)}$, $1\leqslant k\leqslant k_0$, where each $V^{(k)}$ is defined by the assumptions of cases 1–5 above.

Now consider the general case where it is possible that $p_i\in \{2, q\}$. We define $\overline{p}^N=(p_1^N,\dots,p_d^N)$ as follows. If $p_j\notin \{2,q\}$, then we set $p_j^N=p_j$. If $p_j=q$, then we set $p_j^N=q+1/N$, and if $p_j=2$, then $p_j^N=2 \pm 1/N$ (the sign is the same for all $j$; the sign is negative for $K=\{1,\dots,d\}$, otherwise we take the positive sign). For large $N$ we have $p_j^N\notin \{2, q\}$, $1\leqslant j\leqslant d$. Let the function $\widetilde h^N$ be defined like $\widetilde h$, but with $p_j$ replaced by $p_j^N$. Then $\widetilde h^N$ converges uniformly to $\widetilde h$ on $\widetilde D$. If $\overline{p}$ satisfies condition (1.3), then $\overline{p}^N$ for large $N$ also does.

Let $\xi_t$ ($1\leqslant t\leqslant 4$) be given by (5.2)–(5.4). We define a set $T\subset \{1,2,3\}$ as follows: if $I=\{1,\dots,d\}$, then $T=\{1\}$; if $I \ne \{1,\dots,d\}$ and $I\cup J=\{1,\dots,d\}\ne K$, then $T=\{1,3\}$; if $K= \{1,\dots,d\}$, then $T=\{2,3\}$; otherwise, $T=\{1,2,3\}$. Note that all points $\xi_t$, $t\in T$, are distinct.

We claim that $\min_{\widetilde D} \widetilde h=\min_{t\in T} \widetilde h(\xi_t)$. Indeed, let $\xi_t^N$ and $T_N$ be defined similarly to $\xi_t$ and $T$ with $\overline{p}$ replaced by $\overline{p}^N$. Then $\xi_t^N\underset{N\to \infty}{\to} \xi_t$, and for large $N$ we have $T_N=T$ (we consider only such $N$ in what follows). By the above $\min_{\widetilde D} \widetilde h^N=\min_{t\in T}\widetilde h^N(\xi_t^N)$. There exist $t_*\in T$ and a subsequence $\{N_m\}_{m\in \mathbb N}$ such that $\min_{t\in T}\widetilde h^{N_m}(\xi_t^{N_m})=\widetilde h^{N_m}(\xi_{t_*}^{N_m})$. The functions $\widetilde h^N$ converge uniformly to $\widetilde h$ on $\widetilde D$ and $\xi_t^N\underset{N\to \infty}{\to} \xi_t$, and thus $\min_{\widetilde D} \widetilde h=\widetilde h(\xi_{t_*})$. The explicit form of $\widetilde h(\xi_{t_*})$ also follows from the formulae for $\widetilde h^{N_m}(\xi_{t_*}^{N_m})$; see (5.13), (5.16), (5.18), (5.22), and (5.23).

Let us now show that $\xi_{t_*}$ is a unique minimum point of $\widetilde h$. It suffices to verify that there exists a positive constant $c=c(\overline{p},q,\overline{r},d)$ such that for large $m\in \mathbb N$,

$$ \begin{equation} \widetilde h^{N_m}(\xi) -\widetilde h^{N_m}(\xi^{N_m}_{t_*}) \geqslant c|\xi- \xi^{N_m}_{t_*}|, \qquad \xi \in \widetilde D \end{equation} \tag{5.24} $$
(here $|\cdot|$ is the Euclidean norm on $\mathbb R^{d+1}$). Letting $m\to \infty$, this establishes the inequality $\widetilde h(\xi)-\widetilde h(\xi_{t_*})\geqslant c|\xi-\xi_{t_*}|$, $\xi\in \widetilde D$.

Let us prove (5.24). Again, we consider the polytope $V=V(m)$ containing the vertex $\xi_{t_*}^{N_m}$; the function $\widetilde h^{N_m}$ is affine on this polytope (see the above analysis of cases 1–5). It suffices to show that (5.24) holds for points $\xi$ on each edge going out of the vertex $\xi_{t_*}^{N_m}$. Indeed, by the assumptions of the theorem $\widetilde h(\xi_{t_*}) < \widetilde h(\xi_t)$, $t\in T\setminus \{t_*\}$. Hence, since $\widetilde h^N$ converges uniformly to $\widetilde h$ on $\widetilde D$ and $\xi_t^N$ converges to $\xi_t$, for large $m$ we have

$$ \begin{equation*} \widetilde h^{N_m}(\xi_t^{N_m})-\widetilde h^{N_m}(\xi_{t_*}^{N_m}) \underset{\overline{p},q,\overline{r},d}{\gtrsim} |\xi_t^{N_m}-\xi_{t_*}^{N_m}|. \end{equation*} \notag $$
Hence (5.24) holds on the edge between $\xi_{t_*}^{N_m}$ and $\xi^{N_m}_t$, $t\in T \setminus \{t_*\}$. There can also be an edge from $\xi_{t_*}^{N_m}$ connecting it with $\xi_4^{N_m}$ (in this case $\xi_1^{N_m} \in V$; see the analysis of cases 1 and 4); we also have
$$ \begin{equation*} \widetilde h^{N_m}(\xi_4^{N_m})=\frac q2 \widetilde h^{N_m}(\xi_1^{N_m})=\frac q2 \cdot \frac{\langle \overline{r} \rangle}{d}. \end{equation*} \notag $$
Hence (5.24) also holds on this edge. Note also that the edge from $\xi^{N_m}_{t_*}$ can coincide with $l_k^m$, $\widetilde l_k^m$ or $\widehat l_k^m$ (these line segments are given by formulae similar to (5.5), (5.7) and (5.6), with $\overline{p}$ replaced by $\overline{p}^{N_m}$). In the analysis of these cases it was shown that the function $\widetilde h^{N_m}$ has the form $\alpha_jr_j+\mathrm{const}$ on $l_k^m$, $\widetilde l_k^m$ and $\widehat l_k^m$, and $s\in \{1, q/2\}$ on these edges. In view of (5.8), (5.10) and (5.11) we see that (5.24) holds on the edges $l_k^m$, $\widetilde l_k^m$ and $\widehat l_k^m$ going out of $\xi^{N_m}_{t_*}$.

Theorem 1 is proved.

§ 6. Proof of Theorems 3 and 4

First we show that if $\frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{r}\circ \overline{p}\rangle}\leqslant 0$, then $W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d)$ is not compact in $L_q(\mathbb{T}^d)$. We use induction on $d$. For $d=1$ this result is known.

For the step of induction from $d-1$ to $d$ we assume that

$$ \begin{equation*} \frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{r}\circ \overline{p}\rangle}\leqslant 0. \end{equation*} \notag $$
Then there exists $j\in \{1,\dots,d\}$ such that $p_j<q$.

First assume that (1.3) holds. Since $p_j<q$ for some $j$, we have

$$ \begin{equation*} \min_D h\leqslant \frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{r}\circ \overline{p}\rangle}\leqslant 0 \quad\text{for } q\leqslant 2 \end{equation*} \notag $$
and
$$ \begin{equation*} \min_{\widetilde D} \widetilde h \leqslant \frac q2 \biggl( \frac{\langle \overline{r}\rangle}{d}+\frac 1q- \frac{\langle \overline{r}\rangle}{\langle \overline{r}\circ \overline{p}\rangle}\biggr)\leqslant 0 \quad\text{for } q>2 \end{equation*} \notag $$
(see Theorems 1 and 2). Hence
$$ \begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{r}, d, \overline{p}, q}{\gtrsim} 1 \end{equation*} \notag $$
(see Remark 2), and so there is no compact embedding.

Now assume that condition (1.3) is not met, that is, there exists $j\in \{1,\dots,d\}$ such that

$$ \begin{equation} \sum_{i=1}^d \frac{1}{r_i} \biggl(\frac{1}{p_i}-\frac{1}{p_j}\biggr) \geqslant 1. \end{equation} \tag{6.1} $$
We set $\overline{r}_j=(r_1,\dots,r_{j-1},r_{j+1},\dots,r_d)$ and $\overline{p}_j=(p_1, \dots,p_{j-1},p_{j+1},\dots,p_d)$. Condition (6.1) is equivalent to the inequality
$$ \begin{equation} \frac{\langle \overline{r}_j\rangle}{d-1}+\frac{1}{p_j}-\frac{\langle \overline{r}_j\rangle}{\langle \overline{r}_j\circ \overline{p}_j\rangle}\leqslant 0. \end{equation} \tag{6.2} $$

If $p_j\leqslant q$, then

$$ \begin{equation} \frac{\langle \overline{r}_j\rangle}{d-1}+\frac{1}{q}-\frac{\langle \overline{r}_j\rangle}{\langle \overline{r}_j\circ \overline{p}_j\rangle}\leqslant 0 \end{equation} \tag{6.3} $$
and $W^{\overline{r}_j}_{\overline{p}_j}(\mathbb{T}^{d-1})$ is not compactly embedded in $L_q(\mathbb{T}^{d-1})$ by the induction assumption. Hence $W^{\overline{r}}_{\overline{p}}(\mathbb{T}^{d})$ is not compactly embedded in $L_q(\mathbb{T}^{d})$.

Let $p_j>q$. Setting $\overline{p}^*=(p_1,\dots,p_{j-1},q,p_{j+1},\dots,p_d)$ we have

$$ \begin{equation*} \frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{r}\circ \overline{p}^*\rangle}<0. \end{equation*} \notag $$
Therefore, $W^{\overline{r}}_{\overline{p}^*}(\mathbb{T}^{d})$ is not bounded in $L_q(\mathbb{T}^{d})$ (see [1], Theorem 1). On the other hand
$$ \begin{equation*} W^{\overline{r}}_{\overline{p}^*}(\mathbb{T}^{d}) \subset\{f\in \mathring{\mathcal S}'(\mathbb{T}^d)\colon \|\partial_j^{r_j}f\|_{L_{q}(\mathbb{T}^d)}\leqslant 1\} \end{equation*} \notag $$
(the right-hand side is a bounded set in $L_q(\mathbb{T}^{d})$). This contradiction shows that the case $p_j>q$ is impossible.

Proof of Theorem 3. Let $j\in \{1,\dots,d\}$ satisfy (6.1) (which is equivalent to (6.2)). By the assumptions of the theorem $p_j\leqslant q$. Hence (6.3) holds. By the above $W^{\overline{r}_j}_{\overline{p}_j}(\mathbb{T}^{d-1})$ is not compactly embedded in $L_q(\mathbb{T}^{d-1})$; hence $W^{\overline{r}}_{\overline{p}}(\mathbb{T}^{d})$ is not compactly embedded in $L_q(\mathbb{T}^{d})$.

The theorem is proved.

Proof of Theorem 4. We apply Theorem 2 and write out the function $h$ for $q\leqslant 2$ and $\widetilde h$ for $q>2$.

Let $q\leqslant 2$. Then by (1.10) and (1.6),

$$ \begin{equation*} h(\alpha_1,\alpha_2)= \begin{cases} r_1\alpha_1 & \text{for } r_1\alpha_1-r_2\alpha_2\geqslant 0, \\ (1-\lambda)r_1\alpha_1+\lambda r_2\alpha_2 & \text{for } r_1\alpha_1-r_2\alpha_2\leqslant 0 \end{cases} \end{equation*} \notag $$
(the case $h(\alpha_1,\alpha_2)=r_2\alpha_2+1/q-1/p_2 > (1-\lambda)r_1\alpha_1+\lambda r_2\alpha_2$ is possible only for $r_2\alpha_2> r_1\alpha_1+1/p_2-1/p_1$, which contradicts (1.10)). We have ${\langle \overline{r}\rangle}/{2}\ne \lambda r_2$, and so the function $h$ attains its minimum either at
$$ \begin{equation*} \biggl(\frac{1/r_1}{1/r_1+1/r_2}, \frac{1/r_2}{1/r_1+1/r_2}\biggr) \end{equation*} \notag $$
or at $(0,1)$. This implies (1.11).

Let $q>2$. Note that $\widehat s\in [1,q/2]$.

In the case $p_2\geqslant 2$, by Lemma 8,

$$ \begin{equation*} \widetilde h(\alpha_1,\alpha_2,s)= \begin{cases} r_1\alpha_1 & \text{for } r_1\alpha_1-r_2\alpha_2\geqslant 0, \\ (1-\lambda)r_1\alpha_1+\lambda r_2\alpha_2 & \text{for } \dfrac12\cdot \dfrac{1/p_1-1/p_2}{1/2-1/q}(s-1) \\ &\qquad \leqslant r_1\alpha_1-r_2\alpha_2\leqslant 0, \\ r_2\alpha_2-\dfrac 12\cdot \dfrac{1/p_2-1/q}{1/2-1/q}(s-1) & \text{for } r_1\alpha_1-r_2\alpha_2 \\ &\qquad\leqslant \dfrac12\cdot \dfrac{1/p_1-1/p_2}{1/2-1/q}(s-1). \end{cases} \end{equation*} \notag $$
Since $\theta_1\ne \theta_2$, the minimum of this function can either be attained at
$$ \begin{equation*} \biggl(\frac{1/r_1}{1/r_1+1/r_2},\frac{1/r_2}{1/r_1+1/r_2},1\biggr) \end{equation*} \notag $$
or at $(0,\widehat s,\widehat s)$.

In the case $p_2< 2$, by Lemma 8,

$$ \begin{equation*} \widetilde h(\alpha_1,\alpha_2,s)= \begin{cases} r_1\alpha_1 & \text{for } r_1\alpha_1-r_2\alpha_2\geqslant 0, \\ (1-\lambda)r_1\alpha_1+\lambda r_2\alpha_2 & \text{for } \dfrac12\cdot \dfrac{1/p_1-1/p_2}{1/2-1/q}(s-1) \\ &\qquad \leqslant r_1\alpha_1-r_2\alpha_2\leqslant 0, \\ (1-\mu)r_1\alpha_1+\mu r_2\alpha_2-\dfrac 12 (s-1) & \text{for } r_1\alpha_1-r_2\alpha_2 \\ &\qquad \leqslant \dfrac12\cdot \dfrac{1/p_1-1/p_2}{1/2-1/q}(s-1); \end{cases} \end{equation*} \notag $$
the case
$$ \begin{equation*} \widetilde h(\alpha_1,\alpha_2,s)=r_2\alpha_2+\frac 12- \frac{s}{p_2}>(1-\mu)r_1\alpha_1+\mu r_2\alpha_2-\frac 12 (s-1) \end{equation*} \notag $$
is impossible by (1.10).

By (1.12) the function $\widetilde h$ attains its minimum at one of the points $(0,\widehat s,\widehat s)$, $(0,1,1)$ and

$$ \begin{equation*} \biggl(\frac{1/r_1}{1/r_1+1/r_2}, \frac{1/r_2}{1/r_1+1/r_2},1\biggr). \end{equation*} \notag $$

This implies the estimates claimed in part 2 of the theorem.

Theorem 4 is proved.

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Citation: A. A. Vasil'eva, “Kolmogorov widths of a Sobolev class with constraints on derivatives in different metrics”, Sb. Math., 215:11 (2024), 1468–1498
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\paper Kolmogorov widths of a~Sobolev class with constraints on derivatives in different metrics
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\vol 215
\issue 11
\pages 1468--1498
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