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This article is cited in 2 scientific papers (total in 2 papers) Kolmogorov widths of an intersection of a finite family of Sobolev classes A. A. Vasil'evaab a Lomonosov Moscow State University, Faculty of Mechanics and Mathematics b Moscow Center for Fundamental and Applied Mathematics
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2024, Volume 88, Issue 1, DOI: https://doi.org/10.4213/im9398 
Bibliographic databases:
   § 1. IntroductionGaleev [1]–[3] studied the problem of the Kolmogorov widths of an intersection of periodic Sobolev and Nikol’skii classes in $L_q$-spaces. For $q\leqslant 2$, and in the case of “great smoothness” for $q>2$, the problem was reduced to estimating the Kolmogorov $n$-widths of an intersection of $l_{p_j}^{2n}$- balls with different radii (see the notation in § 2); the last problem was solved in [1]. No estimates of the widths were given for $q>2$ in the case of “small smoothness”. Recently in [4], estimates for the Kolmogorov $n$-widths of an intersection of $N$-dimensional balls for $N\geqslant 2n$ were obtained. This allows us to obtain estimates for the widths of an intersection of different function classes. In the present paper, order estimates for the Kolmogorov widths of an intersection of a finite family of Sobolev classes on a John domain (see the definition below) will be obtained; the Sobolev class $W^r_p(\Omega)$ will be defined by restrictions on all partial derivatives of order $r$ (in contrast to the papers [1]–[3], where the smoothness of functions on a multi-dimensional torus was defined by means of one mixed derivative $\partial^{r_1+\dots+r_d}/\partial x_1^{r_1}\cdots \partial x_d^{r_d}$). In addition, we generalize Theorem 1 from [2] on widths of an intersection of Sobolev classes on the one-dimensional torus to the case of “small smoothness”; an answer for “great smoothness” and for the case $q\leqslant 2$ will also be given, but in different terms. In [5], [6] a problem of widths of an intersection of two weighted Sobolev classes (of zeroth and $r$th smoothness) on a John domain was studied. The orders of the widths were found in an explicit form via consideration of a fairly great number of cases. Here, we generalize this result, but the order will be given in terms of the minimum point of some special piecewise-affine function on a polyhedral set. Recall the necessary definitions. Let $X$ be a normed space, and let $C\subset X$, $n\in \mathbb{Z}_+$. The Kolmogorov width of $C$ in the space $X$ is defined by
$$
\begin{equation*}
d_n(C,X)=\inf_{L\in \mathcal{L}_n(X)}\sup_{x\in C}\, \inf_{y\in L}\|x-y\|,
\end{equation*}
\notag
$$
where $\mathcal{L}_n(X)$ is the family of all linear subspaces in $X$ of dimension at most $n$. For more information about widths, see [7]–[9]. Let $\Omega \subset \mathbb{R}^d$ be a bounded domain, and let $1\leqslant p\leqslant \infty$, $r\in \mathbb{Z}_+$. For $f\in L_1^{\mathrm{loc}}(\Omega)$ we denote by $\nabla^r f$ the vector of all generalized partial derivatives of order $r$. If all its components belong to the space $L_p(\Omega)$, then $f$ lies in the Sobolev space $\mathcal{W}^r_p(\Omega)$. The Sobolev class is defined by
$$
\begin{equation*}
W^r_p(\Omega) = \{f\in \mathcal{W}^r_p(\Omega)\colon \|\nabla^r f\|_{L_p(\Omega)}\leqslant 1\},
\end{equation*}
\notag
$$
where $\|\nabla^r f\|_{L_p(\Omega)}$ is the $L_p$-norm of the function $|\nabla^r f(\,{\cdot}\,)|$. If $\Omega$ has Lipschitz boundary, then $\mathcal{W}^r_p(\Omega)$ is compactly embedded into $L_q(\Omega)$ if and only if $r/d+1/q-1/p>0$ (see [10]). If $\Omega$ is a John domain, the embedding condition is the same [11], [12]. Let us give the definition of such domains. We denote by $B_a(x)$ the Euclidean ball of radius $a$ with centre at a point $x$. Definition 1. Let $\Omega\subset\mathbb{R}^d$ be a bounded domain, and let $a>0$. We say that $\Omega \in \mathbf{FC}(a)$ if there is a point $x_*=x_*(\Omega)\in \Omega$ such that, for each $x\in \Omega$, there is a number $T(x)>0$ and a curve $\gamma_x\colon [0, T(x)] \to\Omega$ with the following properties: 1) $\gamma_x$ has the natural parametrization with respect to the Euclidean norm on $\mathbb{R}^d$; 2) $\gamma_x(0)=x$, $\gamma_x(T(x))=x_*$; 3) $B_{at}(\gamma_x(t))\subset \Omega$ for all $t\in [0, T(x)]$. We say that $\Omega$ is a John domain if $\Omega\in \mathbf{FC}(a)$ for some $a>0$. Any domain with Lipschitz boundary is a John domain. The Koch snowflake is another example. For $\sigma >1$, the set
$$
\begin{equation*}
\{(y, z)\colon y\in \mathbb{R}^{d-1},\, z\in \mathbb{R}, \, 0< z<1, \, |y|< z^\sigma\}
\end{equation*}
\notag
$$
is not a John domain. Let $a>0$, let $\Omega \in \mathbf{FC}(a)$ be a domain in $\mathbb{R}^d$, and let $s\geqslant 2$, $r_i\in \mathbb{Z}_+$, $1\leqslant i\leqslant s$, $1<p_i\leqslant \infty$, $1\leqslant i\leqslant s$.We set (that is, $M$ is the intersection of a finite family of Sobolev classes).One of the results of the present paper gives order estimates for the Kolmogorov widths $d_n(M, L_q(\Omega))$. First, we assume that
$$
\begin{equation}
\frac{r_j}{d}-\frac{1}{p_j} < \frac{r_i}{d} -\frac{1}{p_i} \quad \text{for}\quad j>i
\end{equation}
\tag{3}
$$
(in § 3, we will show how the general case can be reduced to this case). From (1) and (3) it follows that Let us introduce the notation for order equalities and inequalities. Let $X$, $Y$ 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 $x\in X$; $f_1(x,y)\underset{y}{\asymp} f_2(x, y)$ means that $f_1(x, y) \underset{y}{\lesssim} f_2(x, y)$ and $f_2(x,y)\underset{y}{\lesssim} f_1(x, y)$. We set
$$
\begin{equation*}
\mathfrak{Z} = \{r_1, \dots, r_s, p_1, \dots, p_s, q, d, a, R\},
\end{equation*}
\notag
$$
where $R=\operatorname{diam} \Omega$. Theorem 1. Let $\Omega \subset \mathbb{R}^d$, $\Omega \in \mathbf{FC}(a)$, $s\geqslant 2$, $1\leqslant q<\infty$, $r_j\in \mathbb{Z}_+$, $1<p_j\leqslant \infty$, $1\leqslant j\leqslant s$, and let (1) and (3) hold. We also suppose that $r_1/d+1/q-1/p_1>0$. 1. If $p_i\geqslant q$ for all $i\in \{1, \dots, s\}$, then
$$
\begin{equation*}
d_n(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\asymp} n^{-r_s/d}.
\end{equation*}
\notag
$$
2. If $q\leqslant 2$, $p_i\leqslant q$ for all $i\in \{1, \dots, s\}$, then
$$
\begin{equation*}
d_n(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\asymp} n^{-r_1/d-1/q+1/p_1}.
\end{equation*}
\notag
$$
3. If $q>2$, $p_i\leqslant 2$ for all $i\in \{1, \dots, s\}$ and $r_1/d\ne 1/p_1$, then
$$
\begin{equation*}
d_n(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\asymp} n^{-\min\{r_1/d+1/2-1/p_1,\, (q/2)(r_1/d + 1/q -1/p_1)\}}.
\end{equation*}
\notag
$$
4. Let $q\leqslant 2$, and let
$$
\begin{equation*}
\begin{aligned} \, I &:=\bigl\{i\in \{1,\dots,s\}\colon p_i\geqslant q\bigr\}\ne \{1, \dots, s\}, \\ J &:=\bigl\{i\in \{1,\dots,s\}\colon p_i\leqslant q\bigr\}\ne \{1, \dots, s\}. \end{aligned}
\end{equation*}
\notag
$$
For $i\in I$, $j\in J$, we define the numbers $\lambda_{ij}$ by the equations
$$
\begin{equation}
\frac 1q = \frac{1-\lambda_{ij}}{p_j} + \frac{\lambda_{ij}}{p_i}.
\end{equation}
\tag{5}
$$
Let
$$
\begin{equation}
(i_0, j_0) = \operatorname*{arg\,max}_{i\in I,\, j\in J} \biggl(\frac{(1-\lambda_{ij})r_j}{d}+\frac{\lambda_{ij}r_i}{d}\biggr).
\end{equation}
\tag{6}
$$
Then
$$
\begin{equation*}
d_n(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\asymp} n^{-((1-\lambda_{i_0j_0})r_{j_0}/d +\lambda_{i_0j_0}r_{i_0}/d)}.
\end{equation*}
\notag
$$
5. Let $q>2$, and let
$$
\begin{equation}
I :=\bigl\{i\in \{1,\dots,s\}\colon p_i\geqslant q\bigr\} \ne \{1, \dots, s\},
\end{equation}
\tag{7}
$$
$$
\begin{equation}
J :=\bigl\{i\in \{1,\dots,s\}\colon 2\leqslant p_i\leqslant q\bigr\},
\end{equation}
\tag{8}
$$
$$
\begin{equation}
K :=\bigl\{i\in \{1,\dots,s\}\colon p_i\leqslant 2\bigr\} \ne \{1, \dots, s\}.
\end{equation}
\tag{9}
$$
We define the numbers $\lambda_{ij}$ and $\widetilde \lambda_{ij}$ by the equations
$$
\begin{equation}
\frac 1q = \frac{1-\lambda_{ij}}{p_j} +\frac{\lambda_{ij}}{p_i}, \qquad i\in I, \quad j\in J\cup K,
\end{equation}
\tag{10}
$$
$$
\begin{equation}
\frac 12 = \frac{1-\widetilde\lambda_{ij}}{p_j} +\frac{\widetilde\lambda_{ij}}{p_i}, \qquad i\in I\cup J, \quad j\in K;
\end{equation}
\tag{11}
$$
we also set
$$
\begin{equation}
(i_0, j_0) = \operatorname*{arg\,max}_{i\in I,\, j\in J\cup K} \biggl(\frac{(1-\lambda_{ij})r_j}{d}+\frac{\lambda_{ij}r_i}{d}\biggr),
\end{equation}
\tag{12}
$$
$$
\begin{equation}
(i_1, j_1) = \operatorname*{arg\,max}_{i\in I\cup J,\, j\in K} \biggl(\frac{(1-\widetilde\lambda_{ij})r_j}{d}+\frac{\widetilde\lambda_{ij}r_i}{d}\biggr).
\end{equation}
\tag{13}
$$
Let the functions $h_0,h_1,h_2\colon [1, q/2] \to \mathbb{R}\cup\{-\infty\}$ be defined by
$$
\begin{equation}
h_0(t) = \begin{cases} t\biggl((1-\lambda_{i_0j_0})\dfrac{r_{j_0}}{d} + \lambda_{i_0j_0} \dfrac{r_{i_0}}{d}\biggr) &\textit{if } I\ne \varnothing, \\ -\infty &\textit{if } I= \varnothing, \end{cases}
\end{equation}
\tag{14}
$$
$$
\begin{equation}
h_1(t) = \begin{cases} t\biggl((1-\widetilde\lambda_{i_1j_1})\dfrac{r_{j_1}}{d} + \widetilde\lambda_{i_1j_1} \dfrac{r_{i_1}}{d} -\dfrac 12\biggr)+\dfrac 12 &\textit{if } K\ne \varnothing, \\ -\infty &\textit{if } K= \varnothing, \end{cases}
\end{equation}
\tag{15}
$$
$$
\begin{equation}
h_2(t) = \begin{cases} {\displaystyle\max_{j\in J} \varphi_j(t)} &\textit{if } J\ne \varnothing, \\ -\infty &\textit{if } J= \varnothing, \end{cases}
\end{equation}
\tag{16}
$$
where We set Assume that the function $h$ has a unique minimum point $t_*$. Then In addition, we will show that if $r_1/d+1/q-1/p_1\leqslant 0$, then $d_n(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\gtrsim} 1$. In the end of § 3 we will explain how to find a point $t_*$ from assertion 5 of Theorem 1. In addition, there are two particular cases, in which an explicit expression for $h(t_*)$ can be given. Theorem 2. Under the conditions of Theorem 1, let $q > 2$, $\{1, \dots, s\} = I\cup K$, where the sets $I$ and $K$ are defined by (7), (9), let $I\ne \{1, \dots, s\}$, $K\ne \{1, \dots, s\}$, and let $ (i_0, j_0), $ $(i_1, j_1)$ be defined by (12) and (13), respectively. 1. If $(1-\widetilde \lambda_{i_1j_1})r_{j_1}/d + \widetilde \lambda_{i_1j_1} r_{i_1}/d - 1/2 >0$, then
$$
\begin{equation*}
d_n(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\asymp} n^{-(1-\widetilde \lambda_{i_1j_1})r_{j_1}/d-\widetilde \lambda_{i_1j_1}r_{i_1}/d}.
\end{equation*}
\notag
$$
2. If $(1-\widetilde \lambda_{i_1j_1})r_{j_1}/d + \widetilde \lambda_{i_1j_1} r_{i_1}/d - 1/2 <0$, then whereTheorem 3. Under the conditions of Theorem 1, let $q>2$, $\{1, \dots, s\} = J$ (see (8)), and let $r_1/d+1/q-1/p_1>0$. 1. Let
$$
\begin{equation*}
\frac{r_s}{d}> \frac 12 \cdot \frac{1/p_s-1/q}{1/2-1/q}.
\end{equation*}
\notag
$$
Then
$$
\begin{equation*}
d_n(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\asymp} n^{-r_s/d}.
\end{equation*}
\notag
$$
2. Let
$$
\begin{equation*}
\frac{r_1}{d} < \frac 12 \cdot \frac{1/p_1-1/q}{1/2-1/q}.
\end{equation*}
\notag
$$
Then
$$
\begin{equation*}
d_n(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\asymp} n^{-(q/2)(r_1/d+1/q-1/p_1)}.
\end{equation*}
\notag
$$
3. Let
$$
\begin{equation*}
\frac{r_s}{d}< \frac 12 \cdot \frac{1/p_s-1/q}{1/2-1/q},\qquad \frac{r_1}{d} > \frac 12 \cdot \frac{1/p_1-1/q}{1/2-1/q},
\end{equation*}
\notag
$$
let the functions $\varphi_j$ be defined by (17), and let $h(t) = \max_{1\leqslant j\leqslant s} \varphi_j(t)$. Suppose that the function $h$ has a unique minimum point $t_*$ on $[1, q/2]$. Then where and the indices $i_*\ne j_*$ are such that $\varphi_{i_*}(t_*) = \varphi_{j_*}(t_*)=h(t_*)$. The algorithm of searching the indices $i_*$ and $j_*$ from assertion 3 of Theorem 3 will be described after its proof. Theorems 1–3 will be verified in § 2. In § 3, we show how the general case (when condition (3) may fail) is reduced to consideration of the intersection of a subfamily of the Sobolev classes which satisfies (3). In § 4, order estimates for the Kolmogorov widths of an intersection of one-dimensional periodic Sobolev classes are obtained. We introduce the necessary notation. Let $\mathbb{T}=[0, 2\pi]$, and let $\mathcal{S}'(\mathbb{T})$ be the space of distributions on $\mathbb{T}$. Let $x= \sum_{k\in \mathbb{Z} \setminus \{0\}} x_k e^{ikt}$ (the series converges in the space $\mathcal{S}'(\mathbb{T})$), $r\in \mathbb{R}$. The Weyl derivative of order $r$ of the function $x$ is defined by
$$
\begin{equation*}
x^{(r)}(t) = \sum_{k\in \mathbb{Z} \setminus \{0\}} (ik)^r x_k e^{ikt}, \qquad (ik)^r = |k|^r\exp\biggl\{\frac{i\pi}{2}\, r\cdot \operatorname{sgn} k\biggr\}.
\end{equation*}
\notag
$$
The Sobolev space $\widetilde{\mathcal{W}}^r_p(\mathbb{T})$ is defined as follows:
$$
\begin{equation*}
\widetilde{\mathcal{W}}^r_p(\mathbb{T}) =\{x\in \mathcal{S}'(\mathbb{T})\colon x^{(r)}\in L_p(\mathbb{T})\};
\end{equation*}
\notag
$$
the Sobolev class $\widetilde W^r_p(\mathbb{T})$ is the set of functions $x\in \widetilde{\mathcal{W}}^r_p(\mathbb{T})$ such that $\|x^{(r)}\|_{L_p(\mathbb{T})}\leqslant 1$. Let $r_j\in \mathbb{R}$, $1<p_j<\infty$, $1\leqslant j\leqslant s$, and let $r_1<\dots<r_s$. We set
$$
\begin{equation*}
\widetilde M = \bigcap_{j=1}^s \widetilde W^{r_j}_{p_j}(\mathbb{T}).
\end{equation*}
\notag
$$
Let $1<q<\infty$. Galeev [13] obtained necessary and sufficient condition for set $\widetilde M$ to lie the space $L_q(\mathbb{T})$; in [2], the problem of estimating the widths $d_n(\widetilde M, L_q(\mathbb{T}))$ was studied. Order estimates were obtained, except for the case of “small smoothness” and $q>2$. Here, we obtain order estimates for all parameters, except some “limiting” cases. Let $\mathfrak{Z} = (r_1, \dots, r_s, p_1, \dots, p_s, q)$, and let the sets $I$, $J$, $K$, the numbers $\lambda_{ij}$, the indices $i_0$, $j_0$, and the function $h$ be as in Theorem 1. Theorem 4. Let $1<q<\infty$, $1<p_j<\infty$, $r_j\in \mathbb{R}$, $1\leqslant j\leqslant s$, $r_1<\dots<r_s$, $r_s-1/p_s<\dots< r_1-1/p_1$. Suppose that – if $p_1\leqslant q$, then $r_1+1/q-1/p_1>0$; – if $p_s\geqslant q$, then $r_s>0$; – if $p_1>q$, $p_s<q$, then $(1-\lambda_{i_0j_0})r_{j_0} + \lambda_{i_0j_0} r_{i_0}>0$. If $q>2$, $\{1,\dots,s\}=K$, we suppose that $r_1\ne 1/p_1$; if $q>2$, $\{1,\dots,s\}\ne K$, $\{1,\dots,s\}\ne I$, we suppose that the function $h$ has a unique minimum point on $[1, q/2]$. Then
$$
\begin{equation*}
d_n(\widetilde{M}, L_q(\mathbb{T})) \underset{\mathfrak{Z}}{\asymp} n^{-\beta},
\end{equation*}
\notag
$$
where $\beta$ is defined as in Theorem 1 for $d=1$. In the general case, when the condition $r_s-1/p_s<\dots< r_1-1/p_1$ may fail as in § 3, the problem can be reduced to the study of the intersection of a smaller number of Sobolev classes, for which this condition is satisfied. In § 5, we formulate a generalization of Theorem 1 to the weighted case. The proof of this result is a generalization of the arguments from § 2 of this paper and from [6]. § 2. Proofs of Theorems 1–3Let $N\in \mathbb{N}$. We denote by $l_p^N$ the space $\mathbb{R}^N$ with the norm
$$
\begin{equation*}
\|(x_1, \dots, x_N)\|_{l_p^N} = \begin{cases} \biggl({\displaystyle\sum_{j=1}^N |x_j|^p}\biggr)^{1/p} &\text{if } p<\infty, \\ {\displaystyle\max_{1\leqslant j\leqslant N}|x_j|} &\text{if }p=\infty. \end{cases}
\end{equation*}
\notag
$$
The widths $d_n(B_p^N, l_q^N)$ were estimated by Pietsch, Stesin, Kashin, Gluskin and Garnaev [14]–[19]. Let us formulate the results on width estimation for the cases that will be considered below. Theorem A (see [18]). Let $1\leqslant p\leqslant q<\infty$, $0\leqslant n\leqslant N/2$. 1. Let $1\leqslant q\leqslant 2$. Then $d_n(B_p^N, l_q^N) \asymp 1$. 2. Let $2<q<\infty$, ThenTheorem B (see [14], [15]). Let $1\leqslant q\leqslant p\leqslant \infty$, $0\leqslant n\leqslant N$. Then Let $A$ be a non-empty set, let $1\leqslant p_\alpha \leqslant \infty$, $\nu_\alpha>0$ for each $\alpha \in A$ and $p_\alpha \ne p_\beta$ for $\alpha \ne \beta$. We set
$$
\begin{equation}
M_0 = \bigcap_{\alpha \in A} \nu_\alpha B_{p_\alpha}^N.
\end{equation}
\tag{19}
$$
For $N=2n$, Galeev [1] estimated the widths $d_n(M_0, l_q^{2n})$. This result was generalized in [4] to all $N\geqslant 2n$. For $\alpha,\beta\in A$, we set
$$
\begin{equation}
\varkappa_{\alpha, \beta} = \begin{cases} \biggl(\dfrac{\nu_\beta}{\nu_\alpha}\biggr)^{p_\alpha p_\beta/(p_\alpha -p_\beta)} &\text{if } \alpha\ne \beta, \\ 1 &\text{if }\alpha=\beta. \end{cases}
\end{equation}
\tag{20}
$$
Theorem C (see [4]). Let $n, N\in \mathbb{N}$, $n\leqslant N/2$, let the set $M_0$ be defined by (19). Suppose that 1. Let $p_\alpha \geqslant q$ for all $\alpha \in A$. Then
$$
\begin{equation}
d_n(M_0, l_q^N) \asymp \inf_{\alpha \in A} \nu_\alpha N^{1/q-1/p_\alpha}.
\end{equation}
\tag{22}
$$
2. Let $q\leqslant 2$, $p_\alpha \leqslant q$ for all $\alpha \in A$. Then
$$
\begin{equation}
d_n(M_0, l_q^N) \asymp \inf_{\alpha \in A} \nu_\alpha.
\end{equation}
\tag{23}
$$
3. Let $2<q<\infty$, and let $p_\alpha\leqslant 2$ for all $\alpha \in A$. Then
$$
\begin{equation}
d_n(M_0, l_q^N) \underset{q}{\asymp} \inf_{\alpha \in A} \nu_\alpha \min \{1,\, n^{-1/2}N^{1/q}\}.
\end{equation}
\tag{24}
$$
4. Let $q\leqslant 2$, $A_1' = \{\alpha \in A\colon p_\alpha\geqslant q\}$, $A_2' = \{\alpha \in A\colon p_\alpha\leqslant q\}$, $A_1'\ne A$, $A_2'\ne A$. Then
$$
\begin{equation}
d_n(M_0, l_q^N) \asymp \inf \{\nu_\alpha \varkappa_{\alpha,\beta}^{1/q-1/p_\alpha}\colon \alpha \in A_1', \, \beta \in A_2' \}.
\end{equation}
\tag{25}
$$
5. Let $2<q<\infty$, $A_1' =\{\alpha \in A\colon p_\alpha\geqslant q\}$, $A_2' = \{\alpha \in A\colon 2\leqslant p_\alpha\leqslant q\}$, $A_3' = \{\alpha \in A\colon p_\alpha \leqslant 2\}$. In addition, we suppose that $A \ne A_1'$, $A \ne A_3'$. Let $K_0=\{(\nu_\alpha, 1/p_\alpha)\}_{\alpha \in A}$. Then
$$
\begin{equation}
\begin{aligned} \, &d_n(M_0, l_q^N) \underset{q}{\asymp} \Phi(n, N, q, K_0) \nonumber \\ &\qquad:= \min \{\Phi_1(n, N, q, K_0),\, \Phi_2(n, N, q, K_0),\, \Phi_3(n, N, q, K_0)\}, \end{aligned}
\end{equation}
\tag{26}
$$
where
$$
\begin{equation}
\Phi_1(n, N, q, K_0) =\inf \bigl\{\nu_\alpha \varkappa_{\alpha,\beta}^{1/q-1/p_\alpha}\colon \alpha \in A_1', \, \beta \in A_2'\cup A_3'\bigr\},
\end{equation}
\tag{27}
$$
(the infimum of the empty set is by definition $+\infty$). Given $\alpha,\beta\in A$, we define the numbers $\lambda_{\alpha,\beta}$ and $\widetilde \lambda_{\alpha,\beta}$ by the equations
$$
\begin{equation*}
\frac 1q = \frac{1-\lambda_{\alpha,\beta}}{p_\beta} + \frac{\lambda_{\alpha,\beta}}{p_\alpha}, \qquad \frac 12 = \frac{1-\widetilde\lambda_{\alpha,\beta}}{p_\beta} + \frac{\widetilde\lambda_{\alpha,\beta}}{p_\alpha}.
\end{equation*}
\notag
$$
Then
$$
\begin{equation}
\nu_\alpha \varkappa_{\alpha,\beta}^{1/q-1/p_\alpha} \stackrel{(20)}{=} \nu_\alpha^{\lambda_{\alpha,\beta}} \nu_\beta^{1-\lambda_{\alpha,\beta}}, \qquad \nu_\alpha\varkappa_{\alpha,\beta}^{1/2-1/p_\alpha} \stackrel{(20)}{=} \nu_\alpha^{\widetilde\lambda_{\alpha,\beta}} \nu_\beta^{1-\widetilde\lambda_{\alpha,\beta}}.
\end{equation}
\tag{30}
$$
In the cases when (21) may fail and the set $A$ is finite, the orders of the widths of the set $M_0$ can also be calculated. Proposition 1. Let $A$ be a finite set, let $M_0$ be defined by (19), and let $N\geqslant 2n$. Then, for $q\leqslant 2$, Proof. The upper estimate follows from the inclusions $M_0\,{\subset}\, \nu_\alpha B^N_{p_\alpha}$ ($\alpha \in A$), $M_0\subset \nu_\alpha B^N_{p_\alpha}\cap \nu_\beta B_{p_\beta}^N \subset \nu_\alpha^{\lambda_{\alpha,\beta}} \nu_\beta^{1-\lambda_{\alpha,\beta}}B_q^N$ ($p_\alpha\geqslant q$, $p_\beta\leqslant q$), $M_0\subset \nu_\alpha B^N_{p_\alpha}\cap \nu_\beta B_{p_\beta}^N \subset \nu_\alpha^{\widetilde\lambda_{\alpha,\beta}} \nu_\beta^{1-\widetilde\lambda_{\alpha,\beta}}B_2^N$ ($p_\alpha\geqslant 2$, $p_\beta\leqslant 2$); the inclusions into the balls of the spaces $l_q^N$ and $l_2^N$ follow from Hölder’s inequality or can be considered as a particular case of Theorem 2 in [1].
Let us prove the lower estimate. Since the set $A$ is finite, there is $A' \subset A$ such that $M_0 = \bigcap_{\alpha\in A'} \nu_\alpha B_{p_\alpha}^N$ and $1\leqslant \varkappa_{\alpha,\beta}\leqslant N$ for $\alpha$, $\beta\in A'$. Indeed, if $\varkappa_{\alpha,\beta}<1$ or $\varkappa_{\alpha,\beta}>N$ for some $\alpha$, $\beta \in A$, then $\nu_\alpha B_{p_\alpha}^N \subset \nu_\beta B_{p_\beta}^N$ or $\nu_\alpha B_{p_\alpha}^N \supset \nu_\beta B_{p_\beta}^N$; the greater ball can be excluded from the family. The required set $A'$ is obtained by excluding a finite number of “unnecessary” balls. Now we apply Theorem C and (30) for $\bigcap_{\alpha\in A'} \nu_\alpha B_{p_\alpha}^N$ and note that the right-hand sides in the estimates have the same orders as $\nu_\alpha d_n(B_{p_\alpha}^N, l_q^N)$, $\nu_\alpha^{\lambda_{\alpha,\beta}}\nu_\beta^{1-\lambda_{\alpha,\beta}}$ and $\nu_\alpha^{\widetilde\lambda_{\alpha,\beta}} \nu_\beta^{1-\widetilde\lambda_{\alpha,\beta}} d_n(B_2^N, l_q^N)$ (see Theorems A and B). Hence, for $q\leqslant 2$,
$$
\begin{equation*}
\begin{aligned} \, d_n(M_0, l_q^N) &=d_n\biggl(\bigcap_{\alpha\in A'} \nu_\alpha B_{p_\alpha}^N, l_q^N\biggr) \\ &\gtrsim \min \Bigl\{ \min_{\alpha \in A'}d_n(\nu_\alpha B_{p_\alpha}^N, l_q^N), \min_{p_\alpha\geqslant q,\, p_\beta\leqslant q,\, \alpha,\beta\in A'} \nu_\alpha^{\lambda_{\alpha,\beta}}\nu_\beta^{1-\lambda_{\alpha,\beta}}\Bigr\} \\ &\geqslant \min \Bigl\{ \min_{\alpha \in A}d_n(\nu_\alpha B_{p_\alpha}^N, l_q^N), \min_{p_\alpha\geqslant q,\, p_\beta\leqslant q,\, \alpha,\beta\in A} \nu_\alpha^{\lambda_{\alpha,\beta}}\nu_\beta^{1-\lambda_{\alpha,\beta}}\Bigr\}. \end{aligned}
\end{equation*}
\notag
$$
The estimate for $q>2$ can be obtained similarly. We also need the following estimate for the widths of the Sobolev class on a John domain (see [20]). Theorem D (see [20]). Let $\Omega \subset \mathbb{R}^d$, $\Omega\in \mathbf{FC}(a)$, $a>0$, $R=\operatorname{diam} \Omega$, $r\in \mathbb{N}$, $1\leqslant p\leqslant \infty$, $1\leqslant q \leqslant \infty$, $r/d+1/q-1/p>0$. Then Let $r\in \mathbb{N}$, $\Omega \subset \mathbb{R}^d$, $\Omega \in \mathbf{FC}(a)$. We denote by $\mathcal{P}_{r-1}(\Omega)$ the space of algebraic polynomials on $\Omega$ of degree at most $r-1$. Let $E_1,E_2\subset \Omega$ be measurable subsets. We say that $E_1$ and $E_2$ do not overlap if $\operatorname{mes}(E_1 \cap E_2) = 0$ (here and in what follows, $\operatorname{mes}$ is the standard Lebesgue measure on $\mathbb{R}^d$). Let $G_1, \dots, G_k$ be pairwise non-overlapping measurable subsets of $\Omega$, and let $T=\{G_j\}_{j=1}^k$. By $\mathcal{S}_T(\Omega)$ we denote the set of functions of the form $\sum_{j=1}^k P_j\cdot \chi_{G_j}$, where $P_j\in \mathcal{P}_{r-1}(\Omega)$. Given $1\leqslant p, q\leqslant \infty$, $f\in L_q(\Omega)$, we write
$$
\begin{equation*}
\|f\|_{p,q,T} = \biggl(\sum_{j=1}^k \|f\|^p_{L_q(G_j)}\biggr)^{1/p}.
\end{equation*}
\notag
$$
Lemma 1. Let $G_1, \dots, G_k$ be pairwise non-overlapping measurable subsets of $\Omega$, and let $T=\{G_j\}_{j=1}^k$, $\nu = \dim \mathcal{S}_T(\Omega)$. Then there is an isomorphism $A\colon \mathcal{S}_T(\Omega) \to \mathbb{R}^\nu$ such that, for all $1\leqslant p, q\leqslant \infty$, The proof is similar to that of Proposition 2 in [5]. Lemma 2. Let the set $M$ be given by (2). Then there exist $c=c(\mathfrak{Z})\geqslant 1$ and a sequence of partitions $\{T_m\}_{m\in \mathbb{Z}_+}$ of the domain $\Omega$ with the following properties: 1) the inequality holds;2) for each $E\in T_m$
$$
\begin{equation}
\operatorname{card} \{E'\in T_{m\pm 1}\colon \operatorname{mes}(E\cap E')>0\} \leqslant c;
\end{equation}
\tag{32}
$$
3) for each $E\in T_m$ there is a projection $P_E\colon L_q(\Omega) \to \mathcal{S}_{\{E\}}(\Omega)$ such that The partition $T_m$ was constructed in [21], in which properties 1) and 2) were also proved. Properties 3) and 4) were established in § 3 of [6] (where a more general weighted case is considered). Lemma 3 (see [9], Ch. VII, § 2). For each $m\in \mathbb{Z}_+$ there exist functions $\psi_{m,j}\in M$ $(1\leqslant j\leqslant 2^m)$ with pairwise non-overlapping supports such that where $c=c(\mathfrak{Z})\geqslant 1$. Let $r_1/d+1/q-1/p_1>0$ (see the conditions of Theorem 1). Then $W^{r_1}_{p_1}(\Omega)\subset L_q(\Omega)$ by the embedding theorem [11], [12] (if $r_1\in \mathbb{N}$) or by Hölder’s inequality (if $r_1=0$). Hence $M\subset L_q(\Omega)$. For $m\in \mathbb{Z}_+$ we define the operator $P_m\colon L_q(\Omega) \to \mathcal{S}_{T_m}(\Omega)$ by the formula
$$
\begin{equation}
P_mf = \sum_{E\in T_m} P_E(f\cdot \chi_E), \qquad f\in L_q(\Omega).
\end{equation}
\tag{36}
$$
By (33), for each function $f\in L_q(\Omega)$ (in particular, for each function $f\in M$) and for each $m_*\in \mathbb{Z}_+$, we have (the series converges in $L_q(\Omega)$).Let From (31), (32) and (34) we have $\operatorname{card} T'_m\underset{\mathfrak{Z}}{\lesssim}2^m$, and, for each function $f\in M$,
$$
\begin{equation}
\|P_{m+1}f-P_mf\|_{p_j,q,T_m'} \underset{\mathfrak{Z}}{\lesssim} 2^{-m(r_j/d+1/q-1/p_j)}, \qquad 1\leqslant j\leqslant s
\end{equation}
\tag{38}
$$
(the last estimate was proved in details in [6], formula (26)). In addition, from (31), (32) and (36) we find that
$$
\begin{equation}
\operatorname{rk} P_m \underset{\mathfrak{Z}}{\lesssim} 2^m, \qquad \nu_m:=\operatorname{rk} (P_{m+1}-P_m) \underset{\mathfrak{Z}}{\lesssim} 2^m
\end{equation}
\tag{39}
$$
(here $\operatorname{rk}$ is the range of an operator). From Lemma 1, (38) and (39) it follows that, for all $k\in \mathbb{Z}_+$, $m\in \mathbb{Z}_+$
$$
\begin{equation*}
d_k\bigl((P_{m+1}-P_m)M, L_q(\Omega)\bigr) \underset{\mathfrak{Z}}{\lesssim} d_k\biggl(\bigcap_{j=1}^s 2^{-m(r_j/d+1/q-1/p_j)} B^{\nu_m}_{p_j}, l_q^{\nu_m}\biggr).
\end{equation*}
\notag
$$
Using this together with (37), (39) we get the following result. Lemma 4. Let $n\in \mathbb{N}$, $k_m\in \mathbb{Z}_+$, $\sum_{2^m\geqslant 2n} k_m \leqslant Cn$, where $C\in \mathbb{N}$. Then there is a number $C_1=C_1(\mathfrak{Z})\in \mathbb{N}$ such that Proof. Let the functions $\psi_{m,j}$ be as in Lemma 3. We set $L=\operatorname{span} \{\psi_{m,j}\}_{j=1}^{2^m}$. Since the functions $\psi_{m,j}$ have non-overlapping supports, there is a linear projection $P\colon L_q(\Omega) \to L$ such that $\|P\|=1$. Let
$$
\begin{equation*}
W = \biggl\{ \sum_{j=1}^{2^m} c_j\psi_{m,j}\colon (c_j)_{j=1}^{2^m} \in \bigcap_{j=1}^s 2^{-m(r_j/d+1/q-1/p_j)}B_{p_j}^{2^m}\biggr\}.
\end{equation*}
\notag
$$
Then
$$
\begin{equation*}
\begin{aligned} \, d_n(M, L_q(\Omega)) &\stackrel{(35)}{\underset{\mathfrak{Z}}{\gtrsim}} d_n(W, L_q(\Omega)) \geqslant d_n(PW, L) \\ &\,=d_n(W, L) \stackrel{(35)}{=} d_n \biggl(\bigcap_{j=1}^s 2^{-m(r_j/d+1/q-1/p_j)}B_{p_j}^{2^m}, l_q^{2^m}\biggr), \end{aligned}
\end{equation*}
\notag
$$
proving the lemma. Remark 1. If $r_1/d+1/q-1/p_1\leqslant 0$, then $d_n(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\gtrsim} 1$. Indeed, applying Lemma 5, we obtain, for all $m\in \mathbb{Z}_+$,
$$
\begin{equation*}
\begin{aligned} \, d_n(M, L_q(\Omega)) &\underset{\mathfrak{Z}}{\gtrsim} d_n\biggl(\bigcap_{j=1}^s 2^{-m(r_j/d+1/q-1/p_j)}B_{p_j}^{2^m}, l_q^{2^m}\biggr) \\ &\geqslant \min_{1\leqslant j\leqslant s} 2^{-m(r_j/d+1/q-1/p_j)} d_n(B_1^{2^m}, l_q^{2^m}) \\ &\!\stackrel{(3)}{=} 2^{-m(r_1/d+1/q-1/p_1)} d_n(B_1^{2^m}, l_q^{2^m})\geqslant d_n(B_1^{2^m}, l_q^{2^m}). \end{aligned}
\end{equation*}
\notag
$$
If $2^m\geqslant \max\{ 2n, n^{q/2}\}$, then $d_n(B_1^{2^m}, l_q^{2^m}) \underset{q}{\gtrsim} 1$ (see Theorem A). Proof of Theorem 1. In what follows, we denote $\log x := \log_2 x$.
Let $j>i$. By (4), we have $p_j<p_i$. Further,
$$
\begin{equation*}
1\stackrel{(3)}{<}\frac{2^{-m(r_j/d+1/q-1/p_j)}}{2^{-m(r_i/d+1/q-1/p_i)}} \stackrel{(1)}{<}2^{m(1/p_j-1/p_i)}.
\end{equation*}
\notag
$$
Hence (21) holds, and for estimating the widths of the intersection of finite-dimensional balls from Lemmas 4 and 5, we can apply Theorem C.
In cases 1–3, in order to estimate from above $d_n(M, L_q(\Omega))$, we use Theorem D and the inclusions $M\subset W^{r_s}_{p_s}(\Omega)$ and $M\subset W^{r_1}_{p_1}(\Omega)$. In estimating from below, from Lemma 5 for $m =\lceil \log(2n)\rceil$ we get
$$
\begin{equation}
d_n(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\gtrsim} d_n \biggl(\bigcap_{j=1}^s n^{-r_j/d-1/q+1/p_j} B_{p_j}^{2n}, l_q^{2n}\biggr).
\end{equation}
\tag{41}
$$
In case 1, by (22), the right-hand side of (41) is equal, up to the order, to
$$
\begin{equation*}
\min_{1\leqslant j\leqslant s} \bigl(n^{-r_j/d-1/q+1/p_j}\cdot n^{1/q-1/p_j}\bigr) \stackrel{(1)}{=} n^{-r_s}.
\end{equation*}
\notag
$$
In cases 2 and 3, by (23) and (24), the right-hand side of (41) has the order
$$
\begin{equation*}
\min_{1\leqslant j\leqslant s} \bigl(n^{-r_j/d-1/q+1/p_j} \cdot n^{-(1/2-1/q)_+}\bigr) \stackrel{(3)}{=} n^{-r_1/d-\max\{1/q, 1/2\}+1/p_1}.
\end{equation*}
\notag
$$
In addition, in case 3, applying Lemma 5 for $m = \lceil (q/2) \log n \rceil$, we get
$$
\begin{equation*}
\begin{aligned} \, d_n(M, L_q(\Omega)) &\underset{\mathfrak{Z}}{\gtrsim} d_n\biggl(\bigcap_{j=1}^s n^{-(q/2) (r_j/d-1/q+1/p_j)} B_{p_j}^{\lceil n^{q/2}\rceil}, l_q^{\lceil n^{q/2}\rceil}\biggr) \\ &\!\!\!\!\!\!\stackrel{(3), \ (24)}{\underset{\mathfrak{Z}}{\gtrsim}} n^{-(q/2) (r_1/d-1/q+1/p_1)}. \end{aligned}
\end{equation*}
\notag
$$
Now we consider case 4. In estimating from above, we will use Lemma 4 with $k_m=0$, and apply (25) and (30). Notice that
$$
\begin{equation*}
2^{-m((1-\lambda_{ij})(r_j/d+1/q-1/p_j)+\lambda_{ij}(r_i/d+1/q-1/p_i))} \stackrel{(5)}{=} 2^{-m((1-\lambda_{ij})r_j/d+\lambda_{ij}r_i/d)}.
\end{equation*}
\notag
$$
We claim that
$$
\begin{equation}
(1-\lambda_{i_0j_0})\frac{r_{j_0}}{d}+\lambda_{i_0j_0}\frac{r_{i_0}}{d}>0.
\end{equation}
\tag{42}
$$
Indeed, $r_i>0$ for $2\leqslant i\leqslant s$, $r_1\geqslant 0$. By the conditions of assertion 4 of Theorem 1, there exist $i$, $j$ such that $\lambda_{ij}\in (0, 1)$. This together with the definition of $(i_0, j_0)$ implies (42).
We get
$$
\begin{equation*}
\begin{aligned} \, d_{C_1n}(M, L_q(\Omega)) &\stackrel{(6), (40)}{\underset{\mathfrak{Z}}{\lesssim}} \sum_{2^m \geqslant 2n} 2^{-m((1-\lambda_{i_0j_0})r_{j_0}/d+\lambda_{i_0j_0}r_{i_0}/d)} \\ &\,\,\,\,\stackrel{(42)}{\underset{\mathfrak{Z}}{\asymp}} n^{-(1-\lambda_{i_0j_0})r_{j_0}/d-\lambda_{i_0j_0}r_{i_0}/d}. \end{aligned}
\end{equation*}
\notag
$$
This implies that
$$
\begin{equation*}
d_n(M, L_q(\Omega))\underset{\mathfrak{Z}}{\lesssim} n^{-(1-\lambda_{i_0j_0})r_{j_0}/d-\lambda_{i_0j_0}r_{i_0}/d}.
\end{equation*}
\notag
$$
The lower estimate is obtained similarly via Lemma 5. Consider case 5. Let $\varepsilon >0$, $m_*(n) \in [\log n, (q/2)\log n]$ (these numbers will be defined later from $\mathfrak{Z}$). Setting $k_m = \lfloor n\cdot 2^{-\varepsilon|m-m_*(n)|}\rfloor$ for $\log (2n)\leqslant m\leqslant (q/2) \log n$, $k_m =0$ for $m> (q/2) \log n$, we have $\sum_{m\geqslant \log (2n)} k_m \leqslant Cn$, where $C=C(\mathfrak{Z}, \varepsilon)$, and hence Lemma 4 applies. Let us use (26)–(30) to estimate from above the summands in the right-hand side of (40). First, for the order estimates for the widths with $k_m=n$ for each $m\geqslant \log(2n)$, we have
$$
\begin{equation*}
d_n\biggl(\bigcap_{j=1}^s 2^{-m(r_j/d+1/q-1/p_j)} B_{p_j}^{2^m}, l_q^{2^m}\biggr) \underset{\mathfrak{Z}}{\asymp} 2^{-h_*(m, n)},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
h_*(m, n) = \max\{h_{0, n}(m), h_{1,n}(m), h_{2,n}(m)\},
\end{equation*}
\notag
$$
and the functions $h_{0, n},h_{1,n},h_{2,n}\colon [\log n, +\infty) \to \mathbb{R}\cup \{-\infty\}$ have the following form. If $I= \varnothing$, then $h_{0,n}\equiv -\infty$; if $I\ne \varnothing$, then
$$
\begin{equation}
h_{0,n}(m) = m\biggl((1-\lambda_{i_0j_0})\frac{r_{j_0}}{d} + \lambda_{i_0j_0} \frac{r_{i_0}}{d}\biggr),
\end{equation}
\tag{43}
$$
where $(i_0, j_0)$ is defined by (12). If $K= \varnothing$, then $h_{1,n} \equiv -\infty$; if $K \ne \varnothing$, then
$$
\begin{equation}
h_{1,n}(m) = \begin{cases} m\biggl((1-\widetilde\lambda_{i_1j_1})\dfrac{r_{j_1}}{d} + \widetilde\lambda_{i_1j_1} \dfrac{r_{i_1}}{d} -\dfrac 12\biggr)+\dfrac 12 \cdot \log n &\text{for } m\leqslant \dfrac q2\log n, \\ m\biggl((1-\widetilde \lambda_{i_1j_1})\dfrac{r_{j_1}}{d} + \widetilde \lambda_{i_1j_1} \dfrac{r_{i_1}}{d} -\dfrac 12 + \dfrac 1q\biggr) &\text{for } m> \dfrac q2\log n, \end{cases}
\end{equation}
\tag{44}
$$
where $(i_1, j_1)$ is defined by (13); if $J=\varnothing$, then $h_{2,n}\equiv -\infty$; if $J\ne \varnothing$, then where
$$
\begin{equation}
\varphi_{j,n}(m) = m\biggl(\frac{r_j}{d} -\frac 12 \cdot \frac{1/p_j-1/q}{1/2-1/q}\biggr) +\frac 12 \cdot \frac{1/p_j-1/q}{1/2-1/q}\cdot \log n
\end{equation}
\tag{46}
$$
for $m\leqslant (q/2) \log n$,
$$
\begin{equation}
\varphi_{j,n}(m) = m\biggl(\frac{r_j}{d} + \frac 1q -\frac{1}{p_j}\biggr)
\end{equation}
\tag{47}
$$
for $m> (q/2) \log n$.
So $h_*(\,{\cdot}\,, n)$ is a continuous convex piecewise-linear function on $[\log n, +\infty)$, and
$$
\begin{equation}
2^{-h_*(m, n)} \underset{\mathfrak{Z}}{\asymp} 2^{-h_*(\log n, n)} \quad \text{for}\quad \log n\leqslant m\leqslant \log (2n).
\end{equation}
\tag{48}
$$
If $h_*(\,{\cdot}\,, n)$ has a unique minimum point on $[\log n, +\infty)$ (we denote it by $m_n$), then
$$
\begin{equation*}
\sum_{m\geqslant \log (2n)} 2^{-h_*(m, n)} \underset{\mathfrak{Z}}{\asymp} 2^{-h_*(m_n, n)}
\end{equation*}
\notag
$$
(for $m\leqslant m_n$, strictly increasing geometric progressions are summed, and for $m\geqslant m_n$, strictly decreasing geometric progressions are summed).
Now we set $m_*(n) = m_n$. From (26)–(29) and the definition of $k_m$ we have
$$
\begin{equation*}
d_{k_m}\biggl(\bigcap_{j=1}^s 2^{-m(r_j/d+1/q-1/p_j)} B_{p_j}^{2^m}, l_q^{2^m}\biggr) \underset{\mathfrak{Z}}{\lesssim} 2^{-h_*(m, n)}\cdot 2^{\varepsilon c_1|m-m_n|},
\end{equation*}
\notag
$$
where $c_1=c_1(\mathfrak{Z})$. Hence, if $\varepsilon >0$ is sufficiently small, then
$$
\begin{equation}
\sum_{m\geqslant \log(2n)} d_{k_m}\biggl(\bigcap_{j=1}^s 2^{-m(r_j/d+1/q-1/p_j)} B_{p_j}^{2^m}, l_q^{2^m}\biggr) \underset{\mathfrak{Z}}{\lesssim} 2^{-h_*(m_n, n)}.
\end{equation}
\tag{49}
$$
If $2^{-h_*(m_n, n)} \underset{\mathfrak{Z}}{\asymp} n^{-\beta_*}$, where $\beta_*=\beta_*(\mathfrak{Z})$, then from (40) and (49) we get the estimate
$$
\begin{equation*}
d_{C_1Cn}(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\lesssim} n^{-\beta_*},
\end{equation*}
\notag
$$
which yields
$$
\begin{equation*}
d_n(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\lesssim} n^{-\beta_*}.
\end{equation*}
\notag
$$
Applying Lemma 5 and using (48), we get, for $\widetilde m_n=\max\{ \lceil m_n\rceil,\lceil\log(2n)\rceil\}$,
$$
\begin{equation*}
d_n(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\gtrsim} d_n\biggl(\bigcap_{j=1}^s 2^{-\widetilde m_n(r_j/d+1/q-1/p_j)} B_{p_j}^{2^{\widetilde m_n}}, l_q^{2^{\widetilde m_n}}\biggr) \underset{\mathfrak{Z}}{\asymp} 2^{-h_*(m_n, n)} \underset{\mathfrak{Z}}{\asymp} n^{-\beta_*}.
\end{equation*}
\notag
$$
Hence it remains to prove that the function $h_*(\,{\cdot}\,, n)$ has a unique minimum point on $[\log n, +\infty)$, and to find the number $\beta_*$. First, let $m\geqslant (q/2) \log n$. From (43)–(45), (47) we have $h_*(\,{\cdot}\,, m) = \alpha m$. We claim that $\alpha >0$. Indeed, if $p_1>q$, then $\alpha \geqslant (1-\lambda_{i_0j_0})r_{j_0}/d + \lambda_{i_0j_0} r_{i_0}/d>0$ (the positivity can be proved as in case 4). If $p_1\leqslant q$, then, by (4), we have $1\in J$, hence $\alpha \geqslant r_1/d+1/q-1/p_1>0$ by conditions of the theorem. Therefore,
$$
\begin{equation*}
\inf_{m\geqslant \log n} h_*(m, n) = \min_{\log n\leqslant m \leqslant (q/2)\log n} h_*(m, n).
\end{equation*}
\notag
$$
By (43)–(46), we have $h_*(m, n) = \max_{1\leqslant k\leqslant l} (\alpha_k m + \beta_k \log n)$ for $\log n \leqslant m \leqslant q/2 \log n$, where $\alpha_k$, $\beta_k$, $l$ depend only on $\mathfrak{Z}$. Substituting $t = m/\log n$, we get
$$
\begin{equation*}
\min_{\log n \leqslant m \leqslant (q/2) \log n} \, \max_{1\leqslant k\leqslant l} (\alpha_k m + \beta_k \log n) = \log n \cdot \min_{1 \leqslant t \leqslant q/2}\, \max_{1\leqslant k\leqslant l} (\alpha_k t + \beta_k).
\end{equation*}
\notag
$$
Recall that the function $h$ from the statement of the theorem has a unique minimum point by the assumption. Comparing the form of the functions $h$ and $h_*(\,{\cdot}\,, n)$, we get that a minimum point for $h_*(\,{\cdot}\,, n)$ is unique and
$$
\begin{equation*}
2^{-h_*(m_n, n)} = n^{-\min_{1 \leqslant t \leqslant q/2}\, \max_{1\leqslant k\leqslant l} (\alpha_k t + \beta_k)} = n^{-\min_{1 \leqslant t \leqslant q/2} h(t)}.
\end{equation*}
\notag
$$
Therefore, $\beta_*=h(t_*)$. This proves Theorem 1. Proof of Theorem 2. We have (see (14)–(16))
Let us find $\min_{1\leqslant t\leqslant q/2} h(t)$. We show that if $t$ is close to $1$, then
$$
\begin{equation*}
h(t) = t\biggl((1-\widetilde\lambda_{i_1j_1})\frac{r_{j_1}}{d} + \widetilde\lambda_{i_1j_1} \frac{r_{i_1}}{d}-\frac 12\biggr)+\frac 12.
\end{equation*}
\notag
$$
To this end, we check that
$$
\begin{equation*}
(1-\lambda_{i_0j_0})\frac{r_{j_0}}{d} + \lambda_{i_0j_0} \frac{r_{i_0}}{d} < (1-\widetilde\lambda_{i_1j_1})\frac{r_{j_1}}{d} + \widetilde\lambda_{i_1j_1} \frac{r_{i_1}}{d}.
\end{equation*}
\notag
$$
It suffices to show that, for all $i\in I$, $k\in K$
$$
\begin{equation*}
(1-\lambda_{ik})\frac{r_{k}}{d} + \lambda_{ik} \frac{r_i}{d} < (1-\widetilde\lambda_{ik})\frac{r_{k}}{d} + \widetilde\lambda_{ik} \frac{r_i}{d}.
\end{equation*}
\notag
$$
This is equivalent to saying that Since $i\in I$, $k\in K$, we have $p_i\geqslant q>2 \geqslant p_k$ (see (7), (9)), hence $k > i$, $r_k > r_i$ by (1) and (4). Therefore, it is sufficient to show that $\widetilde \lambda_{ik}-\lambda_{ik} <0$. We have
$$
\begin{equation}
0<\frac 12 - \frac 1q \stackrel{(10), (11)}{=}(\widetilde \lambda_{ik} -\lambda_{ik}) \biggl(\frac{1}{p_i} - \frac{1}{p_k}\biggr);
\end{equation}
\tag{50}
$$
and now it remains to apply the inequality $p_k<p_i$.
If $(1-\widetilde \lambda_{i_1j_1})r_{j_1}/d + \widetilde \lambda_{i_1j_1} r_{i_1}/d - 1/2 >0$, then
$$
\begin{equation*}
\min_{1\leqslant t\leqslant q/2}h(t) = h(1) = (1-\widetilde \lambda_{i_1j_1})\frac{r_{j_1}}{d} + \widetilde \lambda_{i_1j_1} \frac{r_{i_1}}{d}.
\end{equation*}
\notag
$$
Now we consider the case $(1-\widetilde \lambda_{i_1j_1})r_{j_1}/d + \widetilde \lambda_{i_1j_1} r_{i_1}/d - 1/2 <0$. We prove that if $t$ is close to $q/2$, then $h(t) = t((1-\lambda_{i_0j_0})r_{j_0}/d + \lambda_{i_0j_0} r_{i_0}/d)$. To this end, we check that
$$
\begin{equation}
\biggl((1-\lambda_{i_0j_0})\frac{r_{j_0}}{d} + \lambda_{i_0j_0}\frac{r_{i_0}}{d}\biggr)\frac{q}{2} > \biggl((1-\widetilde\lambda_{i_1j_1})\frac{r_{j_1}}{d} + \widetilde \lambda_{i_1j_1}\frac{r_{i_1}}{d}\biggr)\frac{q}{2} -\frac q4 + \frac 12.
\end{equation}
\tag{51}
$$
It suffices to show that, for all $i\in I$, $k\in K$,
$$
\begin{equation}
\biggl((1-\lambda_{ik})\frac{r_{k}}{d} + \lambda_{ik}\frac{r_i}{d}\biggr)\frac{q}{2} > \biggl((1-\widetilde\lambda_{ik})\frac{r_{k}}{d} + \widetilde \lambda_{ik}\frac{r_i}{d}\biggr)\frac{q}{2} -\frac q4 + \frac 12.
\end{equation}
\tag{52}
$$
This is equivalent to saying that
$$
\begin{equation*}
(\widetilde \lambda_{ik} -\lambda_{ik}) \biggl(\frac{r_k}{d} - \frac{r_i}{d}\biggr) > -\frac 12 +\frac 1q,
\end{equation*}
\notag
$$
or, what is the same (see (50)),
$$
\begin{equation*}
\frac{r_k}{d}-\frac{r_i}{d}<\frac{1}{p_k}-\frac{1}{p_i};
\end{equation*}
\notag
$$
this inequality holds by (3), (4). This completes the proof of (51).
Since $((1-\lambda_{i_0j_0})r_{j_0}/d + \lambda_{i_0j_0}r_{i_0}/d)>0$, the minimum point of the function $h$ belongs to the interval $(1, q/2)$ and satisfies the equation
$$
\begin{equation*}
t_*\biggl((1-\lambda_{i_0j_0})\frac{r_{j_0}}{d} + \lambda_{i_0j_0}\frac{r_{i_0}}{d}\biggr) = t_*\biggl((1-\widetilde\lambda_{i_1j_1})\frac{r_{j_1}}{d} + \widetilde\lambda_{i_1j_1} \frac{r_{i_1}}{d} -\frac 12\biggr)+\frac 12.
\end{equation*}
\notag
$$
Expressing $t_*$ from this equation and substituting it to $t((1-\lambda_{i_0j_0})r_{j_0}/d + \lambda_{i_0j_0}r_{i_0}/d)$, we get
$$
\begin{equation*}
h(t_*) = \frac 12 \cdot\frac{(1-\lambda_{i_0j_0}) r_{j_0}/d + \lambda_{i_0j_0} r_{i_0}/d}{1/2 -(1-\widetilde \lambda_{i_1j_1}) r_{j_1}/d - \widetilde\lambda_{i_1j_1} r_{i_1}/d + (1-\lambda_{i_0j_0}) r_{j_0}/d + \lambda_{i_0j_0} r_{i_0}/d}.
\end{equation*}
\notag
$$
This proves Theorem 2. Proof of Theorem 3. We have $h(t) = \max_{1\leqslant j\leqslant s}\varphi_j(t)$, Hence and the maximum is attained only at $s$. So, if $t$ is close to $1$, then $h(t) = \varphi_s(t)$.
Further,
$$
\begin{equation*}
h\biggl(\frac{q}2\biggr) = \max_{1\leqslant j\leqslant s} \frac q2\biggl(\frac{r_j}{d}+ \frac 1q-\frac{1}{p_j}\biggr) \stackrel{(3)}{=}\frac q2\biggl(\frac{r_1}{d} + \frac 1q-\frac{1}{p_1}\biggr),
\end{equation*}
\notag
$$
and the maximum is attained only at $1$. Hence, if $t$ is close to $q/2$, then $h(t) = \varphi_1(t)$.
We have the following cases to consider. 1. If $r_s/d > (1/2) (1/p_s-1/q)/(1/2-1/q)$, then $\min_{1\leqslant t\leqslant q/2} h(t) = h(1) = r_s/d$. 2. If $r_1/d < (1/2)(1/p_1-1/q)/(1/2-1/q)$, then $\min_{1\leqslant t\leqslant q/2} h(t) = h(q/2) = (q/2)(r_1/d+ 1/q -1/p_1)$. 3. Let $r_s/d < (1/2)(1/p_s-1/q)/(1/2-1/q)$, $r_1/d > (1/2)(1/p_1-1/q)/(1/2-1/q)$. We denote by $t_*$ the minimum point of the function $h$. We have $1<t_*<q/2$, and further, since the minimum point of the function $h$ is unique, there exist $i_*,j_*\in \{1, \dots, s\}$, $i_*\ne j_*$, such that $\varphi_{i_*}(t_*) = \varphi_{j_*}(t_*)=h(t_*)$. Let us find $\varphi_{i_*}(t_*)$. Let $\varphi_i(t)=\varphi_j(t)$. Then
$$
\begin{equation*}
\biggl(\frac{r_j}{d}-\frac{r_i}{d} +\frac 12\cdot \frac{1/p_i-1/p_j}{1/2-1/q}\biggr) t = \frac 12 \cdot \frac{1/p_i-1/p_j}{1/2-1/q}.
\end{equation*}
\notag
$$
Hence $\varphi_i(t) =\theta_{ij}$ (see (18)). Therefore $h(t_*) = \theta_{i_*j_*}$. This proves Theorem 3. Construction of the function $h$ from Theorem 3Recall that $\varphi_s(1) = r_s/d>r_j/d=\varphi_j(1)$, $1\leqslant j\leqslant s-1$, hence $h(t)=\varphi_s(t)$ in a neighbourhood of $1$. Notice that if $j>i$, then
$$
\begin{equation}
\varphi_j\biggl(\frac{q}2\biggr)=\frac q2 \biggl(\frac{r_j}{d} +\frac 1q-\frac{1}{p_j}\biggr)\stackrel{(3)}{<}\frac q2 \biggl(\frac{r_i}{d}+\frac 1q-\frac{1}{p_i}\biggr) =\varphi_i\biggl(\frac{q}2\biggr).
\end{equation}
\tag{53}
$$
Now we describe the induction step. Assume that the function $h(t)$ is already constructed on the interval $[1, t_{l-1}]$, where $l\in \mathbb{N}$, $t_{l-1}\in [1, q/2)$,
$$
\begin{equation}
h(t) = \max_{1\leqslant j\leqslant j_l} \varphi_j(t)\quad \text{for}\quad t\in \biggl[t_{l-1}, \frac{q}2\biggr],
\end{equation}
\tag{54}
$$
$$
\begin{equation}
\varphi_{j_l}(t_{l-1}) > \varphi_j(t_{l-1}), \qquad 1\leqslant j<j_l.
\end{equation}
\tag{55}
$$
Then $h(t) = \varphi_{j_l}(t)$ in a right semi-neighbourhood of $t_{l-1}$. If $j_l=1$, then $h(t)=\varphi_1(t)$ for $t\in [t_{l-1}, q/2]$, which completes the construction. Let $j_l>1$. By (53), $\varphi_j(q/2)>\varphi_{j_l}(q/2)$ for $1\leqslant j<j_l$. This together with (55) yields that, for all $j\in \{1, \dots, j_l-1\}$, there is a point $t^j_l\in (t_{l-1}, q/2)$ such that $\varphi_{j_l}(t^j_l)=\varphi_j(t^j_l)$. We set
$$
\begin{equation}
t_l = \min \{t^j_l\}_{1\leqslant j\leqslant j_l-1}, \qquad j_{l+1} = \min \bigl\{j\in \{1,\dots, j_l-1\}\colon \varphi_{j_l}(t_l) = \varphi_j(t_l)\bigr\}.
\end{equation}
\tag{56}
$$
Now $h(t) = \varphi_{j_l}(t)$ for $t_{l-1}\leqslant t\leqslant t_l$. Let us show that
$$
\begin{equation}
h(t) = \max_{1\leqslant j\leqslant j_{l+1}} \varphi_j(t) \quad \text{for}\quad t\in \biggl[t_l, \frac{q}2\biggr],
\end{equation}
\tag{57}
$$
$$
\begin{equation}
\varphi_{j_{l+1}}(t_l)> \varphi_j(t_l), \qquad 1\leqslant j< j_{l+1}.
\end{equation}
\tag{58}
$$
Indeed, by (53), for each $j\in \{j_{l+1}+1,\dots, j_l\}$ we have $\varphi_j(q/2)< \varphi_{j_{l+1}}(q/2)$. In addition, $\varphi_j(t_l)\leqslant \varphi_{j_{l+1}}(t_l)$ for $j\in \{j_{l+1}+1,\dots, j_l\}$ (otherwise, if $\varphi_j(t_l)> \varphi_{j_{l+1}}(t_l)$, then $t_l^j\in (t_{l-1}, t_l)$ by (55) and by the equality $\varphi_{j_{l+1}}(t_l)=\varphi_{j_l}(t_l)$, this, however, contradicts (56)). Hence $\varphi_j(t)< \varphi_{j_{l+1}}(t)$ for $j_{l+1}+1\leqslant j\leqslant j_l$, $t> t_l$. This together with (54) yields (57). Now let us prove (58). Indeed, if $\varphi_{j_{l+1}}(t_l)< \varphi_j(t_l)$, then $t_l^j\in (t_{l-1}, t_l)$ by (55) and by the equality $\varphi_{j_{l+1}}(t_l)=\varphi_{j_l}(t_l)$; this again contradicts the definition of $t_l$ in (56). If $\varphi_{j_{l+1}}(t_l)= \varphi_j(t_l)$, we obtain a contradiction with the definition of $j_{l+1}$ in (56). Searching the indices $i_*$, $j_*$ in assertion 3 of Theorem 3To this end, we use the described algorithm of construction of the function $h$. We have: $t_0=1$, $j_1=s$, $\varphi'_s<0$, $\varphi'_1>0$ by conditions of assertion 3 of the theorem. Let us find a point $t_1$ and an index $j_2$. If $\varphi'_{j_2}>0$, then $t_1$ is the minimum point of the function $h$, $i_*=j_1$, $j_*=j_2$. The case $\varphi'_{j_2}=0$ is impossible, since the minimum point of the function $h$ is unique. If $\varphi'_{j_2}<0$, we find a point $t_2$ and an index $j_3$; if $\varphi'_{j_3}>0$, then $t_2$ is the minimun point of the function $h$, $i_*=j_2$, $j_*=j_3$. If $\varphi'_{j_3}<0$, then we similarly find a point $t_3$ and an index $j_4$, and so on. Searching the minimum point of the function $h$ in Theorem 1In the case when $J\ne \varnothing$, $J\ne \{1,\dots,s\}$, we construct the function $h_2(t)$ on the whole interval $[1, q/2]$ by the algorithm described above. Then, comparing the values of $h_2$ and $h_0$, $h_1$ at the constructed partition points and at the endpoints of the interval, we find the function $h$ explicitly, after which it is easy to find its minimum point. § 3. The general caseLet now condition (3) fail. In this case, $d_n(M, L_q(\Omega))$ can also be estimated by applying Lemmas 4 and 5, but we cannot use Theorem C directly. Notice that if $j>i$, $r_j/d-1/p_j \geqslant r_i/d-1/p_i$, then
$$
\begin{equation*}
2^{-m(r_j/d+1/q-1/p_j)}B_{p_j}^{2^m} \subset 2^{-m(r_i/d+1/q-1/p_i)}B_{p_i}^{2^m}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\bigcap_{k=1}^s 2^{-m(r_k/d+1/q-1/p_k)}B_{p_k}^{2^m} = \bigcap_{k\ne i} 2^{-m(r_k/d+1/q-1/p_k)}B_{p_k}^{2^m},
\end{equation*}
\notag
$$
and the problem can be reduced to estimating the widths of an intersection of a smaller number of balls. Therefore, the orders of $d_n(M, L_q(\Omega))$ can be evaluated as follows. We choose the numbers $r_{i_1}<r_{i_2}<\dots < r_{i_k}=r_s$ such that
$$
\begin{equation*}
\bigcap_{l=1}^k 2^{-m(r_{i_l}/d+1/q-1/p_{i_l})} B_{p_{i_l}}^{2^m} = \bigcap_{j=1}^s 2^{-m(r_j/d+1/q-1/p_j)}B_{p_j}^{2^m}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\frac{r_{i_j}}{d}-\frac{1}{p_{i_j}} < \frac{r_{i_l}}{d}-\frac{1}{p_{i_l}}, \qquad j>l.
\end{equation*}
\notag
$$
These numbers can be constructed as follows. First, we find the set
$$
\begin{equation*}
I_1= \{s\} \cup \biggl\{j\in \{1,\dots, s-1\} \colon \frac{r_s}{d}-\frac{1}{p_s} < \frac{r_j}{d}-\frac{1}{p_j}\biggr\}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\bigcap_{k=1}^s 2^{-m(r_k/d+1/q-1/p_k)}B_{p_k}^{2^m} = \bigcap_{k\in I_1} 2^{-m(r_k/d+1/q-1/p_k)}B_{p_k}^{2^m}.
\end{equation*}
\notag
$$
Let $j_1=\max I_1\setminus \{s\}$. Setting
$$
\begin{equation*}
I_2= \{s, j_1\} \cup \biggl\{j\in \{1,\dots, j_1-1\} \colon \frac{r_{j_1}}{d}-\frac{1}{p_{j_1}} < \frac{r_j}{d}-\frac{1}{p_j}\biggr\},
\end{equation*}
\notag
$$
we have
$$
\begin{equation*}
\bigcap_{k=1}^s 2^{-m(r_k/d+1/q-1/p_k)}B_{p_k}^{2^m} = \bigcap_{k\in I_2} 2^{-m(r_k/d+1/q-1/p_k)}B_{p_k}^{2^m}.
\end{equation*}
\notag
$$
Let $j_2=\max I_2\setminus \{s, j_1\}$. We set
$$
\begin{equation*}
I_3= \{s, j_1, j_2\} \cup \biggl\{j\in \{1,\dots, j_2-1\} \colon \frac{r_{j_2}}{d}-\frac{1}{p_{j_2}} < \frac{r_j}{d}-\frac{1}{p_j}\biggr\},
\end{equation*}
\notag
$$
and so on. We denote $\widehat M=\bigcap_{j=1}^k W^{r_{i_j}}_{p_{i_j}}(\Omega)$. For $\widehat M$, condition (3) is met. Let the set $\widehat M$ satisfy all the conditions of Theorem 1. Then
$$
\begin{equation*}
d_n(M, L_q(\Omega)) \underset{\mathfrak{Z}}{\asymp} d_n\bigl(\widehat M, L_q(\Omega)\bigr) \underset{\mathfrak{Z}}{\asymp} n^{-\theta},
\end{equation*}
\notag
$$
where $\theta$ depends on $r_{i_1}, \dots, r_{i_k}$, $p_{i_1}, \dots, p_{i_k}$, $q$, $d$ as in Theorem 1.
§ 4. Proof of Theorem 4First, let us obtain an analogue of Lemmas 4 and 5: if $C\in \mathbb{N}$, $k_m\in \mathbb{Z}_+$, and $\sum_{2^m\geqslant 2n}k_m\leqslant Cn$, then
$$
\begin{equation}
d_{C_1Cn}\bigl(\widetilde{M}, L_q(\mathbb{T})\bigr) \underset{\mathfrak{Z}}{\lesssim} \sum_{2^m\geqslant 2n} d_{k_m} \biggl(\bigcap_{j=1}^s 2^{-m(r_j+1/q-1/p_j)} B^{2^m}_{p_j}, l_q^{2^m}\biggr)
\end{equation}
\tag{59}
$$
(here $C_1$ is an absolute constant);
$$
\begin{equation}
d_n\bigl(\widetilde{M}, L_q(\mathbb{T})\bigr) \underset{\mathfrak{Z}}{\gtrsim} d_n \biggl(\bigcap_{j=1}^s 2^{-m(r_j+1/q-1/p_j)} B^{2^m}_{p_j}, l_q^{2^m}\biggr), \qquad m\in \mathbb{N}.
\end{equation}
\tag{60}
$$
Let $m \in \mathbb{N}$. We have $\square_m{=}\,\{k \in \mathbb{Z}\colon 2^{m-1}{\leqslant}\, |k|\,{<}\,2^m\}$, $\mathcal{T}_m {=} \operatorname{span} \{e^{ikt}\}_{k\in \square_m}$. Given $x= \sum_{k\in \mathbb{Z} \setminus \{0\}} x_k e^{ikt}$, we set In order to prove estimates (59), (60) we will use the following well-known facts. Theorem E. Let $1<p<\infty$. Then $\|x^{(r)}\|_{L_p(\mathbb{T})} \underset{p,r}{\asymp} 2^{mr} \|x\|_{L_p(\mathbb{T})}$ for $x\in \mathcal{T}_m$. This theorem follows directly from Marcinkiewicz multiplier theorem (see [22], § 1.5.3, [23], Ch. III, § 15.3; and also [8], Ch. 2, § 2.3, Theorem 18, for $r \geqslant 0$). Theorem F (see [2], Theorem B). There is an isomorphism $A\colon \mathcal{T}_m \to \mathbb{R}^{2^m}$ such that $\|x\|_{L_p(\mathbb{T})} \underset{p}{\asymp} 2^{-m/p} \|Ax\|_{l_p^{2^m}}$ for all $p\in (1, \infty)$, $x\in \mathcal{T}_m$. Given $x\in \mathcal{S}'(\mathbb{T})$, we denote
$$
\begin{equation*}
Px(t)= \biggl(\sum_{m\in \mathbb{N}}|\delta_mx(t)|^2\biggr)^{1/2}.
\end{equation*}
\notag
$$
Theorem G (the Littlewood–Paley theorem; see [23], Ch. III, § 15.2 [8], Ch. 2, § 2.3, Theorem 15). Let $1<q<\infty$. Then $x\in L_q(\mathbb{T})$ if and only if $Px\in L_q(\mathbb{T})$; in addition, $\|x\|_{L_q(\mathbb{T})} \underset{q}{\asymp} \|Px\|_{L_q(\mathbb{T})}$. Let $x=\sum_{k\in \mathbb{Z}\setminus\{0\}}x_ke^{ikt}$. We set $S_nx = \sum_{1\leqslant |k|\leqslant n}x_ke^{ikt}$. Theorem H (see [8], Ch. 2, § 2.3). Let $1<q<\infty$, $x\in L_q(\mathbb{T})$. Then $S_n \to x$ as $n\to \infty$ in the space $L_q(\mathbb{T})$. In [13], a criterion for the set $\widetilde{M}$ to lie in $L_q(\mathbb{T})$ was obtained. Here, we formulate a sufficient condition for embedding in a particular case. From Besov’s theorem (see [23], Ch. III, § 15.6), we have the following result. Theorem I. The following assertions hold. 1. If $1<p, q<\infty$, $r\in \mathbb{R}$, $r-(1/p-1/q)_+\geqslant 0$, then $\widetilde{W}^r_p(\mathbb{T})\subset L_q(\mathbb{T})$. 2. Let $1<p_2\leqslant q\leqslant p_1<\infty$, $r_1,r_2\in \mathbb{R}$, and let $\lambda\in [0, 1]$ be defined by the equation $1/q =(1-\lambda)/(p_2)+\lambda/p_1$. Assume that $(1-\lambda)r_2+\lambda r_1\geqslant 0$. Then $\widetilde{W}^{r_1}_{p_1}(\mathbb{T}) \cap \widetilde{W}^{r_2}_{p_2}(\mathbb{T})\subset L_q(\mathbb{T})$. Corollary 1. Under conditions of Theorem 4, the inclusion $\widetilde{M}\subset L_q(\mathbb{T})$ holds. Estimate (59) follows from Theorems E, F, G, H and Corollary 1. We have
$$
\begin{equation*}
\begin{aligned} \, d_{C_1Cn}\bigl(\widetilde{M}, L_q(\mathbb{T})\bigr) &\leqslant \sum_{2^m\geqslant 2n} d_{k_m} \bigl(\delta_m (\widetilde{M}), L_q(\mathbb{T})\cap \mathcal{T}_m\bigr) \\ &\underset{q}{\lesssim} \sum_{2^m \geqslant 2n} 2^{-m/q}d_{k_m} \bigl(A\delta_m (\widetilde{M}), l_q^{2^m}\bigr) \\ &\underset{\mathfrak{Z}}{\lesssim} \sum_{m\in \mathbb{N}} d_{k_m}\biggl(\bigcap_{j=1}^s 2^{-m(r_j+1/q-1/p_j)}B_{p_j}^{2^m}, l_q^{2^m}\biggr). \end{aligned}
\end{equation*}
\notag
$$
Estimate (60) can be proved as in Theorem 1 of [2]. Applying Theorems E, F and G, we obtain the inequalities
$$
\begin{equation*}
d_n\bigl(\widetilde{M}, L_q(\mathbb{T})\bigr) \underset{\mathfrak{Z}}{\gtrsim} d_n\bigl(\widetilde{M}\cap \mathcal{T}_m, L_q(\mathbb{T}) \cap \mathcal{T}_m\bigr) \underset{\mathfrak{Z}}{\gtrsim} d_n\biggl(\bigcap_{j=1}^s 2^{-m(r_j+1/q-1/p_j)}B_{p_j}^{2^m}, l_q^{2^m}\biggr).
\end{equation*}
\notag
$$
Now we argue as in the proof of Theorem 1 for $d=1$. Proposition 2. Let $r_1<\dots <r_s$, $r_s-1/p_s<\dots< r_1-1/p_1$. Suppose that one of the following conditions holds: 1) $p_1\leqslant q$, $r_1+1/q-1/p_1\leqslant 0$; 2) $p_s\geqslant q$, $r_s\leqslant 0$; 3) $p_1>q$, $p_s<q$, $(1-\lambda_{i_0j_0})r_{j_0}+ \lambda_{i_0j_0}r_{i_0}\leqslant 0$ (see the notation in Theorem 1). Then Proof. Notice that $p_1\geqslant \dots\geqslant p_s$.
If $r_1+1/q-1/p_1\leqslant 0$, then we can prove (61) proceeding as for the intersection of Sobolev classes on a John domain (for $p_1\leqslant q$, as well as for $p_1\geqslant q$). In case 2) using (60) for $2^m\geqslant 2n$ and (22), we get
$$
\begin{equation*}
d_n\bigl(\widetilde{M}, L_q(\mathbb{T})\bigr) \underset{\mathfrak{Z}}{\gtrsim} \min_{1\leqslant j\leqslant s} 2^{-m(r_j+1/q-1/p_j)}\cdot 2^{m(1/q-1/p_j)} = 2^{-mr_s}\geqslant 1.
\end{equation*}
\notag
$$
It remains to consider case 3) with $r_1+1/q-1/p_1>0$. We again use estimate (60); if $q\leqslant 2$, we take $2^m\geqslant 2n$, and if $q>2$, we take $2^m\geqslant n^{q/2}$. If $q\leqslant 2$, then by (25), (30) and by the definition of indices $i_0$, $j_0$,
$$
\begin{equation*}
d_n\bigl(\widetilde{M}, L_q(\mathbb{T})\bigr) \underset{\mathfrak{Z}}{\gtrsim} 2^{-m((1-\lambda_{i_0j_0})r_{j_0} + \lambda_{i_0j_0}r_{i_0})} \geqslant 1.
\end{equation*}
\notag
$$
Let $q>2$. We define the numbers $\widetilde \lambda_{ij}\in [0, 1]$ and the indices $i_1\in I\cup J$, $j_1\in K$ as in assertion 5 of Theorem 1. Applying (26)–(29), we get
$$
\begin{equation*}
d_n\bigl(\widetilde{M}, L_q(\mathbb{T})\bigr) \underset{\mathfrak{Z}}{\gtrsim} 2^{-m\beta_0},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, \beta_0 &=\max \biggl\{ (1-\lambda_{i_0j_0})r_{j_0}+ \lambda_{i_0j_0}r_{i_0}, \max_{j\in J}\biggl(r_j +\frac1{q} -\frac1{p_j}\biggr), \\ &\qquad\qquad(1-\widetilde\lambda_{i_1j_1})r_{j_1} + \widetilde\lambda_{i_1j_1}r_{i_1} + \frac1{q} -\frac12\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
It suffices to show that $\beta_0\leqslant 0$.
By the conditions of the proposition, $(1-\lambda_{i_0j_0})r_{j_0}+ \lambda_{i_0j_0}r_{i_0}\leqslant 0$. We claim that
$$
\begin{equation}
r_j+\frac 1q -\frac{1}{p_j} \leqslant 0, \qquad j\in J\cup K.
\end{equation}
\tag{62}
$$
Indeed, if $r_j+1/q-1/p_j>0$ for some $j\in J\cup K$, then
$$
\begin{equation*}
\begin{aligned} \, 0 &< (1-\lambda_{1j})\biggl(r_j+\frac 1q -\frac{1}{p_j}\biggr) + \lambda_{1j}\biggl(r_1+\frac 1q -\frac{1}{p_1}\biggr) \\ &=(1-\lambda_{1j})r_j+\lambda_{1j}r_1 \leqslant (1-\lambda_{i_0j_0})r_{j_0}+ \lambda_{i_0j_0}r_{i_0}\leqslant 0 \end{aligned}
\end{equation*}
\notag
$$
(here, we used the definition of $i_0$, $j_0$ and the condition $r_1+1/q-1/p_1> 0$). We get a contradiction.
From (62) it follows that $\max_{j\in J}(r_j+1/q-1/p_j)\leqslant 0$. Now we prove that $(1-\widetilde\lambda_{i_1j_1})r_{j_1} + \widetilde\lambda_{i_1j_1}r_{i_1} + 1/q - 1/2\leqslant 0$. For $i_1\in J$, this result follows from (62) and from the equality
$$
\begin{equation*}
\begin{aligned} \, &(1-\widetilde\lambda_{i_1j_1}) \biggl(r_{j_1} +\frac 1q -\frac{1}{p_{j_1}}\biggr) + \widetilde\lambda_{i_1j_1}\biggl(r_{i_1} +\frac 1q -\frac{1}{p_{i_1}}\biggr) \\ &\qquad=(1-\widetilde\lambda_{i_1j_1})r_{j_1} + \widetilde\lambda_{i_1j_1}r_{i_1} + \frac 1q - \frac 12. \end{aligned}
\end{equation*}
\notag
$$
Let $i_1\in I$. By the conditions of the proposition, (52) holds with $d=1$. Therefore,
$$
\begin{equation*}
\begin{aligned} \, (1-\widetilde\lambda_{i_1j_1})r_{j_1}+ \widetilde\lambda_{i_1j_1}r_{i_1} + \frac 1q - \frac 12 &\leqslant (1-\lambda_{i_1j_1}) r_{j_1}+ \lambda_{i_1j_1}r_{i_1} \\ &\leqslant (1-\lambda_{i_0j_0})r_{j_0}+ \lambda_{i_0j_0}r_{i_0}\leqslant 0 \end{aligned}
\end{equation*}
\notag
$$
(here we used the definition of the indices $i_0$, $j_0$ once again). Hence $\beta_0\leqslant 0$, proving the proposition. § 5. A generalization to a weighted caseFirst, we give the definition of an $h$-set. Definition 2 (see [24]). Let $\Gamma\subset \mathbb{R}^d$ be a non-empty compact set, and let $h\colon (0, 1] \to (0,\infty)$ be a nondecreasing function. We say that $\Gamma$ is an $h$-set if there exist a constant $c_*\geqslant 1$ and a finite countably-additive measure $\mu$ on $\mathbb{R}^d$ such that $\operatorname{supp}\mu=\Gamma$ and for all $x\in \Gamma$ and $t\in (0, 1]$. As examples of an $h$-set, we mention $k$-dimensional Lipschitz surfaces, the Koch curve, and some Cantor-type sets. Let $d\in \mathbb{N}$, $a>0$, $\Omega \subset (-1/2, 1/2)^d$, $\Omega \in \mathbf{FC}(a)$, $\Gamma \subset \partial \Omega$ be an $h$-set with $h(t) = t^\theta$, where $0\leqslant \theta <d$; let $r_j\in \mathbb{Z}_+$, $1<p_j\leqslant \infty$, $\beta_j\in \mathbb{R}$, $1\leqslant j\leqslant s$, $r_1<\dots<r_s$, $1\leqslant q<\infty$, $\lambda \in \mathbb{R}$;
$$
\begin{equation*}
\begin{gathered} \, g_j(x)=(\operatorname{dist} (x, \Gamma))^{-\beta_j}, \quad g_j^0(x) = (\operatorname{dist} (x, \Gamma))^{r_j-\beta_j}, \qquad 1\leqslant j\leqslant s, \\ v(x) = (\operatorname{dist} (x, \Gamma))^{-\lambda}. \end{gathered}
\end{equation*}
\notag
$$
The weighted Sobolev class $\widehat W^{r_j}_{p_j,g_j}(\Omega)$ is defined by the equation
$$
\begin{equation*}
\widehat W^{r_j}_{p_j,g_j}(\Omega) =\biggl\{f\in L_1^\mathrm{loc}(\Omega)\colon \biggl\|\frac{\nabla^{r_j}f}{g_j}\biggr\|^{p_j} + \biggl\|\frac{f}{g^0_j}\biggr\|^{p_j} \leqslant 1\biggr\},
\end{equation*}
\notag
$$
$1\leqslant j\leqslant s$. The space $L_{q,v}(\Omega)$ consists of the functions $f$ such that $vf\in L_q(\Omega)$; we also set $\|f\|_{L_{q,v}(\Omega)} :=\|vf\|_{L_q(\Omega)}$. Let
$$
\begin{equation*}
M = \bigcap_{j=1}^s \widehat W^{r_j}_{p_j,g_j}(\Omega).
\end{equation*}
\notag
$$
The sets $I$, $J$, $K$ and the numbers $\lambda_{ij}$, $\widetilde \lambda_{ij}$ are defined as in Theorem 1 (for $q\leqslant 2$ as in assertion 4, and for $q>2$, as in assertion 5). We set $\mu_j=\beta_j+\lambda-r_j-d/q+d/p_j$, $1\leqslant j\leqslant s$;
$$
\begin{equation*}
\mathfrak{Z}_0=(d, r_1, \dots, r_s, q, p_1, \dots, p_s, a, c_*, \beta_1, \dots, \beta_s, \lambda, R),
\end{equation*}
\notag
$$
where $R=\operatorname{diam} \Omega$. Theorem 5. The following estimates hold. 1. Let $q\leqslant 2$. We set
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, h_1(t, m) &= \max_{j\in I} \biggl(-t\biggl(\mu_j+\frac{\theta}{q} -\frac{\theta}{p_j}\biggr) +m\frac{r_j}{d}\biggr), \\ h_2(t, m) &= \max_{j\in J}\biggl(-t\cdot \mu_j + m\biggl(\frac{r_j}{d} +\frac 1q-\frac{1}{p_j}\biggr)\biggr), \\ h_3(t, m) &= \max_{i\in I, j\in J} \biggl(-t\bigl((1-\lambda_{ij})\mu_j+\lambda_{ij}\mu_i\bigr) +m \biggl((1-\lambda_{ij})\frac{r_j}{d}+\lambda_{ij}\frac{r_i}{d}\biggr)\biggr), \end{aligned} \\ h=\max\{h_1, h_2, h_3\}, \qquad \varphi(t) = h(t, 1-\theta t), \quad 0\leqslant t\leqslant \frac{1}{\theta}. \end{gathered}
\end{equation*}
\notag
$$
Let the function $\varphi$ have a unique minimum point $t_*$ on $[0, 1/\theta]$, and let $\varphi(t_*)>0$. Then
$$
\begin{equation*}
d_n(M, L_{q,v}(\Omega)) \underset{\mathfrak{Z}_0}{\asymp} n^{-\varphi(t_*)}.
\end{equation*}
\notag
$$
2. Let $2<q<\infty$. We set
$$
\begin{equation*}
\begin{aligned} \, h_1(t, m) &= \max_{j\in I}\biggl(-t\biggl(\mu_j+\frac{\theta}{q} -\frac{\theta}{p_j}\biggr) +m\frac{r_j}{d}\biggr), \\ h_2(t, m) &= \max_{j\in J}\biggl(-t\cdot \biggl(\mu_j +\frac{\theta}{q}\cdot \frac{1/p_j-1/q}{1/2-1/q}\biggr) \\ &\qquad\qquad + m\biggl(\frac{r_j}{d}-\frac 12 \cdot \frac{1/p_j-1/q}{1/2-1/q}\biggr) + \frac 12 \cdot \frac{1/p_j-1/q}{1/2-1/q}\biggr), \\ h_3(t, m) &= \max_{j\in K}\biggl(-t\biggl(\mu_j +\frac{\theta}{q}\biggr) + m\biggl(\frac{r_j}{d}-\frac{1}{p_j}\biggr)+\frac 12\biggr), \\ h_4(t, m) &= \max_{i\in I,\, j\in J\cup K} \biggl(-t\bigl((1-\lambda_{ij})\mu_j +\lambda_{ij}\mu_i\bigr) +m \biggl((1-\lambda_{ij})\frac{r_j}{d} +\lambda_{ij}\frac{r_i}{d}\biggr)\biggr), \\ h_5(t, m) &= \max_{i\in I\cup J,\, j\in K} \biggl(-t\biggl((1-\widetilde\lambda_{ij})\mu_j +\widetilde\lambda_{ij}\mu_i+\frac{\theta}{q}\biggr) \\ &\qquad\qquad\qquad +m \biggl((1-\widetilde\lambda_{ij})\frac{r_j}{d} +\widetilde\lambda_{ij}\frac{r_i}{d}-\frac 12\biggr)+\frac 12\biggr), \end{aligned}
\end{equation*}
\notag
$$
Let the function $h$ have a unique minimum point $(t_*, m_*)$ on the set $A$, and let $h(t_*, m_*)> 0$. Then The proof generalizes the above arguments from § 3 and from [6]. We notice that the upper estimate can be reduced to evaluation of the sum
$$
\begin{equation*}
\sum_{t, m\in \mathbb{Z}_+\colon 2^{\theta t+m} \geqslant 2n\cdot 2^{-\varepsilon|t-t_n|}} d_{k_{t,m}} \biggl(\bigcap_{j=1}^s 2^{\mu_jt-m(r_j/d+1/q-1/p_j)}B_{p_j}^{\lceil 2^{\theta t+m}\rceil}, l_q^{\lceil 2^{\theta t+m}\rceil}\biggr),
\end{equation*}
\notag
$$
where $\varepsilon=\varepsilon(\mathfrak{Z}_0)$, $t_n=t_n(\mathfrak{Z}_0)$, $k_{t,m}\in \mathbb{Z}_+$, $\sum_{t,m\in \mathbb{Z}_+\colon 2^{\theta t+m} \geqslant 2n \cdot 2^{-\varepsilon|t-t_n|}}k_{t,m}\underset{\mathfrak{Z}_0}{\lesssim} n$, and the lower estimate, to evaluation of
$$
\begin{equation*}
d_n\biggl(\bigcap_{j=1}^s 2^{\mu_jt-m(r_j/d+1/q-1/p_j)}B_{p_j}^{\lceil 2^{\theta t+m}\rceil}, l_q^{\lceil 2^{\theta t+m}\rceil}\biggr).
\end{equation*}
\notag
$$
In estimating these widths, we use Proposition 1 and Theorems A, B.
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