A. A. Vasil'eva, “Kolmogorov widths of a Sobolev class with constraints on derivatives in different metrics”, Sb. Math., 215:11 (2024), 1468

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$\begin{equation} \biggl\| \frac{\partial^{r_j}}{\partial x^{r_j}_j}f\biggr\|_{L_{p_j}(\mathbb{T}^d)} \leqslant 1, \qquad j=1,\dots,d, \end{equation} \tag{1.1}$

$\begin{equation} 1-\sum_{k=1}^d \frac{1}{p_kr_k}>0\quad\text{and} \quad q\geqslant \max_{1\leqslant k\leqslant d} p_k. \end{equation} \tag{1.2}$

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

$\leqslant n$ .

$\begin{equation*} \mathring{\mathbb Z}^d=\{(k_1,k_2,\dots,k_d)\in \mathbb Z^d\colon k_1k_2\dotsb k_d\ne 0\} \end{equation*} \notag$

$\begin{equation*} \mathring{\mathcal S}'(\mathbb{T}^d)=\biggl\{ f\in \mathcal S'(\mathbb{T}^d)\colon f=\sum_{\overline{k}\in \mathring{\mathbb Z}^d} c_{\overline{k}}(f) e^{i(\overline{k}, \cdot)}\biggr\}. \end{equation*} \notag$

$\begin{equation*} \partial_j^{r_j}f:=\frac{\partial^{r_j}f}{\partial x_j^{r_j}} :=\sum_{\overline{k}\in \mathring{\mathbb Z}^d} c_{\overline{k}}(f) (ik_j)^{r_j} e^{i(\overline{k},\cdot)}, \end{equation*} \notag$

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

$\begin{equation*} \langle \overline{a}\rangle=\frac{d}{1/a_1+\dots+1/a_d}. \end{equation*} \notag$

$\begin{equation*} \frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ\overline{r}\rangle} \geqslant 0. \end{equation*} \notag$

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 $1<q\leqslant 2$ .

- (a) If $p_i\leqslant q$ for each $i=1,\dots,d$ , then
  $\begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{r}, \overline{p}, q, d}{\asymp} n^{-\langle \overline{r}\rangle /d-1/q+\langle \overline{r}\rangle/\langle \overline{p} \circ\overline{r}\rangle}. \end{equation*} \notag$
- (b) Let $\{i\in 1,\dots,d\colon p_i>q\}\ne \varnothing$ and $\{i\in 1,\dots,d\colon p_i<q\}\ne \varnothing$ , and let $\langle \overline{r}\rangle/\langle \overline{p} \circ\overline{r}\rangle \ne 1/q$ . Then
  $\begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{r}, \overline{p}, q, d}{\asymp} n^{-\min \{\langle \overline{r}\rangle /d,\langle \overline{r}\rangle /d+1/q-\langle \overline{r}\rangle/\langle \overline{p} \circ\overline{r}\rangle \}}. \end{equation*} \notag$

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 $\theta_3=\frac q2\bigl( \frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ \overline{r}\rangle}\bigr)$ .

- (a) Let $p_i\leqslant 2$ , $1\leqslant i\leqslant d$ , and let $\theta_2\ne \theta_3$ . Then
  $\begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{r}, \overline{p}, q, d}{\asymp} n^{-\min \{\theta_2,\theta_3\}}. \end{equation*} \notag$
- (b) Let $p_i\geqslant 2$ , $1\leqslant i\leqslant d$ , and let $\theta_1\ne \theta_3$ . Then
  $\begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{r}, \overline{p}, q, d}{\asymp} n^{-\min \{\theta_1,\theta_3\}}. \end{equation*} \notag$
- (c) Let $\{i\in 1,\dots,d\colon p_i<2\}\ne \varnothing$ and $\{i\in 1,\dots,d\colon p_i>2\}\ne \varnothing$ . Assume that there exists $j_*\in \{1,2,3\}$ such that $\theta_{j_*}<\min_{j\ne j_*} \theta_j$ . Then
  $\begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{r}, \overline{p}, q, d}{\asymp} n^{-\theta_{j_*}}. \end{equation*} \notag$

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).

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

$\begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{r},\overline{p},q,d}{\asymp} n^{-\widetilde h(\widehat \alpha_1,\dots,\widehat \alpha_d,\widehat s)}. \end{equation*} \notag$

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

$L_q(\mathbb{T}^d)$ .

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 $\widehat s\bigl(1-\frac{r_2(1-2/q)}{1/p_2-1/p_1}\bigr)=1$ .

- (a) Let $p_2\geqslant 2$ . Set $\theta_1=\frac{\langle \overline{r}\rangle}{2}$ , $\theta_2=\widehat s\lambda r_2$ , and assume that $\theta_1\ne \theta_2$ . Then
  $\begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^2),L_q(\mathbb{T}^2)) \underset{\overline{r},\overline{p},q}{\asymp} n^{-\min \{\theta_1,\theta_2\}}. \end{equation*} \notag$
- (b) Let $p_2<2$ . Let $\mu\in (0,1)$ be defined by $\frac 12= \frac{1-\mu}{p_1}+\frac{\mu}{p_2}$ . Set $\theta_1=\frac{\langle \overline{r}\rangle}{2}$ , $\theta_2=\widehat s\lambda r_2$ and $\theta_3=\mu r_2$ . Assume that there exists $j_*\in \{1,2,3\}$ such that
  $\begin{equation} \theta_{j_*}=\min_{j\ne j_*} \theta_j. \end{equation} \tag{1.12}$
  Then
  $\begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^2),L_q(\mathbb{T}^2)) \underset{\overline{r},\overline{p},q}{\asymp} n^{-\theta_{j_*}}. \end{equation*} \notag$

$\begin{equation} \begin{gathered} \, m=m_1+\dots+m_d, \\ \square_{\overline{m}}=\{k\in \mathbb Z^d\colon 2^{m_j-1}\leqslant |k_j|<2^{m_j}, \, 1\leqslant j\leqslant d\} \notag \end{gathered} \end{equation} \tag{2.1}$

$\begin{equation*} \delta_{\overline{m}} f(\cdot)=\sum_{\overline{k}\in \square_{\overline{m}}} c_{\overline{k}}(f)e^{i(\overline{k},\, \cdot)}. \end{equation*} \notag$

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

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 $f\in \mathcal T_{\overline{m}}$ ,

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

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 $f\in \mathcal T_{\overline{m}}$ ,

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

$\begin{equation*} \|(x_i)_{i=1}^N\|_{l_q^N}=\biggl(\sum_{i=1}^N |x_i|^q\biggr)^{1/q} \quad\text{for } q<\infty \end{equation*} \notag$

$\begin{equation*} \|(x_i)_{i=1}^N\|_{l_q^N}=\max_{1\leqslant i\leqslant N}|x_i| \quad\text{for } q=\infty. \end{equation*} \notag$

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

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

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

$q>2$

$\begin{equation} \begin{aligned} \, d_n(M_0,l_q^N) &\underset{q}{\asymp} \min \Bigl\{ \min_{\alpha \in A}d_n(\nu_\alpha B_{p_\alpha}^N,l_q^N),\min_{p_\alpha>q,p_\beta< q} \nu_\alpha ^{1-\lambda_{\alpha,\beta}}\nu_\beta^{\lambda_{\alpha,\beta}}, \nonumber \\ &\qquad\qquad \min_{p_\alpha> 2,p_\beta< 2} \nu_\alpha ^{1-\widetilde\lambda_{\alpha,\beta}}\nu_\beta^{\widetilde\lambda_{\alpha,\beta}}d_n(B_2^N, l_q^N)\Bigr\}. \end{aligned} \end{equation} \tag{2.6}$

$\begin{equation*} V_k=\operatorname{conv}\{(\varepsilon_1 \widehat{x}_{\sigma(1)},\dots, \varepsilon_N \widehat{x}_{\sigma(N)})\colon \varepsilon_j=\pm 1,\,1\leqslant j\leqslant N,\, \sigma \in S_N\}, \end{equation*} \notag$

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

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

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

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

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

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.

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

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.

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

$\begin{equation*} \begin{aligned} \, d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) &\underset{\overline{p},q,\overline{r},d}{\gtrsim} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d)\cap \mathcal T_{\overline{m}}, L_q(\mathbb{T}^d)\cap \mathcal T_{\overline{m}}) \\ &\underset{\overline{p},q,\overline{r},d}{\gtrsim} d_n \biggl(\bigcap_{j=1}^d 2^{-m_jr_j-m/q+m/p_j} B^{2^m}_{p_j},l_q^{2^m}\biggr). \end{aligned} \end{equation*} \notag$

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

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

$\begin{equation} \|\delta_{\overline{m}}f\|_{L_q(\mathbb{T}^d)} \underset{\overline{p},q,\overline{r},d}{\lesssim} d_0 \biggl(\bigcap_{j=1}^d 2^{-m_jr_j-m/q+m/p_j} B^{2^m}_{p_j},l_q^{2^m}\biggr)=:C_{\overline{m}}. \end{equation} \tag{4.3}$

$\begin{equation*} C_{\overline{m}}\lesssim \min \Bigl\{ \min_{j\in I_0} 2^{-m_jr_j}, \,\min_{j\in J_0} 2^{-r_jm_j-m/q+m/p_j}, \,\min_{i\in I_0',j\in J_0'} 2^{-(1-\lambda_{i,j})r_im_i-\lambda_{i,j}r_jm_j}\Bigr\}. \end{equation*} \notag$

$\begin{equation*} \begin{aligned} \, C_{\overline{m}} &\underset{q}{\lesssim} \min \Bigl\{ \min_{j\in I} 2^{-m_jr_j}, \,\min_{j\in J\cup K} 2^{-r_jm_j-m/q+m/p_j}, \\ &\qquad\qquad \min_{i\in I',j\in J'\cup K} 2^{-(1-\lambda_{i,j})r_im_i-\lambda_{i,j}r_jm_j}, \\ &\qquad\qquad \min_{i\in I\cup J',j\in K'} 2^{-(1-\mu_{i,j})r_im_i-\mu_{i,j}r_jm_j -m/q+m/2}\Bigr\} \\ &=\min \Bigl\{ \min_{j\in I} 2^{-m_jr_j},\,\min_{j\in J\cup K} 2^{-r_jm_j-m/q+m/p_j}, \\ &\qquad\qquad \min_{i\in I',j\in J'\cup K} 2^{-(1-\lambda_{i,j})r_im_i-\lambda_{i,j}r_jm_j}\Bigr\}; \end{aligned} \end{equation*} \notag$

$\begin{equation} \begin{aligned} \, \notag &(1-\mu_{i,j})r_im_i+\mu_{i,j}r_jm_j+\frac mq-\frac m2 \\ &\qquad \leqslant\max \biggl\{(1-\lambda_{i,j})r_im_i+\lambda_{i,j}r_jm_j ,r_jm_j+\frac mq-\frac{m}{p_j} \biggr\} \quad\text{if } p_i> q \end{aligned} \end{equation} \tag{4.4}$

$\begin{equation} \begin{aligned} \, \notag &(1-\mu_{i,j})r_im_i+\mu_{i,j}r_jm_j+\frac mq-\frac m2 \\ &\qquad \leqslant\max \biggl\{ r_im_i+\frac mq-\frac{m}{p_i}, r_jm_j+\frac mq-\frac{m}{p_j}\biggr\} \quad \text{if } 2< p_i\leqslant q. \end{aligned} \end{equation} \tag{4.5}$

$\begin{equation} C_{\overline{m}} \underset{\overline{p},q,\overline{r},d}{\lesssim} 2^{-\varphi(m_1,\dots,m_d)}, \end{equation} \tag{4.6}$

$\begin{equation} \begin{aligned} \, \notag \varphi(t_1,\dots,t_d) &=\max \biggl\{\max_{p_j\geqslant q} t_jr_j,\max_{p_j\leqslant q} \biggl(t_jr_j+\frac tq-\frac{t}{p_j}\biggr), \\ &\qquad\qquad\max_{p_i> q,p_j< q} \bigl((1-\lambda_{i,j})r_it_i+\lambda_{i,j}r_jt_j\bigr)\biggr\}, \qquad t=t_1+\dots+t_d. \end{aligned} \end{equation} \tag{4.7}$

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 $N\in \mathbb N$ ,

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

$h$ has a unique minimum point on

$D$ , then

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.

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

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

$\begin{equation*} S_1 \underset{\overline{p},q,\overline{r},d}{\lesssim} 2^{-\frac{q \log n}{2}\min_D h}(\log n)^{d-1} \stackrel{(4.11)}{=} n^{- c_*}(\log n)^{d-1}. \end{equation*} \notag$

$c_*> \widetilde h(\widehat \alpha_1,\dots,\widehat\alpha_d,\widehat s)$ , 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

$\min_{\widetilde D} \widetilde h\leqslant 0$ .

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

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.

• If $I=\{1,\dots,d\}$ , then only case 1 is realized and
$\begin{equation*} \min_{\widetilde D} \widetilde h=\min_{1\leqslant k\leqslant k_0}\min_{V^{(k)}} \widetilde h \stackrel{(5.13)}{=} \frac{\langle \overline{r}\rangle}{d}; \end{equation*} \notag$
note that our arguments in this case do not depend on condition (1.3) (see Remark 1).
• If $I \ne \{1,\dots,d\}$ and $I\cup J=\{1,\dots,d\}$ , then only cases 1, 2 and 4 can be realized. By (5.13), (5.16) and (5.22),
$\begin{equation*} \min_{\widetilde D} \widetilde h=\min_{1\leqslant k\leqslant k_0}\min_{V^{(k)}} \widetilde h=\min \{\theta_1,\theta_3\}, \end{equation*} \notag$
and the minimum on $\widetilde D$ is attained at a unique point (since $\theta_1\ne \theta_3$ by the assumptions of the theorem).
• If $K=\{1,\dots,d\}$ , then only case 3 is realized. By (5.18)
$\begin{equation*} \min_{\widetilde D} \widetilde h=\min_{1\leqslant k\leqslant k_0}\min_{V^{(k)}} \widetilde h= \min \{\theta_2,\theta_3\}, \end{equation*} \notag$
and the minimum on $\widetilde D$ is attained at a unique point since $\theta_2\ne \theta_3$ by the assumptions of the theorem.
• It remains to consider the case where $K\ne \varnothing$ and $I\cup J \ne \varnothing$ . By the assumptions of the theorem there exists $j_*$ such that $\theta_{j_*} < \min_{j\ne j_*} \theta_j$ . If $V^{(k)}$ contains the vertex $\xi_{j_*}$ , then by (5.13), (5.16), (5.18), (5.22) and (5.23) we have $\min_{V^{(k)}} \widetilde h= \theta_{j_*}$ , and the minimum on $V^{(k)}$ is attained only at $\xi_{j_*}$ . If $\xi_{j_*} \notin V^{(k)}$ , then $\min_{V^{(k)}} \widetilde h > \theta_{j_*}$ . Hence
$\begin{equation*} \min_{\widetilde D} \widetilde h=\theta_{j_*} \end{equation*} \notag$
and the minimum point is unique.

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.

$\begin{equation*} \frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{r}\circ \overline{p}\rangle}\leqslant 0. \end{equation*} \notag$

$\begin{equation*} \min_D h\leqslant \frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{r}\circ \overline{p}\rangle}\leqslant 0 \quad\text{for } q\leqslant 2 \end{equation*} \notag$

$\begin{equation*} \min_{\widetilde D} \widetilde h \leqslant \frac q2 \biggl( \frac{\langle \overline{r}\rangle}{d}+\frac 1q- \frac{\langle \overline{r}\rangle}{\langle \overline{r}\circ \overline{p}\rangle}\biggr)\leqslant 0 \quad\text{for } q>2 \end{equation*} \notag$

$\begin{equation*} d_n(W^{\overline{r}}_{\overline{p}}(\mathbb{T}^d),L_q(\mathbb{T}^d)) \underset{\overline{r}, d, \overline{p}, q}{\gtrsim} 1 \end{equation*} \notag$

$\begin{equation} \sum_{i=1}^d \frac{1}{r_i} \biggl(\frac{1}{p_i}-\frac{1}{p_j}\biggr) \geqslant 1. \end{equation} \tag{6.1}$

$\begin{equation} \frac{\langle \overline{r}_j\rangle}{d-1}+\frac{1}{p_j}-\frac{\langle \overline{r}_j\rangle}{\langle \overline{r}_j\circ \overline{p}_j\rangle}\leqslant 0. \end{equation} \tag{6.2}$

$\begin{equation} \frac{\langle \overline{r}_j\rangle}{d-1}+\frac{1}{q}-\frac{\langle \overline{r}_j\rangle}{\langle \overline{r}_j\circ \overline{p}_j\rangle}\leqslant 0 \end{equation} \tag{6.3}$

$\begin{equation*} \frac{\langle \overline{r}\rangle}{d}+\frac 1q-\frac{\langle \overline{r}\rangle}{\langle \overline{r}\circ \overline{p}^*\rangle}<0. \end{equation*} \notag$

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

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.

\Bibitem{Vas24}

\by A.~A.~Vasil'eva

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

\jour Sb. Math.

\yr 2024

\vol 215

\issue 11

\pages 1468--1498

\mathnet{http://mi.mathnet.ru/eng/sm10067}

\crossref{https://doi.org/10.4213/sm10067e}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4858981}

\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2024SbMat.215.1468V}

\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=001419785700003}

\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85217979169}

§ 1. Introduction

§ 2. Preliminaries

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

§ 4. Proof of Theorem 2

§ 5. Proof of Theorem 1

§ 6. Proof of Theorems 3 and 4

Bibliography