| Sbornik: Mathematics |
|
|
|
|
|
This article is cited in 1 scientific paper (total in 1 paper) Dense weakly lacunary subsystems of orthogonal systems and maximal partial sum operator I. V. Limonovaab a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia b Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
Bibliographic databases:
     § 1. IntroductionIn this paper we refine and prove the results announced in [15] (also see [16], Ch. 2). The problem under consideration concerns finding subsystems of orthogonal systems that have the property of lacunarity. The investigations of various classes of lacunary orthonormal systems began in Banach’s works of the 1930s (see [4] and [9]). Such mathematicians as Banach, Marcinkiewicz, Erdős, Agaev, Astashkin, Balykbaev, Bourgain, Vilenkin, Gaposhkin (see the survey [7]), Kashin, Karagulyan, Pisier, Rudin, Sidon, Sepp, Stechkin, Talagrand and others worked in this area in different periods of time. Let $2<p<\infty$. Recall that an orthonormal system of functions $\Phi=\{\varphi_k\}_{k=1}^{\infty}$ is called a $p$-lacunary system, or an $S_p$-system, if for some constant $K$, for each polynomial $P=\sum_{k=1}^N a_k\varphi_k$ in this system we have (see details in [9]). To indicate the dependence on $K$ we also say that $\Phi$ is a $S_p(K)$-system. The following result was presented in [9] (with a reference to Banach [4]).Theorem A. Let $p>2$, and let $\Phi=\{\varphi_k\}_{k=1}^{\infty}$ be an orthonormal system such that Then there exists an infinite set of positive integers $\Lambda$ such that $\{\varphi_k\}_{k\in\Lambda}$ is an $S_p$-system. Let $(X,\mu)$ be a probability space. Below we consider the scale of Orlicz spaces $L_{\psi_{\alpha}}(X)$, where
$$
\begin{equation}
\psi_{\alpha}(t)=t^2\frac{\ln^{\alpha}(e+|t|)}{\ln^{\alpha}(e+1/|t|)}\equiv t^2u_{\alpha}(t), \qquad \alpha>0,
\end{equation}
\tag{1.2}
$$
and the Luxemburg norm of a function $f\in L_{\psi_{\alpha}}(X)$ is defined by
$$
\begin{equation*}
\|f\|_{\psi_{\alpha}}=\inf \biggl\{\lambda>0\colon \int_{X}\psi_{\alpha}\biggl(\frac{f(x)}{\lambda}\biggr)\,d\mu\leqslant 1\biggr\}.
\end{equation*}
\notag
$$
An orthonormal system $\Phi=\{\varphi_k\}_{k=1}^{\infty}$ is said to be $\psi_{\alpha}$-lacunary if for some constant $K$ the inequality $\| P\|_{L_{\psi_{\alpha}}}\leqslant K \| P\|_{L_2}$ holds for each polynomial $P=\sum_{k=1}^N a_k\varphi_k$ with respect to the system $\Phi$. Analogues of Theorem A for orthonormal systems with elements bounded uniformly in the norm of an Orlicz space $L_{\psi_{\alpha}}$ (or a more general space) were established by Balykbaev [2], [3]. Karagulyan [10] showed that for all $\lambda>1$ and $2<q<p$, under the assumptions of Theorem A there exists an $S_q$-subsystem $\{\varphi_{n_k}\}_{k=1}^{\infty}$ such that $n_k<\lambda^k$ for $k>k_0(\lambda)$. The natural question of the maximum density of the sequence $\Lambda$ in Theorem A turns out to be quite complicated. For arbitrary $p>2$ it had remained open even in the case of the trigonometric system until Bourgain published his breakthrough paper [5], where he established the following result. Theorem B. Let $p>2$, and let $\Phi=\{\varphi_k\}_{k=1}^N$ be an orthonormal system such that Here and below we let $\langle N\rangle$ denote the set $\{1, 2,\dots, N \}$, and let $|\Lambda|$ denote the cardinality of the finite set $\Lambda$; in what follows $\log$ denotes $\log_2$. For $\Lambda\subset \langle N\rangle$ we denote by $S_{\Lambda}$ the operator acting by the formula
$$
\begin{equation*}
S_{\Lambda}(\{a_k\}_{k\in\Lambda})=\sum_{k\in\Lambda}a_k\varphi_k.
\end{equation*}
\notag
$$
Note that for even integers $p>2$, provided that the functions have a bounded norm in $L_{p+\delta}$, $0<\delta\leqslant p-2$, a definitive result is due to Agaev (see [1], Theorem 1). Subsequently, Talagrand [17], who used another method, generalized Bourgain’s result to $L_{p,1}$-spaces and obtained some quantitative results for $\theta$-smooth spaces, where $1< \theta\leqslant 2$. Also note the paper [8], where the authors were looking for subsystems $\Phi_{\Lambda}$ such that the operator $S_{\Lambda}\colon l_2(\Lambda)\to L_p$ has a controllable norm in the case when $|\Lambda|$ is of order $N^{2/p}\log^{\beta} N$, $\beta>0$. It is clear that for the set $\Lambda$, whose existence was established in Theorem B, we have
$$
\begin{equation}
\|S_{\Lambda}\colon l_{\infty}(\Lambda) \to L_p(X) \|\leqslant |\Lambda|^{1/2}\cdot K(M, p).
\end{equation}
\tag{1.3}
$$
By weakly lacunary systems we mean ones for which estimates characteristic for ‘sparse’ systems hold, which, however, are weaker than (1.1) — for example, $\psi_{\alpha}$-lacunary systems. In [13], Theorem 1 (also see [12]), using a modification of the method in [5], the authors established Theorem C, which is an analogue of estimate (1.3) for Orlicz spaces $L_{\psi_{\alpha}}$ (see (1.2)) and holds for arbitrary orthogonal systems with uniformly bounded elements. Of course, in this case one can ensure a larger density of the set $\Lambda$ than in Theorem B. Theorem C. Fix $\alpha>0$ and $\rho>0$. Then for an arbitrary orthogonal system $\Phi=\{\varphi_k\}_{k=1}^N$ satisfying the following inequality holds with probability greater than $1-C(\rho)N^{-9}$ for the random set $\Lambda=\Lambda(\omega) =\{i\in\langle N\rangle\colon \xi_i(\omega)=1\}$ generated by a system of independent random variables $\{\xi_i(\omega)\}_{i=1}^N$ taking values $0$ or $1$ such that $\mathbb{E}\xi_i=\log^{-\rho}(N+3)$, $1\leqslant i\leqslant N$: In [1], Theorem 5, Agaev reproduced an example due to Gaposhkin which shows that if the condition of uniform boundedness of functions is replaced by the boundedness of their norms in $L_p$, $p>2$, then, as concerns the number of functions in an $S_p$-subsystem, the situation changes significantly. Theorem D. For $p\geqslant 2$ and $\delta\geqslant 0$ there exists $M(\delta, p)>0$ such that for each $N\geqslant 1$ there exists an orthonormal system $\Phi=\{\varphi_k(x)\}_{k=1}^N$ on $[0,1]$ with the following properties: (1) $\|\varphi_k\|_{p+\delta}\leqslant M(\delta, p)$, $k=1,2,\dots, N$; (2) each $S_p(C)$-subsystem of the system $\Phi$ contains at most $[2C^2N^{\alpha}]$ functions, where In particular, in the case when the $L_p$-norms of functions are uniformly bounded (but no other constraints are imposed) one cannot guarantee the existence of an $S_p$-subsystem of cardinality growing as $N\to\infty$. As concerns the density of weakly lacunary subsystems in $L_{\psi_{\alpha}}$, we can replace the condition that their $L_{\infty}$-norms are bounded by $1$ by the condition
$$
\begin{equation}
\|\varphi_k\|_{L_{p}}\leqslant 1, \qquad k=1, 2,\dots, N,
\end{equation}
\tag{1.5}
$$
for some $p>2$, and Theorem C will still be valid (see Remark 5). Theorem 2 below is a generalization of Theorem C to the case of $S_{\Lambda}$ acting from $l_2(\Lambda)$ to $L_{\psi_{\alpha}}(X)$ and of functions satisfying the weaker condition (1.5) for $p>2$. Note that in [15] we announced Theorem 2 for systems of functions satisfying condition (1.5) for $p>4$. Consider the maximal partial sum operator $S_{\Phi}^*$, which assigns to a vector $\{a_k\}_{k=1}^N$ in $\mathbb{R}^N$ the function
$$
\begin{equation*}
S_{\Phi}^*(\{a_k\}_{k=1}^N)(x)=\sup_{1\leqslant M\leqslant N}\biggl |\sum_{k=1}^M a_k\varphi_k(x)\biggr|.
\end{equation*}
\notag
$$
It is well known (see [14]) that the lacunarity property allows one to improve estimates for the norm of $S_{\Phi}^*$. For example, Stechkin generalized a result of Erdős for the trigonometric system by showing (see [14], Theorem 9.8) that if $\Phi=\{\varphi_k\}_{k=1}^{\infty}$ is an $S_p(K)$-system, then $\|S_{\Phi}^*\colon l_2\to L_p(X)\|\leqslant C(K)$. It follows from Balykbaev’s results in [3] that $S_{\Phi}^*$ is a bounded operator from $l_2$ to $L_{\psi_{\alpha}}(X)$ (and therefore also to $L_2(X)$) in the case of a $\psi_{\alpha}$-lacunary orthonormal system $\Phi$ for $\alpha>4$. It was shown in [13] that for $\rho>4$ each orthogonal system $\{\varphi_k\}_{k=1}^N$ with property (1.4) has a subsystem $\Phi_{\Lambda}$ of $N/\log^{\rho}(N+3)$ functions such that $\|S_{\Phi_{\Lambda}}^{*}\colon l_{\infty}(\Lambda) \to L_{2}(X)\|\leqslant C(\rho)\sqrt{|\Lambda|}$. Theorem 3 below claims that for the subsystem found in Theorem 2 the norm of the maximal partial sum operator from $l_2(\Lambda)$ to $L_2(X)$ has a better estimate than the classical Menshov-Rademacher theorem (for instance, see [14], Theorem 9.1) guarantees for general orthogonal systems. Also note a deep result of Bourgain [6], who showed that under the assumptions of Theorem B the system $\Phi$ can be rearranged so that the norm of the maximal partial sum operator with respect to the resulting system as an operator from $l_2$ to $L_{2}$ is bounded by $C(M)\log\log N$. The main results of the paper, namely, Theorems 2 and 3, are stated and proved in § 4. § 2. Auxiliary results2.1. Estimates related to the norm of the space $L_{\psi_{\alpha}}$In this subsection we prove Lemmas 1 and 2, which we require for the proof of Lemma 8, our main lemma. Corollary 1 will be needed in the proof of Theorem 1. Recall that the function $u_{\alpha}$ was defined in (1.2). Proof. Let $g\in L_p(X)$, $\|g\|_{L_2(X)}\leqslant 1$ and $\|g\|_{L_p(X)}\leqslant K$. Set $C_0=K^{p/(p-2)}>e$. We represent $X$ as a union $X=X_1\cup X_2$, where $X_1=\{x\in X\colon |g(x)|<C_0\}$ and $X_2=\{x\in X\colon |g(x)|\geqslant C_0\}$. Let $g_1=gI_{X_1}$ and $g_2=gI_{X_2}$, where $I_S$ denotes the indicator function of the set $S$. Then we have
$$
\begin{equation}
\biggl\|gu_{\alpha}\biggl(\frac{g}{\lambda}\biggr)\biggr\|_{L_2(X)} \leqslant \biggl\|g_1u_{\alpha}\biggl(\frac{g_1}{\lambda}\biggr)\biggr\|_{L_2(X)} +\biggl\|g_2u_{\alpha}\biggl(\frac{g_2}{\lambda}\biggr)\biggr\|_{L_2(X)}.
\end{equation}
\tag{2.1}
$$
Since $\|g\|_{L_2(X)}\leqslant 1$, it follows that $\|g_1\|_{L_2(X)}\leqslant 1$ and
$$
\begin{equation}
\biggl\|g_1u_{\alpha}\biggl(\frac{g_1}{\lambda}\biggr)\biggr\|_{L_2(X)}\leqslant \|g_1\|_{L_2(X)}\biggl\|u_{\alpha}\biggl(\frac{g_1}\lambda\biggr)\biggr\|_{L_{\infty}(X)}\leqslant 1\cdot \ln^{\alpha}(e+C_0)\leqslant C_1(\alpha, p)\ln^{\alpha}K.
\end{equation}
\tag{2.2}
$$
Because $\|g\|_{L_p(X)}\leqslant K$, we have $\|g_2\|_{L_p(X)}\leqslant K$ and
$$
\begin{equation}
\begin{aligned} \, \!\!\biggl\|g_2u_{\alpha}\biggr(\frac{g_2}{\lambda}\biggr)\biggr\|_{L_2(X)}^2 &\leqslant\sum_{k=0}^{\infty}2^{2k+2}C_0^2\mu\{x\colon2^k C_0\,{\leqslant}\, |g_2(x)|\,{\leqslant}\, 2^{k+1}C_0\}\ln^{2\alpha}(e\,{+}\,2^{k+1}C_0) \nonumber \\ &\leqslant\sum_{k=0}^{\infty}\frac{2^{2k+2}C_0^2K^p\ln^{2\alpha}(e^{2(k+1)}C_0)}{2^{kp}C_0^p} \nonumber \\ &\leqslant\sum_{k=0}^{\infty}2^{2k-kp+2}C_0^{2-p}2^{2\alpha}(k+1)^{2\alpha}K^p\ln^{2\alpha} (eC_0) \nonumber \\ &=C_2(\alpha, p)K^{-p}K^p\ln^{2\alpha}(eK^{p/(p-2)})\leqslant C_3(\alpha,p)\ln^{2\alpha}K. \end{aligned}
\end{equation}
\tag{2.3}
$$
From (2.1)–(2.3) we obtain the result of Lemma 1 and complete the proof. Corollary 1. Let $\alpha>0$ and $p>2$. Then any function $g$ such that $\|g\|_{L_p(X)}\leqslant K$, where $K>e$, and $\|g\|_{L_2(X)}\leqslant 1$ satisfies the inequality Proof. We can assume that $\|g\|_{\psi_{\alpha}}\geqslant 1$, for otherwise (2.4) is obvious. It follows from the definition of $u_{\alpha}$ that $u_{\alpha}=u_{\alpha/2}^2$, so that by Lemma 1, for $\lambda\geqslant 1$ we have
$$
\begin{equation*}
\int_{X} \biggl(\frac{g}{\lambda}\biggr)^2 u_{\alpha}\biggl(\frac{g}{\lambda}\biggr) d\mu= \frac{1}{\lambda^2}\int_{X} g^2 u_{\alpha/2}^2\biggl(\frac{g}{\lambda}\biggr) d\mu\leqslant \frac{1}{\lambda^2} G_{\alpha/2}^2(\lambda, K)\leqslant C^2\biggl(\frac{\alpha}{2}, p\biggr)\frac{\ln^{\alpha}K}{\lambda^2},
\end{equation*}
\notag
$$
which is equal to $1$ for $\lambda= C(\alpha/2, p)\ln^{\alpha/2}K$. This yields inequality (2.4).
The proof is complete. Lemma 2 is a simple consequence of Lemma A (see [13], Lemma 4). Proof. Let $f\in L_2(X)$ and $\|f\|_{L_2(X)}=1$. It follows from the definition of $u_{\alpha}$ (see (1.2)) and Lemma A that
The lemma is proved. 2.2. An estimate for metric entropyThe aim of this subsection is to prove Lemma 7. Note that in [5] (also see [13], Lemma 8) Lemma 7 was proved for uniformly bounded orthogonal systems but it was mentioned in [6] (without further comments) that it also holds for orthogonal systems of functions with uniformly bounded $L_p$-norms, $p>2$. The proof is as in [5] (and/or [6]), but the transition from (2.10) to (2.12) needs a further justification. Definition. For $S\subset L_q(X)$ the metric entropy $N_{q}(S, t)$ is the minimum number of balls in $L_q(X)$ or radius $t$ with centres in $S$ such that their union covers $S$. We state and prove a modification of a classical result on the cardinality of an $\varepsilon$-net for a unit ball in an $m$-dimensional space. Lemma 3. Let $B$ be a unit ball in an $m$-dimensional normed space $X$ and $M\subset B$ be some subset. Then for each $\varepsilon\leqslant 1$ there exists an $\varepsilon$-net $\mathbb{G}$ of $M$ of cardinality at most $(3/\varepsilon)^m$ such that $\mathbb{G}\subset M$. Proof. We use a standard argument with estimates for volumes. Assume without loss of generality that $B$ has its centre at zero. Let $r\colon X\to \mathbb{R}^m$ be the natural isomorphism. Let $G$ be a maximal $\varepsilon$-distinguishable set of vectors in $M$; then $G$ forms an $\varepsilon$-net for $M$. Now we estimate $|G|$. Let $\operatorname{Vol}$ denote the volume of sets in $\mathbb{R}^m$. Let $B_1, \dots, B_{|G|}$ be open balls of radius $\varepsilon/2$ in $S$ with centres in $G$. Clearly, $B_j\subset (1+\varepsilon/2)B$, $\operatorname{Vol}(r(B_j))=(\varepsilon/2)^m\operatorname{Vol}(r(B))$, $j=1,\dots, |G|$, and the sets $r(B_j)$ with distinct indices $j$ are disjoint. Therefore, which yields $|G|\leqslant (2/\varepsilon+1)^m\leqslant (3/\varepsilon)^m$, so that $G$ is the required net.
The proof is complete. In what follows we often use the following well-known estimate for binomial coefficients: for some absolute positive constant $C$ and $1\leqslant m\leqslant n$
$$
\begin{equation}
\log \binom{n}{m} < Cm\log\biggl(\frac{n}{m}+1\biggr).
\end{equation}
\tag{2.6}
$$
The following simple result is often used for estimates of entropy. Lemma 4. Let $\Phi=\{\varphi_{i}\}_{i=1}^{n}$ be an orthogonal system of functions in $L_q(X)$, $q\geqslant 2$, let $I\subset\{1,\dots, n\}$, $|I|=m_0$, and let $b_1, b_2\in\mathbb{R}$, $0<b_1<b_2$. Then Proof. Set $\mathcal{P}_I = \bigl\{\sum_{i\in I}a_i\varphi_i \colon \|\overline{a}\|_2\leqslant 1\bigr\}$ and $N_{b_2} = N_q(\mathcal{P}_I, b_2)$. Let $B_1, \dots,B_{N_{b_2}}$ be balls of radius $b_2$ in the $m_0$-dimensional space $\bigl\{\sum_{i\in I}a_i\varphi_i \colon a_i\in\mathbb{R}\bigr\}$ which have centres $f_1, \dots, f_{N_{b_2}}\in \mathcal{P}_I$ and cover $\mathcal{P}_I$. Using Lemma 3, for each set $B_j\cap \mathcal{P}_I$, $j=1,\dots, N_{b_2}$, we find a cover consisting of at most $(3b_2/b_1)^{m_0}$ balls of radius $b_1$ with centres in $\mathcal{P}_I$. The union of the covers of the sets $B_1\cap \mathcal{P}_I,\dots, B_{N_{b_2}}\cap \mathcal{P}_I$ is a $b_1$-cover of $\mathcal{P}_I$; it contains at most $N_{b_2}\cdot(3b_2/b_1)^{m_0}$ elements, so that
$$
\begin{equation*}
N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i \colon \|\overline{a}\|_2\leqslant 1\biggr\}, b_1\biggr) \leqslant N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i \colon \|\overline{a}\|_2\leqslant 1\biggr\}, b_2\biggr)\cdot \biggl(\frac{3b_2}{b_1}\biggr)^{m_0},
\end{equation*}
\notag
$$
which yields Lemma 4. Corollary 2. Under the assumptions of Lemma 4 let $\|\varphi_i\|_q\leqslant K$, $i= 1,\dots, n$. Then the following estimate holds for $0<b_1<b_2=\sqrt{m_0}K$: Proof. It is sufficient to observe that for $\|\overline{a}\|_2\leqslant 1$ we have
$$
\begin{equation*}
\biggl\|\sum_{i\in I}a_i\varphi_i\biggr\|_q \leqslant \sum_{i\in I}\|a_i\varphi_i\|_q \leqslant \sum_{i\in I}|a_i|K \leqslant \sqrt{m_0}K,
\end{equation*}
\notag
$$
which means that
$$
\begin{equation*}
N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i \colon \|\overline{a}\|_2\leqslant 1\biggr\}, \sqrt{m_0}K\biggr)=1.
\end{equation*}
\notag
$$
Then we can use Lemma 4.
The corollary is proved. Given a system $\Phi=\{\varphi_{i}\}_{i=1}^{n}$ and a number $m\leqslant n$, we define a set $\mathcal{P}_{m}$:
$$
\begin{equation}
\mathcal{P}_{m}=\biggl\{\sum_{i \in A} a_{i} \varphi_{i}\colon \| \overline{a} \|_2 \leqslant 1 \text{ and } |A| \leqslant m\biggr\}.
\end{equation}
\tag{2.7}
$$
Lemma 5. The following inequality holds under the assumptions of Lemma 4: Proof. For $I\subset\langle n\rangle$ set $\mathcal{P}_I=\{\sum_{i\in I} a_i\varphi_i\colon \|\overline{a}\|_2\leqslant 1\}$. To each $f\in\mathcal{P}_m$ we can assign some set $I(f)\subset\langle n\rangle$ of cardinality $|I(f)|=m$ such that $f\in\mathcal{P}_{I(f)}$. Fix such a correspondence. Let $B_1,\dots, B_{N_q(\mathcal{P}_m, b_2)}$ be balls of radius $b_2$ with centres $f_1, \dots, f_{N_q(\mathcal{P}_m, b_2)}\in \mathcal{P}_m$ that cover $\mathcal{P}_m$. We will construct a cover $\Omega$ of the set $\mathcal{P}_m$ by at most $\binom{n}{m}N_q(\mathcal{P}_m, b_2)$ balls of radius $2b_2$ that has the following property: for each $f\in\mathcal{P}_m$ there is a ball in $\Omega$ with centre $\widetilde f$ in $\mathcal{P}_{I(f)}$ that covers $f$ and, moreover, $I(\widetilde f)=I(f)$. We construct the set $Z$ of centres of balls in $\Omega$. First we put in $Z$ the points $f_1, \dots, f_{N_q(\mathcal{P}_m, b_2)}$. Let $k\in\{1,\dots, N_q(\mathcal{P}_m, b_2)\}$ and $J\subset \langle n\rangle$, $|J|=m$, satisfy $J\neq I(f_k)$. Set $M_{J, k}=\{f\in\mathcal{P}_J\colon \|f-f_k\|_q\leqslant b_2, \ I(f)=J\}$. If $M_{J, k}$ is nonempty, then we add to $Z$ some (arbitrary) element of $M_{J, k}$. Performing the same procedure for all $k=1,\dots, N_q(\mathcal{P}_m, b_2)$ and $J\subset \langle n\rangle$ such that $|J|=m$ and $J\neq I(f_k)$, we obtain the required cover. Let $f\in Z$, and let $B$ be a ball in $\Omega$ with centre $f$. By Lemma 3, any set $B\cap\mathcal{P}_{I(f)}$, which is a subset of a ball of radius $2b_2$ in the $m$-dimensional space $\bigl\{\sum_{i\in I(f)}a_i\varphi_i \colon a_i\in\mathbb{R}\bigr\}$, can be covered by at most $(6b_2/b_1)^m$ balls of radius $b_1$ with centres in $B\cap\mathcal{P}_{I(f)}$. The system of all these balls for all $f\in Z$ is a $b_1$-cover of $\mathcal{P}_m$ with at most $\binom{n}{m}N_q(\mathcal{P}_m, b_2)(6b_2/b_1)^m$ elements, so that
$$
\begin{equation*}
N_q(\mathcal{P}_m, b_1)\leqslant \binom{n}{m}N_q(\mathcal{P}_m, b_2)\biggl(\frac{6b_2}{b_1}\biggr)^m.
\end{equation*}
\notag
$$
This and (2.6) yield Lemma 5. Lemma 6. Let $\Phi=\{\varphi_{i}\}_{i=1}^{n}$ be an orthogonal system of functions such that $\|\varphi_{i}\|_{q_0}\leqslant 1$, $i=1,\dots, n$, for some $q_0>2$. Then for $2< q\leqslant q_0$ there exists $c(q)>1$ such that for $t>c(q)$ Proof. For any $\overline{a}$ and $A\in\langle n\rangle$ such that $\|\overline{a}\|_2\leqslant 1$ and $|A|\leqslant m$ we have
$$
\begin{equation*}
\biggl\|\sum_{i\in A}a_i\varphi_i\biggr\|_q \leqslant \sum_{i\in A}\|a_i\varphi_i\|_q \leqslant \sum_{i\in A}|a_i| \leqslant \sqrt{m},
\end{equation*}
\notag
$$
so for $t\geqslant\sqrt{m}$ inequality (2.8) is obviously true. Let $t<\sqrt{m}$.
Assume that $2^{(k-2)/2}\leqslant t<2^{(k-1)/2}$, $k\geqslant 4$. We consider a function $f=\sum_{i \in A} a_{i} \varphi_{i}$, where $\|\overline{a}\|_2\leqslant 1$ and $|A|\leqslant m$, and write the chain of equalities
$$
\begin{equation}
\begin{aligned} \, &\sum_{i\in A} a_{i} \varphi_{i}(u) =\sum_{i\in A} a_{i} \varepsilon_{i}^{1} \varphi_{i}(u)+\sum_{i\in A} a_{i}(1-\varepsilon_{i}^{1}) \varphi_{i}(u) \nonumber \\ &\qquad =\sum_{i\in A} a_{i} \varepsilon_{i}^{1} \varphi_{i}(u)+\sum_{i\in A} a_{i}(1-\varepsilon_{i}^{1}) \varepsilon_{i}^{2} \varphi_{i}(u)+\sum_{i\in A} a_{i}(1-\varepsilon_{i}^{1})(1-\varepsilon_{i}^{2}) \varphi_{i}(u)=\cdots \nonumber \\ \nonumber &\qquad=\sum_{i\in A} a_{i} \varepsilon_{i}^{1} \varphi_{i}(u)+\sum_{i\in A} a_{i}(1-\varepsilon_{i}^{1}) \varepsilon_{i}^{2} \varphi_{i}(u)+\dotsb \\ \nonumber &\qquad\qquad +\sum_{i\in A} a_{i}(1-\varepsilon_{i}^{1}) \dotsb(1-\varepsilon_{i}^{k-1}) \varepsilon_{i}^{k} \varphi_{i}(u) +\sum_{i\in A} a_{i}(1-\varepsilon_{i}^{1}) \dotsb(1-\varepsilon_{i}^{k}) \varphi_{i}(u) \\ &\qquad\equiv \Phi(\varepsilon, u)+\sum_{i\in A} a_{i}(1-\varepsilon_{i}^{1}) \dotsb(1-\varepsilon_{i}^{k}) \varphi_{i}(u), \end{aligned}
\end{equation}
\tag{2.9}
$$
where $(\varepsilon_i^j)_{1\leqslant i\leqslant n, 1\leqslant j\leqslant k}$ take the values $\pm{1}$ in an arbitrary way.
Note that
$$
\begin{equation*}
\begin{aligned} \, &\int\|\Phi(\varepsilon, u)\|_{L_{q}(d u)} \,d \varepsilon \leqslant \biggl\|\biggl\|\sum_{i\in A} a_{i} \varepsilon_{i}^{1} \varphi_{i}(u)\biggr\|_{L_q(du)}+\biggl\|\sum_{i\in A} a_{i}(1-\varepsilon_{i}^{1}) \varepsilon_{i}^{2} \varphi_{i}(u)\biggr\|_{L_q(du)} \\ &\qquad\qquad+\dots+\biggl\|\sum_{i\in A} a_{i}(1-\varepsilon_{i}^{1}) \dotsb (1-\varepsilon_{i}^{k-1}) \varepsilon_{i}^{k} \varphi_{i}(u)\biggr\|_{L_q(du)}\biggr\|_{L_1(d\varepsilon)} \\ &\qquad\leqslant \biggl\|\sum_{i\in A} a_{i} \varepsilon_{i}^{1} \varphi_{i}(u)\biggr\|_{L_q(du \otimes d\varepsilon^1)} \\ &\qquad\qquad +\sum_{l=2}^k \int\biggl\|\sum_{i\in A} a_{i}(1-\varepsilon_{i}^{1}) \dotsb(1-\varepsilon_{i}^{l-1}) \varepsilon_{i}^{l} \varphi_{i}(u)\biggr\|_{L_{q}(d u \otimes d {\varepsilon}^{l})}\,d \varepsilon^{1} \dotsb d \varepsilon^{l-1}. \end{aligned}
\end{equation*}
\notag
$$
Then by Khinchin’s inequality
$$
\begin{equation}
\begin{aligned} \, \notag &\int\|\Phi(\varepsilon, u)\|_{L_{q}(d u)} \,d \varepsilon \leqslant \biggl(\int\biggl(\biggl(\int \biggl|\sum_{i\in A} a_{i} \varepsilon_{i}^{1} \varphi_{i}(u)\biggr|^q d\varepsilon^1\biggr)^{1/q}\biggr)^q \,du\biggr)^{1/q} \\ \notag &\ {+}\sum_{l=2}^k \int\biggl( \int\biggl(\int\biggl|\sum_{i\in A} a_{i}(1-\varepsilon_{i}^{1}) \dotsb(1-\varepsilon_{i}^{l-1}) \varepsilon_{i}^{l} \varphi_{i}(u)\biggr|^q \,d\varepsilon^l\biggr)^{{1}/{q}\cdot q} \,du\biggr)^{1/q} d \varepsilon^{1} \dotsb d \varepsilon^{l-1} \\ \notag &\leqslant C(q)\biggl(\int\biggl(\sum_{i\in A} a_i^2\varphi_i^2(u)\biggr)^{q/2}\, du\biggr)^{1/q} \\ &\ + C(q) \sum_{l=2}^k \int\biggl(\int\biggl(\sum_{i\in A}a_i^2(1-\varepsilon_i^1)^2\dotsb(1-\varepsilon_i^{l-1})^2 \varphi_i^2(u)\biggr)^{q/2}\,du\biggr)^{1/q}\,d\varepsilon. \end{aligned}
\end{equation}
\tag{2.10}
$$
We estimate the first term in (2.10) taking the inequalities $\|\varphi_i\|_q\leqslant \|\varphi_i\|_{q_0}\leqslant 1$, $i\in\langle n\rangle$, into account:
$$
\begin{equation*}
\begin{aligned} \, &\biggl(\int\biggl(\sum_{i\in A} a_i^2\varphi_i^2(u)\biggr)^{q/2} \,du\biggr)^{1/q} {=}\,\biggl\|\sum_{i\in A} a_i^2\varphi_i^2(u)\biggr\|_{L_{q/2}(du)}^{1/2} {\leqslant}\, \biggl(\sum_{i\in A} \|a_i^2\varphi_i^2(u)\|_{L_{q/2}(du)}\biggr)^{1/2} \\ &\qquad =\biggl(\sum_{i\in A} a_i^2\biggl(\int |\varphi_i(u)|^q \,du\biggr)^{2/q}\biggr)^{1/2}\leqslant \biggl(\sum_{i\in A} a_i^2\biggr)^{1/2}\leqslant 1. \end{aligned}
\end{equation*}
\notag
$$
In a similar way we estimate the second term in (2.10): for $1\leqslant l\leqslant k$ we have
$$
\begin{equation}
\begin{aligned} \, \nonumber &\int\biggl(\int\biggl(\sum_{i\in A}{a_i^2(1-\varepsilon_i^1)^2 \dotsb(1-\varepsilon_i^l)^2}\varphi_i^2(u)\biggr)^{q/2}\,du\biggr)^{1/q}\,d\varepsilon \\ \nonumber &\ =\biggl\|\biggl(\biggl\|\sum_{i\in A}{a_i^2(1-\varepsilon_i^1)^2\cdots(1-\varepsilon_i^l)^2}\varphi_i^2(u) \biggr\|_{L_{q/2}(d u)}\biggr)^{1/2}\biggr\|_{L_1(d\varepsilon)} \\ \nonumber &\ \leqslant\biggl\|\biggl(\sum_{i\in A}\|{a_i^2(1-\varepsilon_i^1)^2 \dotsb(1-\varepsilon_i^l)^2}\varphi_i^2(u)\|_{L_{q/2}(d u)}\biggr)^{1/2}\biggr\|_{L_1(d\varepsilon)} \\ \nonumber &\ =\biggl\|\biggl(\sum_{i\in A}a_i^2(1-\varepsilon_i^1)^2 \dotsb(1-\varepsilon_i^l)^2\|\varphi_i^2(u) \|_{L_{q/2}(d u)}\biggr)^{1/2}\biggr\|_{L_1(d\varepsilon)} \\ \nonumber &\ \leqslant\int\biggl(\sum_{i\in A}{a_i^2(1-\varepsilon_i^1)^2 \dotsb(1-\varepsilon_i^l)^2}\biggr)^{1/2}\,d\varepsilon\,{\leqslant}\,\biggl(\int\sum_{i\in A} a_i^2(1-\varepsilon_i^1)^2 \dotsb(1-\varepsilon_i^l)^2\,d\varepsilon\!\biggr)^{1/2} \\ &\ =\biggl(\sum_{i\in A} a_i^2\int(1-\varepsilon_i^1)^2 \dotsb(1-\varepsilon_i^l)^2\,d\varepsilon\biggr)^{1/2} =\biggl(\sum_{i\in A} \frac{a_i^2 2^{2l}}{2^l}\biggr)^{1/2}\leqslant 2^{l/2}. \end{aligned}
\end{equation}
\tag{2.11}
$$
Continuing (2.10) we obtain
$$
\begin{equation}
\int\|\Phi(\varepsilon, u)\|_{L_{q}(d u)}\, d \varepsilon \leqslant C(q) \biggl(1+\sum_{l=1}^{k-1} 2^{l/2}\biggr)<C_1(q) 2^{k/2}<c_1 t
\end{equation}
\tag{2.12}
$$
(throughout this proof $c_1=c_1(q)$).
Set
$$
\begin{equation*}
A_{\varepsilon}=\{i\in A\colon\varepsilon_i^1= \dots=\varepsilon_i^k=-1\};
\end{equation*}
\notag
$$
then $|A_{\varepsilon}|=2^{-k}\sum_{i\in A}(1-\varepsilon_i^1)\dotsb(1-\varepsilon_i^k)$, and therefore
$$
\begin{equation}
\int|A_{\varepsilon}|\,d\varepsilon=2^{-k}\sum_{i\in A}\int(1-\varepsilon_i^1)\dotsb(1-\varepsilon_i^k)\,d\varepsilon\leqslant 2^{-k}m<\frac{m}{2t^2}.
\end{equation}
\tag{2.13}
$$
Moreover, from (2.11), for $l=k$ we have
$$
\begin{equation}
\int\biggl(\sum_{i\in A}{a_i^2(1-\varepsilon_i^1)^2\dotsb(1-\varepsilon_i^k)^2}\biggr)^{1/2}\,d\varepsilon\leqslant 2^{k/2}<c_2t.
\end{equation}
\tag{2.14}
$$
Taking (2.9) and (2.12)–(2.14) into account we can find a sufficiently large $\widetilde{c}(c_1, c_2)\equiv \widetilde{c}>3$ and can select the signs $\varepsilon_i^j$ so that the function
$$
\begin{equation*}
\varphi\equiv\sum_{i\in A} a_i(1-\varepsilon_i^1)\dotsb(1-\varepsilon_i^k)\varphi_i
\end{equation*}
\notag
$$
satisfies the conditions
$$
\begin{equation}
\biggl\| \sum_{i\in A} a_i\varphi_i-\varphi\biggr\|_q\leqslant \frac{\widetilde{c}}{2}t\quad\text{and} \quad \varphi\in \frac{\widetilde{c}}{2}t\mathcal{P}_{[{m}/{t^2}]}.
\end{equation}
\tag{2.15}
$$
Let $\widetilde{N}_q(\mathcal{P}_m, r)$ be the minimum number of balls of radius $r$ in $L_q(X)$ (not necessarily with centres in $\mathcal{P}_m$) such that their union covers $\mathcal{P}_m$. It follows from (2.15) that each $\widetilde{c}t/2$-net for $\frac{\widetilde{c}}{2}t\mathcal{P}_{[{m}/{t^2}]}$ is a $\widetilde{c}t$-net for $\mathcal{P}_m$. Therefore,
$$
\begin{equation*}
\begin{aligned} \, \log \widetilde{N}_q(\mathcal{P}_m, \widetilde{c}t) &\leqslant \log \binom{n}{[{m}/{t^2}]}+\sup_{|I|=[{m}/{t^2}]} \log N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i\colon \| \overline{a} \|_2\leqslant \frac{\widetilde{c}}{2}t \biggr\}, \frac{\widetilde{c}}{2}t \biggr) \\ &\leqslant C\frac{m}{t^2}\log\frac{nt^2}{m}+\sup_{|I|=[{m}/{t^2}]} \log N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i\colon \| \overline{a} \|_2\leqslant 1\biggr\}, 1\biggr). \end{aligned}
\end{equation*}
\notag
$$
Set $c=2\widetilde{c}$. Clearly, $N_q(\mathcal{P}_m, ct)\leqslant \widetilde{N}_q(\mathcal{P}_m, \widetilde{c}t)$, so that
$$
\begin{equation}
\log {N}_q(\mathcal{P}_m, ct)\leqslant C\frac{m}{t^2}\log\frac{nt^2}{m}+\sup_{|I|=[{m}/{t^2}]} \log N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i\colon \| \overline{a} \|_2\leqslant 1\biggr\}, 1\biggr).
\end{equation}
\tag{2.16}
$$
If $m/t^4\leqslant 1$, then we use Corollary 2 and obtain the following estimate:
$$
\begin{equation*}
\sup_{|I|=[{m}/{t^2}]}\log N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i\colon \| \overline{a} \|_2\leqslant 1\biggr\}, 1\biggr) \leqslant \frac{m}{t^2}\log\biggl(3\sqrt\frac{m}{t^2}\biggr) \leqslant \frac{m}{t^2}\log (3t).
\end{equation*}
\notag
$$
Now suppose that ${m}/{t^{2r+2}}\leqslant 1<{m}/{t^{2r}}$, where $r\in\mathbb{N}$, $r>1$. We estimate the second term in (2.16) using the same method involving a reduction of the support as in establishing (2.16); it is important here that $c_1$, $c_2$ and $c$ depend only on $q$. The same arguments as above show that for $t>2$ and each $I\subset\langle n\rangle$ we have
$$
\begin{equation}
\begin{aligned} \, \nonumber &\log N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i\colon \| \overline{a} \|_2\leqslant 1\biggr\}, ct\biggr) \leqslant \log \widetilde{N}_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i\colon \| \overline{a} \|_2\leqslant 1\biggr\}, \widetilde{c}t\biggr) \\ \nonumber &\qquad \leqslant\log \binom{|I|}{[|I|/t^2]} +\sup_{J\subset I\colon |J|=[|I|/t^2]}\log N_q\biggl(\biggl\{\sum_{i\in J}a_i\varphi_i\colon \| \overline{a} \|_2\leqslant\frac{\widetilde{c}t}2\biggr\}, \frac{\widetilde{c}t}2\biggr) \\ &\qquad \leqslant C\frac{|I|}{t^2}\log t^2+\sup_{J\subset I\colon |J|=[|I|/t^2]} \log N_q\biggl(\biggl\{\sum_{i\in J}a_i\varphi_i\colon \| \overline{a} \|_2\leqslant 1\biggr\}, 1\biggr) \end{aligned}
\end{equation}
\tag{2.17}
$$
(if $|I|/t^2<1$, then the last inequality is obvious; for $|I|/t^2\geqslant 1$ we have used (2.6) and the fact that $|I|/[|I|/t^2]\leqslant 2t^2\leqslant t^3$). Note that in (2.17) we can take the greatest integer function in place of $[|I|/t^2]$.
Using Lemma 4 and inequality (2.17) $r-1$ times sequentially, we obtain
$$
\begin{equation*}
\begin{aligned} \, &\sup_{|I|=[{m}/{t^2}]} \log N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i\colon \| \overline{a} \|_2\leqslant 1\biggr\}, 1\biggr) \\ &\ \ \leqslant \frac{m}{t^2}\log(3ct)+ \sup_{|I|=[{m}/{t^2}]} \log N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i\colon\|\overline{a}\|_2\leqslant 1\biggr\}, ct\biggr) \\ &\ \ \leqslant \frac{m}{t^2}\log(3ct)+ C\frac{m}{t^4}\log {t^2} +\sup_{|I|=[{m}/{t^4}]}\log N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i\colon \|\overline{a}\|_2\leqslant 1\biggr\}, 1\biggr)\leqslant\dotsb \\ &\ \ \leqslant \biggl(\frac{m}{t^2}+\frac{m}{t^4} +\dots+\frac{m}{t^{2{r-2}}}\biggr)\log(3ct) + C\biggl(\frac{m}{t^4}+\frac{m}{t^6} +\dots+\frac{m}{t^{2r}}\biggr)\log t^2 \\ &\ \ \qquad +\sup_{|I|=[{m}/{t^{2r}}]}\log N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i\colon \|\overline{a}\|_2\leqslant 1\biggr\}, 1\biggr) \\ &\ \ \leqslant\biggl(\frac{m}{t^2}+\frac{m}{t^4}+\frac{m}{t^6} +\dotsb\biggr)\log(3ct) + C\biggl(\frac{m}{t^4}+\frac{m}{t^6} +\dotsb\biggr)\log t^2+ \frac{m}{t^{2r}}\log\biggl(3\sqrt\frac{m}{t^{2r}}\biggr) \\ &\ \ <C_1\frac{m}{t^2}\log t; \end{aligned}
\end{equation*}
\notag
$$
here in the next to the last inequality we used Corollary 2, while the last inequality holds because $t>2$. Each $t_1>2c$ can be represented as $t_1=ct$ for $t>2$, so that combining the estimates obtained we have
$$
\begin{equation*}
\begin{aligned} \, &\log N_q(\mathcal{P}_m, t_1)=\log N_q(\mathcal{P}_m, ct)\leqslant C\frac{m}{t^2}\log\frac{nt^2}{m}+C_2\frac{m}{t^2}\log t \\ &\qquad \leqslant C_3 m\biggl(\log \biggl(\frac{n}{m}+1\biggr)+\log{t}\biggr) t^{-2}\leqslant C(q) m\log \biggl(\frac{n}{m}+1\biggr)\cdot t_1^{-2}\log{t_1}. \end{aligned}
\end{equation*}
\notag
$$
Lemma 6 is proved. Lemma 7. Let $\Phi=\{\varphi_{i}\}_{i=1}^{n}$ be an orthogonal system of functions such that $\|\varphi_{i}\|_{q_0}\leqslant 1$, $i=1,\dots, n$, for some $q_0>2$. Then for $2< q< q_0$ there exist $C(q_0, q)>0$ and $\eta=\eta(q_0,q)>2$ such that Proof. First we prove the first inequality in (2.18) for $t$ greater than some ${c(q_0,q)\!>\!1}$. Let
$$
\begin{equation*}
\frac{1}{q}=\frac{1-\theta}{2}+\frac{\theta}{q_0}, \qquad 0<\theta<1, \quad \theta=\theta(q_0,q),
\end{equation*}
\notag
$$
where $c(q_0)$ is the constant in Lemma 6. For any $f, g \in \mathcal{P}_{m}$ we have $\|f\|_2\leqslant 1$ and $\|g\|_2\leqslant 1$, so that by Hölder’s inequality
$$
\begin{equation*}
\|f-g\|_{q} \leqslant\|f-g\|_{2}^{1-\theta}\|f-g\|_{q_0}^{\theta} \leqslant 2\|f-g\|_{q_0}^{\theta}.
\end{equation*}
\notag
$$
Hence for $t>2c(q_0)^{\theta}$, from Lemma 6 we have
$$
\begin{equation*}
\begin{aligned} \, \log N_{q}(\mathcal{P}_{m}, t) &\leqslant \log N_{q_0}\biggl(\mathcal{P}_{m},\biggl(\frac{t}{2}\biggr)^{1 / \theta}\biggr) \\ &\leqslant C(q_0) m\log \biggl(\frac{n}{m}+1\biggr)\cdot \biggl(\frac{t}{2}\biggr)^{-2 / \theta} \log \biggl(\frac{t}{2}\biggr)^{1 / \theta}, \end{aligned}
\end{equation*}
\notag
$$
where $t^{-2 / \theta} \log ({t}/{2})^{1 / \theta}<t^{-\eta}$ (for $t>c(q_0,q)$) for some $\eta=\eta(\theta)>2$.
For $1/2<t \leqslant c(q_0,q)$, from Lemma 5 and inequality (2.18) for $2c(q_0,q)t$ we obtain
$$
\begin{equation*}
\begin{aligned} \, &\log N_{q}(\mathcal{P}_{m}, t) \leqslant Cm\log\biggl(\frac{n}{m}+1\biggr)+C m \log (2c(q_0,q))+\log N_{q} (\mathcal{P}_{m}, 2c(q_0,q)t) \\ &\qquad \leqslant Cm\log\biggl(\frac{n}{m}+1\biggr)+ C_1(q_0,q)m+ C(q_0,q) m\log \biggl(\frac{n}{m}+1\biggr)\cdot (2c(q_0,q)t)^{-\eta} \\ &\qquad \leqslant \widetilde{C}(q_0,q) m\log \biggl(\frac{n}{m}+1\biggr)\cdot t^{-\eta} \end{aligned}
\end{equation*}
\notag
$$
(we can satisfy the last inequality by taking a larger constant $\widetilde{C}(q_0,q)$). Hence the first inequality in (2.18) is proved.
To obtain the second inequality in (2.18) we use Lemma 4:
$$
\begin{equation*}
\begin{aligned} \, &\log N_{q}(\mathcal{P}_{m}, t) \leqslant \log \binom{n}{m}+\sup _{|A| \leqslant m} \log N_{q}\biggl(\biggl\{\sum_{i \in A} a_{i} \varphi_{i}\colon \| \overline{a} \|_2 \leqslant 1\biggr\}, t\biggr) \\ &\qquad\leqslant C m \log \biggl(\frac{n}{m}+1\biggr)+ m \log \frac{3}{t}+\sup _{|A| \leqslant m} \log N_{q}\biggl(\biggl\{\sum_{i \in A} a_{i} \varphi_{i}\colon \| \overline{a} \|_2 \leqslant 1\biggr\}, 1\biggr) \\ &\qquad\leqslant C(q_0,q)\biggl(\log \biggl(\frac{n}{m}+1\biggr)+\log \frac{1}{t}\biggr) m \leqslant 2C(q_0,q) m\log \biggl(\frac{n}{m}+1\biggr)\cdot \log \frac{1}{t} \end{aligned}
\end{equation*}
\notag
$$
(apart from Lemma 4, in the second inequality we also used (2.6), and in the next to the last inequality we used Lemma 7 for $n=m$ and $t=1$).
The proof is complete. § 3. Main lemmaLet $U_2(\Lambda)\subset\mathbb{R}^N$ be the set of vectors of Euclidean norm $1$ and with support in $\Lambda\subset\langle N\rangle$ such that all their nonzero components have the same modulus:
$$
\begin{equation}
U_2(\Lambda)=\biggl\{\overline{a}=\{a_i\}_{i=1}^N\colon \operatorname{supp}(\overline{a})\subset \Lambda \text{ and for } i\in\operatorname{supp}(\overline{a})\ \ |a_i|=\frac{1}{\sqrt{|\operatorname{supp}(\overline{a})|}}\biggr\},
\end{equation}
\tag{3.1}
$$
where $\operatorname{supp}(\overline{a})=\{i\in\langle N\rangle\colon a_i\neq 0\}$. For $m_0\leqslant |\Lambda|$ we set $U_2(\Lambda, m_0) =\{\overline{a}\in U_2(\Lambda)\colon |{\operatorname{supp}(\overline{a})}|=m_0\}$. For $m\in\langle N\rangle$ let $H_{m}$ denote the family of subsets $A$ of $ \langle N\rangle$ such that $|A|\leqslant m$. Let $\{\xi_i(\omega)\}_{i=1}^N$ be a system of independent random variables (selectors) defined on the probability space $(\Omega, \nu)$ such that
$$
\begin{equation}
\xi_i(\omega)=0 \quad\text{or}\quad \xi_i(\omega)=1,\quad\text{and} \quad \mathbb{E}\xi_i=\delta, \quad 1\leqslant i\leqslant N.
\end{equation}
\tag{3.2}
$$
For $\omega\in\Omega$ set
$$
\begin{equation}
\Lambda_{\omega}=\{i\in\langle N\rangle\colon \xi_i(\omega)=1\}.
\end{equation}
\tag{3.3}
$$
Also for $m_0\leqslant m$ let (here $\alpha$ is assumed to be fixed)
$$
\begin{equation}
J_{m, m_0, \rho}(\omega)= \sup_{\substack{A\in H_{m} \\ \overline{a}\in U_2(A, m_0)}} \biggl\|\sum_{i\in A}{a_i\xi_i(\omega)\varphi_i(x)}\biggr\|_{\psi_{\alpha}}
\end{equation}
\tag{3.4}
$$
(in what follows $\delta=\log^{-\rho}(N+3)$, so we write $J_{m, m_0, \rho}$). Proceeding as in [13] we find an upper estimate for $\|J_{m, m_0, \rho}(\omega)\|_{q_0}$ in the case when $q_0=\log N$. Taking this value of $q_0$ has an advantage: if $\|f\|_{q_0}=1$ for $q_0=\log N$, then for some absolute constant $K$ we have Lemma 8 (main lemma). Let $\Phi=\{\varphi_k\}_{k=1}^N$ be an orthogonal system that for some $p>2$ has the following property: Remark 1. It follows from (3.18) that under the assumptions of Lemma 8, if ${m_0>N/\log^b (N+3)>\log^4(N+3)}$ for some $b>0$, then To prove Lemma 8 we introduce some notation and establish Lemma 9. For ${A\in H_{m}}$, $\overline{a}\in U_2(A, m_0)$ and $m_0\leqslant|A|$ set
$$
\begin{equation}
f_{m, A, \overline{a}}(\omega, x)=\sum_{i\in A}a_i\xi_i(\omega)\varphi_i(x),
\end{equation}
\tag{3.8}
$$
and for $\lambda>0$ let
$$
\begin{equation}
F_{m,m_0,\lambda}(\omega)\equiv F_{\lambda}(\omega)=\sup_{\substack{A\in H_{m}\\ \overline{a}\in U_2(A, m_0)}}\int_X{\psi_{\alpha}\biggl(\frac{f_{m, A,\overline{a}}(\omega, x)}{\lambda}\biggr) d\mu}
\end{equation}
\tag{3.9}
$$
(here $m>0$, $m_0\leqslant m$ and $\rho>0$ are assumed to be fixed). Lemma 9. For some $\lambda\geqslant 1$, $q_0 \geqslant 1$, $m\in\langle N\rangle$ and $m_0\leqslant m$ let $\|F_{\lambda}(\omega)\|_{q_0}\leqslant 1$. Then $\|J_{m, m_0, \rho}(\omega)\|_{q_0}\leqslant 2\lambda$. Proof. We represent $\Omega$ in the form $\Omega=B\cup C$, where
$$
\begin{equation*}
B\equiv \{\omega\in\Omega\colon J_{m,m_0,\rho}(\omega)<\lambda\}\quad\text{and} \quad C\equiv\{\omega\in\Omega\colon J_{m,m_0,\rho}(\omega) \geqslant \lambda\}.
\end{equation*}
\notag
$$
Let $\omega\in C$. We denote by $\widetilde{A}, \widetilde{\overline{a}}$ a pair providing the supremum in (3.4) for this $\omega$. Then, since $\psi_{\alpha}$ is convex (see [3]) and $\psi_{\alpha}(0)=0$, for $t\geqslant 1$ and $z>0$ we have ${\psi_{\alpha}(tz)\geqslant t\psi_{\alpha}(z)}$ and
$$
\begin{equation*}
\begin{aligned} \, F_{\lambda}(\omega) &=\sup_{\substack{A\in H_{m}\\ \overline{a}\in U_2(A, m_0)}} \int_X \psi_{\alpha}\biggl(\frac{f_{m, A,\overline{a}}(\omega, x)}{\lambda}\biggr)d\mu \geqslant \int_X\psi_{\alpha} \biggl(\frac{f_{m, \widetilde{A},\widetilde{\overline{a}}}(\omega, x)}{\lambda}\biggr)d\mu \\ &\geqslant\frac{\| f_{m, \widetilde{A},\widetilde{\overline{a}}}(\omega, x)\|_{\psi_{\alpha}}}{\lambda} \int_X\psi_{\alpha}\biggl(\frac{f_{m, \widetilde{A},\widetilde{\overline{a}}}(\omega, x)} {\| f_{m, \widetilde{A},\widetilde{\overline{a}}}(\omega, x)\|_{\psi_{\alpha}}}\biggr)d\mu \\ &=\frac{\| f_{m, \widetilde{A},\widetilde{\overline{a}}}(\omega, x)\|_{\psi_{\alpha}}}{\lambda}=\frac{J_{m,m_0,\rho}(\omega)}{\lambda}, \end{aligned}
\end{equation*}
\notag
$$
so that (here we denote by $\chi_C$ the characteristic function of the set $C$). Then we have
The lemma is proved. Remark 3. We see from the proof that Lemma 9 remains valid if in the definitions of $F_{\lambda}$ and $J_{m,m_0,\rho}$ we replace the supremum with respect to $A\in H_{m}$ and $\overline{a}\in U_2(A, m_0)$ by the supremum with respect to $A$ and $\overline{a}$ from arbitrary finite sets. Of the function $\psi_{\alpha}$ we only need to be convex and satisfy $\psi_{\alpha}(0)=0$. Now we need Lemma B (Lemma 1 in [5]; see the full proof in [13], Lemma 5) and Lemma C (Lemma 3 in [13]). Lemma B. Let $\mathcal E\subset \mathbb{R}^N_+$ and $B=\sup_{x\in \mathcal E}\|x\|_2$. Let $0<\delta<1$, and let $\{\xi_i\}_{i=1}^N$ be independent random variables (see (3.2)); let $1\leqslant m\leqslant N$ and $q_0\geqslant 1$. Then Lemma C. There exists $C(\alpha)>0$ such that for $u_{\alpha}(t)$ defined in (1.2) and arbitrary $s, s'\in\mathbb{R}$ the following inequality holds: Proof of Lemma 8. We use a modification of Bourgain’s method: we fix $\lambda\geqslant 1$, $A\in H_{m}$ and $\overline{a}\in U_2(A, m_0)$ and set (see (1.2))
$$
\begin{equation}
L(A, \overline{a}, \lambda, \omega)=\int_X \psi_{\alpha} \biggl(\frac{f_{m, A, \overline{a}}(\omega, x)}{\lambda}\biggr)d\mu =\int_X \frac{f^2_{m, A, \overline{a}}(\omega, x)}{\lambda^2}u_{\alpha} \biggl(\frac{f_{m, A, \overline{a}}(\omega, x)}{\lambda}\biggr)d\mu.
\end{equation}
\tag{3.10}
$$
From the definition of $f_{m, A,\overline{a}}$ (see (3.8)) and (3.10) we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber 0 &\leqslant L(A, \overline{a}, \lambda, \omega) =\frac{1}{\lambda^2}\biggl\langle\sum_{i\in A}a_i\xi_i(\omega)\varphi_i, \sum_{i\in A}{a_i\xi_i(\omega)\varphi_i}\cdot u_{\alpha} \biggl(\frac{\sum_{i\in A}a_i\xi_i(\omega)\varphi_i}{\lambda} \biggr)\biggr\rangle \\ \nonumber &=\frac{1}{\lambda^2} \sum_{i\in A}a_i\xi_i(\omega) \biggl\langle\varphi_i, \sum_{j\in A}{a_j\xi_j(\omega)\varphi_j}\cdot u_{\alpha} \biggl(\frac{\sum_{j\in A}a_j\xi_j(\omega)\varphi_j}{\lambda} \biggr)\biggr\rangle \\ \nonumber &\leqslant\frac{1}{\lambda^2}\sup_{g\in \widetilde{\mathcal{P}}_{m_0}} \sum_{i\in A}|a_i| \xi_i(\omega) \biggl|\biggl\langle\varphi_i, gu_{\alpha}\biggl(\frac g\lambda\biggr)\biggr\rangle\biggr| \\ &=\frac{1}{\lambda^2}\sup_{g\in \widetilde{\mathcal{P}}_{m_0}} \sum_{i\in A\cap \operatorname{supp}{\overline{a}}}\frac{1}{\sqrt{m_0}}\, \xi_i(\omega) \biggl|\biggl\langle\varphi_i, gu_{\alpha}\biggl(\frac g\lambda\biggr)\biggr\rangle\biggr|, \end{aligned}
\end{equation}
\tag{3.11}
$$
where $\langle\,\cdot\,{,}\,\cdot\,\rangle$ is the inner product in $L_2(X)$ and
$$
\begin{equation*}
\widetilde{\mathcal{P}}_{m_0}=\biggl\{ g\colon g=\sum_{i\in B} b_i\varphi_i,\ \sum_{i=1}^N b_i^2\leqslant 1, \ |B|\leqslant m_0\biggr\}.
\end{equation*}
\notag
$$
In the proof of (3.11) we also used the fact that for fixed $\omega$, $A$ and $\overline{a}\in U_2(A, m_0)$ we have
$$
\begin{equation*}
\sum_{j\in A}{a_j\xi_j(\omega)\varphi_j}=\sum_{j\in A\cap \operatorname{supp}{\overline{a}}}{a_j\xi_j(\omega)\varphi_j}\in \widetilde{\mathcal{P}}_{m_0}.
\end{equation*}
\notag
$$
Note that for functions $g$ of the form $g=\sum_{i\in B} b_i\varphi_i$, where $\sum_{i=1}^N b_i^2\leqslant 1$ and $|B|\leqslant m_0$, we have the estimates
$$
\begin{equation}
\biggl\|\sum_{i\in B}b_i\varphi_i(x)\biggr\|_2 =\sqrt{\sum_{i\in B} b_i^2\|\varphi_i\|_2^2}\leqslant 1\quad\text{and} \quad \biggl\|\sum_{i\in B}b_i\varphi_i(x)\biggr\|_p \leqslant \sum_{i\in B}|b_i|\,\|\varphi_i\|_p\leqslant \sqrt{m_0}.
\end{equation}
\tag{3.12}
$$
It is clear from the definition of $F_{\lambda}$ (see (3.9)) and (3.11) that
$$
\begin{equation}
\begin{aligned} \, \nonumber \|F_{\lambda}(\omega)\|_{q_0} &=\Bigl \|\sup_{\substack{A\in H_{m} \\ \overline{a}\in U_2(A, m_0)}}L(A, \overline{a}, \lambda, \omega)\Bigr\|_{q_0} \\ &\leqslant\frac{1}{\lambda^2\sqrt{m_0}}\biggl\|\sup_{B\colon |B|=m_0}\sup_{g\in\widetilde{\mathcal{P}}_{m_0}} \sum_{i\in B}\xi_i(\omega) \biggl|\biggl\langle\varphi_i, gu_{\alpha}\biggl(\frac g\lambda\biggr)\biggr\rangle\biggr|\biggr\|_{q_0} \nonumber \\ &\equiv \frac{1}{\lambda^2\sqrt{m_0}} R_{\lambda, \delta, m_0}. \end{aligned}
\end{equation}
\tag{3.13}
$$
We find an upper estimate for $R_{\lambda, \delta, m_0}$. Consider the set
$$
\begin{equation*}
\mathcal E=\mathcal E_{m_0}=\biggl\{ \biggl\{ \biggl|\biggl\langle \varphi_i, gu_{\alpha}\biggl(\frac g\lambda\biggr)\biggr\rangle \biggr| \biggr\}_{i=1}^N,\ g\in\widetilde{\mathcal{P}}_{m_0}\biggr\}
\end{equation*}
\notag
$$
in $\mathbb{R}^N$. It follows from Lemma B that
$$
\begin{equation}
R_{\lambda, \delta, m_0}\,{\leqslant}\, C\biggl(\delta m_0+\frac{q_0}{\log(1/\delta)} \biggr)^{1/2}B +C\biggl( \log\frac{1}{\delta}\biggr)^{-1/2}\int_0^B (\log N_2(\mathcal E, t))^{1/2}\,dt,
\end{equation}
\tag{3.14}
$$
where $B=\sup_{z\in \mathcal E}\|z\|_2$. We estimate $B$ using Bessel’s inequality and Lemma 1, bearing (3.12) in mind:
$$
\begin{equation}
\biggl( \sum_{i=1}^N\biggl|\biggl\langle\varphi_i, gu_{\alpha}\biggl(\frac g\lambda\biggr)\biggr\rangle\biggr|^2\biggr)^{1/2} \leqslant \biggl\|gu_{\alpha}\biggl(\frac g\lambda\biggr) \biggr\|_2\leqslant C(\alpha, p)\ln^{\alpha}m_0.
\end{equation}
\tag{3.15}
$$
Set $p_0=(p+2)/2$; then $2<p_0<p$. It follows from Lemmas C and 2 and Hölder’s inequality that for any $h$, $g\in L_p(X)$ such that $\|h\|_2\leqslant 1$ and $\|g\|_2\leqslant 1$ and any $\lambda\geqslant 1$
$$
\begin{equation}
\begin{aligned} \, \nonumber &\biggl\|hu_{\alpha}\biggl(\frac{h}{\lambda}\biggr) -gu_{\alpha}\biggl(\frac{g}{\lambda}\biggr)\biggr\|_2 \leqslant C_{\alpha}\biggl(\int_X|h-g|^2\biggl(u_{\alpha}\biggl(\frac{h}{\lambda}\biggr) +u_{\alpha}\biggl(\frac{g}{\lambda}\biggr)\biggr)^2 \,d \mu\biggr)^{1/2} \\ &\qquad \leqslant C_{\alpha}\|h-g\|_{p_0} \biggl\|u_{\alpha}\biggl(\frac{h}{\lambda}\biggr)+u_{\alpha} \biggl(\frac{g}{\lambda}\biggr)\biggr\|_{2(p+2)/(p-2)} \leqslant \|h-g\|_{p_0} \frac{2C(\alpha, p)}{\ln^{\alpha}(\lambda+1)}. \end{aligned}
\end{equation}
\tag{3.16}
$$
To estimate the entropy numbers we use again Bessel’s inequality, (3.16) and (2.5), and for $g,h\in \widetilde{\mathcal{P}}_{m_0}$ we obtain
$$
\begin{equation*}
\begin{aligned} \, &\biggl(\sum_{i=1}^N \biggl( \biggl| \biggl\langle\varphi_i, gu_{\alpha}\biggl(\frac g\lambda\biggr)\biggr\rangle\biggr| -\biggl|\biggl\langle\varphi_i, hu_{\alpha} \biggl(\frac h\lambda \biggr) \biggr\rangle \biggr| \biggr)^2 \biggr)^{1/2} \\ &\qquad\leqslant \biggl\|gu_{\alpha}\biggl(\frac{g}{\lambda}\biggr)-hu_{\alpha} \biggl(\frac h\lambda \biggr) \biggr\|_2 \leqslant 2C(\alpha,p)\|h-g\|_{p_0}Q_{\alpha}(\lambda). \end{aligned}
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
N_2(\mathcal E, t)\leqslant N_{p_0} \biggl(\widetilde{P}_{m_0}, \frac{Q^{-1}_{\alpha}(\lambda)}{2C({\alpha},p)}\cdot t \biggr).
\end{equation}
\tag{3.17}
$$
From Lemma 7 we find that
$$
\begin{equation*}
\log N_{p_0}(\widetilde{\mathcal{P}}_{m_0}, t)\leqslant \begin{cases} C(p)m_0\log\biggl(\dfrac{N}{m_0}+1\biggr)\cdot t^{-\eta}, &t>\dfrac{1}{2}, \\ C(p)m_0\log\biggl(\dfrac{N}{m_0}+1\biggr)\cdot \log \dfrac{1}{t}, &t\leqslant \dfrac{1}{2}, \end{cases}
\end{equation*}
\notag
$$
where $\eta=\eta(p)>2$. From (3.14), (3.17) and (3.15) we obtain
$$
\begin{equation*}
\begin{aligned} \, &R_{\lambda, \delta, m_0} \leqslant C_1(\alpha,p)\biggl(\delta m_0+\frac{q_0}{\log(1/\delta)}\biggr)^{1/2}\ln^{\alpha}m_0 \\ &\qquad\qquad\quad +C\biggl(\log\frac{1}{\delta}\biggr)^{-1/2}\int_0^B \biggl(\log N_{p_0}\biggl(\widetilde{\mathcal{P}}_{m_0}, \frac{Q_{\alpha}^{-1}(\lambda)}{2C(\alpha,p)}t\biggr)\biggr)^{1/2}\,dt \\ &\qquad \leqslant C_1(\alpha,p)\biggl((\delta m_0)^{1/2}\ln^{\alpha}m_0 +\frac{q_0^{1/2}}{\log^{1/2}(1/\delta)} \ln^{\alpha}m_0\biggr) \\ &\qquad\qquad +C_1(p)\frac{\sqrt{m_0}\log^{1/2}(1+N/m_0)}{\log^{1/2}(1/\delta)} \biggl( \int_0^{C(\alpha,p)Q_{\alpha}(\lambda)} \biggl(\log\frac{2C(\alpha,p)}{tQ_{\alpha}^{-1}(\lambda)}\biggr)^{1/2}\,dt \\ &\qquad\qquad +(2C(\alpha,p))^{\eta/2}\int_{C(\alpha,p)Q_{\alpha}(\lambda)}^\infty (Q_{\alpha}(\lambda))^{\eta/2}t^{-\eta/2}\,dt\biggr). \end{aligned}
\end{equation*}
\notag
$$
Here we have
$$
\begin{equation*}
\begin{aligned} \, &\int_0^{C(\alpha,p)Q_{\alpha}(\lambda)} \biggl(\log\frac{2C({\alpha},p)}{tQ_{\alpha}^{-1}(\lambda)}\biggr)^{1/2}\,dt \\ &\qquad=C({\alpha},p)Q_{\alpha}(\lambda)\int_0^1\biggl(\log\frac{2}{x}\biggr)^{1/2}\,dx =C_2({\alpha},p)Q_{\alpha}(\lambda) \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
(2C({\alpha},p))^{\eta/2}\int_{C({\alpha,p})Q_{\alpha}(\lambda)}^\infty (Q_{\alpha}(\lambda))^{\eta/2}t^{-\eta/2}\,dt =C_3({\alpha},p)Q_{\alpha}(\lambda).
\end{equation*}
\notag
$$
As a result (see (3.13) and (2.5)), we have
$$
\begin{equation}
\begin{aligned} \, \nonumber \|F_{\lambda}(\omega)\|_{q_0} &\leqslant C(\alpha, p) \biggl(\delta^{1/2}\lambda^{-2}\ln^{\alpha}m_0 +\frac{q_0^{1/2}}{\log^{1/2}(1/\delta)}\, \frac{\ln^{\alpha}m_0}{\lambda^2\sqrt{m_0}} \\ &\qquad +\frac{\log^{1/2}(1+N/m_0)}{\log^{1/2}(1/\delta)\lambda^2\ln^{\alpha}(\lambda+1)} \biggr). \end{aligned}
\end{equation}
\tag{3.18}
$$
Hence for $m_0\geqslant \log^4 (N+3)$, $q_0=\log N$ and $\lambda$ satisfying the three conditions
$$
\begin{equation*}
\| J_{m, m_0, \rho}(\omega)\|_{q_0}\leqslant \widetilde{C}(\alpha,\rho,p)D(\rho, \alpha, N).
\end{equation*}
\notag
$$
Lemma 8 is proved. § 4. Main resultsLet $W(\Lambda)$ be the normed space (namely, the discrete Lorentz space) such that its unit ball is the convex hull of the vectors in $U_2(\Lambda)$ (see (3.1)). Recall that $S_{\Lambda}(\{a_k\}_{k\in\Lambda})=\sum_{k\in\Lambda}a_k\varphi_k$. We consider orthogonal (but not necessarily normalized) systems of functions $\Phi=\{\varphi_k\}_{k=1}^N$ such that (3.6) holds for $p>2$. Dividing all functions in such a system by $M>0$ we obtain that Theorems 1–4 below also hold for the orthonormal system $\Phi$ of functions satisfying $\|\varphi_k\|_{L_{p}}\leqslant M$, $k=1, 2,\dots, N$; in this case the right-hand sides of (4.1), (4.2) and (4.7) must be multiplied by $M$. Theorem 1. Let $\alpha>1/2$, $\rho>0$ and some $p>2$. Given an orthogonal system ${\Phi=\{\varphi_k\}_{k=1}^N}$ satisfying (3.6), the following inequality holds with probability greater than $1-C(\rho)N^{-9}$ for the random set $\Lambda=\Lambda_{\omega}$ generated by a system of random variables $\{\xi_i(\omega)\}_{i=1}^N$ (see (3.2)) such that $\mathbb{E}\xi_i=\log^{-\rho}(N+3)$ for $1\leqslant i\leqslant N$: Remark 4. It follows from the proof of Theorem 1 and Remark 2 that the quantity $\log^{\beta}(N+3)$ can be replaced in (4.1) by the largest of the numbers Remark 5. It follows from Theorem 1 and Remark 1 that Theorem C also holds when (1.4) is replaced by (3.6) for $p>2$ and $K(\alpha, \rho)$ is replaced by $K(\alpha, \rho,p)$. It follows from the comment to Lemma $3$ in [11] that for each $\overline{a}\in\mathbb{R}^N$ such that $\operatorname{supp}(\overline{a})\subset\Lambda$ we have the estimate
$$
\begin{equation*}
\|\overline{a}\|_{W(\Lambda)}^2\leqslant C \|\overline{a}\|_2^2\cdot \ln(|\Lambda|+3).
\end{equation*}
\notag
$$
Thus, the following result is a consequence of Theorem 1. Theorem 2. Let $\alpha>3/2$, $\rho>2$ and $p>2$. Then, given an orthogonal system $\Phi=\{\varphi_k\}_{k=1}^N$ satisfying (3.6), the following inequality holds with probability greater than $1-C(\rho)N^{-9}$ for the random set $\Lambda=\Lambda_{\omega}$ generated by a system of random variables $\{\xi_i(\omega)\}_{i=1}^N$ (see (3.2)) such that $\mathbb{E}\xi_i=\log^{-\rho}(N+3)$ for $1\leqslant i\leqslant N$: Remark 6. From estimate (3.12) and Corollary 1, for each system $\Phi$ satisfying (3.6), any set $\Lambda\subset\langle N\rangle$ and any vector $\overline{a}$ such that $\|\overline{a}\|_2\leqslant 1$ we obtain Remark 7. It was noted in [13] that it cannot be expected from the Orlicz space generated by a function $\psi_{\alpha}$ (see (1.2)) that a randomly chosen subsystem of cardinality $N/\log^{\beta}N$ (where $\beta$ is an arbitrarily large constant) is $\psi_{\alpha}$-lacunary. We explain this now. For an arbitrary positive constant $C $ and large $N=N(C)$ consider the orthogonal system $\Phi=\{\varphi_k\}_{k=1}^m$ of $m=\sqrt{\log N}$ functions on $[0,1]$ satisfying (3.6) that is not $\psi_{\alpha}(C)$-lacunary. We can assume that Deducing Theorem 1 from Lemma 8Since the unit ball of $W(\Lambda)$ is the convex hull of the vectors in $U_2(\Lambda)$, to prove Theorem 1 it is sufficient to look at vectors in $U_2(\Lambda)$. For $\overline{a}\in U_2(\Lambda, m_0)$, where $\Lambda\subset \langle N\rangle$, we have
$$
\begin{equation*}
\biggl\|\sum_{j\in\Lambda}a_j\varphi_j(x)\biggr\|_2 =\sqrt{\sum_{j\in\Lambda} a_j^2\|\varphi_j\|_2^2}\leqslant 1
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\biggl\|\sum_{j\in\Lambda}a_j\varphi_j(x)\biggr\|_p\leqslant \sum_{j\in\Lambda}|a_j|\, \|\varphi_j\|_p\leqslant \sqrt{m_0};
\end{equation*}
\notag
$$
therefore, for $m_0\leqslant \log^4(N+3)$, using Corollary 1 we obtain
$$
\begin{equation*}
\begin{aligned} \, \biggl\|\sum_{j\in\Lambda}a_j\varphi_j(x)\biggr\|_{\psi_{\alpha}} &\leqslant C(\alpha, p)\log^{\alpha/2}{\sqrt{e^2m_0}}\leqslant C_1(\alpha, p)\log^{\alpha/2}\log (N+3) \\ &\leqslant \widetilde{C}(\alpha, p)\log^{1/4}(N+3). \end{aligned}
\end{equation*}
\notag
$$
Once we have an estimate of $\|J_{N, m_0, \rho}(\omega)\|_{q_0}$ for $q_0=\log N$ and $m_0>\log^{4}(N+3)$ (see Lemma 8), taking (3.5) into account, we can claim that
$$
\begin{equation}
\begin{aligned} \, \nonumber \mathbb{P}(E) &\equiv \mathbb{P}\bigl\{ \omega\colon\forall\, m_0\in [\log^{4}(N+3), N] \ \ J_{N, m_0, \rho}(\omega)\leqslant C(\alpha, \rho, p)D(\rho, \alpha, N) \bigr\} \\ &\geqslant 1-\frac{1}{N^{9}}. \end{aligned}
\end{equation}
\tag{4.3}
$$
Recall Hoeffding’s inequality: if $\{X_i\}_{i=1}^m$ is a system of independent random variables on a probability space such that $a_i\leqslant X_i\leqslant b_i$ for $i\in\langle m\rangle$, then the following estimate for probabilities holds for $S=\sum_{i=1}^m(X_i-EX_i)$ and $t>0$:
$$
\begin{equation}
\mathbb{P}\{|S|\geqslant t\}\leqslant 2\exp -\frac{2t^2}{\sum_{i=1}^m(b_i-a_i)^2}.
\end{equation}
\tag{4.4}
$$
Fix $\rho>0$ and $\delta=\log^{-\rho}(N+3)$. Take the system of functions $\{\xi_i(\omega)\}_{i=1}^N$ as $\{X_i\}_{i=1}^m$; then
$$
\begin{equation*}
m=N\quad\text{and} \quad \sum_{i=1}^m X_i(\omega)=\sum_{i=1}^N \xi_i(\omega)= |\Lambda_{\omega}|
\end{equation*}
\notag
$$
(see (3.3)). Substituting $t=\delta \cdot N/3$ into (4.4), we obtain the following inequality for all points $\omega \in\Omega$ outside a set of measure $\leqslant \exp (-C(\rho)N^{1/2})$:
$$
\begin{equation}
\bigl||\Lambda_{\omega}|-\delta N\bigr|\leqslant \delta \frac{N}{3}.
\end{equation}
\tag{4.5}
$$
Let $W$ denote the set of $\omega\in\Omega$ such that (4.5) holds. Let $\widetilde{E}=E\cap W$, where $E$ was defined in (4.3); then $1-\nu\widetilde{E}<C(\rho)N^{-9}$. We verify that for each $\omega\in\widetilde{E}$ the subsystem of functions $\Phi_{\Lambda}=\{\varphi_i(x)\}_{i\in\Lambda}$, where $\Lambda=\Lambda_{\omega}$, satisfies (4.1). In fact, it follows from (4.5) that for each $\omega\in W$ the quantity $s(\omega)=s$ (the cardinality of the system $\Phi_{\Lambda}$) is of order $N\log^{-\rho}(N+3)$, or, more precisely,
$$
\begin{equation*}
\frac{2}{3}N\log^{-\rho}(N+3)\leqslant s\leqslant \frac{4}{3}N\log^{-\rho}(N+3).
\end{equation*}
\notag
$$
It follows from the definitions of $\Lambda_{\omega}$ and $\widetilde E$ that for each vector $\overline{a}\in U_2(\Lambda_{\omega})$ satisfying $|\operatorname{supp}\overline{a}|>\log^{4}(N+3)$ we have
$$
\begin{equation}
\biggl\| \sum_{i\in \Lambda_{\omega}}a_i\varphi_i(x)\biggr\|_{\psi_{\alpha}} =\biggl\| \sum_{i\in \Lambda_{\omega}}a_i\xi_i(\omega)\varphi_i(x)\biggr\|_{\psi_{\alpha}} \leqslant C(\alpha, \rho, p)D(\rho, \alpha, N).
\end{equation}
\tag{4.6}
$$
Theorem 1 is proved. Theorem 3. If $\rho>2$, then for any orthogonal system $\Phi=\{\varphi_k\}_{k=1}^N$ satisfying (3.6) for $p>2$ there exists $\Lambda\subset\langle N\rangle$, $|\Lambda|\geqslant N\log^{-\rho}(N+3)$, such that Remark 8. Let $\rho>7$. Then for orthonormal systems Theorem 3 can be deduced as a consequence of Theorem 2 above, Theorem 3 and Assertion 3 in [3], which show that if $\alpha>4$, then for a $\psi_{\alpha}$-lacunary system the maximal partial sum operator is bounded from $l_2$ to $L_2(X)$. To prove Theorem 3 we need Lemma D, which is well known in this context (for instance, see [14], Lemma 9.1). Lemma D. Given a vector $\overline{a}=\{a_n\}_{n=1}^M\in\mathbb{R}^M$, there exist an integer $l$, $1\leqslant l\leqslant M$, and two vectors $\overline{a}'$ and $\overline{a}''$ of the form such that the following relations hold: (1) $\overline{a}'+\overline{a}''=\overline{a}$; (2) $\|\overline{a}'\|_2^2\leqslant \frac{1}{2}\|\overline{a}\|_2^2$ and $\|\overline{a}''\|_2^2\leqslant \frac{1}{2}\|\overline{a}\|_2^2$; (3) $|a_l'|\leqslant |a_l|$ and $|a_l''|\leqslant |a_l|$. Proof of Theorem 3. For $\rho>3$ we take $\alpha=1/2+\rho/2$ (in this case $2<\alpha$ and $\alpha/2-\rho/4+1/2= 3/4$) and set $\gamma=3/4$; for $2<\rho\leqslant 3$ and $\varepsilon> 0$ we take $\alpha= 2+2\varepsilon$ (in this case $2<\alpha$ and $\alpha/2-\rho/4+1/2> 3/4$) and set
Let $\Phi_{\Lambda}$ be the subsystem constructed in the proof of Theorem 1 for $\alpha$ as indicated (that is, $\Lambda=\Lambda_{\omega}$, $\omega\in\widetilde{E}$). Then it satisfies inequality (4.2) for $\beta+ 1/2= \gamma$, and we can assume that $M\equiv |\Lambda|>N/\log^{\rho}(N+3)$ (it follows from the proof of Theorem 1 that $|\Lambda|>2N/(3\log^{\rho}(N+3))$, and in order to show that a required subsystem of cardinality greater than $N/\log^{\rho}(N+3)$ exists we choose the missing functions in a similar way from the remaining functions $\varphi_k$, $k\in \langle N\rangle\setminus \Lambda$). We change the notation for functions in $\Phi_{\Lambda}$: let $\Phi_{\Lambda}=\{u_j(x)\}_{j=1}^M$. We will use the standard binary decomposition procedure (for instance, see [14], Theorem 9.8). Fix an arbitrary vector $\overline{a}=\{a_n\}_{n=1}^M$ such that $\|\overline{a}\|_2=1$. For $\rho>3$ we want to show that
$$
\begin{equation}
\|S_{\Phi_{\Lambda}}^*(\overline{a})\|_{L_2(X)}\leqslant C(\rho, p)\log^{\gamma}(N+3).
\end{equation}
\tag{4.9}
$$
In the case when $2<\rho\leqslant 3$ we need to prove (4.9) for $C(\rho, \varepsilon, p)$ in place of $C(\rho,p)$. We assume that $\varepsilon>0$ is fixed, and do not indicate the dependence on it. Also, since $\alpha$ is constructed from $\rho$ and $\varepsilon$, instead of indicating a dependence of $\alpha$, we indicate below a dependence of $\rho$.
Since the operator $S_{\Phi_{\Lambda}}^*$ is continuous, all coordinates $a_n$, $n=1,\dots, M$, can be assumed to be distinct from zero. For each $s=0,\dots, s_0$ (we select $s_0$ below) we represent the vector $\overline{a}$ as a sum of $2^s$ vectors: here $r_0^0=\overline{a}$, and we construct the vectors $r_{\nu}^s$ in turn: once $r_{\nu}^{s-1}$ has been constructed, using Lemma D for $\overline{a}=r_{\nu}^{s-1}$ we represent it in the form $r_{\nu}^{s-1}=r_{2\nu}^s+r_{2\nu+1}^s$, where the vectors $r_{2\nu}^s$ and $r_{2\nu+1}^s$ have the form (4.8) and satisfy conditions 1)–3) in Lemma D, where, furthermore, $a_l''\neq 0$. Then the vectors $r_{\nu}^s$, $\nu=0, 1,\dots, 2^s-1$, have the form
$$
\begin{equation}
r_{\nu}^s=(0,\dots, 0, a'_{l_{\nu}^s}, a_{l_{\nu}^s+1},\dots, a_{l_{\nu+1}^s-1}, a''_{l_{\nu+1}^s}, 0,\dots, 0), \qquad a'_{l_{\nu}^s}\neq 0,
\end{equation}
\tag{4.11}
$$
where $1=l_0^s\leqslant l_1^s\,{\leqslant}\,{\cdots}\,{\leqslant}\, l_{2^s}^s=M$. It follows from Lemma D that for $\nu=0,1,\dots, 2^{s-1}- 1$ we have
$$
\begin{equation}
\begin{gathered} \, \max\{\|r_{2\nu}^s\|_{2}^2, \|r_{2\nu+1}^s\|_{2}^2\}\leqslant \frac{1}{2}\|r_{\nu}^{s-1}\|_{2}^2, \\ \max\{\|r_{2\nu}^s\|_{{\infty}}, \|r_{2\nu+1}^s\|_{{\infty}}\}\leqslant \|r_{\nu}^{s-1}\|_{{\infty}}. \end{gathered}
\end{equation}
\tag{4.12}
$$
By (4.12) we can select $s_0$ such that at most two coordinates in each vector $r_{\nu}^{s_0}$, $\nu=0,1,\dots, 2^{s_0}-1$, are distinct from zero. Since the coordinates $\overline{a}$ are nonzero and $a'_{l_{\nu}^s}\neq 0$, it follows that $l_{\nu+1}^{s_0}\leqslant l_{\nu}^{s_0}+2$ for $\nu=0,1,\dots,2^s-1$ and $s=1,\dots,s_0$. Set $r_{2^s}^s=\overline{0}$, $s=1,2,\dots, s_0$.
It follows from (4.10)–(4.12) and relations 1) and 3) in Lemma D that
$$
\begin{equation*}
\sum_{\nu=0}^{j-1}r_{\nu}^s=(a_1,\dots, a_{l_j^s-1}, \widetilde{a_{l_j^s}}, 0,\dots, 0), \qquad |\widetilde{a_{l_j^s}}|\leqslant |a_{l_j^s}|.
\end{equation*}
\notag
$$
For each $M_0=1,2,\dots, M-1$ we select $j\equiv j(M_0)$, $j\in\{1,\dots, 2^{s_0}\}$, so that $l_{j-1}^{s_0}\leqslant M_0<l_j^{s_0}$, and for $M_0=M$ we set $j(M_0)=2^{s_0}$. Then for $\overline{a}(M_0):=(a_1,\dots, a_{M_0},0,\dots,0)$, $M_0=1,2,\dots, M$, we obtain the expansion
$$
\begin{equation*}
\overline{a}(M_0)=\sum_{\nu=0}^{j-1}r_{\nu}^{s_0}+\overline{b}, \qquad \overline{b}=\{b_n\}_{n=1}^M, \quad |b_n|\leqslant |a_n|, \quad 1\leqslant n\leqslant M,
\end{equation*}
\notag
$$
where the vector $\overline{b}$ has at most two nonzero coordinates. Taking the equality $r_{2\nu}^s+r_{2\nu+1}^s=r_{\nu}^{s-1}$ into account, for $M_0=1,2,\dots, M$ we obtain
$$
\begin{equation*}
\overline{a}(M_0)=\sum_{s=1}^{s_0}r_{\nu(s, M_0)}^s+\overline{b}, \qquad 0\leqslant \nu(s, M_0)\leqslant 2^s.
\end{equation*}
\notag
$$
Hence for any numbers $\{y_n\}_{n=1}^{\infty}$ we can represent the sum $\sum_{n=1}^{M_0} a_ny_n$ in the following form:
$$
\begin{equation*}
\begin{gathered} \, \sum_{n=1}^{M_0} a_ny_n=\sum_{n=1}^M(\overline{a}(M_0))_ny_n =\sum_{s=1}^{s_0}\sum_{n=1}^{M}(r_{\nu(s, M_0)}^s)_ny_n+\Delta, \\ |\Delta|\leqslant 2\max_{1\leqslant n\leqslant M}|a_ny_n|. \end{gathered}
\end{equation*}
\notag
$$
Using this representation for $y_n=u_n(x)$ and for $M_0=M_0(x)$ satisfying
$$
\begin{equation*}
\biggl|\sum_{n=1}^{M_0}a_nu_n(x)\biggr|=\delta(x) :=S_{\Phi_{\Lambda}}^*(\overline{a})(x)=\max_{1\leqslant N'\leqslant M}\biggl|\sum_{n=1}^{N'}a_nu_n(x)\biggr|,
\end{equation*}
\notag
$$
we find that
$$
\begin{equation*}
\begin{aligned} \, \delta(x) &\leqslant \sum_{s=1}^{s_0} \biggl|\sum_{n=1}^{M}\bigl(r_{\nu(s, M_0(x))}^s\bigr)_n u_n(x)\biggr|+2\max_{1\leqslant n\leqslant M}|a_nu_n(x)| \\ &\leqslant\sum_{s=1}^{s_0}\sum_{\nu=0}^{2^s}\chi_{Z(s, \nu)}(x)\biggl|\sum_{n=1}^{M}(r_{\nu}^s)_nu_n(x)\biggr| +2\sqrt{\sum_{n=1}^M(a_nu_n(x))^2}, \end{aligned}
\end{equation*}
\notag
$$
where $Z(s, \nu):=\{x\in X \colon \nu(s, M_0(x))=\nu\}$ and $\|\overline{a}\|_2=1$, so that
$$
\begin{equation}
\|\delta(x)\|_2\leqslant \sum_{s=1}^{s_0}\biggl\|\sum_{\nu=0}^{2^s}\chi_{Z(s, \nu)}(x)\biggl|\sum_{n=1}^{M}(r_{\nu}^s)_nu_n(x)\biggr|\biggr\|_2+2.
\end{equation}
\tag{4.13}
$$
Since $\|r_{\nu}^s\|_2\leqslant 2^{-s/2}$ and we have (4.2) for $\Phi_{\Lambda}$, the following inequality holds:
$$
\begin{equation*}
\begin{gathered} \, \biggl\|\sum_{n=1}^{M}(r_{\nu}^s)_nu_n(x)\biggr\|_{\psi_{\alpha}}\leqslant 2^{-s/2} \widetilde{D}(\rho, p, N), \\ \widetilde{D}(\rho, p, N)=K'(\alpha, \rho, p)\log^{\gamma}(N+3). \end{gathered}
\end{equation*}
\notag
$$
We need Lemmas E and F below (which are Lemmas 9 and 10 in [13]). Lemma E. Let $f\in L_2(X)$ satisfy $\mu (\operatorname{supp} f)=a\leqslant 1$ and $\|f\|_{\psi_{\alpha}}=1$. Then Lemma F. Let $\gamma>0$ and $a_{\eta}\geqslant 0$ for $\eta=1, \dots, N$, and let $\sum_{\eta=1}^N a_{\eta}\leqslant 1$. Then From Lemma E we obtain
$$
\begin{equation}
\biggl\|\chi_{Z(s, \nu)}(x)\sum_{n=1}^{M}(r_{\nu}^s)_nu_n(x)\biggr\|_{2}\leqslant 2^{-s/2} C(\rho)\widetilde{D}(\rho, p, N) \ln^{-\alpha/2}\biggl(e+\frac{1}{\mu(Z(s, \nu))}\biggr).
\end{equation}
\tag{4.14}
$$
For fixed $s$, $s=1,\dots, s_0$, and different $\nu$ the sets $Z(s, \nu)$ are disjoint, and therefore $\sum_{\nu=0}^{2^s}\mu(Z(s, \nu))\leqslant 1$ and
$$
\begin{equation}
\biggl\|\sum_{\nu=0}^{2^s}\chi_{Z(s, \nu)}(x)\biggl|\sum_{n=1}^{M}(r_{\nu}^s)_nu_n(x)\biggr|\biggr\|_2^2 =\sum_{\nu=0}^{2^s}\biggl\|\chi_{Z(s, \nu)}(x)\biggl|\sum_{n=1}^{M}(r_{\nu}^s)_nu_n(x)\biggr|\biggr\|_2^2.
\end{equation}
\tag{4.15}
$$
Hence it follows from (4.13)–(4.15), Lemma F and the inequality $\alpha>2$ that
$$
\begin{equation*}
\begin{aligned} \, \|S_{\Phi_{\Lambda}}^*(\overline{a})\|_{L_2(X)} &\leqslant \sum_{s=1}^{s_0}C(\rho)\widetilde{D}(\rho, p, N)\sqrt{\sum_{\nu=0}^{2^s}2^{-s}\ln^{-\alpha}\biggl(e+\frac{1}{\mu(Z(s, \nu))}\biggr)} +2 \\ &\leqslant \sum_{s=1}^{s_0} C_1(\rho)\widetilde{D}(\rho, p, N)\ln^{-\alpha/2}(e+2^s)+2\leqslant \widetilde{C}(\rho)\widetilde{D}(\rho, p, N). \end{aligned}
\end{equation*}
\notag
$$
Thus, (4.9) holds. The proof of Theorem 3 is complete. Remark 9. Theorem $2$ in [13] can be extended to systems of functions satisfying condition (3.6) for $p>2$ instead of the condition of uniform boundedness. The following result holds: for $\rho>4$, for each orthogonal system $\Phi=\{\varphi_k\}_{k=1}^N$ satisfying the property (3.6) for $p>2$, there exists $\Lambda\subset\langle N\rangle$, $|\Lambda|\geqslant N\log^{-\rho}(N+3)$, such that Splitting a system into several subsystemsTheorems 1–3 claim that from an orthogonal system $\Phi=\{\varphi_k\}_{k=1}^N$ of functions satisfying (3.6) for $p>2$, we can extract a subsystem $\Phi_{\Lambda}$ of sufficiently large density such that the norms of the operators $S_{\Lambda}$ and $S_{\Phi_{\Lambda}}^{*}$ have nice estimates. In addition, estimates (4.1), (4.2) and (4.7) hold with large probability for an arbitrary set $\Lambda=\Lambda_{\omega}$ generated by a system $\{\xi_i(\omega)\}_{i=1}^N$ (see (3.2)) such that $\delta=\log^{-\rho}(N+3)$. It is clear from the proofs that we can take $\omega$ in some $\widetilde{E}$ such that $1-\nu\widetilde{E}<C(\rho)N^{-9}$, and for $\omega\in \widetilde{E}$ the cardinality of $\Lambda_{\omega}$ has the estimate $2N\delta\leqslant3|\Lambda_{\omega}|\leqslant 4N\delta$ (see (4.5)). It is clear that in place of $\delta=\log^{-\rho}(N+3)$ we can take $\delta_0=1/M$, where $M=[\log^{\rho}(N+3)]$. Consider a system of independent random variables $\zeta_i$, $i=1,\dots, N$, on a probability space $(\Omega, \nu)$ which take the values $1,2,\dots, M$ with equal probability. Then for each $j\in\langle M\rangle$ the variables $\xi_i^j(\omega)=\chi_{\{\zeta_i=j\}}(\omega)$, $i=1,\dots, N$, are independent selectors satisfying $\mathbb{E}\xi_i^j=\delta_0$. Hence for $j\in\langle M\rangle$ there exists $\widetilde{E_j}$ such that ${1-\nu\widetilde{E_j}<C(\rho)N^{-9}}$, and for $\omega\in \widetilde{E_j}$ we have estimates (4.2) and (4.7) for the set $\Lambda_{\omega}^j\equiv \{i\in\langle N\rangle\colon \xi_i^j(\omega)=1\}=\{i\in\langle N\rangle\colon \zeta_i(\omega)=j\}$. Then for $\omega\in E_0\equiv\bigcap_{j=1}^M\widetilde{E_j}$ we have these estimates for each $j\in\langle M\rangle$ and, moreover, $\nu(E_0)>1-C(\rho)N^{-8}$ and $\langle N\rangle=\bigsqcup_{j=1}^M\Lambda_{\omega}^j$. We have the following result. Theorem 4. Let $\rho>2$. Then each orthogonal system $\Phi=\{\varphi_k\}_{k=1}^N$ satisfying (3.6) for $p>2$ can be split into $M\equiv [\log^{\rho}(N+3)]$ subsystems $\Phi_{\Lambda_j}$ so that estimates (4.2) for $\alpha> 3/2$ and (4.7) hold for the sets $\Lambda_j$, $j=1,\dots,M$, and, furthermore, $2N/M\leqslant3|\Lambda_j|\leqslant 4N/M$. AcknowledgementThe author is grateful to B. S. Kashin for valuable discussions and for a reference to [11].
Citation in format AMSBIB
\Bibitem{Lim23} |
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|