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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
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     § 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<∞. Recall that an orthonormal system of functions Φ={φk}∞k=1 is called a p-lacunary system, or an Sp-system, if for some constant K, for each polynomial P=∑Nk=1akφk in this system we have (see details in [9]). To indicate the dependence on K we also say that Φ is a Sp(K)-system. The following result was presented in [9] (with a reference to Banach [4]).Theorem A. Let p>2, and let Φ={φk}∞k=1 be an orthonormal system such that Then there exists an infinite set of positive integers Λ such that {φk}k∈Λ is an Sp-system.Let (X,μ) be a probability space. Below we consider the scale of Orlicz spaces Lψα(X), where and the Luxemburg norm of a function f∈Lψα(X) is defined byAn 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
\begin{equation*}
\|\varphi_k\|_{L_{\infty}}\leqslant M, \qquad k=1, 2,\dots, N.
\end{equation*}
\notag
Then there exists a set \Lambda\subset\langle N\rangle such that |\Lambda|\geqslant N^{2/p} and for each polynomial P=\sum_{k\in\Lambda} a_k\varphi_k estimate (1.1) holds for K=K(M, p). 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
\begin{equation}
\|\varphi_k\|_{L_{\infty}}\leqslant 1, \qquad k=1, 2,\dots, N,
\end{equation}
\tag{1.4}
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 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,
\begin{equation*}
|G|\cdot\biggl(\frac\varepsilon2\biggr)^m\operatorname{Vol}(r(B))\leqslant \biggl(1+\frac\varepsilon2\biggr)^m\operatorname{Vol}(r(B)),
\end{equation*}
\notag
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 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
\begin{equation*}
\begin{aligned} \, &\log N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i \colon \|\overline{a}\|_2\leqslant 1\biggr\}, b_1\biggr) \\ &\qquad \leqslant m_0\log \frac{3b_2}{b_1} + \log N_q\biggl(\biggl\{\sum_{i\in I}a_i\varphi_i \colon \|\overline{a}\|_2\leqslant 1\biggr\}, b_2\biggr), \end{aligned}
\end{equation*}
\notag
where \overline{a}=(a_1,\dots, a_n)\in\mathbb{R}^n. 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
\begin{equation}
\log N_{q}(\mathcal{P}_{m}, t) \leqslant \begin{cases} C(q_0, q) m\log \biggl(\dfrac{n}{m}+1\biggr)\cdot t^{-\eta} &\textit{for } t>\dfrac{1}{2}, \\ C(q_0, q) m\log \biggl(\dfrac{n}{m}+1\biggr) \cdot \log \dfrac{1}{t} &\textit{for } 0<t \leqslant \dfrac{1}{2}. \end{cases}
\end{equation}
\tag{2.18}
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: Then for q_0=\log N, \delta=\log^{-\rho}(N+3) (see (3.2)), m_0\geqslant \log^4 (N+3) and m\geqslant m_0,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
\begin{equation*}
1\geqslant \|F_{\lambda}(\omega)\|_{q_0}\geqslant \bigl\|\chi_C(\omega)F_{\lambda}(\omega)\bigr\|_{q_0} \geqslant \biggl\| \frac{\chi_C(\omega)}{\lambda}J_{m,m_0,\rho}(\omega) \biggr\|_{q_0}
\end{equation*}
\notag
(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
\begin{equation*}
\begin{aligned} \, &\biggl\|\sup_{x\in \mathcal E, |A|\leqslant m} \biggl(\sum_{i\in A}\xi_i(\omega)x_i\biggr)\biggr\|_{L_{q_0}(d\nu)} \\ &\qquad \leqslant C\biggl(\delta m+\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{aligned}
\end{equation*}
\notag
Here N_2(\mathcal E, t) denotes the minimum number of Euclidean balls of radius t in \mathbb{R}^N such that their union covers \mathcal E, and C is an absolute constant. 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
\begin{equation*}
\log^{\alpha/2-\rho/4}(N+3)\quad\text{and} \quad \frac{\log^{1/4}(N+3)}{\log^{1/4+\alpha/2}\log(N+3)}.
\end{equation*}
\notag
The same holds for Theorem 2. 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
\begin{equation*}
\biggl\|\sum_{j\in\Lambda}a_j\varphi_j(x)\biggr\|_{\psi_{\alpha}} \leqslant C(\alpha, p)\log^{\alpha/2}\sqrt{e^2|\Lambda|}.
\end{equation*}
\notag
We write \alpha>1/2 in the statement of Theorem 1 because otherwise \beta>\alpha/2, and estimate (4.1) is trivial. For the same reason we assume that \alpha>3/2 in the statement of Theorem 2. 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 (it is sufficient to replace the \varphi_k(x) by the functions \varphi_k(2x)\chi_{[0,1/2]}-\varphi_k(2x- 1)\chi_{[1/2,1]}). Consider the systems of functions \Phi_n=\{\varphi^n_k\}_{k=1}^m, n\in\langle M\rangle, {M\equiv N/m}, on [0,1]^M constructed as follows: \varphi^n_k(\overline{x})=\varphi_k(x_n) and \overline{x}\!=\!(x_1,\dots, x_M). Clearly, the \Phi_n, n\in\langle M\rangle, are not \psi_{\alpha}(C)-lacunary either, and \Phi_0 — the union of the \Phi_n, n\in\langle M\rangle, — is an orthogonal system of functions on [0,1]^M with property (3.6). The probability of the event that none of the systems \Phi_n, n\in\langle M\rangle, lies in a subsystem of \Phi_0 of density \delta N\equiv N/\log^{\beta} N is equal to the quantity (1-\delta^m)^{N/m}, which tends to zero for large N. Hence a random subsystem of cardinality N/\log^{\beta} N contains the whole of at least one system \Phi_n, n\in\langle M\rangle, so that it is not \psi_{\alpha}(C)-lacunary. Thus, it is natural that even for large \rho the estimate in Theorem 2 contains a factor increasing with N.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
\begin{equation}
\begin{gathered} \, \overline{a}'=\{a_1,\dots, a_{l-1}, a_l', 0,\dots, 0\}, \\ \overline{a}''=\{0,\dots, 0, a_l'', a_{l+1}, \dots, a_M\} \end{gathered}
\end{equation}
\tag{4.8}
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}
This publication is cited in the following 1 articles:
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