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This article is cited in 1 scientific paper (total in 1 paper) Brief Communications Existence of dense subsystems with lacunarity property in orthogonal systems I. V. Limonovaab a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow b Lomonosov Moscow State University
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     Let $\Phi=\{\varphi_k\}_{k=1}^{N}$ be an orthogonal system of functions on a probability space $(X,\mu)$. Put $\langle N\rangle\equiv\{1,2,\dots, N\}$, and for $\Lambda\subset \langle N\rangle$ let $S_{\Lambda}$ be the operator defined by $S_{\Lambda}(\{a_k\}_{k\in\Lambda})= \sum_{k\in\Lambda}a_k\varphi_k(x)$. Put $\Phi_{\Lambda}=\{\varphi_k\}_{k\in\Lambda}$. Let $\delta\in(0,1)$, and let $\{\xi_i(\omega)\}_{i=1}^N$ be a system of independent random variables on a probability space $(\Omega,\nu)$ such that $\xi_i(\omega)=0$ or 1 and $\mathsf{E}\xi_i=\delta$, $1\leqslant i\leqslant N$. For $\omega\in\Omega$ set $\Lambda(\omega)\equiv\Lambda(\omega,N)\equiv \{i\in\langle N\rangle\colon\xi_i(\omega)=1\}$. Below we consider the scale of Orlicz spaces $L_{\psi_{\alpha}}$, where $\alpha>0$,
$$
\begin{equation}
\psi_{\alpha}(t)=t^2\,\frac{\ln^{\alpha}(e+|t|)}{\ln^{\alpha}(e+1/|t|)}\,,\text{ and } \|f\|_{\psi_{\alpha}}=\inf\biggl\{\lambda>0\colon \int_{X}\psi_{\alpha} \biggl(\frac{f(x)}{\lambda}\biggr)\,d\mu\leqslant 1\biggr\}.
\end{equation}
\tag{1}
$$
In his fundamental work [1] Bourgain solved the problem of the existence of $p$-lacunary ($p>2$) subsystems of size $N^{2/p}$ in an arbitrary uniformly bounded orthogonal system $\{\varphi_k\}_{k=1}^N$. Using a modification of the method in [1] the following result was obtained in [2], Theorem 1. Theorem A. Let $\alpha>0$ and $\rho>0$ be fixed, and let $\Phi=\{\varphi_k\}_{k=1}^N$ be an arbitrary orthogonal system with the property Then for a random set $\Lambda=\Lambda(\omega)$ generated by a set of random variables $\{\xi_i(\omega)\}_{i=1}^N$ such that $\mathsf{E}\xi_i=\delta=(\log_2 (N+3))^{-\rho}$, $1\leqslant i\leqslant N$, the inequality holds with probability greater than $1-C(\rho)N^{-9}$. This note is a continuation of [2] (also see [3]). Theorem 2 below is a generalization of Theorem A to $S_{\Lambda}$ acting from the space $l_2(\Lambda)$ to $L_{\psi_{\alpha}}(X)$, and to functions $\varphi_k$ satisfying only the weaker condition $\|\varphi_k\|_{p}\leqslant 1$, $1 \leqslant k \leqslant N$, where $p>4$. As mentioned in [2], we cannot expect that for the Orlicz space generated by (1) a random subsystem with cardinality $N/(\log_2 N)^{\beta}$ (where $\beta$ is an arbitrarily large constant) is $\psi_{\alpha}$-lacunary (that is, such that the inequality $\|\sum_{k=1}^N a_k\varphi_k\|_{\psi_{\alpha}}\leqslant C\|\overline{a}\|_2$ holds for each $\overline{a}=\{a_k\}_{k=1}^N\in\mathbb{R}^N$), so even for large $\rho$ the inequality in Theorem 2 below contains a factor increasing with $N$. Let $U_2(\Lambda)$ be vectors with Euclidean norm $1$ and supports in $\Lambda$ such that all their non-zero components have the same modulus:
$$
\begin{equation*}
U_2(\Lambda)=\bigl\{{\overline{a}}=\{a_i\}_{i=1}^N\colon \operatorname{supp}(\overline{a})\subset \Lambda;\, |a_i|=|\operatorname{supp}(\overline{a})|^{-1/2} \text{ for } i\in\operatorname{supp}(\overline{a})\bigr\},
\end{equation*}
\notag
$$
where $\operatorname{supp}(\overline{a})= \{i\in\langle N\rangle\colon a_i\ne 0\}$. Let $W(\Lambda)$ denote the normed space (the discrete Lorentz space) with unit ball equal to the convex hull of the vectors in $U_2(\Lambda)$. Set $\beta=\max\{\alpha/2-\rho/4,1/4\}$.
Theorem 1. Let $\alpha>1/2$, $\rho>0$, and $p>4$. Then, given an arbitrary orthogonal system $\Phi=\{\varphi_k\}_{k=1}^N$ such that for a random set $\Lambda=\Lambda(\omega)$ generated by a set of random variables $\{\xi_i(\omega)\}_{i=1}^N$ such that $\mathsf{E}\xi_i=\delta=(\log_2 (N+3))^{-\rho}$, $1\leqslant i\leqslant N$, the inequality holds with probability greater than $1-C(\rho)N^{-9}$. Remark 1. It follows from the proof of Theorem 1 that Theorem A also holds in the case when condition (2) is replaced by (3) (for $p>4$) and $K(\alpha,\rho)$ by $K(\alpha,\rho,p)$. It follows from a comment to Lemma 3 in [4] that if $\overline{a}\in\mathbb{R}^N$ satisfies $\operatorname{supp}(\overline{a})\subset\Lambda$, then $\|\overline{a}\|_{W(\Lambda)}^2\leqslant C\|\overline{a}\|_2^2 \ln(|\Lambda|+3)$. Thus, Theorem 1 yields the following result. Theorem 2. Let $\alpha>3/2$, $\rho>2$, and $p>4$. Then, given an arbitrary orthogonal system $\Phi=\{\varphi_k\}_{k=1}^N$ with property (3), for a random set $\Lambda=\Lambda(\omega)$ generated by a set $\{\xi_i(\omega)\}_{i=1}^N$ such that $\mathsf{E}\xi_i=\delta=(\log_2 (N+3))^{-\rho}$, $1\leqslant i\leqslant N$, the inequality holds with probability greater than $1-C(\rho)N^{-9}$. Consider the maximal partial sum operator $S_{\Phi}^*$ that, for $\{a_k\}_{k=1}^N\in\mathbb{R}^N$, is defined by $S_{\Phi}^*(\{a_k\}_{k=1}^N)(x)=\sup_{1\leqslant M\leqslant N} \bigl|\,\sum_{k=1}^M a_k\varphi_k(x)\bigr|$. It is well known (see [5]) that lacunarity enables one to improve estimates for the norm of $S_{\Phi}^*$. In [6] Balykbaev showed that for $\alpha>4$, given a $\psi_{\alpha}$-lacunary system, the maximal partial sum operator is bounded from $l_2^N$ to $L_{\psi_{\alpha}}(X)$ (and therefore, also to $L_2(X)$). It was shown in [2] that for $\rho>4$, in any orthogonal system $\{\varphi_k\}_{k=1}^N$ with property (2) we can find a subsystem $\Phi_{\Lambda}$ of $N/(\log_2(N+3))^{\rho}$ functions such that the estimate $\|S_{\Phi_{\Lambda}}^{*}\colon l_{\infty}(\Lambda) \to L_{2}(X)\|\leqslant C(\rho)\sqrt{|\Lambda|}$ holds. It turns out that for the subsystem $\Phi_{\Lambda}$ of the system $\Phi$ 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 Menshov–Rademacher theorem guarantees. Theorem 3. Given $\rho>2$ and any orthogonal system $\Phi=\{\varphi_k\}_{k=1}^N$ with property (3) for $p>4$, there exists $\Lambda\subset\langle N\rangle$, $|\Lambda|\geqslant N(\log_2(N+3))^{-\rho}$, such that 
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\Bibitem{Lim22} |
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