Local rearrangement invariant spaces and distribution of Rademacher series
type: Item
instance of: scholarly article
title: Local rearrangement invariant spaces and distribution of Rademacher series
zbMATH Open document ID: 1407.46024
author: Javier Carrillo-Alanís
published in: Positivity
publication date: 2018-04-19T00:00:00Z
review text: Since the original Khinchin inequalities for the Rademacher sequence in \(L^p[0,1]\) were established, the question about local versions of these inequalities has been studied by several authors, including Zygmund, Rodin, Semenov, Zhou, Astashkin and Curbera, among others. The class of spaces that were initially considered are the rearrangement invariant Banach function spaces. For such a space \(X\), a local version means that the interval \([0,1]\) is replaced by a measurable subset \(E\). In order to study the local versions, this restriction in the support has to be translated to the definition of local rearrangement invariant space: the notation \(X|E\) is used. Some classes of Orlicz spaces are also needed. Let \( G\) be the closure of \(L^\infty\) in the Orlicz space \(L^{M_2}\) generated by the function \(M_2(t) = \exp{t^2} - 1\). Proposition 1 states that for an r.i. space on \([0,1]\), \(X \supset G\), there exist constants \(A'_X, B'_X >0\) such that, for every \(E \subset [0,1]\) of positive measure, there exists \(N=N(E)\) such that \[ A'_X \Big( \sum_{k=1}^\infty a_k^2 \Big)^{1/2} \leq \Big\|_X \leq \Big\| \sum_{k=1}^\infty a_k r_{N+k} \Big\|_{X|E} \leq B'_X \Big( \sum_{k=1}^\infty a_k^2 \Big)^{1/2} \] for all \((a_k) \in \ell^2\). In fact, it is shown in Corollary~4 that this is equivalent to \(G \subseteq X\). This information is completed in Theorem 3 that provides the following result: for an r.i. space on \([0,1]\) and a non-empty set \(E \subset [0,1]\), there exists \(N=N(E)\) such that, if \(\sum_{k=1}^\infty a_k r_{k} \in X\), \[ C_1 \Big\| \sum_{k=1}^\infty a_k r_{k} \Big\|_X \leq \Big\| \sum_{k=1}^\infty a_k r_{N+k} \Big\|_{X|E} \leq C_2 \Big\| \sum_{k=1}^\infty a_k r_{k} \Big\|_X \] for absolute constants \(C_1, C_2 >0\). More comments and partial results that are interesting by themselves are also given.
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 60E99
zbMATH DE Number: 6861651
zbMATH Keywords: function spaces
zbMATH Keywords: Rademacher series
zbMATH Keywords: rearrangement invariant spaces
zbMATH Keywords: distribution function
zbMATH Keywords: Khintchine inequalities
reviewed by: E. A. Sánchez-Pérez
MaRDI profile type: Publication
full work available at URL: s11117-017-0500-2
OpenAlex ID: W2613956724
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DOI: 10.1007/S11117-017-0500-2
description: scientific article; zbMATH DE number 6861651