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Funktsional. Anal. i Prilozhen., 2014, Volume 48, Issue 4, Pages 1–8 (Mi faa3169)  

This article is cited in 1 scientific paper (total in 1 paper)

Khintchine Inequality for Sets of Small Measure

S. V. Astashkin

Samara State University

Abstract: The following theorem is proved. Let $r_i$ be the Rademacher functions, i.e., $r_i(t):=\operatorname{sign}\sin(2^i\pi t)$, $t\in[0,1]$, $i\in\mathbb{N}$. If a set $E\subset [0,1]$ satisfies the condition $m(E\cap (a,b))>0$ for any interval $(a,b)\subset [0,1]$, then, for some constant $\gamma=\gamma(E)>0$ depending only on $E$ and for all sequences $a=(a_k)_{k=1}^\infty\in\ell^2$,
$$ \int_E|\sum_{i=1}^\infty a_ir_i(t)| dt\ge \gamma (\sum_{i=1}^\infty a_i^2)^{1/2}. $$
As a consequence of this result, a version of the weighted Khintchine inequality is obtained.

Keywords: Rademacher functions, Khintchine inequality, $L_p$-spaces, Paley–Zygmund inequality

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00198


DOI: https://doi.org/10.4213/faa3169

Full text: PDF file (158 kB)
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English version:
Functional Analysis and Its Applications, 2014, 48:4, 235–241

Bibliographic databases:

UDC: 517.982.22+517.521
Received: 04.03.2013

Citation: S. V. Astashkin, “Khintchine Inequality for Sets of Small Measure”, Funktsional. Anal. i Prilozhen., 48:4 (2014), 1–8; Funct. Anal. Appl., 48:4 (2014), 235–241

Citation in format AMSBIB
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\transl
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\pages 235--241
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. J. Carrillo-Alanis, “Local rearrangement invariant spaces and distribution of Rademacher series”, Positivity, 22:1 (2018), 63–81  crossref  mathscinet  zmath  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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