Weighted maximal inequalities for the Haar system
type: Item
instance of: scholarly article
title: Weighted maximal inequalities for the Haar system
zbMATH Open document ID: 1404.42038
published in: Monatshefte für Mathematik
publication date: 2018-06-27T00:00:00Z
review text: Let \(h=(h_n)_{n\geq0}\) be the collection of Haar functions on \([0, 1)\), given by \[ \begin{aligned} h_0&=\chi_{[0, 1)}, \\ h_1&=\chi_{[0, 1/2)}-\chi_{[1/2, 1)}, \\ h_2&=\chi_{[0, 1/4)}-\chi_{[1/4,1/2)}, \\ h_3&=\chi_{[1/2, 3/4)}-\chi_{[3/4,1)}, \\ h_4&=\chi_{[0, 1/8)}-\chi_{[1/8,1/4)} \\ h_5&=\chi_{[1/4, 3/8)}-\chi_{[3/8,1/2)}, \\ h_6&=\chi_{[1/2, 5/8)}-\chi_{[5/8,3/4)} \\ h_7&=\chi_{[3/4, 7/8)}-\chi_{[7/8,1)} \end{aligned} \] and so on. A weight \(\omega\) is called a dyadic \(A_p\) weight with \(p\in(1, \infty)\), if \[ [\omega]_{A_p}=\sup_I\left(\frac1{|I|}\int_I\omega\right)\left(\frac1{|I|}\int_I \omega^{-1/(p-1)}\right)^{p-1}<\infty; \] and \(\omega\) is called a dyadic \(A_1\) weight, if \[ [\omega]_{A_1}=\sup_I\left(\frac1{|I|}\int_I\omega\right)/{\mathop{\operatorname{essinf}}_I}\omega<\infty. \] Here in the both conditions, the suprema are taken over all dyadic subintervals \(I\) of \([0, 1)\). This paper studies the weighted maximal \(L^p\)-inequalities for the Haar system \(h=(h_n)_{n\geq0}\), with the optimal dependence on the characteristics of the weights involved. The author shows that for \(p\in(1, \infty)\), if \(\omega\) is a dyadic \(A_p\) weight, \(N\) is a non-negative integer, \(a_0, a_1,\dots, a_N\) are real numbers and \(\varepsilon_0, \varepsilon_1,\dots, \varepsilon_N\) is a sequence of signs, then \[ \left\|\max_{0\leq n\leq N}\left|\sum_{k=0}^n \varepsilon_ka_kh_k\right|\right\|_{L^p(\omega)} \leq 2^{1+1/p}C_p[\omega]_{A_p}^{\max\{1, 1/(p-1)\}}\left\|\sum_{k=0}^N a_kh_k\right\|_{L^p(\omega)}. \] Moreover, the exponent \(\max\{1, 1/(p-1)\}\) is the best possible. The author also shows that for any parameters \(p,q\in[1, \infty)\), there is a constant \(C_{p,q}\) depending only on \(p\) and \(q\) which has the following property: \[ \left\|\max_{0\leq n\leq N}\left|\sum_{k=0}^n \varepsilon_ka_kh_k\right|\right\|_{L^q(\omega)} \leq C_{p,q}[\omega]_{A_p}\left\|\max_{0\leq n\leq N}\left|\sum_{k=0}^n a_kh_k\right|\right\|_{L^q(\omega)}. \]…
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 60G42
zbMATH DE Number: 6894850
zbMATH Keywords: maximal
zbMATH Keywords: dyadic
zbMATH Keywords: Bellman function
zbMATH Keywords: best constants
author: Adam Osȩkowski
reviewed by: Dongyong Yang
MaRDI profile type: Publication
full work available at URL: s00605-018-1184-9
OpenAlex ID: W2801710041
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Wikidata QID: Q129963308
DOI: 10.1007/S00605-018-1184-9
description: scientific article; zbMATH DE number 6894850