On the binary analogs of the Hardy and Hardy--Littlewood operators
type: Item
instance of: scholarly article
title: On the binary analogs of the Hardy and Hardy--Littlewood operators
zbMATH Open document ID: 0941.44003
DOI: 10.1007/BF02677528
published in: Siberian Mathematical Journal
publication date: 2000-05-04T00:00:00Z
review text: Given \(x\in\mathbb R^+\) and an integer \(n\) such that \(2^n\leq x <2^{n+1}\), put \[ H_{\omega}(f)(x)= \sum_{m=n+1}^{\infty}2^{-m}\int_{2^m}^{2^{m+1}} f(y) dy,\quad B_{\omega}(f)(y)=2^{-n} \int_{0}^{2^{n}}f(y) dy,\; \] where \(f\in L_p({\mathbb R}^+)\) with \(1\leq p<\infty\). The operators \(H_{\omega}\) and \(B_{\omega}\) are binary analogs of the Hardy and Hardy-Littlewood operators. Recall the definition of the Walsh transform. Given \(x\in\mathbb R^+\) and a positive integer \(n\), put \(x_n\equiv [2^n x]\pmod 2\) and \(x_{-n}\equiv [2^{1-n} x]\pmod 2\), i.e., by definition, the numbers \(x_n\) and \(x_{-n}\) belong to \(\{0,1\}\). We can define the functions \[ t(x,y)=\sum_{n=1}^{\infty}(x_ny_{-n}+x_{-n}y_n),\quad X(x,y)=(-1)^{t(x,y)} \] with \(x,y\in\mathbb R^+\). The Walsh transform is defined by the equality \[ \widehat{f}(x)=\int_{{\mathbb R}^+}X(x,y)f(y) dy. \] The main result of the article is the following analog of the Titchmarsh theorem: Theorem. Let \(f\in L_p({\mathbb R}^+)\). Then \[ \widehat{H}_{\omega}(f)=B_{\omega}(\widehat{f})\quad (1\leq p\leq 2),\qquad \widehat{B}_{\omega}(f)=H_{\omega}(\widehat{f})\quad (1< p\leq 2). \]
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 65T50
zbMATH DE Number: 1438941
zbMATH Keywords: Hardy's operator
zbMATH Keywords: Hardy-Littlewood operator
zbMATH Keywords: Walsh transform
zbMATH Keywords: Titchmarsh theorem
zbMATH Keywords: binary analogs
author: Boris Golubov
MaRDI profile type: Publication
cites work: Q5339461
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cites work: The Generalized Walsh Functions
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description: scientific article; zbMATH DE number 1438941