Transplanting maximal inequalities between Laguerre and Hankel multipliers
type: Item
instance of: scholarly article
title: Transplanting maximal inequalities between Laguerre and Hankel multipliers
zbMATH Open document ID: 0866.42006
DOI: 10.1007/BF01297769
published in: Monatshefte für Mathematik
publication date: 1997-07-07T00:00:00Z
full work available at URL: https://eudml.org/doc/178741
review text: Suppose that \(m\) is a function on \(R^+\). For a function \(f\) on \(R^+\), with a Laguerre expansion \(\sum^\infty_{n=0}c_n\phi^a_n\), define \(L^*_mf(x)= \sup_{\varepsilon>0}|\sum^\infty_{n=0}c_nm (\varepsilon n^{1/2})\phi^a_n(x)|\). Similarly, define the maximal function \(H^*_nf(x)= \sup_{\varepsilon>0}|[m(\varepsilon\cdot) \widehat{f}(\infty) ]^\vee|\), where the Hankel transform and its inverse are denoted by \({}^\wedge\) and \({}^\vee\), respectively. The paper under review modifies techniques of \textit{Y. Kanjin} [Proc. Am. Math. Soc. 103, No. 4, 1063-1069 (1988; Zbl 0671.42016)] to show that, if \(L^*_m\) is bounded on \(L^p(R^+,x^{2a+1}dx)\), then so is \(H^*_m\). Some extensions and consequences of this result are described.
Mathematics Subject Classification ID: 42A45
Mathematics Subject Classification ID: 33C55
Mathematics Subject Classification ID: 42C10
zbMATH DE Number: 936844
zbMATH Keywords: maximal inequalities
zbMATH Keywords: Laguerre multipliers
zbMATH Keywords: Hankel multipliers
zbMATH Keywords: maximal function
zbMATH Keywords: Hankel transform
author: Krzysztof Stempak
reviewed by: Michael G. Cowling
MaRDI profile type: Publication
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OpenAlex ID: W1974732984
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description: scientific article; zbMATH DE number 936844