On maximal functions with rough kernels in \(L(\log L)^{1/2} (\mathbb S^{n-1})\)
type: Item
instance of: scholarly article
title: On maximal functions with rough kernels in \(L(\log L)^{1/2} (\mathbb S^{n-1})\)
zbMATH Open document ID: 1086.42008
published in: Collectanea Mathematica
publication date: 2005-03-30T00:00:00Z
full work available at URL: https://eudml.org/doc/41820
review text: Let \(\Omega\) be a homogeneous function of degree zero with cancellation property \(\int_{S^{n-1}}\Omega(x')\,d\sigma(x')=0\), and \(h\) be a suitable function on \((0,\infty)\). The author considers the following singular integrals \[ T_{\Omega,h}f(x)=\roman{p.v.}\int_{\mathbb R^n}f(x-y)| y| ^{-n}h(| y| )\Omega(y/| y| ) \,dy, \] and their maximal operator \(\mathcal M_{\Omega}(f)(x)=\sup_{h\in H; \| h\| _H\leq1}| T_{\Omega,h}f(x)| \), where \(H=L^2(\mathbb R^+, dr/r)\). \textit{L. K. Chen} and \textit{H. Lin} [Ill. J. Math. 34, No.~1, 120--126 (1990; Zbl 0682.42012)] gave that \(\mathcal M_{\Omega}\) is bounded on \(L^p\) for \(p>2n/(2n-1)\) under the assumption that \(\Omega\) is continuous. The author's main theorem is: \(\mathcal M_{\Omega}\) is bounded on \(L^p\) for \(p\geq2\) under the assumption \(\Omega\in L(\log L)^{1/2}\), and the exponent \(1/2\) is optimal. This improves Chen and Lin's result. He also gives related results on \(T_{\Omega,h}\) and the Marcinkiewicz integral.
reviewed by: Kôzô Yabuta
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 2149099
zbMATH Keywords: singular integrals
zbMATH Keywords: maximal operator
zbMATH Keywords: Marcinkiewicz integral
zbMATH Keywords: \(L^p\) boundedness
author: Ahmad Al-Salman
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2149099