[Lp] ESTIMATES FOR A ROUGH MAXIMAL OPERATOR ON PRODUCT SPACES

Title & Authors
[Lp] ESTIMATES FOR A ROUGH MAXIMAL OPERATOR ON PRODUCT SPACES
AL-QASSEM HUSSAIN MOHAMMED;

Abstract
We establish appropriate $\small{L^p}$ estimates for a class of maximal operators $\small{S_{\Omega}^{(\gamma)}}$ on the product space $\small{R^n\;\times\;R^m\;when\;\Omega}$ lacks regularity and $\small{1\;\le\;\gamma\;\le\;2.\;Also,\;when\;\gamma\;=\;2}$, we prove the $L^p\;(2\;{\le}\;P\;<\;\infty)\;boundedness\;of\;S_{\Omega}^{(\gamma)}\;whenever\;\Omega$ is a function in a certain block space $\small{B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})}$ (for some q > 1). Moreover, we show that the condition $\small{\Omega\;{\in}\;B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})}$ is nearly optimal in the sense that the operator $\small{S_{\Omega}^{(2)}}$ may fail to be bounded on $\small{L^2}$ if the condition $\small{\Omega\;{\in}\;B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})}$ is replaced by the weaker conditions $\Omega\;{\in}\;B_q^{(0,\varepsilon)}(S^{n-1}\;\times\;S^{m-1})\;for\;any\;-1\;<\;\varepsilon\;<\;0.$
Keywords
rough kernel;block space;singular integral;maximal operator;product domains;
Language
English
Cited by
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