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ENDPOINT MAPPING PROPERTIES OF SPHERICAL MAXIMAL OPERATORS

Published online by Cambridge University Press:  27 January 2003

Andreas Seeger ,
Terence Tao  and
James Wright
Andreas Seeger
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA (seeger@math.wisc.edu)
Terence Tao
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA (tao@math.ucla.edu)
James Wright
Affiliation:
Department of Mathematics and Statistics, University of Edinburgh, King’s Building, Mayfield Road, Edinburgh EH3 9JZ, UK (jimw@maths.unsw.edu.au)
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Abstract

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For a function $f\in L^p(\mathbb{R}d)$, $d\ge 2$, let $A_tf(x)$ be the mean of $f$ over the sphere of radius $t$ centred at $x$. Given a set $E\subset(0,\infty)$ of dilations we prove various endpoint bounds for the maximal operator $M_E$ defined by $M_E f(x)=\sup_{t\in E}|A_tf(x)|$, under some regularity assumptions on $E$.

AMS 2000 Mathematics subject classification: Primary 42B25. Secondary 42B20

Keywords

spherical meansmaximal operators
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 2 , Issue 1 , January 2003 , pp. 109 - 144
Copyright
2003 Cambridge University Press