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Winter 2002 Weighted inequalities for some spherical maximal operators
Javier Duoandikoetxea, Edurne Seijo
Illinois J. Math. 46(4): 1299-1312 (Winter 2002). DOI: 10.1215/ijm/1258138481


Given a set $E\subset (0,\infty)$, the spherical maximal operator associated to the parameter set $E$ is defined as the supremum of the spherical means of a function when the radii of the spheres are in $E$. The aim of the paper is to study boundedness properties of these operators on the spaces $L^p(|x|^{\alpha})$. It is shown that the range of values of $\alpha$ for which boundedness holds behaves essentially as follows: (i) for $p > n/(n-1)$ and negative $\alpha$ the range does not depend on $E$; (ii) when $\alpha$ is positive it depends only on the Minkowski dimension of $E$; (iii) if $p < n/(n-1)$ and $\alpha$ is negative, sets with the same Minkowski dimension can give different ranges of boundedness.


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Javier Duoandikoetxea. Edurne Seijo. "Weighted inequalities for some spherical maximal operators." Illinois J. Math. 46 (4) 1299 - 1312, Winter 2002.


Published: Winter 2002
First available in Project Euclid: 13 November 2009

zbMATH: 1026.42019
MathSciNet: MR1988265
Digital Object Identifier: 10.1215/ijm/1258138481

Primary: 42B25
Secondary: 28A80

Rights: Copyright © 2002 University of Illinois at Urbana-Champaign


Vol.46 • No. 4 • Winter 2002
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