Weighted inequalities for some spherical maximal operators

Abstract

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.