Power weight inequalities for spherical maximal functions

TL;DR

This paper characterizes the boundedness of spherical maximal functions with general dilations on weighted L^p spaces, solving an open problem by providing a complete range description using the Legendre--Assouad function.

Contribution

It offers a comprehensive description of the allowable p and alpha ranges for spherical maximal functions with general dilation sets, extending previous partial results.

Findings

Complete characterization of boundedness ranges for spherical maximal functions

Resolution of an open problem by Duoandikoetxea and Seijo

Use of Legendre--Assouad function to describe dilation set properties

Abstract

This paper is about spherical maximal functions with general dilation sets acting on functions in weighted $L^p(|x|^\alpha)$ spaces. Aside from endpoint cases, a complete description of the allowable ranges of $p$, $\alpha$ is given in terms of the Legendre--Assouad function of the dilation set. This settles, up to endpoints, an open problem of Duoandikoetxea and Seijo.