Spherical maximal operators with fractal sets of dilations on radial functions

TL;DR

This paper investigates the behavior of spherical maximal operators with fractal dilation sets on radial functions, revealing dimension-dependent properties and providing complete endpoint results in higher dimensions and detailed spectra in two dimensions.

Contribution

It introduces a comprehensive analysis of Lebesgue space mapping properties for these operators, linking the type set to fractal dimensions like Minkowski and Assouad spectra.

Findings

Type set depends on upper Minkowski dimension in higher dimensions.

Complete endpoint results are established in higher dimensions.

Closure of the $L^p\to L^q$ type set characterized in 2D using the upper Assouad spectrum.

Abstract

For a given set of dilations $E\subset [1,2]$, Lebesgue space mapping properties of the spherical maximal operator with dilations restricted to $E$ are studied when acting on radial functions. In higher dimensions, the type set only depends on the upper Minkowski dimension of $E$, and in this case complete endpoint results are obtained. In two dimensions we determine the closure of the $L^p\to L^q$ type set for every given set $E$ in terms of a dimensional spectrum closely related to the upper Assouad spectrum of $E$.