$L^p\to L^q$ bounds for spherical maximal operators

TL;DR

This paper establishes near-optimal $L^p$ to $L^q$ bounds for spherical maximal operators over subsets of radii, highlighting the influence of fractal dimensions of the set on the bounds and applying these results to sparse domination.

Contribution

It introduces bounds for spherical maximal functions that depend on the Minkowski and Assouad dimensions of the radius set, advancing understanding of fractal influences on harmonic analysis operators.

Findings

Derived near-sharp $L^p o L^q$ estimates for spherical maximal functions.

Showed the dependence of bounds on Minkowski and Assouad dimensions of the radius set.

Applied results to obtain sparse domination bounds for related operators.

Abstract

Let $f\in L^p(\mathbb{R}^d)$, $d\ge 3$, and let $A_t f(x)$ the average of $f$ over the sphere with radius $t$ centered at $x$. For a subset $E$ of $[1,2]$ we prove close to sharp $L^p\to L^q$ estimates for the maximal function $\sup_{t\in E} |A_t f|$. A new feature is the dependence of the results on both the upper Minkowski dimension of $E$ and the Assouad dimension of $E$. The result can be applied to prove sparse domination bounds for a related global spherical maximal function.