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

19 pages, 1 figure. Final version to appear in Mathematische Zeitschrift

Cited by in corpus (3)

• Multi-scale sparse domination
• Spherical maximal operators on Heisenberg groups: Restricted dilation sets
• Lebesgue space estimates for spherical maximal functions on Heisenberg groups