Spherical maximal functions and fractal dimensions of dilation sets

Abstract

For the spherical mean operators ${\mathcal{A}}_t$ in ${\mathbb{R}}^d$, $d\ge 2$, we consider the maximal functions $M_{E}f={\sup }_{t\in E}|{\mathcal{A}}_{t}f|$, with dilation sets $E\subset [1,2]$. In this paper we give a surprising characterization of the closed convex sets which can occur as closure of the sharp $L^p$ improving region of $M_E$ for some $E$. This region depends on the Minkowski dimension of $E$, but also other properties of the fractal geometry such as the Assouad spectrum of $E$ and subsets of $E$. A key ingredient is an essentially sharp result on $M_E$ for a class of sets called (quasi-)Assouad regular which is new in two dimensions.

30 pages, 3 figures. Fixed a typo, final version

Cited by in corpus (8)

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• Assouad type dimensions of infinitely generated self-conformal sets
• Spherical maximal operators on Heisenberg groups: Restricted dilation sets
• Interpolating with generalized Assouad dimensions
• Lebesgue space estimates for spherical maximal functions on Heisenberg groups
• $L^p-L^q$ estimates for the circular maximal operators on Heisenberg radial functions
• Assouad spectrum of Gatzouras-Lalley carpets