Lower bounds and fixed points for the centered Hardy--Littlewood maximal operator

TL;DR

This paper investigates lower bounds and fixed point properties of the centered Hardy--Littlewood maximal operator across different dimensions, revealing dimension-dependent behaviors and generic shape effects.

Contribution

It establishes new lower bounds for the maximal operator in low dimensions and analyzes fixed point thresholds for generic convex bodies in higher dimensions.

Findings

Lower bounds for the maximal operator in dimensions 1 and 2.

Existence of fixed point thresholds for convex bodies in dimensions 3 and higher.

Generic shapes have higher fixed point thresholds than the Euclidean ball.

Abstract

For all $p>1$ and all centrally symmetric convex bodies $K\subset \mathbb{R}^d$ define $Mf$ as the centered maximal function associated to $K$. We show that when $d=1$ or $d=2$, we have $||Mf||_p\ge (1+\epsilon(p,K))||f||_p$. For $d\ge 3$, let $q_0(K)$ be the infimum value of $p$ for which $M$ has a fixed point. We show that for generic shapes $K$, we have $q_0(K)>q_0(B(0,1))$.