Lower bounds for uncentered maximal functions in any dimension

TL;DR

This paper establishes lower bounds for uncentered maximal functions in any dimension, demonstrating that certain inequalities hold for these operators but not for centered ones, and provides explicit bounds for parallelepipeds.

Contribution

It proves the existence of dimension-dependent lower bounds for uncentered maximal functions and applies the Bellman function method to various set families, including parallelepipeds.

Findings

Lower bounds exist for uncentered maximal functions in all dimensions.

The Bellman function approach is effective for analyzing maximal operators.

Explicit bound for parallelepipeds: A(n,p)=(p/(p-1))^{1/p}.

Abstract

In this paper we address the following question: given $ p\in (1,\infty)$, $n \geq 1$, does there exists a constant $A(p,n)>1$ such that $\| M f\|_{L^{p}}\geq A(n,p) \| f\|_{L^{p}}$ for any nonnegative $f \in L^{p}(\mathbb{R}^{n})$, where $Mf$ is a maximal function operator defined over the family of shifts and dilates of a centrally symmetric convex body. The inequality fails in general for the centered maximal function operator, but nevertheless we give an affirmative answer to the question for the uncentered maximal function operator and the almost centered maximal function operator. In addition, we also present the Bellman function approach of Melas, Nikolidakis and Stavropoulos to maximal function operators defined over various types of families of sets, and in case of parallelepipeds we will show that $A(n,p)=\left(\frac{p}{p-1}\right)^{1/p}$.