Uncentred maximal operators with respect to half balls on Damek--Ricci spaces

TL;DR

This paper introduces a modified uncentred Hardy--Littlewood maximal operator on Damek--Ricci spaces using half balls, which exhibits improved boundedness properties including an $L ext{log}L$ endpoint estimate and boundedness on all $L^p$ spaces for p > 1.

Contribution

It presents a novel variant of the maximal operator with better boundedness properties on Damek--Ricci spaces compared to the classical operator.

Findings

The modified maximal operator satisfies an $L ext{log}L$ endpoint estimate.

It is bounded on $L^p$ for all $p$ in $(1, olinebreak\infty]$.

Abstract

In this paper we study a variant of the uncentred Hardy--Littlewood maximal operator on Damek--Ricci spaces in which balls are replaced by suitable half balls. Perhaps surprisingly, such modified maximal operator has better boundedness properties than the classical one. In particular, it satisfies an $L\log L$ endpoint estimate and it is bounded on $L^p$ for every $p$ in $(1,\infty]$.