A modification of Hardy-Littlewood maximal-function on Lie groups

TL;DR

This paper introduces a modified Hardy-Littlewood maximal-function on Lie groups using an -norm with weights, demonstrating convergence to the classical maximal-function and analyzing boundedness and continuity in this setting.

Contribution

It proposes a new -norm based integral-function on Lie groups and studies its convergence, boundedness, and continuity properties, extending classical maximal-function theory.

Findings

$I_{p,w}f$ converges pointwise to $Mf$ as $p o \u221e$

Boundedness of the operator $I_{p,w}$ is established

Continuity of $I_{p,w}f$ is shown on Lie groups with invariant metrics

Abstract

For a real-valued function $f$ on a metric measure space $(X,d,\mu)$ the Hardy-Littlewood maximal-function of $f$ is given by the following `supremum-norm': $$Mf(x):=\sup_{r>0}\frac{1}{\mu(\mathcal{B}_{x,r})}\int_{\mathcal{B}_{x,r}}|f|d\mu.$$ In this note, we replace the supremum-norm on parameters $r$ by $\mathcal{L}_p$-norm with weight $w$ on parameters $r$ and define Hardy-Littlewood integral-function $I_{p,w}f$. It is shown that $I_{p,w}f$ converges pointwise to $Mf$ as $p\to\infty$. Boundedness of the sublinear operator $I_{p,w}$ and continuity of the function $I_{p,w}f$ in case that $X$ is a Lie group, $d$ is a left-invariant metric, and $\mu$ is a left Haar-measure (resp. right Haar-measure) are studied.