Estimates for the full maximal function on graded Lie groups

TL;DR

This paper extends Bourgain's classical maximal function inequalities from Euclidean space to graded Lie groups, using group Fourier analysis and Rockland operators to establish new bounds for spherical averages.

Contribution

It introduces a novel extension of maximal function estimates to graded Lie groups employing Fourier transform techniques and hypoelliptic operators.

Findings

Established restricted weak inequalities for maximal functions on graded Lie groups.

Unified Euclidean and non-Euclidean harmonic analysis approaches.

Provided a framework for further analysis of maximal functions in non-commutative settings.

Abstract

On $\mathbb{R}^n,$ a classical result due to Bourgain establishes the restricted weak $(\frac{n}{n-1},1)$ inequality for the full maximal function $M_F^{d\sigma}$ associated to the spherical averages. In this work we present an extension to Bourgain's result on graded Lie groups for a family of full maximal operators. We formulate this extension using the group Fourier transform of the measures under consideration and the symbols of (positive Rockland operators which are) positive left-invariant hypoelliptic partial differential operators on graded Lie groups.