Heat maximal function on a Lie group of exponential growth

TL;DR

This paper investigates the heat maximal function on a specific Lie group with exponential growth, establishing its boundedness from Hardy space H^1 to L^1 but also showing it doesn't characterize H^1 entirely.

Contribution

It proves the boundedness of the heat maximal function from H^1 to L^1 and demonstrates its limitations in characterizing Hardy space H^1 on the Lie group.

Findings

Heat maximal function is bounded from H^1 to L^1.

Heat maximal function does not characterize H^1.

Results apply to a Lie group with exponential growth.

Abstract

Let G be the Lie group R^2\rtimes R^+ endowed with the Riemannian symmetric space structure. Let X_0, X_1, X_2 be a distinguished basis of left-invariant vector fields of the Lie algebra of G and define the Laplacian \Delta=-(X_0^2+X_1^2+X_2^2). In this paper, we show that the maximal function associated with the heat kernel of the Laplacian \Delta is bounded from the Hardy space H^1 to L^1. We also prove that the heat maximal function does not provide a maximal characterization of the Hardy space H^1.