A maximal function characterisation of the Hardy space for the Gauss measure

TL;DR

This paper characterizes the Hardy space H^1 for the Gauss measure using maximal functions in one dimension and describes nonnegative functions in higher dimensions, establishing L^p spaces inclusion.

Contribution

It provides a maximal function characterization of H^1(g) in one dimension and extends understanding of nonnegative functions in higher dimensions for the Gauss measure.

Findings

Maximal function characterization of H^1(g) in dimension one.

Description of nonnegative functions in H^1(g) in arbitrary dimensions.

L^p(g) is contained in H^1(g) for 1<p≤∞.

Abstract

In dimension one we give a maximal function characterisation of the Hardy space H^1(g) for the Gauss measure g, introduced by G. Mauceri and S. Meda. In arbitrary dimension, we give a description of the nonnegative functions in H^1(g) and use it to prove that L^p(g) is a contained in H^1(g) for 1<p\le\infty.