Maximal function characterizations for Hardy spaces on spaces of homogeneous type with finite measure and applications

TL;DR

This paper establishes maximal function characterizations for Hardy spaces on spaces of homogeneous type with finite measure, addressing open questions and extending to various differential operators.

Contribution

It provides the first complete maximal function characterizations for Hardy spaces in finite measure spaces and applies these results to multiple classes of differential operators.

Findings

Proved nontangential and radial maximal function characterizations.

Addressed an open problem in the literature.

Extended results to elliptic, Schrödinger, and Fourier-Bessel operators.

Abstract

We prove nontangential and radial maximal function characterizations for Hardy spaces associated to a non-negative self-adjoint operator satisfying Gaussian estimates on a space of homogeneous type with finite measure. This not only addresses an open point in the literature, but also gives a complete answer to the question posed by Coifman and Weiss in the case of finite measure. We then apply our results to give maximal function characterizations for Hardy spaces associated to second order elliptic operators with Neumann and Dirichlet boundary conditions, Schr\"odinger operators with Dirichlet boundary conditions, and Fourier--Bessel operators.