Wavelet Characterizations of the Atomic Hardy Space $H^1$ on Spaces of Homogeneous Type

TL;DR

This paper develops wavelet-based characterizations of the atomic Hardy space on spaces of homogeneous type, providing new tools for analysis in these complex metric measure spaces.

Contribution

It introduces an unconditional wavelet basis for the atomic Hardy space and establishes multiple equivalent characterizations using wavelets and spline functions.

Findings

Unconditional wavelet basis for $H^1_{at}({ m extbf{X}})$

Equivalent characterizations of $H^1_{at}({ m extbf{X}})$ in terms of wavelets

Crucial lower bounds for regular wavelets obtained

Abstract

Let $({\mathcal X},d,\mu)$ be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $H^1_{\rm at}({\mathcal X})$ be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hyt\"onen, together with obtaining some crucial lower bounds for regular wavelets, the authors give an unconditional basis of $H^1_{\rm at}({\mathcal X})$ and several equivalent characterizations of $H^1_{\rm at}({\mathcal X})$ in terms of wavelets, which are proved useful.