Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases

TL;DR

This paper develops a comprehensive Hardy space theory on product spaces of homogeneous type using orthonormal wavelet bases, extending classical analysis tools without additional assumptions on the underlying spaces.

Contribution

It introduces a new Littlewood--Paley theory on product spaces of homogeneous type based on wavelet coefficients, broadening the scope of harmonic analysis in these settings.

Findings

Established product Hardy spaces and their duals on spaces of homogeneous type.

Proved Calderón--Zygmund decomposition and interpolation theorems in this context.

Extended wavelet analysis techniques to general product spaces without extra assumptions.

Abstract

In this paper, using the remarkable orthonormal wavelet basis constructed recently by Auscher and Hyt\"onen, we establish the theory of product Hardy spaces on spaces ${\widetilde X} = X_1\times X_2\times\cdot \cdot\cdot\times X_n$, where each factor $X_i$ is a space of homogeneous type in the sense of Coifman and Weiss. The main tool we develop is the Littlewood--Paley theory on $\widetilde X$, which in turn is a consequence of a corresponding theory on each factor space. We define the square function for this theory in terms of the wavelet coefficients. The Hardy space theory developed in this paper includes product~$H^p$, the dual $\cmo^p$ of $H^p$ with the special case $\bmo = \cmo^1$, and the predual $\vmo$ of $H^1$. We also use the wavelet expansion to establish the Calder\'on--Zygmund decomposition for product $H^p$, and deduce an interpolation theorem. We make no additional assumptions on the quasi-metric or the doubling measure for each factor space, and thus we extend to the full generality of product spaces of homogeneous type the aspects of both one-parameter and multiparameter theory involving the Littlewood--Paley theory and function spaces. Moreover, our methods would be expected to be a powerful tool for developing wavelet analysis on spaces of homogeneous type.