Criterion of the boundedness of singular integrals on spaces of homogeneous type

TL;DR

This paper establishes a criterion for the boundedness of singular integrals on Hardy spaces and their duals in spaces of homogeneous type, extending previous results without additional geometric assumptions.

Contribution

It provides a general boundedness criterion for singular integrals on Hardy spaces in spaces of homogeneous type without extra geometric conditions.

Findings

Atomic Hardy spaces coincide with wavelet-based Hardy spaces.

Develops molecule theory in this general setting.

Extends boundedness results to all spaces of homogeneous type with doubling measures.

Abstract

It was well known that geometric considerations enter in a decisive way in many questions of harmonic analysis. The main purpose of this paper is to provide the criterion of the boundedness for singular integrals on the Hardy spaces and as well as on its dual, particularly on $\bmo$ for spaces of homogeneous type $(X, d,\mu)$ in the sense of Coifman and Weiss, where the quasi-metric $d$ may have no regularity and the measure $\mu$ satisfies only the doubling property. We make no additional geometric assumptions on the quasi-metric or the doubling measure and thus, the results of this paper extend to the full generality of all related previous ones, in which the extra geometric assumptions were made on both the quasi-metric $d$ and the measure $\mu.$ To achieve our goal, we prove that the atomic Hardy spaces introduced by Coifman and Weiss coincide with the Hardy spaces defined in terms of wavelet coefficients and develop the molecule theory for this general setting. The main tools used in this paper are atomic decomposition, the orthonormal wavelet basis constructed recently by Auscher and Hyt\"onen, the discrete Calder\'on-type reproducing formula, the almost orthogonal estimates, implement various stopping time arguments and the duality of the Hardy spaces with the Carleson measure spaces.