Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type

TL;DR

This paper develops the theory of product Hardy spaces linked to operators with heat kernel bounds on spaces of homogeneous type, establishing key decompositions and showing their equivalence to Lebesgue spaces under certain conditions.

Contribution

It introduces a new framework for product Hardy spaces associated with operators satisfying Davies--Gaffney estimates on spaces of homogeneous type, including a Calderón--Zygmund decomposition and interpolation results.

Findings

Established Calderón--Zygmund decomposition on product spaces

Proved the equivalence of product Hardy spaces and Lebesgue spaces under Gaussian estimates

Extended Hardy space theory to operators with weak heat kernel bounds

Abstract

The aim of this article is to develop the theory of product Hardy spaces associated with operators which possess the weak assumption of Davies--Gaffney heat kernel estimates, in the setting of spaces of homogeneous type. We also establish a Calder\'on--Zygmund decomposition on product spaces, which is of independent interest, and use it to study the interpolation of these product Hardy spaces. We then show that under the assumption of generalized Gaussian estimates, the product Hardy spaces coincide with the Lebesgue spaces, for an appropriate range of~$p$.