Product of functions in $\BMO$ and $\H^{1}$ in non-homogeneous spaces

TL;DR

This paper explores the product of functions in BMO and H^1 spaces within non-homogeneous metric measure spaces, demonstrating a decomposition into L^1 and Hardy-Orlicz components under minimal measure assumptions.

Contribution

It introduces a new definition of the product in non-doubling spaces and establishes its decomposition into integrable and Hardy-Orlicz parts.

Findings

Product decomposes into L^1 and Hardy-Orlicz parts

Applicable to non-doubling measures with growth conditions

Extends classical BMO-H^1 product theory to non-homogeneous spaces

Abstract

Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition and may not be doubling, we define the product of functions in the regular $BMO$ and the atomic block $\H^{1}$ in the sense of distribution, and show that this product may be split into two parts, one in $L^{1}$ and the other in some Hardy-Orlicz space.