Products of functions in $\BMO$ and $\H^{1}$ spaces on spaces of homogeneous type

TL;DR

This paper extends the theory of products of functions in BMO and H^1 spaces, as well as Lipschitz and H^p spaces, to RD-spaces with reverse doubling property, broadening their applicability in harmonic analysis.

Contribution

It generalizes the definitions and properties of these function products to RD-spaces, which include spaces of homogeneous type with reverse doubling, a significant extension beyond classical settings.

Findings

Extended product definitions to RD-spaces with reverse doubling

Established properties of these products in the new setting

Connected Lipschitz and H^p spaces within this framework

Abstract

We give an extension to certain \textit{RD-space} $\X$, i.e space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property, of the definition and various properties of the product of functions in $\BMO(\X)$ and $\H^{1}(\X)$, and functions in Lipschitz space $\Lambda_{\frac{1}{p}-1}(\X)$ and $\H^{p}(\X)$ for $p\in(\frac{\n}{\n+\theta},1]$, where $\n$ and $\theta$ denote respectively the "dimension" and the order of $\X$