A q-atomic decomposition of weighted tent spaces on spaces of homogeneous type and its application

TL;DR

This paper extends the atomic decomposition theory of tent spaces to weighted versions on spaces of homogeneous type, enabling new analysis tools for weighted Hardy spaces linked to self-adjoint operators.

Contribution

It introduces a $q$-atomic decomposition for weighted tent spaces on spaces of homogeneous type, generalizing previous unweighted results.

Findings

Established a $q$-atomic decomposition for $T^p_{2,w}(X)$.

Provided an atomic decomposition for weighted Hardy spaces.

Extended classical tent space theory to weighted and more general settings.

Abstract

The theory of tent spaces on $\mathbb{R}^n$ was introduced by Coifman, Meyer and Stein, including atomic decomposition, duality theory and so on. Russ generalized the atomic decomposition for tent spaces to the case of spaces of homogeneous type $(X,d,\mu)$. The main purpose of this paper is to extend the results of Coifman, Meyer, Stein and Russ to weighted version. More precisely, we obtain a $q$-atomic decomposition for the weighted tent spaces $T^p_{2,w}(X)$, where $0<p\leq 1, 1<q<\infty,$ and $w\in A_\infty$. As an application, we give an atomic decomposition for weighted Hardy spaces associated to nonnegative self-adjoint operators on $X$.