New Atomic Decompositions of Weighted Local Hardy Spaces

TL;DR

This paper introduces a new atomic decomposition framework for weighted local Hardy spaces, enabling better analysis of operators like Calderón-Zygmund in weighted settings.

Contribution

It develops a novel class of weighted local approximate atoms and establishes atomic decompositions for weighted local Hardy spaces with applications to operator boundedness.

Findings

Established weighted local atomic decompositions for $h_{ ext{omega}}^p(R^n)$

Proved boundedness of inhomogeneous Calderón-Zygmund operators on these spaces

Extended classical atomic theory to weighted local contexts

Abstract

We introduce a new class of weighted local approximate atoms including classical weighted local atoms. Then we further obtain the weighted local approximate atomic decompositions of weighted local Hardy spaces $h_{\omega} ^p(R^n)$ with $0<p\leq 1$ and weight $\omega\in A_1(R^n)$. As an application, we prove the boundedness of inhomogeneous Calder\'on-Zygmund operators on $h_{\omega}^p(R^n)$ via weighted local approximate atoms and molecules.