Inhomogeneous cancellation conditions and Calder\'on-Zygmund type operators on $h^p$

TL;DR

This paper introduces a new approach to molecules on local Hardy spaces, establishing Hardy's inequality and enhanced continuity results for inhomogeneous Calderón-Zygmund operators under specific cancellation conditions.

Contribution

It presents a novel method for molecules on Goldberg's local Hardy spaces with an appropriate cancellation condition, leading to improved operator continuity results.

Findings

Proved a version of Hardy's inequality for local Hardy spaces.

Established improved continuity results for inhomogeneous Calderón-Zygmund operators.

Developed a new approach to molecules on $h^p$ spaces.

Abstract

In this work we present a new approach to molecules on Goldberg's local Hardy spaces $h^p(\mathbb{R}^n)$, $0<p\leq1$, assuming an appropriate cancellation condition. As applications, we prove a version of Hardy's inequality and improved continuity results for inhomogeneous Calder\'on-Zygmund operators on these spaces.