Necessary cancellation conditions for the boundedness of operators on local Hardy spaces

TL;DR

This paper establishes necessary cancellation conditions for the boundedness of certain linear operators on local Hardy spaces, matching previously known sufficient conditions, thus providing a complete characterization.

Contribution

It introduces a necessary cancellation condition for operators on local Hardy spaces that aligns with existing sufficient conditions, completing the boundedness criteria.

Findings

Necessary and sufficient cancellation condition identified

Condition expressed via the $T^{ ext{*}}$ criterion

Applicable to inhomogeneous Calderón--Zygmund operators

Abstract

In this work we present necessary cancellation conditions for the continuity of linear operators in $h^p(\mathbb{R}^n)$, $0<p\leq 1$, that map atoms into pseudo-molecules. Our necessary condition, expressed in terms of the $T^{\ast}$ condition, is the same as the one recently proved sufficient in [3], thus providing a necessary and sufficient cancellation condition for the boundedness of inhomogeneous Calder\'on--Zygmund type operators