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

Abstract

In this work we present necessary cancellation conditions for the continuity of linear operators in $h^p({\mathbb{R}}^n)$, $0<p\le 1$, 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 Calderón--Zygmund type operators

8 pages