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

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