  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
