Necessary cancellation conditions for the boundedness of operators on local Hardy spaces (Q6564503)

Let \(0<p\le 1\). Denote by \(h^p(\mathbb{R}^n)\) the local Hardy space on \(\mathbb{R}^n\). In this paper, the authors obtain necessary cancellation conditions, in terms of the \(T^\ast\) condition, for the continuity of linear operators on \(h^p(\mathbb{R}^n)\) that map atoms in \(h^p(\mathbb{R}^n)\) into pseudo-molecules. As an application, the authors further provide a necessary and sufficient cancellation condition for the boundedness of the inhomogeneous Calderon--Zygmund type operator on \(h^p(\mathbb{R}^n)\).