An atomic decomposition characterization of flag Hardy spaces $H_F^p({\text{rr}}^n\times {\text{rr}}^m)$ with applications

In this paper, we give an atomic decomposition characterization of flag Hardy spaces $H_F^p({\text{rr}}^n\times {\text{rr}}^m)$ for $0<p\le 1$, which were introduced in \cite{hl1}. A remarkable feature of atoms of such flag Hardy spaces is that these atoms have only partial cancellation conditions. As an application, we prove a boundedness criterion for operators on flag Hardy spaces.