Decomposition theorems for Hardy spaces on products of Siegel upper half spaces and bi-parameter Hardy spaces

Abstract

Products of Siegel upper half spaces are Siegel domains, whose Silov boundaries have the structure of products ${\mathcal{H}}_1\times {\mathcal{H}}_2$ of Heisenberg groups. By the reproducing formula of bi-parameter heat kernel associated to sub-Laplacians, we show that a function in holomorphic Hardy space $H^1$ on such a domain has boundary value belonging to bi-parameter Hardy space $H^1({\mathcal{H}}_1\times {\mathcal{H}}_2)$. With the help of atomic decomposition of $H^1({\mathcal{H}}_1\times {\mathcal{H}}_2)$ and bi-paramete rharmonic analysis, we show that the Cauchy-Szeg\H o projection is a bounded operator from $H^1({\mathcal{H}}_1\times {\mathcal{H}}_2)$ to holomorphic Hardy space $H^1$, and any holomorphic $H^1$ function can be decomposed as a sum of holomorphic atoms. Bi-parameter atoms on ${\mathcal{H}}_1\times {\mathcal{H}}_2$ are more complicated than $1$-parameter ones, and so are holomorphic atoms.

26 pages