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

TL;DR

This paper establishes decomposition theorems for Hardy spaces on products of Siegel upper half spaces, linking boundary values, atomic decompositions, and boundedness of the Cauchy-Szegő projection using bi-parameter harmonic analysis.

Contribution

It introduces new atomic decomposition results for bi-parameter Hardy spaces on Siegel domain boundaries and proves the boundedness of the Cauchy-Szegő projection in this setting.

Findings

Boundary values of holomorphic Hardy space functions lie in bi-parameter Hardy spaces.

Cauchy-Szegő projection is bounded from bi-parameter Hardy spaces to holomorphic Hardy spaces.

Holomorphic functions in Hardy spaces can be decomposed into sums of holomorphic atoms.

Abstract

Products of Siegel upper half spaces are Siegel domains, whose Silov boundaries have the structure of products $\mathscr H_1\times\mathscr H_2$ 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 (\mathscr H_1\times \mathscr H_2)$. With the help of atomic decomposition of $ H^1 (\mathscr H_1\times \mathscr H_2)$ and bi-paramete rharmonic analysis, we show that the Cauchy-Szeg\H o projection is a bounded operator from $ H^1 (\mathscr H_1\times \mathscr 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 $\mathscr H_1\times\mathscr H_2$ are more complicated than $1$-parameter ones, and so are holomorphic atoms.