Atomic Decomposition Theorem for Hardy spaces on Products of Siegel Upper Half Spaces and Bi-parameter Hardy Spaces

Products of Siegel upper half spaces are Siegel domains whose Silov boundaries have the structure of products ℋ_1×ℋ_2 of Heisenberg groups. By the reproducing formula of bi-parameter heat kernel associated to sub-Laplacians, it is established that a function in holomorphic Hardy space H^1 on such a domain has boundary value belonging to bi-parameter Hardy space H^1 (ℋ_1×ℋ_2) . With the help of atomic decomposition of H^1 (ℋ_1×ℋ_2) and bi-parameter harmonic analysis, we show that the Cauchy–Szegő projection is a bounded operator from H^1 (ℋ_1×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 ℋ_1×ℋ_2 are more complicated than 1-parameter ones, and so are holomorphic atoms.