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

2023/02/01 by Wei Wang, Wang, Wei, Qingyan Wu +1
Mathematics · #32A35 #32A40 #42B30 #43A85 #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2302.00490

openalex publication_date 2023/02/01 · openalex created_date 2023/02/04 · openalex updated_date 2026/07/28

Abstract

Products of Siegel upper half spaces are Siegel domains, whose Silov boundaries have the structure of products \mathscr H₁×\mathscr H₂ 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 H¹ on such a domain has boundary value belonging to bi-parameter Hardy space H¹ (\mathscr H₁× \mathscr H₂). With the help of atomic decomposition of H¹ (\mathscr H₁× \mathscr H₂) and bi-paramete rharmonic analysis, we show that the Cauchy-Szeg\H o projection is a bounded operator from H¹ (\mathscr H₁× \mathscr H₂) to holomorphic Hardy space H¹, and any holomorphic H¹ function can be decomposed as a sum of holomorphic atoms. Bi-parameter atoms on \mathscr H₁×\mathscr H₂ are more complicated than 1-parameter ones, and so are holomorphic atoms.

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