2024/12/18 by Bo-Yong Chen, Chen, Bo-Yong, Yuanpu Xiong +1
Mathematics · #Algebraic and Geometric Analysis
paper · pdf · doi:10.48550/arxiv.2412.13854
In this article, we investigate the connection between certain real variable things and the Bergman theory. We first use Hardy-type inequalities to give an L2 Hartogs-type extension theorem and an Lp integrability theorem for the Bergman kernel KΩ(⋅,w). We then use the Sobolev-Morrey inequality to show the absolute continuity of Bergman kernels on planar domains with respect to logarithmic capacities. Finally, we give lower bounds of the minimum κ(Ω) of the Bergman kernel KΩ(z) in terms of the interior capacity radius for planar domains and the volume density for bounded pseudoconvex domains in \mathbb Cn. As a consequence, we show that κ(Ω)≥ c0 λ1(Ω) holds on planar domains, where c0 is a numerical constant and λ1(Ω) is the first Dirichlet eigenvalue of -Δ.