Pointwise estimates of the Bergman kernel with an exponential weight on the unit ball

2024/07/01 by Cho, Hong Rae, Park, Soohyun
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.00988

Abstract

We consider the weighted Bergman space A²_ψ(\Bn) of all holomorphic functions on \Bn square integrable with respect to a particular exponential weight measure e^-ψ dV on \Bn, where ψ(z):=(1)/(1-|z|²). We prove the following estimate for the Bergman kernel K_ψ(z,w) of A²_ψ(\Bn): |K_ψ(z,w)|²≤ C\frace^ψ(z)+ψ(w)\rm Vol(B_ψ(z,1))\rm Vol(B_ψ(w, 1))e^{-ε d_ψ(z,w)}, z, w∈\Bn, where d_ψ is the Riemannian distance induced by the potential function ψ and B_ψ(z,1) is the d_ψ-ball of center z and radius 1. The result is motivated by Christ \citeChr.

Related

• New characterizations of Bergman spaces
• Weighted Bergman kernels and ⋆-products
• On the Bergman projections acting on L^∞ in the unit ball \mathbb B_n
• Some properties of the p-Bergman kernel and metric
• Weighted Bergman kernels for nearly holomorphic functions on bounded symmetric domains