1.1. Definitions Throughout this paper, λ denotes the Lebesgue measure on C and ωo = dd|z| the Euclidean Kähler form in C, where d = √ −1 4 (∂̄ − ∂). Let φ ∈ C (C) be a function, μ a measure in C, and p ∈ [1,∞). One can define the spaces Lp(e−pφdμ) and F (μ, φ) := Lp(e−pφdμ) ∩ O(C). If the measure μ is Lebesgue measure, we simply write F (λ, φ) =: F (φ). Similarly one can define L∞(e−φ, μ) = {f ...

Herein, the theory of Bergman kernel is developed to weighted case. A general form reproducing obtained, by which we can calculate concrete functions for specific weights and domains.

In this paper we characterize the kernel of an intermediate Hankel operator on the Bergman space in terms of the inner divisors and obtain a characterization for finite rank intermediate Hankel operators.

This survey article, intended to be accessible to students, discusses results about the presence or absence of zeroes of the Bergman kernel function of a bounded domain in C. Six open problems are stated.

We prove the asymptotic of logarithmic Bergman kernel. And as an application, we calculate conditional expectation density zeros Gaussian random sections powers a positive line bundle that vanish along fixed smooth subvariety.