Pointwise estimates for the Bergman kernel of the weighted Fock space

TL;DR

This paper establishes upper pointwise estimates for the Bergman kernel in weighted Fock spaces with subharmonic weights, and characterizes the compactness of related operators based on the Laplacian measure.

Contribution

It provides new upper bounds for the Bergman kernel and characterizes the compactness of the canonical solution operator in weighted Fock spaces with doubling measure weights.

Findings

Derived explicit upper pointwise estimates for the Bergman kernel.

Characterized the compactness of the solution operator in terms of the Laplacian measure.

Connected kernel estimates with operator compactness criteria.

Abstract

We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in $L^2(e^{-2\phi})$ where $\phi$ is a subharmonic function with $\Delta \phi$ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of $\Delta \phi$.