Pointwise estimates of weighted Bergman kernels in several complex variables

TL;DR

This paper establishes new pointwise bounds for weighted Bergman kernels in several complex variables by developing an Agmon theory approach for weighted Kohn Laplacians, extending previous results from one complex variable.

Contribution

It introduces a novel Agmon theory framework for weighted Kohn Laplacians on (0,1)-forms in several complex variables, generalizing earlier one-variable results.

Findings

Derived new pointwise bounds for weighted Bergman kernels in $\, ext{C}^n$

Extended Christ's results from $\, ext{C}$ to higher dimensions

Developed a Schrödinger operator-inspired approach for Kohn Laplacians

Abstract

We prove new pointwise bounds for weighted Bergman kernels in $\mathbb{C}^n$, whenever a coercivity condition is satisfied by the associated weighted Kohn Laplacian on $(0,1)$-forms. Our results extend the ones obtained in $\mathbb{C}$ by Christ. Our main idea is to develop a version of Agmon theory (originally introduced to deal with Schr\"odinger operators) for weighted Kohn Laplacians on $(0,1)$-forms, inspired by the fact that these are unitarily equivalent to certain generalized Schr\"odinger operators.