A note on $L^2$ boundary integrals of the Bergman kernel

TL;DR

This paper establishes bounds relating the boundary integrals of the Bergman kernel to its values inside convex domains with smooth boundaries, revealing a precise quantitative relationship involving the kernel and boundary distance.

Contribution

It provides the first explicit bounds connecting $L^2$ boundary integrals of the Bergman kernel to its pointwise values and boundary distance in convex domains.

Findings

Boundaries of the Bergman kernel are controlled by its interior values.

Explicit constants relate boundary integrals to kernel and boundary distance.

Results hold for convex domains with $C^2$ boundary.

Abstract

For any bounded convex domain $\Omega$ with $C^{2}$ boundary in $\mathbb{C}^{n}$, we show that there exist positive constants $C_{1}$ and $C_{2}$ such that \[ C_{1}\sqrt{\dfrac{K\left(w,w\right)}{\delta\left(w\right)}}\leq\left\Vert K\left(\cdot,w\right)\right\Vert _{L^{2}\left(\partial\Omega\right)}\leq C_{2}\sqrt{\dfrac{K\left(w,w\right)}{\delta\left(w\right)}}, \] for any $w\in\Omega$. Here $K$ is the Bergman kernel of $\Omega$, and $\delta$ is the distance-to-boundary function.