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

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 \[ C_{1}\sqrt{\dfrac{K\left(w,w\right)}{δ\left(w\right)}}\leq\left\Vert K\left(\cdot,w\right)\right\Vert _{L^{2}\left(\partialΩ\right)}\leq C_{2}\sqrt{\dfrac{K\left(w,w\right)}{δ\left(w\right)}}, \] for any $w\in \Omega$. Here $K$ is the Bergman kernel of $\Omega$, and $\delta$ is the distance-to-boundary function.