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)}{\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.