Boundary behavior of the Szeg\"o kernel

TL;DR

This paper investigates the boundary behavior of the Szeg"o kernel in smooth bounded pseudoconvex domains in ${ m C}^2$, establishing a localization principle and a lower bound near boundary points.

Contribution

It introduces a H"ormander-type localization principle for the Szeg"o kernel and proves a specific boundary lower bound in complex two-dimensional domains.

Findings

Established a localization principle for the Szeg"o kernel.

Proved a boundary lower bound of |z - z_0|^{-1/3} for the kernel.

Results apply to smooth bounded pseudoconvex domains in ${ m C}^2$.

Abstract

We give a H\"ormander-type localization principle for the Szeg\"o kernel $S_\Omega(z)$. We also show that for each boundary point $z_0$, $S_\Omega(z)\gtrsim|z-z_0|^{-\frac{1}{3}}$ holds non-tangentially for any bounded pseudoconvex domain with smooth boundary in ${\mathbb C}^2$.