The Szeg\"o kernel for non-pseudoconvex tube domains in C^2

TL;DR

This paper investigates the Szeg"o kernel on non-pseudoconvex tube domains in C^2 defined by polynomial inequalities, revealing off-diagonal singularities and boundary behavior.

Contribution

It extends previous work by analyzing the Szeg"o kernel for more general non-pseudoconvex domains with polynomial boundary functions.

Findings

Szeg"o kernel exhibits off-diagonal boundary singularities.

Identifies points on the boundary where the kernel remains finite.

Generalizes earlier results from degree 4 to higher even degrees.

Abstract

We consider the Szeg\"o kernel for non-pseudoconvex domains in C^2 given by \Omega = {(z,w): Im w > b(Re z)} for b a non-convex even-degree polynomial with positive leading coefficient. This is an extension of results previously obtained by the authors for the case in which b has degree 4. We show that the Szeg\"o kernel has singularities off the diagonal of the boundary of \bar{\Omega} \times \bar{\Omega} for all such domains, as well as points on the diagonal of the boundary at which it is finite.