The Szegö kernel for certain non-pseudoconvex domains in ${\mathbb C}^{2}$

Abstract

We consider the Szegö kernel for domains in ${\mathbb C}^{2}$ given by \[ \Omega = \bigl\{(z,w) : \mathrm{Im} (w) > b\bigl(\mathrm{Re} (z)\bigr)\bigr\} \] for $b$ a non-convex quartic polynomial with positive leading coefficient. Such domains are not pseudoconvex. We describe a subset of $\overline{\Omega} \times \overline{\Omega}$ on which the kernel and all of its derivatives are finite. We show that there are points off the diagonal of $\partial \Omega \times \partial \Omega$ at which the Szegö kernel is infinite as well as points on the diagonal at which it is finite.