An application of John ellipsoids to the Szego kernel on unbounded convex domains

Abstract

We use convex geometry tools, in particular John ellipsoids, to obtain a size estimate for the Szegő kernel on the boundary of a class of unbounded convex domains in ${\mathbb{C}}^n.$ Given a polynomial $b:{\mathbb{R}}^n\to \mathbb{R}$ satisfying a certain growth condition, we consider domains of the type ${\Omega }_b=\{z\in {\mathbb{C}}^{n+1}:\mathrm{Im}[z_{n+1}]>b(\mathrm{Re}[z_1],\dots ,\mathrm{Re}[z_n])\}.$