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

TL;DR

This paper employs convex geometry, specifically John ellipsoids, to estimate the Szegő kernel size on the boundary of certain unbounded convex domains in complex space, linking geometric properties to kernel behavior.

Contribution

It introduces a novel approach using John ellipsoids to derive size estimates for the Szegő kernel on unbounded convex domains in complex analysis.

Findings

Established size estimates for the Szegő kernel based on convex geometric tools.

Connected geometric properties of domains with kernel estimates in several complex variables.

Provided a framework for analyzing kernels on unbounded convex domains using convex geometry.

Abstract

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