Bergman kernels for Paley-Wiener spaces and Nazarov's proof of the Bourgain-Milman theorem

TL;DR

This paper establishes a general inequality for Bergman kernels in convex-weighted spaces and explores its application in Nazarov's proof of the Bourgain-Milman theorem, providing an alternative to traditional H"ormander estimates.

Contribution

It introduces a new inequality for Bergman kernels in convex-weighted spaces and applies it to simplify Nazarov's proof of the Bourgain-Milman theorem.

Findings

Derived a general inequality for Bergman kernels in convex-weighted spaces

Demonstrated the application of this inequality in Nazarov's proof of the Bourgain-Milman theorem

Provided an alternative approach to H"ormander's estimates in complex analysis

Abstract

We give a general inequality for Bergman kernels of Bergman spaces defined by convex weights in $\C^n$. We also discuss how this can be used in Nazarov's proof of the Bourgain-Milman theorem, as a substitute for H\"ormander's estimates for the $\dbar$-equation.