Asymptotic expansion of polyanalytic Bergman kernels

TL;DR

This paper derives the asymptotic expansion of bianalytic Bergman kernels for weighted spaces, extending known results from the analytic case to the bianalytic setting and introducing new metrics.

Contribution

It provides the first asymptotic expansion for bianalytic Bergman kernels under standard weight conditions, generalizing classical analytic results.

Findings

Asymptotic expansion formulas for bianalytic Bergman kernels

Application to two new bianalytic Bergman metrics

Extension of analytic kernel results to bianalytic functions

Abstract

We consider mainly the Hilbert space of bianalytic functions on a given domain in the plane, square integrable with respect to a weight. We show how to obtain the asymptotic expansion of the corresponding bianalytic Bergman kernel for power weights, under the standard condition on those weights. This is known only in the analytic setting, from the work of e.g. Tian, Yau, Zelditch, Catlin, et al. We remark that a bianalytic function may be identified with a vector-valued analytic function, supplied with a locally singular metric on the vectors. We also apply our findings to two bianalytic Bergman metrics introduced here.