Homogeneously polyanalytic kernels on the unit ball and the Siegel domain

TL;DR

This paper characterizes homogeneously polyanalytic functions on complex domains, establishes a mean value property, and computes their reproducing kernels, extending known results to higher dimensions and specific domains.

Contribution

It provides a new polynomial representation, a weighted mean value property, and explicit reproducing kernels for homogeneously polyanalytic functions on the unit ball and Siegel domain.

Findings

Polynomial representation of homogeneously polyanalytic functions.

Weighted mean value property established.

Explicit reproducing kernels computed for specific domains.

Abstract

We prove that the homogeneously polyanalytic functions of total order $m$, defined by the system of equations $\overline{D}^{(k_1,\ldots,k_n)} f=0$ with $k_1+\cdots+k_n=m$, can be written as polynomials of total degree $<m$ in variables $\overline{z_1},\ldots,\overline{z_n}$, with some analytic coefficients. We establish a weighted mean value property for such functions, using a reproducing property of Jacobi polynomials. After that, we give a general recipe to transform a reproducing kernel by a weighted change of variables. Applying these tools, we compute the reproducing kernel of the Bergman space of homogeneously polyanalytic functions on the unit ball in $\mathbb{C}^n$ and on the Siegel domain. For the one-dimensional case, analogous results were obtained by Koshelev (1977), Pessoa (2014), Hachadi and Youssfi (2019).