Homogeneously polyanalytic kernels on the unit ball and the Siegel domain

Abstract

We prove that the homogeneously polyanalytic functions of total order $m$, defined by the system of equations ${\overline{D}}^{(k_1,\dots ,k_n)}f=0$ with $k_1+\cdots +k_n=m$, can be written as polynomials of total degree $<m$ in variables $\overline{z_1},\dots ,\overline{z_n}$, 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).

26 pages