The Khavinson-Shapiro Conjecture for the Bergman Projection in One and Several Complex Variables

TL;DR

This paper demonstrates that the Bergman projection on ellipsoids in complex spaces preserves polynomial degree and relates this to the Khavinson-Shapiro conjecture, extending polynomial solution properties from real to complex variables.

Contribution

It establishes a complex analogue of a polynomial solution result for the Bergman projection on ellipsoids, connecting it to the Khavinson-Shapiro conjecture in several complex variables.

Findings

Bergman projection of polynomials on ellipsoids remains polynomial of same degree.

The result extends polynomial solution properties from real to complex domains.

Polyharmonic Bergman projections also map polynomials to polynomials on ellipsoids.

Abstract

We reveal a complex analogue to a result about polynomial solutions to the Dirichlet Problem on ellipsoids in $\mathbb{R}^n$ by showing that the Bergman projection on any ellipsoid in $\mathbb{C}^n$ is such that the projection of any polynomial function of degree at most $N$ is a holomorphic polynomial function of degree at most $N$. The discussion is motivated by a connection between the Bergman projection and the Khavinson-Shapiro conjecture in $\mathbb{C}$. We also relate the Khavinson-Shapiro conjecture to polyharmonic Bergman projections in $\mathbb{R}^n$ by showing that these projections take polynomials to polynomials on ellipsoids.