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arXiv · 1504.04635

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

Alan R. Legg
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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.

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Alan R. Legg. 2015-04-17. The Khavinson-Shapiro Conjecture for the Bergman Projection in One and Several Complex Variables. https://arxiv.org/abs/1504.04635

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