Bergman kernel and projection on the unbounded worm domain

TL;DR

This paper investigates the Bergman kernel and projection on the unbounded worm domain, revealing their extension properties and mapping behavior on Sobolev and Lebesgue spaces, highlighting the domain's complex analysis features.

Contribution

It demonstrates the infinite dimensionality of the Bergman space on the unbounded worm domain and analyzes the boundary behavior and mapping properties of the Bergman kernel and projection.

Findings

Bergman space of alW_ is infinite dimensional

Bergman kernel extends holomorphically near most boundary points

Bergman projection does not preserve Sobolev or L^p spaces for certain parameters

Abstract

In this paper we study the Bergman kernel and projection on the unbounded worm domain $$ \mathcal{W}_\infty = \big\{(z_1,z_2)\in\mathbb{C}^2 : \big|z_1-e^{i\log|z_2|^2}\big|^2<1, z_2\neq0\big\}. $$ We first show that the Bergman space of $\mathcal{W}_\infty$ is infinite dimensional. Then we study Bergman kernel $K$ and Bergman projection $\mathcal{P}$ for $\mathcal{W}_\infty$. We prove that $K(z,w)$ extends holomorphically in $z$ (and antiholomorphically in $w$) near each point of the boundary except for a specific subset that we study in detail. By means of an appropriate asymptotic expansion for $K$, we prove that the Bergman projection $\mathcal{P}:W^s\not\to W^s$ if $s>0$ and $\mathcal{P}:L^p\not\to L^p$ if $p\neq2$, where $W^s$ denotes the classic Sobolev space, and $L^p$ the Lebesgue space, respectively, on $\mathcal{W}_\infty$.