Bergman kernel and projection on the unbounded worm domain

Abstract

In this paper we study the Bergman kernel and projection on the unbounded worm domain $${\mathcal{W}}_{\infty }=\{(z_1,z_2)\in {\mathbb{C}}^2:|z_1-e^{i\log |z_2{|}^2}{|}^2<1,z_2\ne 0\}.$$ 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 $\mathcal{P}:W^s\nrightarrow W^s$ if $s>0$ and $\mathcal{P}:L^p\nrightarrow L^p$ if $p\ne 2$, where $W^s$ denotes the classic Sobolev space, and $L^p$ the Lebesgue space, respectively, on ${\mathcal{W}}_{\infty }$.

23 pages, 3 figures, Author Accepted Manuscript

References in corpus (1)

• The Bergman kernel and projection on non-smooth worm domains

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