Bergman kernel and projection on the unbounded worm domain

In this paper we study the Bergman kernel and projection on the unbounded worm domain $${\mathcal{W}}_{\mathrm{\infty }}=\{(z_1,z_2)\in {\mathbb{C}}^2:\mid z_1-e^{i\log \mathrm{\mid }{z}_2{\mathrm{\mid }}^2}{\mid }^2<1,z_2\mathrm{\ne }0\}\mathrm{.}$$ We first show that the Bergman space of ${\mathcal{W}}_{\mathrm{\infty }}$ is infinite dimensional. Then we study Bergman kernel $K$ and Bergman projection $\mathcal{P}$ for ${\mathcal{W}}_{\mathrm{\infty }}$. 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 ${\mathcal{W}}_{\mathrm{\infty }}$.