Bergman kernel and projection on the unbounded Diederich-Fornæss worm domain

In this paper we study the Bergman kernel and projection on the unbounded worm domain $$ cW_infty = ig{ (z_1,z_2)inbC^2:, ig|z_1-e^{ilog|z_2|^2}ig|^2<1 , z_2 eq0ig} , . $$ We first show that the Bergman space of $cW_infty$ is infinite dimensional. Then we study Bergman kernel $K$ and Bergman projection $cP$ for $cW_infty$. 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 $cP:W^s ot o W^s$ if $s>0$ and $cP:L^p ot o L^p$ if $p eq2$, where $W^s$ denotes the classic Sobolev space, and $L^p$ the Lebesgue space, respectively, on $cW_infty$.