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 W1 = n (z1,z2) 2 C2 : z1 ei log |z2| 2 2 < 1 for z2 6= 0 o . We first show that the Bergman space of W1 is infinite dimensional. Then we study the Bergman kernel K and the Bergman projection P for W1. 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 P : Ws 6! Ws if s > 0 and P : L p 6! L p if p 6= 2, where Ws and L p denote the classic Sobolev space, and the Lebesgue space, respectively, on W1. Mathematics S