New results on the Bergman kernel of the worm domain in complex space

We study the Bergman kernel and projection on the worm domains for β > π. We calculate the kernels explicitly, up to an error term that can be controlled. Denote by P the Bergman projection on Dβ and by P′ the one on D′β. We show that is bounded when 1 < p < ∞, while if and only if 2/(1 + vβ) < p < 2/(1 - vβ), where vβ = π/(2β - π). Along the way, we give a new proof of the failure of Condition R on these worms. Finally, we are able to show that the singularities of the Bergman kernel on the boundary are not contained in the boundary diagonal.