The Bergman kernel and projection on non-smooth worm domains

We study the Bergman kernel and projection on the worm domains D β = {ζ ∈ ℂ2 : Re (ζ 1e-i log|ζ2|2) > 0, | log |ζ 2|2| < β - π/2} and D′β = {z ∈ ℂ2 : |Im z1 - log |z2| 2| < π/2, | log |z2|2| < β - π/2} for β > π. These two domains are biholomorphically equivalent via the mapping D′β ∋ (z1, z2) → (ez1, z2) ∋ Dβ. We calculate the kernels explicitly, up to an error term that can be controlled. As a result, we can determine the Lp-mapping properties of the Bergman projections on these worm domains. Denote by P the Bergman projection on Dβ and by P′ the one on D′β. We calculate the sharp range of p for which the Bergman projection is bounded on Lp. More precisely we show that P′ : LP(D′β) → LP (D′β) boundedly when 1 < p < ∞, while P : Lp(Dβ) → LP(D β) 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.