The Bergman Kernel and Projection on Non-smooth Worm Domains

We study the Bergman kernel and projection on the worm domains Dβ = { ζ ∈ C : Re ( ζ1e −i log |ζ2| 2) > 0, ∣∣ log |ζ2| ∣∣ < β − π 2 } and D β = { z ∈ C : ∣Im z1 − log |z2| ∣∣ < π 2 , | log |z2| | < β − π 2 } for β > π. These two domains are biholomorphically equivalent via the mapping D β ∋ (z1, z2) 7→ (e z1 , z2) ∋ Dβ . We calculate the kernels explicitly, up to an error term that can be controlled. As a result, we can determine the L-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 L. More precisely we show that P ′ : L(D β) −→ L (D β) boundedly when 1 < p < ∞, while P : L(Dβ) −→ L (Dβ) if and only if 2/(1+ νβ) < p < 2/(1− νβ), where νβ = 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. Introduction The (smooth) worm domain was first created by Diederich and Fornæss [DF] to provide counterexamples to certain classical conjectures about the geometry of pseudoconvex domains. Chief among these examples is that the smooth worm is bounded and pseudoconvex and smooth yet its closure does not have a Stein neighborhood basis. More recently, thanks to work of Kiselman [Ki], Barrett [Ba2], and Christ [Chr], the worm has played an important role in the study of the ∂-Neumann 2000 Mathematics Subject Classification: 32A25, 32A36.