Irregularity of the Bergman Projection on Smooth Unbounded Worm Domains

Irregularity of the Bergman Projection on Worm Domains in C

We construct higher-dimensional versions of the Diederich-Fornæss worm domains and show that the Bergman projection operators for these domains are not bounded on high-order Lp-Sobolev spaces for 1 ≤ p < ∞.

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 contr...

IRREGULARITY OF THE BERGMAN PROJECTION ON WORM DOMAINS IN C n

We construct higher-dimensional versions of the Diederich-Fornæss worm domains and show that the Bergman projection operators for these domains are not bounded on high-order Lp-Sobolev spaces for 1 ≤ p < ∞.

The Bergman Projection on Sectorial Domains

Global C∞ Irregularity of the ∂̄–neumann Problem for Worm Domains

where ρ is a defining function for Ω, = ∂̄∂̄∗ + ∂̄∗∂̄, u, f are (0, 1) forms, and denotes the interior product of forms. Under the stated hypotheses on Ω, this problem is uniquely solvable for every f ∈ L(Ω). The Neumann operator N , mapping f to the solution u, is continuous on L(Ω). The Bergman projection B is the orthogonal projection of L(Ω) onto the closed subspace of L holomorphic functions o...