On a higher-dimensional worm domain and its geometric properties

TL;DR

This paper introduces new 3D worm domains that are smooth, pseudoconvex, and have complex boundary properties, revealing limitations of Bergman projections in preserving Sobolev spaces.

Contribution

The paper constructs novel 3D worm domains with specific geometric and functional properties, extending classical 2D examples to higher dimensions.

Findings

Domains are smoothly bounded and pseudoconvex.

They possess nontrivial Nebenhülle.

Bergman projections fail to preserve high-order Sobolev spaces.

Abstract

We construct new $3$-dimensional variants of the classical Diederich-Fornaess worm domain. We show that they are smoothly bounded, pseudoconvex, and have nontrivial Nebenh\"{u}lle. We also show that their Bergman projections do not preserve the Sobolev space for sufficiently large Sobolev indices.