Irregularity of the Bergman projection on smooth unbounded worm domains

Abstract

In this work we consider smooth unbounded worm domains ${\mathcal{Z}}_{\lambda }$ in ${\mathbb{C}}^2$ and show that the Bergman projection, densely defined on the Sobolev spaces $H^{s,p}({\mathcal{Z}}_{\lambda })$, $p\in (1,\infty )$, $s\ge 0$, does not extend to a bounded operator $P_{\lambda }:H^{s,p}({\mathcal{Z}}_{\lambda })\to H^{s,p}({\mathcal{Z}}_{\lambda })$ when $s>0$ or $p\ne 2$. The same irregularity was known in the case of the non-smooth unbounded worm. This improved result shows that the irregularity of the projection is not a consequence of the irregularity of the boundary but instead of the infinite windings of the worm domain.

Revised version. To appear on the Mediterranean Journal of Mathematics

References in corpus (1)

• Hardy Spaces for a Class of Singular Domains

Cited by in corpus (2)

• Nonabelian ramified coverings and $L^p$-boundedness of Bergman projections in ${\mathbb{C}}^2$
• On a higher-dimensional worm domain and its geometric properties