Irregularity of the Bergman projection on smooth unbounded worm domains

TL;DR

This paper demonstrates that the Bergman projection on smooth unbounded worm domains in complex space fails to extend as a bounded operator on Sobolev spaces for certain parameters, highlighting irregularity due to domain winding.

Contribution

It proves the irregularity of the Bergman projection on smooth unbounded worm domains, independent of boundary smoothness, due to infinite windings, extending previous non-smooth results.

Findings

Bergman projection not bounded on Sobolev spaces for s>0 or p≠2

Irregularity caused by domain winding, not boundary smoothness

Extends known irregularity results to 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 $H^{s,p}(\mathcal Z_\lambda)$, $p\in(1,\infty)$, $s\ge0$, 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 $p\neq2$. 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.