Global $C^\nf$ Irregularity of the $\bar\partial$--Neumann Problem for Worm Domains

Irregularity of the Bergman projection on worm domains in C^n

We construct higher-dimensional versions of the Diederich-Fornaess worm domains and show that the Bergman projection operators for these domains are not bounded on high-order $L^p$-Sobolev spaces for $1\leq p<\infty.$

Irregularity of the Bergman projection on smooth unbounded worm domains

In this work we consider smooth unbounded worm domains $\mathcal Z_λ$ in $\mathbb C^2$ and show that the Bergman projection, densely defined on the Sobolev spaces $H^{s,p}(\mathcal Z_λ)$, $p\in(1,\infty)$, $s\ge0$, does not extend to a bounded operator $P_λ:H^{s,p}(\mathcal Z_λ)\to H^{s,p}(\mathcal Z_λ)$ 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.

A remark on irregularity of the dbar-Neumann problem on non-smooth domains

It is an observation due to J.J. Kohn that for a smooth bounded pseudoconvex domain D in $C^n$ there exists s>0 such that the dbar-Neumann operator on D maps $W^s_{(0,1)}(D)$ (the space of $(0,1)$-forms with coefficient functions in $L^2$-Sobolev space of order s) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain D in $C^2$, smooth except at one point, whose dbar-Neumann operator is not bounded on $W^s_{(0,1)}(D)$ for any s>0.

Analysis and geometry on worm domains

We describe recent work on the Bergman kernel of the (non-smooth) worm domain in several complex variables. An asymptotic expansion is obtained for the Bergman kernel. Mapping properties of the Bergman projection are studied. Irregularity properties of the kernal at the boundary are established. This is an expository paper, and considerable background is provided. Discussion of the smooth worm is also included.

Essential norm estimates for the $\bar{\partial}$-Neumann operator on convex domains and worm domains

In the paper we give a lower estimate for the essential norm of the $\bar{\partial}$-Neumann operator on convex domains and worm domains of Diederich and Fornaess.

更高维度的蠕虫区域

Higher dimensional worm domains

本文提出构造一类边界包含给定 Stein 流形的光滑有界伪凸区域的方法，可指定余维数和 D'Angelo 类，并在此类区域下探讨 $\overline\partial$-Neumann 问题正则性理论的开放问题。