模型蠕虫域上Szegö投影的正则性
Regularity of the Szegö projection on model worm domains
摘要：
本文研究了无界模型蠕虫域$D'_β$边界上Lebesgue和Sobolev空间上Szegö投影的正则性.我们考虑Hardy空间$H^2(D'_β)$ .用$bD'_β$表示$D'_β$的边界，经典的是，$H^2(D'_β)$可以用$L^2(bD'_β,dσ)$的闭子空间来标识，用$H^2(bD'_β)$表示，该闭子空间由$H^2(D'_β)$中的函数的边界值组成，其中$dσ$是诱导勒贝格测度。正交希尔伯特空间投影$P: L^2(D'_β,dσ)\to H^2(bD'_β)$称为塞格投影。设$W^{s,p}(bD'_β)$表示$bD'_β$上的Lebesgue-Sobolev空间.证明了初始定义在稠密子空间$W^{s,p}(bD'_β)\cap L^2(bD'_β,dσ)$上的$P$，对于$1<p<\infty$和$s\ge0$，可扩张到有界算子$P: W^{s,p}(bD'_β)\to W^{s,p}(bD'_β)$.
英文摘要：
In this paper we study the regularity of the Szegö projection on Lebesgue and Sobolev spaces on the boundary of the unbounded model worm domain $D'_β$. We consider the Hardy space $H^2(D'_β)$. Denoting by $bD'_β$ the boundary of $D'_β$, it is classical that $H^2(D'_β)$ can be identified with the closed subspace of $L^2(bD'_β,dσ)$, denoted by $H^2(bD'_β)$, consisting of the boundary values of functions in $H^2(D'_β)$, where $dσ$ is the induced Lebesgue measure. The orthogonal Hilbert space projection $P: L^2(D'_β,dσ)\to H^2(bD'_β)$ is called the Szegö projection. Let $W^{s,p}(bD'_β)$ denote the Lebesgue--Sobolev space on $bD'_β$. We prove that $P$, initially defined on the dense subspace $W^{s,p}(bD'_β)\cap L^2(bD'_β,dσ)$, extends to a bounded operator $P: W^{s,p}(bD'_β)\to W^{s,p}(bD'_β)$, for $1<p<\infty$ and $s\ge0$.
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