Regularity of the Szeg\"o projection on model worm domains

TL;DR

This paper investigates the regularity properties of the Szeg"o projection on Lebesgue and Sobolev spaces on the boundary of the unbounded model worm domain, establishing boundedness results for a range of function spaces.

Contribution

It proves that the Szeg"o projection extends as a bounded operator on Sobolev spaces on the boundary of the model worm domain, broadening understanding of its regularity.

Findings

Szeg"o projection is bounded on Sobolev spaces W^{s,p} for 1<p<∞ and s≥0.

Extension of the Szeg"o projection from dense subspaces to full Sobolev spaces.

Regularity results apply to the boundary of the unbounded model worm domain D'_eta.

Abstract

In this paper we study the regularity of the Szeg\"o projection on Lebesgue and Sobolev spaces on the boundary of the unbounded model worm domain $D'_\beta$. We consider the Hardy space $H^2(D'_\beta)$. Denoting by $bD'_\beta$ the boundary of $D'_\beta$, it is classical that $H^2(D'_\beta)$ can be identified with the closed subspace of $L^2(bD'_\beta,d\sigma)$, denoted by $H^2(bD'_\beta)$, consisting of the boundary values of functions in $H^2(D'_\beta)$, where $d\sigma$ is the induced Lebesgue measure. The orthogonal Hilbert space projection $P: L^2(D'_\beta,d\sigma)\to H^2(bD'_\beta)$ is called the Szeg\"o projection. Let $W^{s,p}(bD'_\beta)$ denote the Lebesgue--Sobolev space on $bD'_\beta$. We prove that $P$, initially defined on the dense subspace $W^{s,p}(bD'_\beta)\cap L^2(bD'_\beta,d\sigma)$, extends to a bounded operator $P: W^{s,p}(bD'_\beta)\to W^{s,p}(bD'_\beta)$, for $1<p<\infty$ and $s\ge0$.