Regularity of the Szegö projection on model worm domains

Abstract

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 $D_{\beta }^{\prime }$. We consider the Hardy space $H^2(D_{\beta }^{\prime })$. Denoting by $b{D}_{\beta }^{\prime }$ the boundary of $D_{\beta }^{\prime }$, it is classical that $H^2(D_{\beta }^{\prime })$ can be identified with the closed subspace of $L^2(b{D}_{\beta }^{\prime },d\sigma )$, denoted by $H^2(b{D}_{\beta }^{\prime })$, consisting of the boundary values of functions in $H^2(D_{\beta }^{\prime })$, where $d\sigma$ is the induced Lebesgue measure. The orthogonal Hilbert space projection $P:L^2(D_{\beta }^{\prime },d\sigma )\to H^2(b{D}_{\beta }^{\prime })$ is called the Szegö projection. Let $W^{s,p}(b{D}_{\beta }^{\prime })$ denote the Lebesgue--Sobolev space on $b{D}_{\beta }^{\prime }$. We prove that $P$, initially defined on the dense subspace $W^{s,p}(b{D}_{\beta }^{\prime })\cap L^2(b{D}_{\beta }^{\prime },d\sigma )$, extends to a bounded operator $P:W^{s,p}(b{D}_{\beta }^{\prime })\to W^{s,p}(b{D}_{\beta }^{\prime })$, for $1<p<\infty$ and $s\ge 0$.