Sharp Estimates for the Szegő Projection on the Distinguished Boundary of Model Worm Domains

Abstract

In this paper we study the regularity of the Szeg\H{o} projection on Lebesgue and Sobolev spaces on the distinguished boundary of the unbounded model worm domain $D_\beta$. We denote by $d_b(D_\beta)$ the distinguished boundary of $D_\beta$ and define the corresponding Hardy space $\mathscr{H}^2(D_\beta)$. This can be identified with a closed subspace of $L^2(d_b(D_\beta),d\sigma)$, that we denote by $\mathscr{H}^2(d_b(D_\beta))$, where $d\sigma$ is the naturally induced measure on $d_b(D_\beta)$. The orthogonal Hilbert space projection $\mathscr{P}: L^2(d_b(D_\beta),d\sigma)\to \mathscr{H}^2(d_b(D_\beta))$ is called the Szeg\H{o} projection on the distinguished boundary. We prove that $\mathscr{P}$, initially defined on the dense subspace $L^2(d_b( D_\beta),d\sigma)\cap L^p(d_b(D_\beta), d\sigma)$ extends to a bounded operator $\mathscr{P}: L^p(d_b(D_\beta), d\sigma)\to L^p(d_b(D_\beta), d\sigma)$ if and only if $\textstyle{\frac{2}{1+\nu_\beta}}<p<\textstyle{\frac{2}{1-\nu_\beta}}$ where $\nu_\beta=\textstyle{\frac{\pi}{2\beta-\pi}},\beta>\pi$. Furthermore, we also prove that $\mathscr{P}$ defines a bounded operator $\mathscr{P}: W^{s,2}(d_b(D_\beta),d\sigma)\to W^{s,2}(d_b(D_\beta), d\sigma)$ if and only if $0\leq s<\textstyle{\frac{\nu_\beta}{2}}$ where $W^{s.2}(d_b( D_\beta), d\sigma)$ denotes the Sobolev space of order $s$ and underlying $L^2$-norm. Finally, we prove a necessary condition for the boundedness of $\mathscr{P}$ on $W^{s,p}(d_b(D_\beta), d\sigma)$, $p\in(1,\infty)$, the Sobolev space of order $s$ and underlying $L^p$-norm.