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

Abstract

In this paper we study the regularity of the Szegő 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 ${\mathscr {P}}: L^2(d_b(D_\beta ), d\sigma )\rightarrow {\mathscr {H}}^2(d_b(D_\beta ))$ is called the Szegő projection on the distinguished boundary. We prove that ${\mathscr {P}}$, initially defined on the dense subspace $L^2\cap L^p(d_b( D_\beta ),d\sigma )$ extends to a bounded operator ${\mathscr {P}}: L^p(d_b(D_\beta ), d\sigma )\rightarrow L^p(d_b(D_\beta ), d\sigma )$ 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 ${\mathscr {P}}: W^{s,2}(d_b(D_\beta ),d\sigma )\rightarrow W^{s,2}(d_b(D_\beta ), d\sigma )$ if and only if $0\le s<\textstyle {\frac{\nu _\beta }{2}}$ 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.