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 ${\mathcal{H}}^2(D_{\beta })$. This can be identified with a closed subspace of $L^2(d_b(D_{\beta }),d\sigma )$, that we denote by ${\mathcal{H}}^2(d_b(D_{\beta }))$, where $d\sigma$ is the naturally induced measure on $d_b(D_{\beta })$. The orthogonal Hilbert space projection $\mathcal{P}:L^2(d_b(D_{\beta }),d\sigma )\to {\mathcal{H}}^2(d_b(D_{\beta }))$ is called the Szegő projection on the distinguished boundary. We prove that $\mathcal{P}$, 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 $\mathcal{P}:L^p(d_b(D_{\beta }),d\sigma )\to L^p(d_b(D_{\beta }),d\sigma )$ if and only if $\frac{2}{1+{\nu }_{\beta }}<p<\frac{2}{1-{\nu }_{\beta }}$ where ${\nu }_{\beta }=\frac{\pi }{2\beta -\pi },\beta >\pi$. Furthermore, we also prove that $\mathcal{P}$ defines a bounded operator $\mathcal{P}:W^{s,2}(d_b(D_{\beta }),d\sigma )\to W^{s,2}(d_b(D_{\beta }),d\sigma )$ if and only if $0\le s<\frac{{\nu }_{\beta }}{2}$ 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 $\mathcal{P}$ 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.

27 pages

References in corpus (5)

• The Bergman kernel and projection on non-smooth worm domains
• The Cauchy-Szeg\H o projection for domains in $\mathbb C^n% with minimal smoothness
• Bergman kernel and projection on the unbounded worm domain
• Essential norm estimates for the $\bar{\partial }$-Neumann operator on convex domains and worm domains
• Completeness on the worm domain and the Müntz-Szász problem for the Bergman space

Cited by in corpus (2)

• Completeness on the worm domain and the Müntz-Szász problem for the Bergman space
• Irregularity of the Bergman projection on smooth unbounded worm domains