Revista Matemática Iberoamericana


Full-Text PDF (831 KB) | Metadata | Table of Contents | RMI summary
Volume 34, Issue 1, 2018, pp. 111–193
DOI: 10.4171/RMI/982

Published online: 2018-02-06

Estimates for the Szegő projection on uniformly finite-type subdomains of $\mathbb C^2$

Aaron Peterson[1]

(1) Northwestern University, Evanston, USA

We prove precise growth and cancellation estimates for the Szegő kernel of an unbounded model domain $\Omega\subset\mathbb{C}^2$ under the assumption that b$\Omega$ satisfies a uniform finite-type hypothesis. Such domains have smooth boundaries which are not algebraic varieties, and therefore admit no global homogeneities that allow one to use compactness arguments in order to obtain results. As an application of our estimates, we prove that the Szegő projection $\mathbb{S}$ of $\Omega$ is exactly regular on the non-isotropic Sobolev spaces $NL_k^p({\rm b}\Omega)$ for $1 < p<{+\infty}$ and $k=0,1,\ldots$, and also that $\mathbb{S}\colon \Gamma_\alpha (E)\rightarrow \Gamma_\alpha({\rm b}\Omega)$, for $E\Subset {\rm b}\Omega$ and $0<\alpha<+\infty$, with a bound that depends only on ${\rm diam}(E)$, where $\Gamma_\alpha$ are the non-isotropic Hölder spaces.

Keywords: Szegő projection, finite type, unbounded domain, regularity

Peterson Aaron: Estimates for the Szegő projection on uniformly finite-type subdomains of $\mathbb C^2$. Rev. Mat. Iberoamericana 34 (2018), 111-193. doi: 10.4171/RMI/982