Estimates for the Szegő Projection on Uniformly Finite-Type Subdomains of $\mathbb{C}^2$

Abstract

We prove precise growth and cancellation estimates for the Szegő kernel of an unbounded model domain $Ω\subset\mathbb{C}^2$ under the assumption that ${\rm b}Ω$ 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 $Ω$ is exactly regular on the non-isotropic Sobolev spaces $NL_k^p({\rm b}Ω)$ for $1<p<+\infty$ and $k=0,1,\ldots$, and also that $\mathbb{S}:Γ_α(E)\rightarrow Γ_α({\rm b}Ω)$, for $E\Subset {\rm b}Ω$ and $0<α<+\infty$, with a bound that depends only on ${\rm diam}(E)$, where $Γ_α$ are the non-isotropic Hölder spaces.