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

Abstract

We prove precise growth and cancellation estimates for the Szeg\H{o} kernel of an unbounded model domain $\Omega\subset\mathbb{C}^2$ under the assumption that ${\rm b}\Omega$ 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 Szeg\H{o} 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 $\mathbb{S}:\Gamma_\alpha (E)\rightarrow \Gamma_\alpha({\rm b}\Omega)$, 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 H\"older spaces.