The commutator of the Cauchy--Szeg\H{o} Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted regularity

TL;DR

This paper characterizes when the commutator of the Cauchy--Szegő projection is bounded or compact on weighted Lebesgue spaces for domains with minimal boundary smoothness in complex space.

Contribution

It provides the first characterization of the boundedness and compactness of commutators of the Cauchy--Szegő projection on minimally smooth strongly pseudoconvex domains.

Findings

Characterizes boundedness of commutators on weighted spaces for $1<p< $

Provides explicit bounds for the commutator in Lebesgue spaces

Establishes criteria for compactness of the commutator

Abstract

Let $D\subset\mathbb C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$, and let $S_\omega$ denote the Cauchy--Szeg\H{o} projection defined with respect to (any) positive continuous multiple $\omega$ of induced Lebesgue measure for the boundary of $D$. We characterize compactness and boundedness (the latter with explicit bounds) of the commutator $[b, S_\omega]$ in the Lebesgue space $L^p(bD, \Omega_p)$ where $\Omega_p$ is any measure in the Muckenhoupt class $A_p(bD)$, $1<p<\infty$. We next fix $p =2$ and we let $S_{\Omega_2}$ denote the Cauchy--Szeg\H{o} projection defined with respect to (any) measure $\Omega_2 \in A_2(bD)$, which is the largest class of reference measures for which a meaningful notion of Cauchy-Leray measure may be defined. We characterize boundedness and compactness in $L^2(bD, \Omega_2)$ of the commutator $\displaystyle{[b,S_{\Omega_2}]}$.