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

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 $\displaystyle{[b,S_{\Omega_2}]}$.