The commutator of the Cauchy--Szegő 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--Szegő 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--Szegő 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 $[b,S_{{\Omega }_2}]$.

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