The commutator of the Cauchy--Szegő Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted regularity
Xuan Thinh Duong, Loredana Lanzani, Ji Li, Brett D. Wick
TL;DR
The paper extends the CRW/KL2 framework to domains with minimal boundary smoothness by characterizing when the commutator [b,S_ω] is bounded or compact on weighted L^p spaces on the boundary. Using a robust comparison with explicit Cauchy-type operators C_ε and a Kerzman–Stein-type inversion, it establishes that BMO(bD,σ) controls weighted L^p boundedness for all A_p weights, with explicit bounds, and VMO yields compactness. A parallel p=2 theory is developed for S_{Ω_2}, connecting BMO/VMO to boundedness/compactness without dependence on p. The work relies on extrapolation across p, the structure of Leray Levi-like measures, and detailed kernel decompositions to overcome the lack of sharp Szegő kernel estimates in minimal smoothness.
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 $S_ω$ denote the Cauchy--Szegő projection defined with respect to (any) positive continuous multiple $ω$ of induced Lebesgue measure for the boundary of $D$. We characterize compactness and boundedness (the latter with explicit bounds) of the commutator $[b, S_ω]$ in the Lebesgue space $L^p(bD, Ω_p)$ where $Ω_p$ is any measure in the Muckenhoupt class $A_p(bD)$, $1<p<\infty$. We next fix $p =2$ and we let $S_{Ω_2}$ denote the Cauchy--Szegő projection defined with respect to (any) measure $Ω_2 \in A_2(bD)$, 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 $L^2(bD, Ω_2)$ of the commutator $\displaystyle{[b,S_{Ω_2}]}$.