The Cauchy--Szeg\"{o} Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted theory

Abstract

Let $D\subset\mathbb C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$. A 2017 result of Lanzani \& Stein states that the Cauchy--Szeg\"{o} projection $S_\omega$ defined with respect to a bounded, positive continuous multiple $\omega$ of induced Lebesgue measure, {maps $L^p(bD, \omega)$ to $L^p(bD, \omega)$ continuously} for any $1<p<\infty$. Here we show that $S_\omega$ satisfies explicit quantitative bounds in $L^p(bD, \Omega)$, for any $1<p<\infty$ and for any $\Omega$ in the maximal class of \textit{$A_p$}-measures, that is for $\Omega_p = \psi_p\sigma$ where $\psi_p$ is a Muckenhoupt $A_p$-weight and $\sigma$ is the induced Lebesgue measure (with $\omega$'s as above being a sub-class). Earlier results rely upon an asymptotic expansion and subsequent pointwise estimates of the Cauchy--Szeg\"o kernel, but these are unavailable in our setting of minimal regularity {of $bD$}; at the same time, more recent techniques that allow to handle domains with minimal regularity (Lanzani--Stein 2017) are not applicable to $A_p$-measures. It turns out that the method of {quantitative} extrapolation is an appropriate replacement for the missing tools. To finish, we identify a class of holomorphic Hardy spaces defined with respect to $A_p$-measures for which a meaningful notion of Cauchy--Szeg\"o projection can be defined when $p=2$.