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

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–Szegö 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 $1<p<\mathrm{\infty }$. Here we show that $S_{\omega }$ satisfies explicit quantitative bounds in $L^p(bD,\mathrm{\Omega })$, for any $1<p<\mathrm{\infty }$ and for any $\mathrm{\Omega }$ in the maximal class of $A_p$-measures, that is for ${\mathrm{\Omega }}_p={\psi }_p\sigma$ 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–Szegö 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–Szegö projection can be defined when $p=2$.