The Cauchy-Szegő Projection and its commutator for domains in $\mathbb C^n$ with minimal smoothness: Optimal bounds

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 and Stein states that the Cauchy-Szeg\H{o} projection $\mathcal S$ maps $L^p(bD, \omega)$ to $L^p(bD, \omega)$ continuously for any $1<p<\infty$ whenever the reference measure $\omega$ is a bounded, positive continuous multiple of induced Lebesgue measure. Here we show that $\mathcal S_\omega$ (defined with respect to any measure $\omega$ as above) satisfies explicit, optimal bounds in $L^p(bD, \Omega_p)$, for any $1<p<\infty$ and for any $\Omega_p$ in the maximal class of $A_p$-measures, that is $\Omega_p = \psi_p\sigma$ where $\psi_p$ is a Muckenhoupt $A_p$-weight and $\sigma$ is the induced Lebesgue measure. As an application, we characterize boundedness in $L^p(bD, \Omega_p)$ {with explicit bounds}, and compactness, of the commutator $[b,\mathcal S_\omega]$ for any $A_p$-measure $\Omega_p$, $1<p<\infty$. We next introduce the notion of holomorphic Hardy spaces for $A_p$-measures, and we characterize boundedness and compactness in $L^2(bD, \Omega_2)$ of the commutator $\displaystyle{[b,\mathcal S_{\Omega_2}]}$ where $\mathcal S_{\Omega_2}$ is the Cauchy-Szeg\H{o} projection defined with respect to any given $A_2$-measure $\Omega_2$. Earlier results rely upon an asymptotic expansion and subsequent pointwise estimates of the Cauchy-Szeg\H{o} kernel, but these are unavailable in our setting of minimal regularity {of $bD$}; at the same time, recent techniques in Lanzani-Stein 2017 that allow to handle domains with minimal regularity are not applicable to $A_p$-measures. It turns out that the method of {quantitative} extrapolation is an appropriate replacement for the missing tools.