The Cauchy--SzegH{o} Projection and its commutator for Domains in $mathbb C^n$ with Minimal Smoothness

Let $DsubsetC^n$ 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--SzegH{o} projection $EuScript S_omega$ defined with respect to any textit{Leray Levi-like} measure $omega$ is bounded in $L^p(bD, omega)$ for any $1<p<infty$. (For this class of domains, induced Lebesgue measure $sigma$ is Leray Levi-like.) Here we show that $EuScript S_omega$ is in fact bounded in $L^p(bD, Omega_p)$ for any $1<p<infty$ and for any $Op$ in the far larger class of textit{$A_p$-like} measures (modeled after the Muckenhoupt $A_p$-weights for $sigma$). As an application, we characterize boundedness and compactness in $L^p(bD, Omega_p)$ for $1<p<infty$, of the commutator $[b, EuScript S_omega]$. We next introduce the holomorphic Hardy spaces $H^p(bD, Omega_p)$, $1<p<infty$, and we characterize boundedness and compactness in $L^2(bD, Omega_2)$ of the commutator $displaystyle{[b,EuScript S_{Omega_2}]}$ of the Cauchy--SzegH{o} projection defined with respect to any $A_2$-like measure $Omega_2$. Earlier closely related results rely upon an asymptotic expansion, and subsequent pointwise estimates, of the Cauchy--SzegH o kernel that are not available in the settings of minimal regularity {of $bD$} and/or $A_p$-like measures.