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 [17] states that the Cauchy–Szegő projection \(\mathcal {S}_\omega \) 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 \(\mathcal {S}_\omega \) satisfies explicit quantitative 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 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–Szegő 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 [17] 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-Szegő projection can be defined when \(p=2\) .
