The Cauchy–Szegő Projection for Domains in Cⁿ with Minimal Smoothness: Weighted Theory

Let D⊂Cn be a bounded, strongly pseudoconvex domain whose boundary bD satisfies the minimal regularity condition of class C2. A 2017 result of Lanzani & Stein [17] states that the Cauchy–Szegő projection Sω defined with respect to a bounded, positive continuous multiple ω of induced Lebesgue measure, maps Lp(bD,ω) to Lp(bD,ω) continuously for any 1<∞. Here we show that Sω satisfies explicit quantitative bounds in Lp(bD,Ωp), for any 1<∞ and for any Ωp in the maximal class of Ap-measures, that is for Ωp=ψpσ where ψp is a Muckenhoupt Ap-weight and σ is the induced Lebesgue measure (with ω’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 Ap-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 Ap-measures for which a meaningful notion of Cauchy-Szegő projection can be defined when p=2.