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

TL;DR

The paper addresses the problem of weighted $L^p$-regularity for the Cauchy--Szegő projection on bounded, strongly pseudoconvex domains with minimal boundary smoothness, under $A_p$-type measures. It develops a weighted extrapolation framework that replaces kernel expansion techniques and establishes explicit $A_p$-character dependent bounds for $S_\omega$, including a sharp $L^2$-bound $\|S_ω\|_{L^2(bD, Ω_2)\to L^2(bD, Ω_2)} \lesssim [Ω_2]_{A_2}^3$ and extrapolated $L^p$-bounds $\|S_ω g\|_{L^p(bD, Ω_p)} \lesssim [Ω_p]_{A_p}^{3\cdot\max\{1, 1/(p-1)\}}\|g\|_{L^p(bD, Ω_p)}$. A key innovation is a quantitative cancellation estimate for Cauchy--Leray truncations, enabling the $A_p$-weighted theory to hold in the minimal-smoothness setting. The results also define holomorphic Hardy spaces $H^p(bD, Ω_p)$ for $p=2$ in this weighted framework, and demonstrate that the Cauchy--Szegő projection remains meaningful only at $p=2$ in the $A_p$-context. Overall, the work extends weighted harmonic analysis techniques to several complex variables under minimal boundary regularity and provides precise operator-norm dependencies on $A_p$-characteristics.

Abstract

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 $S_ω$ defined with respect to a bounded, positive continuous multiple $ω$ of induced Lebesgue measure, {maps $L^p(bD, ω)$ to $L^p(bD, ω)$ continuously} for any $1<p<\infty$. Here we show that $S_ω$ satisfies explicit quantitative bounds in $L^p(bD, Ω)$, for any $1<p<\infty$ and for any $Ω$ in the maximal class of \textit{$A_p$}-measures, that is for $Ω_p = ψ_pσ$ where $ψ_p$ is a Muckenhoupt $A_p$-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--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 $p=2$.

Theorems & Definitions (11)

• Theorem 1.1
• Definition 1.2
• Proposition 1.3
• Lemma 2.1
• proof
• proof : Proof of Proposition \ref{['P:Hardy-def-NT']}
• Theorem 3.1
• proof
• Proposition 3.2
• Remark 3.3