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 & E. M. Stein states that the Cauchy–Szegö projection $\mathcal{S}_\omega$ 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 $[b,\mathcal{S}_{\Omega_2}]$ where $\mathcal{S}_{\Omega_2}$ is the Cauchy–Szegö 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–Szegö kernel, but these are unavailable in our setting of minimal regularity of $bD$; at the same time, recent techniques 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.

This is joint work with Xuan Thinh Duong (Macquarie University), Ji Li (Macquarie University) and Brett Wick (Washington University in St. Louis).

The video of this talk is available on the IISc Math Department channel.

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Last updated: 17 May 2024