The Cauchy-Szegö projection and its commutator for domains in $\mathbb C^n$ with minimal smoothness

Loredana Lanzani (Syracuse University, USA)

07-Dec-2021, 14:00-15:00 (2 years ago)

Abstract: Let $D\subset\C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$. A 2017 result of Lanzani and Stein states that the Cauchy-Szegö projection $S_\omega$ defined with respect to any 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 is Leray Levi-like.) Here we show that $S_\omega$ is in fact bounded in $L^p(bD, \Omega_p)$ for any $1 < p< \infty$ and for any $\Omega_p$ in the optimal class of $A_p$ measures, that is $\Omega_p = \psi_p\sigma$ where $\sigma$ is induced Lebesgue measure and $\psi_p$ is any Muckenhoupt $A_p$-weight. As an application, we characterize boundedness and compactness in $L^p(bD, \Omega_p)$ for any $1 < p < \infty$ and for any $A_p$ measure $\Omega_p$, of the commutator $[b, S_p]$ for any Leray Levi-like measure $\omega$. We next introduce the notion of holomorphic Hardy spaces for $A_p$ measures, $1 < p < \infty$, and we characterize boundedness and compactness in $L^2(bD, \Omega_2)$ of the commutator $\displaystyle{[b,S_{\Omega_2}]}$ of the Cauchy-Szegö projection defined with respect to any $A_2$ measure $\Omega_2$. Earlier closely related results rely upon an asymptotic expansion, and subsequent pointwise estimates, of the Cauchy-Szegö kernel, but these are unavailable in the settings of minimal regularity {of $bD$} and/or $A_p$ measures; it turns out that the real harmonic analysis method of 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).

complex variablesdynamical systems

Audience: researchers in the topic


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