The commutator of the Cauchy-Szegő projection for domains in $C^n$ with minimal smoothness

Loredana Lanzani (Syracuse University)

31-Mar-2021, 16:00-17:00 (3 years ago)

Abstract: Let $D\subset\C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$. We characterize boundedness and compactness in $L^p(bD, \omega)$,, for $1< p < \infty$,of the commutator $[b,S_\omega]$ where $S_\omega$ is the Cauchy--Szegő (orthogonal) projection of $L^2(bD, \omega)$ onto the holomorphic Hardy space $H^2(bD, \omega)$ and the measure $\omega$ belongs to a family (the ``Leray Levi-like'' measures) that includes induced Lebesgue measure $\sigma$. We next consider a much larger family of measures $\{\Omega_p\}$ modeled after the Muckenhoupt $A_p$-weights for $\sigma$: we define the holomorphic Hardy spaces $H^p(bD, \Omega_p)$ and we characterize boundedness and compactness of $[b, S_{\Omega_2}]$ in $L^2(bD, \Omega_2)$. Earlier closely related results rely upon an asymptotic expansion, and subsequent pointwise estimates, of the Cauchy--Szegő kernel that are not available in the settings of minimal regularity {of $bD$} and/or $A_p$-like measures.

This is joint work with Xuan Thinh Duong, Ji Li and Brett D. Wick.

classical analysis and ODEsfunctional analysisrepresentation theoryspectral theory

Audience: researchers in the topic


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