The Cauchy--Szegő Projection and its commutator for Domains in $\mathbb C^n$ with Minimal Smoothness

26 May 2020  ·  Xuan Thinh Duong, Loredana Lanzani, Ji Li, Brett D. Wick ·

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 \& Stein states that the Cauchy--Szeg\H{o} projection $\EuScript S_\omega$ defined with respect to any \textit{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 $\sigma$ is Leray Levi-like.) Here we show that $\EuScript S_\omega$ is in fact bounded in $L^p(bD, \Omega_p)$ for any $1<p<\infty$ and for any $\Op$ in the far larger class of \textit{$A_p$-like} measures (modeled after the Muckenhoupt $A_p$-weights for $\sigma$). As an application, we characterize boundedness and compactness in $L^p(bD, \Omega_p)$ for $1<p<\infty$, of the commutator $[b, \EuScript S_\omega]$. We next introduce the holomorphic Hardy spaces $H^p(bD, \Omega_p)$, $1<p<\infty$, and we characterize boundedness and compactness in $L^2(bD, \Omega_2)$ of the commutator $\displaystyle{[b,\EuScript S_{\Omega_2}]}$ of the Cauchy--Szeg\H{o} projection defined with respect to any $A_2$-like measure $\Omega_2$. Earlier closely related results rely upon an asymptotic expansion, and subsequent pointwise estimates, of the Cauchy--Szeg\H o kernel that are not available in the settings of minimal regularity {of $bD$} and/or $A_p$-like measures. read more

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Complex Variables Classical Analysis and ODEs