The commutator of the Cauchy--Szegő Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted regularity
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866913796628414464 |
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| author | Duong, Xuan Thinh Lanzani, Loredana Li, Ji Wick, Brett D. |
| author_facet | Duong, Xuan Thinh Lanzani, Loredana Li, Ji Wick, Brett D. |
| contents | Let $D\subset\mathbb C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$, and let $S_ω$ denote the Cauchy--Szegő projection defined with respect to (any) positive continuous multiple $ω$ of induced Lebesgue measure for the boundary of $D$. We characterize compactness and boundedness (the latter with explicit bounds) of the commutator $[b, S_ω]$ in the Lebesgue space $L^p(bD, Ω_p)$ where $Ω_p$ is any measure in the Muckenhoupt class $A_p(bD)$, $1<p<\infty$. We next fix $p =2$ and we let $S_{Ω_2}$ denote the Cauchy--Szegő projection defined with respect to (any) measure $Ω_2 \in A_2(bD)$, which is the largest class of reference measures for which a meaningful notion of Cauchy-Leray measure may be defined. We characterize boundedness and compactness in $L^2(bD, Ω_2)$ of the commutator $\displaystyle{[b,S_{Ω_2}]}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2005_12740 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The commutator of the Cauchy--Szegő Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted regularity Duong, Xuan Thinh Lanzani, Loredana Li, Ji Wick, Brett D. Complex Variables Classical Analysis and ODEs Let $D\subset\mathbb C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$, and let $S_ω$ denote the Cauchy--Szegő projection defined with respect to (any) positive continuous multiple $ω$ of induced Lebesgue measure for the boundary of $D$. We characterize compactness and boundedness (the latter with explicit bounds) of the commutator $[b, S_ω]$ in the Lebesgue space $L^p(bD, Ω_p)$ where $Ω_p$ is any measure in the Muckenhoupt class $A_p(bD)$, $1<p<\infty$. We next fix $p =2$ and we let $S_{Ω_2}$ denote the Cauchy--Szegő projection defined with respect to (any) measure $Ω_2 \in A_2(bD)$, which is the largest class of reference measures for which a meaningful notion of Cauchy-Leray measure may be defined. We characterize boundedness and compactness in $L^2(bD, Ω_2)$ of the commutator $\displaystyle{[b,S_{Ω_2}]}$. |
| title | The commutator of the Cauchy--Szegő Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted regularity |
| topic | Complex Variables Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2005.12740 |