$\boldsymbol{\text{ℂ}^n}$中最小光滑性域的柯西-塞格投影交换子：加权正则性
The commutator of the Cauchy--Szegő Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted regularity
AI总结：
针对$\boldsymbol{\text{ℂ}^n}$中边界为$C^2$类的有界强拟凸域，研究了加权柯西-塞格投影交换子$[b, S_ω]$在$L^p(bD, Ω_p)$中的紧性与有界性（含显式界），并刻画了$p=2$时交换子$[b,S_{Ω_2}]$在对应空间的相关性质。
摘要：
设$D\boldsymbol{\text{⊂ℂ}^n}$是有界强拟凸域，其边界$bD$满足$C^2$类最小正则性条件，$S_ω$表示关于$D$边界诱导勒贝格测度的（任意）正连续倍数$ω$定义的柯西-塞格投影。我们刻画了交换子$[b, S_ω]$在勒贝格空间$L^p(bD, Ω_p)$中的紧性和有界性（后者含显式界），其中$Ω_p$是Muckenhoupt类$A_p(bD)$中的任意测度，$1<p<\boldsymbol{\text{∞}}$。接着固定$p=2$，令$S_{Ω_2}$表示关于（任意）测度$Ω_2 ∈ A_2(bD)$定义的柯西-塞格投影，$A_2(bD)$是可定义有意义柯西-勒雷测度的最大参考测度类，我们刻画了交换子$[b,S_{Ω_2}]$在$L^2(bD, Ω_2)$中的有界性和紧性。
英文摘要：
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 $\displaystyle{[b,S_{Ω_2}]}$.
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