论文标题
曲线上类似考奇（Cauchy）的内核的对称性
Symmetrization of a Cauchy-like kernel on curves
论文作者
论文摘要
给定具有指定规律性的曲线$γ\ subset \ mathbb c $，我们研究了一个类似cauchy的内核$k_γ$的界限和阳性，其定义由$γ$界定的域的几何学和复杂功能理论决定。我们的结果表明，$ \ mathtt s [\ text {re}k_γ] $和$ \ mathtt s [\ text {im}k_γ] $（即，$k_γ$的真实和想象中的对象的对称性）与以前在cauchy kernel in the of the of the Cremator的文献相对差异。例如，数量$ \ mathtt s [\ text {re}k_γ]（\ mathbf z）$和$ \ mathtt s [\ text {im}k_γ]（\ mathbf z）$可以像$ \ frac32c^2（\ mathbf z）$ and $ - $ - \ - \ frac \ frac \ frac 12c^2（^2（\ frac32c^2） z $是$γ$中的任何三个积分，$ c（\ mathbf z）$是$ \ mathbf z $的menger曲率。对于原始的Cauchy内核而言，M。Melnikov的标志性结果使真实和虚构零件的对称形式分别等于$ \ Mathbb c $中所有三个tuples的$ \ frac12c^2（\ Mathbf Z）$。
Given a curve $Γ\subset \mathbb C$ with specified regularity, we investigate boundedness and positivity for a certain three-point symmetrization of a Cauchy-like kernel $K_Γ$ whose definition is dictated by the geometry and complex function theory of the domains bounded by $Γ$. Our results show that $\mathtt S[\text{Re} K_Γ]$ and $\mathtt S[\text{Im} K_Γ]$ (namely, the symmetrizations of the real and imaginary parts of $K_Γ$) behave very differently from their counterparts for the Cauchy kernel previously studied in the literature. For instance, the quantities $\mathtt S[\text{Re} K_Γ](\mathbf z)$ and $\mathtt S[\text{Im} K_Γ](\mathbf z)$ can behave like $\frac32c^2(\mathbf z)$ and $-\frac12c^2(\mathbf z)$, where $\mathbf z$ is any three-tuple of points in $Γ$ and $c(\mathbf z)$ is the Menger curvature of $\mathbf z$. For the original Cauchy kernel, an iconic result of M. Melnikov gives that the symmetrized forms of the real and imaginary parts are each equal to $\frac12c^2(\mathbf z)$ for all three-tuples in $\mathbb C$.
