Given a curve $\mathrm{\Gamma }\subset \mathbb{C}$ with specified regularity, we investigate boundedness and positivity for a certain three-point symmetrization of a Cauchy-like kernel $K_{\mathrm{\Gamma }}$ whose definition is dictated by the geometry and complex function theory of the domains bounded by Γ. Our results show that $\mathtt{S}[\text{Re}{K}_{\mathrm{\Gamma }}]$ and $\mathtt{S}[\text{Im}{K}_{\mathrm{\Gamma }}]$ (namely, the symmetrizations of the real and imaginary parts of $K_{\mathrm{\Gamma }}$) behave very differently from their counterparts for the Cauchy kernel previously studied in the literature. For instance, the quantities $\mathtt{S}[\text{Re}{K}_{\mathrm{\Gamma }}](\mathbf{z})$ and $\mathtt{S}[\text{Im}{K}_{\mathrm{\Gamma }}](\mathbf{z})$ can behave like $\frac{3}{2}{c}^2(\mathbf{z})$ and $-\frac{1}{2}{c}^2(\mathbf{z})$, where z is any three-tuple of points in Γ and $c(\mathbf{z})$ is the Menger curvature of 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 $\frac{1}{2}{c}^2(\mathbf{z})$ for all three-tuples in $\mathbb{C}$.

给定一条曲线 $\mathrm{\Gamma }\subset \mathbb{C}$ 以指定的规律性，我们研究了类柯西核的某个三点对称化的有界性和正性 ${钾}_{\mathrm{\Gamma }}$其定义由 Γ 所界定域的几何和复函数理论决定。我们的结果表明$\mathtt{秒}[\text{关于}{钾}_{\mathrm{\Gamma }}]$ 和 $\mathtt{秒}[\text{我是}{钾}_{\mathrm{\Gamma }}]$ （即，实部和虚部的对称化 ${钾}_{\mathrm{\Gamma }}$) 的行为与先前在文献中研究的柯西核的对应行为非常不同。例如，数量$\mathtt{秒}[\text{关于}{钾}_{\mathrm{\Gamma }}](\mathbf{z})$ 和 $\mathtt{秒}[\text{我是}{钾}_{\mathrm{\Gamma }}](\mathbf{z})$ 可以表现得像 $\frac{3}{2}{C}^2(\mathbf{z})$ 和 $-\frac{1}{2}{C}^2(\mathbf{z})$，其中z是 Γ 中的任意三元组点，并且$C(\mathbf{z})$是z的门格尔曲率。对于原始的柯西核，M. Melnikov 的一个标志性结果给出了实部和虚部的对称形式分别等于$\frac{1}{2}{C}^2(\mathbf{z})$ 对于所有三元组 $\mathbb{C}$.