Symmetrization of a Cauchy-like kernel on curves

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 $\mathrm{\Gamma }$. Our results show that $\mathtt{S}[\text{Re}{K}_{\mathrm{\Gamma }}]$ and $\mathtt S[$\text{Im}$ K_{\Gamma}]$(namely,thesymmetrizationsoftherealandimaginarypartsof$K_{\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_{\Gamma}](\mathbf z)$canbehavelike$\frac32c^2(\mathbf z)$and$-\frac12c^2(\mathbf z)$,where$\mathbf z$isanythree-tupleofpointsin$\Gamma$and$c(\mathbf z)$istheMengercurvatureof$\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 $\frac{1}{2}{c}^2(\mathbf{z})$ for all three-tuples in $\mathbb{C}$.