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- W3204407419 abstract "Given a curve $Gammasubset mathbb C$ with specified regularity, we investigate boundedness and positivity for a certain three-point symmetrization of a Cauchy-like kernel $K_{Gamma}$ whose definition is dictated by the geometry and complex function theory of the domains bounded by $Gamma$. Our results show that $mathtt S[text{Re} K_{Gamma}]$ and $mathtt S[text{Im} K_{Gamma}]$ (namely, the symmetrizations of the real and imaginary parts of $K_{Gamma}$) behave very differently from their counterparts for the Cauchy kernel previously studied in the literature. For instance, the quantities $mathtt S[text{Re} K_{Gamma}](mathbf z)$ and $mathtt S[text{Im} K_{Gamma}](mathbf z)$ can behave like $frac32c^2(mathbf z)$ and $-frac12c^2(mathbf z)$, where $mathbf z$ is any three-tuple of points in $Gamma$ 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 $mathbb C$." @default.
- W3204407419 created "2021-10-11" @default.
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- W3204407419 date "2020-01-26" @default.
- W3204407419 modified "2023-10-11" @default.
- W3204407419 title "Symmetrization of a Cauchy-like kernel on curves" @default.
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- W3204407419 hasPublicationYear "2020" @default.
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