Symmetrization of a Cauchy-like kernel on curves

Given a curve Γ⊂C with specified regularity, we investigate boundedness and positivity for a certain three-point symmetrization of a Cauchy-like kernel KΓ whose definition is dictated by the geometry and complex function theory of the domains bounded by Γ. Our results show that S[ReKΓ] and S[ImKΓ] (namely, the symmetrizations of the real and imaginary parts of KΓ) behave very differently from their counterparts for the Cauchy kernel previously studied in the literature. For instance, the quantities S[ReKΓ](z) and S[ImKΓ](z) can behave like [Formula presented] and [Formula presented], where z is any three-tuple of points in Γ and c(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 [Formula presented] for all three-tuples in C.