@article{c367e0f3b7d84191bef5e284a187e7ba,

title = "Symmetrization of a Cauchy-like kernel on curves",

abstract = "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.",

keywords = "Cauchy Integral, Double Layer Potential, Kernel symmetrization, Menger curvature",

author = "Loredana Lanzani and Malabika Pramanik",

note = "Funding Information: The authors were supported by awards no. DMS-1503612 and DMS-1901978 from the National Science Foundation USA; and a Discovery grant from the National Science and Engineering Research Council of Canada. Part of this work took place (a) at the Mathematical Sciences Research Institute in Berkeley, California, where the authors were in residence during a thematic program in the spring of 2017; (b) at the Park City Mathematics Institute in July 2018, during the thematic program in harmonic analysis, and (c) at the Isaac Newton Institute for Mathematical Sciences, where the first-named author was in residence in Fall 2019 during the program Complex Analysis: Theory and Applications (EPSRC grant no. EP/R014604/1). We thank the institutes and the programs organizers for their generous support and hospitality. Last but not least, we are very grateful to the reviewer for helpful feedback and for pointing out relevant references. Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

year = "2021",

month = nov,

day = "1",

doi = "10.1016/j.jfa.2021.109202",

language = "English (US)",

volume = "281",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

publisher = "Academic Press Inc.",

number = "9",

}