Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves - Mathematics > Functional AnalysisReport as inadecuate




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Abstract: In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra ofsingular integral operators with piecewise continuous coefficients on Lebesguespaces $L^p\Gamma$ over Lyapunov curves have the shape of circular arcs.About 25 years later, Albrecht B\-ottcher and Yuri Karlovich realized thatthese circular arcs metamorphose to so-called logarithmic leaves with a medianseparating point when Lyapunov curves metamorphose to arbitrary Carlesoncurves. We show that this result remains valid in a more general setting ofvariable Lebesgue spaces $L^{p\cdot}\Gamma$ where $p:\Gamma\to1,\infty$satisfies the Dini-Lipschitz condition. One of the main ingredients of theproof is a new sufficient condition for the boundedness of the Cauchy singularintegral operator on variable Lebesgue spaces with weights related tooscillations of Carleson curves.



Author: Alexei Yu. Karlovich

Source: https://arxiv.org/



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