# Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves - Mathematics > Functional Analysis

Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves - Mathematics > Functional Analysis - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

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.

Autor: Alexei Yu. Karlovich

Fuente: https://arxiv.org/