Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves

Details Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves

2008-10-17

arxiv.org/abs/0810.3110v1

Mathematics

Functional Analysis

14 pages

Scientific paper

In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces $L^p(\Gamma)$ over Lyapunov curves have the shape of circular arcs. About 25 years later, Albrecht B\"ottcher and Yuri Karlovich realized that these circular arcs metamorphose to so-called logarithmic leaves with a median separating point when Lyapunov curves metamorphose to arbitrary Carleson curves. We show that this result remains valid in a more general setting of variable Lebesgue spaces $L^{p(\cdot)}(\Gamma)$ where $p:\Gamma\to(1,\infty)$ satisfies the Dini-Lipschitz condition. One of the main ingredients of the proof is a new sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with weights related to oscillations of Carleson curves.

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