Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves

2008/10/17 by Alexei Yu. Karlovich, Karlovich, Alexei Yu.
Mathematics · #47B35 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.0810.3110

openalex publication_date 2008/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces L^p(Γ) over Lyapunov curves have the shape of circular arcs. About 25 years later, Albrecht Böttcher 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(⋅)(Γ) where p:Γ→(1,∞) 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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