Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves

TL;DR

This paper extends the understanding of singular integral operators' local spectra from classical Lebesgue spaces on Lyapunov and Carleson curves to variable Lebesgue spaces with Dini-Lipschitz continuous exponents, revealing that spectral shapes transform similarly.

Contribution

It generalizes previous spectral shape results to variable Lebesgue spaces on arbitrary Carleson curves, introducing a new boundedness condition for the Cauchy singular integral operator.

Findings

Spectral shapes transform from circular arcs to logarithmic leaves on Carleson curves.

A new sufficient condition for boundedness of the Cauchy singular integral operator on variable Lebesgue spaces.

Extension of spectral analysis to variable exponent spaces with oscillating weights.

Abstract

In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces $L^p(\Gamma)$ 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.