@inbook {7479,
	title = {Singular integral operators on variable Lebesgue spaces over arbitrary Carleson curves},
	booktitle = {Topics in Operator Theory: Operators, Matrices and Analytic Functions, Vol. 1. Operator Theory: Advances and Applications, 202},
	year = {2010},
	pages = {321-336},
	publisher = {Birkh{\"a}user},
	organization = {Birkh{\"a}user},
	address = {Basel},
	abstract = {<script src={\textquoteright}https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML{\textquoteright}></script><p>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{\"o}ttcher 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 condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with weights related to oscillations of Carleson curves.</p>
},
	url = {http://www.springerlink.com/content/wk281h0005423485/},
	author = {Karlovich, Alexei Yu.},
	editor = {Ball, JA and Bolotnikov, V and Helton, JW and Rodman, L and Spitkovsky, IM}
}
