Karlovich, Alexei Yu. "
Singular integral operators on variable Lebesgue spaces over arbitrary Carleson curves."
Topics in Operator Theory: Operators, Matrices and Analytic Functions, Vol. 1. Operator Theory: Advances and Applications, 202. Eds. JA Ball, V. Bolotnikov, JW Helton, L. Rodman, and IM Spitkovsky. Basel: Birkhäuser, 2010. 321-336.
AbstractIn 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 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 \(L^{p(\cdot)}(\Gamma)\) 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.
Karlovich, Alexei Yu. "
Singular integral operators on variable Lebesgue spaces with radial oscillating weights."
Operator Algebras, Operator Theory and Applications.Operator Theory Advances and Applications, 195 . Eds. JJ Grobler, LE Labuschagne, and M. Möller. Basel: Birkhäuser, 2010. 185-212.
AbstractWe prove a Fredholm criterion for operators in the Banach algebra of singular integral operators with matrix piecewise continuous coefficients acting on a variable Lebesgue space with a radial oscillating weight over a logarithmic Carleson curve. The local spectra of these operators are massive and have a shape of spiralic horns depending on the value of the variable exponent, the spirality indices of the curve, and the Matuszewska-Orlicz indices of the weight at each point. These results extend (partially) the results of A. Böttcher, Yu. Karlovich, and V. Rabinovich for standard Lebesgue spaces to the case of variable Lebesgue spaces.