Singular integral operators on variable Lebesgue spaces with radial oscillating weights
Abstract
We 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. B\"ottcher, Yu. Karlovich, and V. Rabinovich for standard Lebesgue spaces to the case of variable Lebesgue spaces.
Cite
@article{arxiv.0708.0778,
title = {Singular integral operators on variable Lebesgue spaces with radial oscillating weights},
author = {Alexei Yu. Karlovich},
journal= {arXiv preprint arXiv:0708.0778},
year = {2009}
}
Comments
24 pages. Theorem 1.1 is stated as a necessary and sufficient condition. The necessity portion is new, its proof is added