Singular integral operators on variable Lebesgue spaces with radial oscillating weights

Details Singular integral operators on variable Lebesgue spaces with radial oscillating weights Singular integral operators on variable Lebesgue spaces with radial oscillating weights

2007-08-06

arxiv.org/abs/0708.0778v3

Mathematics

Functional Analysis

24 pages. Theorem 1.1 is stated as a necessary and sufficient condition. The necessity portion is new, its proof is added

Scientific paper

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.

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