Alexei Yu. Karlovich
Published 2007-08-06, updated 2009-03-03Version 3
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.
Comments: 24 pages. Theorem 1.1 is stated as a necessary and sufficient condition. The necessity portion is new, its proof is added
On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces
Algebras of singular integral operators on Nakano spaces with Khvedelidze weights over Carleson curves with logarithmic whirl points
Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces
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