Singular integral operators on variable Lebesgue spaces with radial oscillating weights

TL;DR

This paper establishes a Fredholm criterion for singular integral operators on variable Lebesgue spaces with radial oscillating weights, revealing complex local spectra shaped like spiralic horns influenced by various indices.

Contribution

It extends existing results on singular integral operators from standard to variable Lebesgue spaces, incorporating radial oscillating weights and detailed spectral analysis.

Findings

Fredholm criterion for the operators is proven.

Local spectra are characterized as spiralic horns.

Results generalize previous work to variable Lebesgue spaces.

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.