Singular integral operators on variable Lebesgue spaces with radial oscillating weights

2007/08/06 by Alexei Yu. Karlovich, Karlovich, Alexei Yu.
Mathematics · Physics and Astronomy · #45E05 #46E30 #47A68 #47B35 #Advanced Differential Geometry Research #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.0708.0778

openalex publication_date 2007/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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. Böttcher, Yu. Karlovich, and V. Rabinovich for standard Lebesgue spaces to the case of variable Lebesgue spaces.

Related

• Algebras of singular integral operators on Nakano spaces with Khvedelidze weights over Carleson curves with logarithmic whirl points
• Remark on the Boundedness of the Cauchy Singular Integral Operator on Variable Lebesgue Spaces with Radial Oscillating Weights
• Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves
• Semi-Fredholm singular integral operators with piecewise continuous coefficients on weighted variable Lebesgue spaces are Fredholm
• Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces