Fredholmness of Singular Integral Operators with Piecewise Continuous Coefficients on Weighted Banach Function Spaces

By A. Karlovich. 2003

59 citations recorded by the source corpus.

We prove necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. These conditions are formulated in terms of indices of submultiplicative functions associated with local properties of the space, of the curve, and of the weight. As an example, we consider weighted Nakano spaces $L^{p(\cdot)}_w$ (weighted Lebesgue spaces with variable exponent). Moreover, our necessary conditions become also sufficient for weighted Nakano spaces over nice curves whenever $w$ is a Khvedelidze weight, and the variable exponent $p(t)$ satisfies the estimate $|p(\tau)-p(t)|\le A/(-\log|\tau-t|)$.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/a74a7492997b8b9d31592ad606346832d1368d52

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In Applied Mathematics

1. S. Samko
2. P. Hasto
3. V. Kokilashvili
4. Jingshi Xu
5. Alexandre Almeida
6. Lars Diening
7. A. Meskhi
8. Kwok-Pun Ho

Citations in the corpus

• On a progress in the theory of lebesgue spaces with variable exponent: maximal and singular operators (2005, background): “A further generalization of Fredholmness results for singular integral operators in Banach function spaces, including the necessity of Fredholmness conditions for the weighted spaces Lp(·)(�, ρ) on ‘bad’ curves may be found in ref. [ 83 ].”