Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces

Details Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces

2003-05-07

arxiv.org/abs/math/0305108v1

Mathematics

Functional Analysis

Scientific paper

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 $|p(\tau)-p(t)|\le A/(-\log|\tau-t|)$.

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