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Singular integral operators in weighted spaces with generalized Hölder condition. (English) Zbl 0946.45006
The author considers the singular integral operator $N=aI+bS$ with piecewise continuous coefficients in the generalized weighted Hölder spaces ${H}_{0}^{\omega }\left({\Gamma },\rho \right)$ , where ${\Gamma }$ is a generalized Lyapunov curve and $\rho \left(t\right)={\prod }_{k=0}^{n}{r}_{k}\left(|t-{t}_{k}|\right)$ with ${r}_{k}\left(t\right)$ from some class ${V}_{\alpha }$. A boundedness condition in this space for the singular operator $S$ in terms of the characteristic $\omega \left(t\right)$ is described. Fredholmness conditions for the operator $N$ and a formula for calculating its index are given in terms of the so called index numbers of the characteristic $\omega \left(t\right)$ as well as some connections between these characteristics and some characteristics of the functions ${r}_{k}\left(t\right)$.

##### MSC:
 45P05 Integral operators 45E05 Integral equations with kernels of Cauchy type 47A53 (Semi-)Fredholm operators; index theories