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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 ω (Γ,ρ) , where Γ is a generalized Lyapunov curve and ρ(t)= k=0 n r k (|t-t k |) with r k (t) from some class V α . A boundedness condition in this space for the singular operator S in terms of the characteristic ω(t) 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 ω(t) as well as some connections between these characteristics and some characteristics of the functions r k (t).

MSC:
45P05Integral operators
45E05Integral equations with kernels of Cauchy type
47A53(Semi-)Fredholm operators; index theories