Singular Integral Equations on Piecewise Smooth Curves in Spaces of Smooth Functions

. We prove the boundedness,of the Cauchy singular integral oper- ator in modified weighted Sobolev KW, p;q(¡;‰) spaces under the assump- tion that the smoothness parameters m;„;s are large. The underlying contour ¡ is piecewise smooth,with angular points and even with cusps. We obtain Fredholm criteria and an index formula for singular integral equations with piecewise smooth coecients,and complex conjugation in these spaces pro- vided the underlying contour has angular points but no cusps. The Fredholm property and the index turn out to be independent of the integer parts of the smoothness parameters m;„;s. The results are applied to an oblique deriv- ative problem (the Poincar´e problem) in plane domains with angular points and peaks on the boundary.