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

Abstract

We prove the boundedness of the Cauchy singular integral operator in modified weighted Sobolev $\mathbb{K}\mathbb{W}_{p}^{m}(\Gamma ,\rho )$, Hölder-Zygmund $\mathbb{K}\mathbb{Z}_{\mu }^{0}(\Gamma ,\rho )$ Bessel potential $\mathbb{K}\mathbb{H}_{p}^{s}(\Gamma ,\rho )$ and Besov $\mathbb{K}\mathbb{B}_{{p,q}}^{s}(\Gamma ,\rho )$ spaces under the assumption that the smoothness parameters m,μ,s are large. The underlying contour Γ 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 coefficients and complex conjugation in these spaces provided 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,µ,s. The results are applied to an oblique derivative problem (the Poincaré problem) in plane domains with angular points and peaks on the boundary.