Singular Integral Operators with Cauchy Kernel in Fractional Spaces

Abstract

We single out the Besov spaces that embed into the class of continuous functions and enjoy the Fredholm theory of linear singular integral equations with Cauchy kernel. We give basic results of this theory in the class of continuous (rather than Holder continuous) functions in terms of Besov spaces. Alongside elliptic operators we consider violations of ellipticity: the degeneration of the symbol of an operator at finitely many points.