Cauchy Singular Integral Operators in Weighted Spaces of Continuous Functions

Abstract.

We study the Cauchy singular integral operator SwI on (−1, 1), where |w| is a generalized Jacobi weight. This operator is considered in pairs of weighted spaces of continuous functions, where the weights u and v are generalized Jacobi weights with nonnegative exponents such that |w| = u/v. We introduce a certain polynomial approximation space which is well appropriate to serve as domain of definition of SwI. A description of this space in terms of smoothness properties shows that it can be viewed as a limit case of weighted Besov spaces of continuous functions. We use our results to prove necessary and sufficient conditions for the continuity of operators awI + SbwI and $\varrho^{-1}(aw\varrho I\,+\,bSw\varrho I),\,\varrho^{-1}\,\in\,b^{-1}\Pi$ (Π: set of all algebraic polynomials), in certain pairs of Ditzian-Totik type Besov spaces.