SINGULAR INTEGRAL EQUATIONS

This chapter discusses the singular integral equations of the case where the symbol is independent of the pole and the case where the symbol is dependent on the pole. The general equation, the case where the symbol is independent of the pole, can be reduced to the equivalent equation of the Riesz–Schauder type. The chapter also discusses the regularization and domains of constancy of the index, and index theorem. Moreover, the chapter reviews the case where a singular integral is taken over not a Euclidean space but any closed m-dimensional Liapounov manifold that one shall make subject to the conditions of Giraud; namely, the manifold Γ can be orientated; it can be covered by a finite number of partly overlapping parts Γ j , each of which permits of a smooth one-to-one mapping onto a finite region of an m -dimensional Euclidean space E m . In addition, it is proved that the index of a singular equation with a symbol and positively bounded below absolutely was equal to zero.