CHAPTER V - SINGULAR INTEGRALS IN Lp SPACES

Abstract

This chapter discusses the simplest singular operators, in which the integral is taken over a Euclidean space Em and where the symbol is independent of the pole. A theorem is stated and proved, given by Mikhlin, that allows one to widen the operator with regard to continuity to the whole space L2 (Em), if it were defined beforehand for a dense set. The chapter also reviews the singular operator, the symbol of which depends on the pole and the fundamental theorem on the boundedness of a singular integral operator in L2. In addition, it is discussed that integral can be reduced to the sum of two integrals, the first of which has a weak singularity, and the second is singular, taken over a finite domain of a Euclidean space. The first of the said integrals defines an operator, completely continuous in Lp (Γ), 1 <p < ∞. From this it is easy to deduce, that the integral operator is bounded in L2 (Γ), correspondingly in Lp(Γ) if the symbol of this integral satisfies the conditions of the theorems discussed in the chapter.