Spaces of smooth functions and operators on them

Abstract

In this chapter we present some very basic facts about spaces of smooth functions (specifically, about Sobolev and Morrey–Campanato spaces). Next, we give some additional information about singular integrals (note that in the stability theorem in the last chapter we employed some facts not covered before). We prove that the adjoints to singular integral operators take L ∞ to BMO and discuss the weak L 1-boundedness condition (3.45). More generally, we shall see that, under some additional assumptions, the adjoints to singular integral operators take some Morrey-Campanato spaces to themselves. Also, we analyze partial sum operators for wavelet expansions from a singular integral theory viewpoint. All this will be of much importance in Part 2 of the book.