Some Spaces of Functions and Distributions

Abstract

This chapter is devoted to some aspects of distribution theory. In Section 1.1 we define various spaces of smooth functions as well as some differential operators; we insist on an invariant definition of these objects, since this will be important in the study of N-body Hamiltonians. Various definitions and facts from distribution theory are reviewed in Section 1.2. The reader should be familiar with the contents of these two sections because we fix notations and terminology which are in some cases not quite standard. Sections 1.3 and 1.4 are more technical and may be skipped temporarily. We prove results relating local regularity properties of a distribution with the behaviour at infinity of its Fourier transform. We also establish an identity due to A.P. Calderón giving a representation of a distribution in terms of its derivatives of a fixed order plus a regular term. Finally, we use this representation in order to prove several facts that will be needed in later chapters.