Petersen B.E. Introduction to the Fourier transform & pseudo-differential operators

Boston: Pitman A dvanced Publishing Program, 1983. - 369p.This book is an introduction to the Fourier transform and to the theory of pseudo-differential operators. As a text it is intended to be used at the second year graduate level. However, care has been taken to keep the text reasonably accessible. Thus large parts of it may profitably be used as
a supplement for a first year course in functional analysis.
Chapter 1 presents Schwartz’ theory of distributions. It is possible to cover much ground without invoking the theory of topological vector spaces. One then misses, however, the chance to illustrate some of the basic theorems of functional analysis. Moreover, eventually the functional
analysis becomes well-nigh indispensable. Therefore in Section 7, prior to considering distributions, we give a very brief introduction to the theory of locally convex spaces. The reader familiar with Banach spaces should have little difficulty making the transition to the more general setting. In Chapter 2 we continue the study of distributions and develop the theory f the Fourier transform on the space of temperate distributions.
The remainder of the book is given over to pseudo-differential operators. In Chapter 3 we construct an operational calculus for the (non-commuting) operators of multiplication by the coordinate functions and differentiation. The Fourier transform is the major tool here. The resulting operators are the pseudo-differential operators. Chapter 4 concerns the continuity of pseudo-differential operators on Sobolev spaces. In Chapter 5 we first discuss the Dirichlet problem in an setting and Garding’s inequality. We then give a proof of the sharp Garding inequality. This result leads to a theorem on propagation of singularities of solutions of pseudo-differential equations. This result together with the
results of Chapter 4 then leads, for example, to a local existence theorem for operators with real principal symbol and simple real characteristics.
Each chapter begins with an introduction which gives a detailed description of the contents of the chapter and, in some cases, some historical remarks. Additional historical remarks and some exercises are scattered throughout the text.