The rest of this course will mainly be concerned with “variable coefficient Fourier analysis”—that is, finding natural variable coefficient versions of the restriction theorem, and so forth. One of our ultimate goals will be to extend these results to the setting of eigenfunction expansions given by the spectral decomposition of a self-adjoint pseudo-differential operator. To state the results, however, and to develop the necessary tools for their study, we need to go over some of the main elements in the theory of pseudo-differential operators. These will be given in Section 1 and our presentation will be a bit sketchy but essentially self-contained. For a more thorough treatment, we refer the reader to the books of Hörmander [7], Taylor [2], and Treves [1]. In Section 2 we present the equivalence of phase function theorem for pseudo-differential operators. This will play an important role in the parametrix construction for the (variable coefficient) half-wave operator. Finally, in Section 3, we present background needed for the study of Fourier analysis on manifolds, such as basic facts about the spectral function. We also present a theorem of Seeley on powers of elliptic differential operators which allows one to reduce questions about the Fourier analysis of higher order elliptic operators to questions about first order operators.

Some Basics

We start out by defining pseudo-differential operators on ℝn.

Many things could be said about the way this book was written but we shall be brief.

It all started with several lecture courses given by N. Varopoulos at Universite Paris VI during the period 1982–87. At the time, Coulhon and SaloffCoste were post-doctoral students and took notes. An early part of these notes appeared for limited circulation in 1986. It was then decided that, when completed, these notes would be published as a set of graduate “Lecture Notes”. The project dragged on for several years; by 1990, through the efforts of Saloff-Coste, enough work had been put into the notes to make them presentable as a real book.

This book is primarily an advanced research monograph. It should be accessible to those graduate students that are prepared to make the personal investment and effort to familiarize themselves with the background material.

N. Varopoulos did very little of the actual writing and did not put any work into the preparation of manuscripts; he is however responsible for most of the new mathematics that is presented here. This mathematical work was done during the 1980s and was built on the following basic material.

Existing semigroup theory, especially Beurling–Deny theory; this is work that was done in the 1950s and 1960s. The work of J. Moser and J. Nash on parabolic equations was also a great inspiration in this context.

The theory of second order subelliptic differential operators and especially the “sum of squares operators”. This is work done in the 1960s by L. Hormander. The Harnack estimates, which are essential for us, were completed by J.-M. Bony a little later. This work has since been further developed by several authors.

At the end of this chapter, we shall know almost everything about the heat kernel and the Sobolev inequalities associated to a family of Hörmander fields on a nilpotent Lie group.

Later we shall obtain related results in much wider settings. Local questions will be studied on manifolds in Chapter V. Global questions will be studied on unimodular groups (Chapters VI, VII and VIII). We shall, in Chapter IX, even consider non-unimodular Lie groups as far as Sobolev inequalities are concerned.

But it is pleasant to see right away how the semigroup machinery of Chapter II and the considerations of Chapter III about sublaplacians yield complete results in the particular setting of nilpotent groups; this is because, in some sense, the geometry of these groups is not too complicated.

In Section 1 we recall general properties of nilpotent Lie groups, and in Section 2, we give examples. Section 3 is simple, but essential: we obtain for free a powerful scaled Harnack inequality. In Section 4, we show how to estimate the heat kernel with respect to the volume growth, thanks to this Harnack inequality. An analysis of the Lie algebra gives in Section 5 an estimate from above and below of the volume of the ball of radius t. In Sections 6 and 7, we draw fairly complete consequences of all this, using Chapter II; we also introduce a device which yields L1 Sobolev inequalities and which we shall use again later.

Introduction

This chapter and the next are not concerned with left invariant sublaplacians and their associated heat kernel ht on unimodular Lie groups. Nevertheless, the matters we shall treat are closely related to the main stream of this book. Indeed, in the previous chapter we investigated the behaviour of ||ht||∞ for 0 ≤ t ≤ 1. We would now like to study ||ht||infin; for t ≥ 1. This will be achieved in Chapter VIII, but we are going to attack this problem from a somewhat more general point of view.

Let F(k) be the kth convolution power of F ∈ L1 ∩ L∞. In order to find out the behaviour of ||ht||∞ for t ≥ 1, it suffices to look at hk = h(k), k = 1,2,… Moreover, the function h1 has a rapid decay at infinity since we know that h1(x) ≤ C exp(−cρ2(x)); see V.4.3. It is thus natural to address ourselves to the more general question of the behaviour of ||F(k)||∞, as k tends to infinity, for symmetric, positive compactly supported functions F of integral one.

Clearly enough, the Lie structure is no longer relevant here. Locally compact, unimodular groups which are compactly generated form the natural setting within which we will work. What we will eventually be able to show is that the decay of ||F(k)||∞ (with F as above) is governed by the volume growth of the group.

In this chapter we shall present some of the results which are central and for which we need the full thrust of our methods.