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.