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.