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.