"The book is very concise and contains a great wealth of ideas and results...Each chapter contains a small section, 'References and comments', in which the authors, in their own way, introduce the reader to the brief history of the subject and its bibliography." A. Hulanicki, Mathematical Reviews

• Preface; Foreword; 1. Introduction; 2. Dimensional inequalities for semigroups of operators on the Lp spaces; 3. Systems of vector fields satisfying Hörmander's condition; 4. The heat kernel on nilpotent Lie groups; 5. Local theory for sums of squares of vector fields; 6. Convolution powers on finitely generated groups; 7. Convolution powers on unimodular compactly generated groups; 8. The heat kernel on unimodular Lie groups; 9. Sobolev inequalities on non-unimodular Lie groups; 10. Geometric applications; Bibliography; Index.