Contents

Abstract ix

1. Introduction and statement of the results 1

2. The control distance and the local Harnack inequality 21

3. The proof of the Harnack inequality from Varopoulos's theorem and

propositions 1.6.3 and 1.6.4 23

4. Holder continuity 38

5. Nilpotent Lie groups 38

6. Sub-Laplacians on nilpotent Lie groups 41

7. A function which grows linearly 44

8. Proof of propositions 1.6.3 and 1.6.4 in the case of nilpotent Lie groups 45

9. Proof of the Gaussian estimate in the case of nilpotent Lie groups 48

10. Polynomials on nilpotent Lie groups 53

11. A Taylor formula for the heat functions on nilpotent Lie groups 54

12. Harnack inequalities for the derivatives of the heat functions on nilpotent

Lie groups 61

13. Harmonic functions of polynomial growth on nilpotent Lie groups 61

14. Proof of the Berry-Esseen estimate in the case of nilpotent Lie groups 62

15. The nil-shadow of a simply connected solvable Lie group 64

16. Connected Lie groups of polynomial volume growth 66

17. Proof of propositions 1.6.3 and 1.6.4 in the general case 73

18. Proof of the Gaussian estimate in the general case 77

19. A Berry-Esseen estimate for the heat kernels on connected Lie groups of

polynomial volume growth 80

20. Polynomials on connected Lie groups of polynomial growth 84