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
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