# Sub-Laplacians with Drift on Lie Groups of Polynomial Volume Growth

Share this page
*Georgios K. Alexopoulos*

We prove a parabolic Harnack inequality for a centered sub-Laplacian \(L\) on a connected Lie group \(G\) of polynomial volume growth by using ideas from Homogenisation theory and by adapting the method of Krylov and Safonov. We use this inequality to obtain a Taylor formula for the heat functions and thus we also obtain Harnack inequalities for their space and time derivatives. We characterise the harmonic functions which grow polynomially. We obtain Gaussian estimates for the heat kernel and estimates similar to the classical Berry-Esseen estimate. Finally, we study the associated Riesz transform operators. If \(L\) is not centered, then we can conjugate \(L\) by a convenient multiplicative function and obtain another centered sub-Laplacian \(L_C\). Thus our results also extend to non-centered sub-Laplacians.

#### Table of Contents

# Table of Contents

## Sub-Laplacians with Drift on Lie Groups of Polynomial Volume Growth

- Contents vii8 free
- Abstract ix10 free
- 1. Introduction and statement of the results 112 free
- 2. The control distance and the local Harnack inequality 2132 free
- 3. The proof of the Harnack inequality from Varopoulos's theorem and propositions 1.6.3 and 1.6.4 2334
- 4. Hölder continuity 3849
- 5. Nilpotent Lie groups 3849
- 6. Sub-Laplacians on nilpotent Lie groups 4152
- 7. A function which grows linearly 4455
- 8. Proof of propositions 1.6.3 and 1.6.4 in the case of nilpotent Lie groups 4556
- 9. Proof of the Gaussian estimate in the case of nilpotent Lie groups 4859
- 10. Polynomials on nilpotent Lie groups 5364
- 11. A Taylor formula for the heat functions on nilpotent Lie groups 5465
- 12. Harnack inequalities for the derivatives of the heat functions on nilpotent Lie groups 6172
- 13. Harmonic functions of polynomial growth on nilpotent Lie groups 6172
- 14. Proof of the Berry-Esseen estimate in the case of nilpotent Lie groups 6273
- 15. The nil-shadow of a simply connected solvable Lie group 6475
- 16. Connected Lie groups of polynomial volume growth 6677
- 17. Proof of propositions 1.6.3 and 1.6.4 in the general case 7384
- 18. Proof of the Gaussian estimate in the general case 7788
- 19. A Berry-Esseen estimate for the heat kernels on connected Lie groups of polynomial volume growth 8091
- 20. Polynomials on connected Lie groups of polynomial growth 8495
- 21. A Taylor formula for the heat functions on connected Lie groups of polynomial volume growth 8697
- 22. Harnack inequalities for the derivatives of the heat functions 92103
- 23. Harmonic functions of polynomial growth 93104
- 24. Berry-Esseen type of estimates for the derivatives of the heat kernel 95106
- 25. Riesz transforms 96107
- Bibliography 99110