eBook ISBN:  9781470403324 
Product Code:  MEMO/155/739.E 
List Price:  $60.00 
MAA Member Price:  $54.00 
AMS Member Price:  $36.00 
eBook ISBN:  9781470403324 
Product Code:  MEMO/155/739.E 
List Price:  $60.00 
MAA Member Price:  $54.00 
AMS Member Price:  $36.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 155; 2002; 101 ppMSC: Primary 22; 43
We prove a parabolic Harnack inequality for a centered subLaplacian \(L\) 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 BerryEsseen 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 subLaplacian \(L_C\). Thus our results also extend to noncentered subLaplacians.
ReadershipGraduate students and research mathematicians interested in topological groups, Lie groups, and harmonic analysis.

Table of Contents

Chapters

1. Introduction and statement of the results

2. The control distance and the local Harnack inequality

3. The proof of the Harnack inequality from Varopoulos’s theorem and Propositions 1.6.3 and 1.6.4

4. Hölder continuity

5. Nilpotent Lie groups

6. SubLaplacians on nilpotent Lie groups

7. A function which grows linearly

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

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

10. Polynomials on nilpotent Lie groups

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

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

13. Harmonic functions of polynomial growth on nilpotent Lie groups

14. Proof of the BerryEsseen estimate in the case of nilpotent Lie groups

15. The nilshadow of a simply connected solvable Lie group

16. Connected Lie groups of polynomial volume growth

17. Proof of Propositions 1.6.3 and 1.6.4 in the general case

18. Proof of the Gaussian estimate in the general case

19. A BerryEsseen estimate for the heat kernels on connected Lie groups of polynomial volume growth

20. Polynomials on connected Lie groups of polynomial growth

21. A Taylor formula for the heat functions on connected Lie groups of polynomial volume growth

22. Harnack inequalities for the derivatives of the heat functions

23. Harmonic functions of polynomial growth

24. BerryEsseen type of estimates for the derivatives of the heat kernel

25. Riesz transforms


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We prove a parabolic Harnack inequality for a centered subLaplacian \(L\) 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 BerryEsseen 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 subLaplacian \(L_C\). Thus our results also extend to noncentered subLaplacians.
Graduate students and research mathematicians interested in topological groups, Lie groups, and harmonic analysis.

Chapters

1. Introduction and statement of the results

2. The control distance and the local Harnack inequality

3. The proof of the Harnack inequality from Varopoulos’s theorem and Propositions 1.6.3 and 1.6.4

4. Hölder continuity

5. Nilpotent Lie groups

6. SubLaplacians on nilpotent Lie groups

7. A function which grows linearly

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

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

10. Polynomials on nilpotent Lie groups

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

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

13. Harmonic functions of polynomial growth on nilpotent Lie groups

14. Proof of the BerryEsseen estimate in the case of nilpotent Lie groups

15. The nilshadow of a simply connected solvable Lie group

16. Connected Lie groups of polynomial volume growth

17. Proof of Propositions 1.6.3 and 1.6.4 in the general case

18. Proof of the Gaussian estimate in the general case

19. A BerryEsseen estimate for the heat kernels on connected Lie groups of polynomial volume growth

20. Polynomials on connected Lie groups of polynomial growth

21. A Taylor formula for the heat functions on connected Lie groups of polynomial volume growth

22. Harnack inequalities for the derivatives of the heat functions

23. Harmonic functions of polynomial growth

24. BerryEsseen type of estimates for the derivatives of the heat kernel

25. Riesz transforms