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Sub-Laplacians with Drift on Lie Groups of Polynomial Volume Growth
Georgios K. Alexopoulos, University of Paris, Orsay, France
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Memoirs of the American Mathematical Society
2002; 101 pp; softcover
Volume: 155
ISBN-10: 0-8218-2764-2
ISBN-13: 978-0-8218-2764-2
List Price: US$60 Individual Members: US$36
Institutional Members: US\$48
Order Code: MEMO/155/739

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.

Graduate students and research mathematicians interested in topological groups, Lie groups, and harmonic analysis.

• Introduction and statement of the results
• The control distance and the local Harnack inequality
• The proof of the Harnack inequality from Varopoulos's theorem and propositions 1.6.3 and 1.6.4
• Hölder continuity
• Nilpotent Lie groups
• Sub-Laplacians on nilpotent Lie groups
• A function which grows linearly
• Proof of propositions 1.6.3 and 1.6.4 in the case of nilpotent Lie groups
• Proof of the Gaussian estimate in the case of nilpotent Lie groups
• Polynomials on nilpotent Lie groups
• A Taylor formula for the heat functions on nilpotent Lie groups
• Harnack inequalities for the derivatives of the heat functions on nilpotent Lie groups
• Harmonic functions of polynomial growth on nilpotent Lie groups
• Proof of the Berry-Esseen estimate in the case of nilpotent Lie groups
• The nil-shadow of a simply connected solvable Lie group
• Connected Lie groups of polynomial volume growth
• Proof of propositions 1.6.3 and 1.6.4 in the general case
• Proof of the Gaussian estimate in the general case
• A Berry-Esseen estimate for the heat kernels on connected Lie groups of polynomial volume growth
• Polynomials on connected Lie groups of polynomial growth
• A Taylor formula for the heat functions on connected Lie groups of polynomial volume growth
• Harnack inequalities for the derivatives of the heat functions
• Harmonic functions of polynomial growth
• Berry-Esseen type of estimates for the derivatives of the heat kernel
• Riesz transforms
• Bibliography