Memoirs of the American Mathematical Society 2002; 101 pp; softcover Volume: 155 ISBN10: 0821827642 ISBN13: 9780821827642 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 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. Readership Graduate students and research mathematicians interested in topological groups, Lie groups, and harmonic analysis. Table of Contents  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
 SubLaplacians 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 BerryEsseen estimate in the case of nilpotent Lie groups
 The nilshadow 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 BerryEsseen 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
 BerryEsseen type of estimates for the derivatives of the heat kernel
 Riesz transforms
 Bibliography
