SUB-LAPLACIANS WIT H DRIF T 17

In the rest of this article, to every monomial P;, i G N, we shall associate and

fix, once and for all, a corrected monomial Qp , satisfying (1.11.1) and (1.11.2).

2. A Taylor formula for the heat functions. The following result is the gener-

alisation of theorem 1.9.2.

THEOREM

1.11.2. For all k G N there is c/~ 0 such that for all n G N and

all functions u satisfying

— + L J u = 0, in (-n 2 , n2) x t/n

i^e have

(1.11.3) s u p ^ | u - ^ ^n-

d e

SH^Q^|

;

(-i,i)xf/ V ^ r r ^ H U ,

where the constants A{ satisfy

\Ai\ Cfcll^Hoo

for all 0 i fk and where

V j - i t " j /

1.12. Harmonic functions of polynomial growth. The following result is

an application of theorem 1.11.2 and generalises theorem 1.10.1.

THEOREM

1.12.1. Let L be a centered left invariant sub-Laplacian on a con-

nected Lie group G of polynomial volume growth. Then every L-harmonic function

u on G (i.e. satisfying Lu = 0 on G) which grows polynomially, is equal to a linear

combination of corrected monomials Qp .

A result of this type has first been proved by Avellaneda and Lin [AL2] in

the case of differential operators with periodic coefficients in Rn. A generalisation

of this result for symmetric sub-Laplacians on Lie groups of polynomial volume

growth was given in [ALo], where it was also used to prove a Sobolev inequality.

Finally, we point out that in the context of Riemmannian manbifolds with non-

negative Ricci curvature, there is a wellknown conjecture of S.T.Yau which says that

the linear space of harmonic functions with polynomial growth of a fixed rate is

finite dimensional. For results and further references related to this conjecture we

refer the reader to the recent paper of Colding and Minicozzi II [CM].

1.13. Harnack inequalities for the derivatives of the heat functions.

1. Nilpotent Lie groups. An consequence of theorem 1.9.2 is the following:

THEOREM

1.13.1. Let L be a centered left invariant sub-Laplacian on a simply

connected nilpotent Lie group N. Let also U be a compact neighborhood of the

identity element e of N and let a, b G N. Then, for all k £ N and all left invariant