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.
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
(-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.
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:
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
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