SUB-LAPLACIANS WIT H DRIF T 15
PROPOSITION
1.9.1. Let L be a centered left invariant sub-Laplacian on N
and let Lo be the associated limit operator. Then to any monomial P(£, x) we can
associate a polynomial P(£, x) satisfying
QP(t,x) = P(t,x) + W(t,x)
(1.9.2) degHWdegHP-l
^ + Lo) P(t,x)=(§
i
+ L) QP(t,x).
Note that the polynomial Qp(t,x) in the above proposition, is not necessarily
unique.
From now on, we shall denote, for all k £ N, by
P0(t,x),P2(t,x),...,PUk(t,x)
the monomials with homogeneous degree k and we shall fix, once and for all,
polynomials
QPo(£, x), QP2(£, x),..., QPuk (£, x)
satisfying (1.9.2).
2. A Taylor formula for the heat functions.
The following result gives a Taylor formula for the heat functions. The proof
is based on ideas from [ALl, A12]. These ideas have already been used in the
context of Lie groups in [Al, A2, ALo]. The interest of the method lies in the
fact that we do not make use of any a priori control on the derivatives of the heat
functions.
THEOREM
1.9.2. For all k e N there is ck 0 such that for all n e N and all
functions u satisfying
+ L J u = 0, in (-n 2 , n2) x Un
we have
(1.9.3) supMu- ] T
Atn-dez»p*QPz\;(-l,l)xu\
ck
where the constants Ai satisfy
Ck 1 1 ^ 1 1 oo -i 0 i vk
and where
n
(/c+1)||d|
1.10. Harmonic functions of polynomial growth on connected nilpo-
tent Lie groups. Let G b e a connected Lie group and let [ / b e a compact neigh-
borhood of the identity element e of G. We shall say that a function u ow G grows
polynomially, if there is c 0 and A 0 such that
(1.10.1) sup {|T/|; Un} cnA, n G N.
An application of theorem 1.9.2 is the following:
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