18 GEORGIOS K. ALEXOPOULOS
vector fields Y\,..., Ym there are a, /?, 6 G N, a j3 0 and c 0 such that for all
n G N and all u 0 satisfying ( ^ + L) u = 0 in (0,
6n2)
x
[/0n;
(1.13.1)
snpl\^Y1Y2...Yrnu(an2Jx)\:xeUan\
cn
2 / e

m
inf
{u(/^n2,x)
: x G t/
6n
} .
An alternative way of proving the above result is by adapting the local theory in
[V3, VSC] using the family of dilations associated with the limit group at infinity
(cf. [NRS]).
2. Lie groups of polynomial volume growth. Unfortunately, the Harnack in
equality (1.13.1) is not necessarily true for m 2 when the group is not nilpotent.
To see this let us consider the group l
9
X M studied in section 1.5 and let us
consider the function
u(x,z) =xj+il;j(z)1 (x,z) eRq\M
where n j q. This function grows linearly, i.e. there is c 0 such that
sup{M
;Ur)}
cr, r 1
and it satisfies Lu = 0. Also, ZiU = Z^, 1 i m. So, if ZiZ^ ^ 0 for some
1 i, j m and U is chosen large enough, so that 0 x M C [/, then the inequality
sup{\ZiZju\ ; U} cr~2sup{\u\ ; Ur}, r 1
is false.
THEOREM
1.13.2. Let L be a centered left invariant subLaplacian on a con
nected Lie group G of polynomial volume growth. Let also U be a compact neigh
borhood of the identity element e of G and a,b G N. Then, for all k G N and all
left invariant vector fields Y there are a, /?, 0 G N, a (5 6 and c 0 such that
for all n G N and all u 0 satisfying (J^ + L) u = 0 in (0, On2) x U0n,
(1.13.2) sup  \ ^ Y u ( a n 2 , x ) \ : x G Uan\ cn'2^1 inf {u((3n2,x) : x G Ubn} .
Nevertherless, we can still controll the higher order spatial derivatives by dif
ferentiating the expression in the left part of (1.11.3). The expression that we shall
obtain though, will involve other spacial derivatives of lower order as well as the
correctors
i\)%
and their derivatives (see for example theorem 1.14.9 below).
1.14. BerryEsseen estimates.
1. Nilpotent Lie groups. Let L be a centered left invariant subLaplacian on a
simply connected nilpotent Lie group N and let LQ be the associated limit operator.
Let also pt{x, y) and Pt(x, y) be the heat kernels of L and LQ respectively.
We have the following BerryEsseen type of estimate (cf. [Fe, Pe]):
THEOREM
1.14.1. There are constants c,
CL
0 such that
(1.14.1) \pt(x,e)p°t{xJe)\
ct~{D+l)/2,
x £ N, t 1.
Since LQ, hence also Pt(x,y), is dilation invariant, there is a constant CL 0
such that
(1.14.2) p?(e,e) = c
L
£
D
/
2
, t 0.