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 sub-Laplacian 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. Berry-Esseen estimates.
1. Nilpotent Lie groups. Let L be a centered left invariant sub-Laplacian 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 Berry-Esseen 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.
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