22 GEORGIOS K. ALEXOPOULOS
This implies that there is c 0 such that
(2.1.2) -\x\Gl + d(e,x)c\x\G, xeG.
In contrast the behavior of the balls Br(x) as r 0 depends on the choice
of the Hormander vector fields Ei, ...,Ep. We can prove in particular (cf. [NSW,
VSC]) that there is d = d(Ei, ...,EP) G N and a constant c 0 such that for all
x G G,
(2.1.3) - r
d
|^
r
(x)| cr
d
, 0 r 1.
c
2.2. A local Harnack inequality. Let L = - ( ^ + ... + £ £ ) + #o be a left
invariant sub-Laplacian on a connected Lie group G. Then we have the following
local Harnack inequality which is due to Bony [Bo] (see also [VSC ch. Ill]):
THEOREM
2.2.1. Let V be a connected open subset of G, K a compact subset
of V, I (a, b) and a t\ t^ si #2 b. Then, for all 0 k G Z and
0 n G Z and a// vector fields Y\, ...,Fn G 0 t/iere zs a constant Ck,n 0 st^c/i /or
even/ positive solution of (-j^ + L) u = 0 in I x V, we have
(2.2.1) sup||^yi,...,y
n
w|;[ti,fe ] x i ^ j cfcninf{w;[si,82] x i ^ } .
This is a local result and the Lie group structure does not play any particular
role in its proof.
Let us denote by ei the linear sub-space of g generated by the vector fields
E\,...,EP and by e2 the linear sub-space of n generated by the Lie brackets
[E^Ej], li,jp.
The following result is proved by using a local scaling and by observing that
the rescaled vector fields satisfy Hormander's condition in a uniform way (cf. [V,
VSC ch. III]).
THEOREM
2.2.2. Let L be as above and let us assume that E$ G t\ + t2- Then,
for all 0 k G Z, 0 n G Z, for all 0 a 1, 0 a 6 1 and all vector fields
Yi, ...,Yn G ei there is a constant Ck,n 0 such that for all t G (0,1] and every
positive solution u of (J^ + L) u 0 in (0,t) x B^(x), we have
(2.2.2) sup {\^Y1,...,Ynu(at,y)\:yeBv^
ck,nt-k-n/2mf{u(bt,y):yeBv^t}.
Previous Page Next Page