SUB-LAPLACIANS WIT H DRIF T 13
When the drift term EQ = 0, then the elliptic version of the above inequality was
proved by Varopoulos [V2] by adapting the method of Moser [Ml]. The parabolic
version, also for sub-Laplacians with drift EG = 0, was proved by Salof-Coste [SC]
by using a Gaussian estimate on the spacial derivatives of the heat kernel.
1. The strategy of the proof. This inequality is proved by the method of Krylov
and Safonov [KS]. This method uses certain information on the growth of the
positive heat functions. To obtain this information we use the following three
results.
The first result is a theorem of Varopoulos [V6], which asserts that the heat
kernel pt(x, y) of L decreases with a certain uniform speed as t oo:
THEOREM
1.6.2 (Varopoulos [V6]). Let L be a non-necessarily centered left
invariant sub-Laplacian on G and letpt(x,y) the heat kernel of L. Then there is a
constant c 0 such that
(1.6.2)
Pt
(x, y) c rD 2, t 1, x,yeG.
The other two results concern the distribution of the mass of the heat kernel
Pt{x,y) as t oo:
PROPOSITION
1.6.3. Let pt(x,y) be the heat kernel of a centered left invariant
sub-Laplacian L on a connected Lie group of polynomial volume growth G and let
U be a compact neighborhood of the identity element e of G. Then for all a G N
there is 5 0 such that
(1.6.3) / pt(x,y)dy5,
for all neN,
a~2n2
t
a2n2
and
xeUan.
PROPOSITION
1.6.4. Let L, pt{x,y) and U be as above. Then for all S 0
there is a G N such that
(1.6.4) / pt(e,y)dy6,
J
for allntN, 0 t
n2.
The above propositions are rather obvious in the case of dilation invariant
sub-Laplacians on stratified nilpotent groups. In the case of general connected
nilpotent Lie groups, they are proved by making use of the associated limit sub-
Laplacian. In the case of more general connected Lie groups of polynomial volume
growth they are proved by making use of the associated homogenised sub-Laplacian.
2. The oscillation of the heat functions. If B C R x G and u is a function
defined on B1 then let us set
Osc(u,B) sup{|w(t,x) u{s,y)\ : (t,x), (s,y) G B}.
In the course of the proof of (1.6.1) we shall prove the following result:
PROPOSITION
1.6.5. There is ceN and a G (0,1) such that
(1.6.5) Osc(tx,(t-n
2
,t) x
Un)
a Osc (u, {t -
c2n2,t)
x
Ucn)
for all n G N and all functions u satisfying
9 ^ . . . .
+ L)u = 0 in
(t-c2n2,t)xUc22
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