3. The proof of the Harnack inequality from
Varopoulos's theorem and propositions 1.6.3 and 1.6.4
In this section we shall give the proof of the following result, from Varopoulos's
theorem 1.6.2 and by assuming propositions 1.6.3 and 1.6.4.
3.1. Let L be a centered left invariant sub-Laplacian on a connected
Lie group of polynomial volume growth G. Then for all a, b 0 there is (3 a 1,
c 0 and A 0 such that for all r 1 and all u 0 satisfying
+L\U = 0 in
x J3cr(e),
we have
(3.1) sup {u;(ar 2 , (a + a2)r2) x Bar(e)} Ainf {u; (f3r2, {(5 + b2)r2) x Bhr{e)} .
In view of (2.2.1) and (2.2.2), theorem 1.6.1 follows from the above result.
The proof is inspired from Krylov and Safonov [KS]. In the first part of the
proof we shall use theorem 1.6.2 and propositions 1.6.3 and 1.6.4, to prove the first
growth lemma 3.1.1.
The second part of the proof consists of the proof of the second growth lemma
3.3.1. For the proof of this lemma we follow closely Krylov and Safonov [KS]. We
use in particular certain covering lemmas.
A direct consequence of the second growth lemma is proposition 1.6.5 on the
oscillation of the heat functions which, by a standard argument (see for example
[Law]) implies (3.1).
The proof is long. This is, in part, due to the fact that we made an effort
to write it in such a way that it can be easily adapted to convolution powers of
densities on connected Lie groups and of probability measures on discrete groups
of polynomial volume growth.
3.1. The first growth lemma. If B C R x G, A C B and (t,x) G B then,
adopting the notation of [KS], we set
*((t,x),A,B) =
inf u(t, x) : u 0, u(s, y) 1 for (5, y) G A and 1—--fL)w = 0 i n i ? .
If A! C B then we set
^(A / ,A,5 ) = inf{^((t,x),A,J5) : (t,x) G i ' } .
3.1.1 (first growth lemma). For alia 1 there is c a and 6,£ G (0,1)
such that
(3.1.1) tf
a V ) x S
(e), A, (0,
x Bcr(e)) S
for all r 1 and every measurable subset A C
x Br(e), satisfying
As an immediate consequence of (3.1.1) and the local Harnack inequality 2.2.1, we
have the following corollary:
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