Brownian motion and harnack inequality for ... - Math.caltech.edu

230 M. AIZENMAN AND 8. SIMON
Combined with (3.20) this reduces the problem to showing that
for h L 0. Since 4 is bounded from above, we only need to show that
Using the duality (3.18) again, we see that this is equivalent to
which is (1.23).
XEW
The following is in many ways the first of the two main results of this paper
(v = 1,2 can be accommodated by using the remark after Proposition 3.6):
THEOREM 3.8. Let v 2 3, V E K,‘”. Then for any xo, there are balls C Q
about xo (depending onb on local norms of V) such that, for f E L’(aQ; du),
Proof: Let A be a closed union of spheres about xo with @ n A = 0, A C Q
(see Figure 1). Let S, be a sphere about xo contained in A and let Qr be the ball
whose boundary is S,. By the strong Markov property of Brownian motion, for
x E Q’,
(Mb’i)(x) = (M3)P)
Figure I. A reference for the proof of Theorem 3.8.