14 GEORGIOS K. ALEXOPOULOS
1.7. Positive harmonic functions. A consequence of the Harnack inequality
(1.6.1) is the following result:
THEOREM
1.7.1. Let L be a centered left invariant sub-Laplacian on a con-
nected Lie group G of polynomial volume growth. Then, every positive function
u 0 satisfying Lu 0 on G is constant.
1.8. Gaussian estimates for the heat kernel. Making use of the Harnack
inequality (1.6.1) we can prove the following upper Gaussian estimate:
THEOREM
1.8.1. Let pt(x,y) be the heat kernel of a centered left invariant
sub-Laplacian L on a connected Lie group G of polynomial volume growth and let
\.\G be as in (1.1). Then there is a constant c 0 such that
(1.8.1) pt(x,y) c
t~D'2
exp U ? - ^ \ , tl,x,yeG.
When EQ G ei + t2 then we also have a small time (i.e. valid for 0 t 1)
Gaussian estimates (cf. [He, V4, V7 appendix A.4]). If EQ £ t\ -f 12 then the
situation is much more complicated (cf. [BL1, BL2]).
We also have the following lower Gaussian estimate:
COROLLARY
1.8.2. Let L and pt(x,y) be as above. Then there is a constant
c 1 such that
(1.8.2) pt(x, y) \ t-D'2 exp ( - c 1 ^ 1 ^ )
for all t 1, and x,y G G.
For time t = 1, the above estimate has been proved by Varopoulos (cf. [V4,
V5, V8]). If \x~1y\c t/c, then (1.8.2) follows be repeating the lower estimate
for t = 1. If \x~1y\c t/c, then (1.8.2) follows from the Harnack inequality (1.6.1)
by arguing as in [VSC, pp. 47-50].
1.9. A Taylor formula for the heat functions on nilpotent Lie groups.
1. Polynomials on nilpotent Lie groups. If T V is a simply connected nilpotent
Lie group, then using the exponential coordinates we can identify AT, as a differential
manifold, with
Rq.
So, by a monomial on N we shall understand a monomial on
Rq.
For every monomial P(x), there is a unique integer k 0 such that for every
compact neighborhood U of e there is c 0 such that
(1.9.1) -nk sup{|P(x)|, xeUn} cnk, n G N.
c
We shall define the homogeneous degree deg# P of P(x) by deg# P k.
We shall say that P(t,x) is a monomial on R x N if P(t,x) = tmQ(x), with
Q(x) a monomial on N. We shall define the homogeneous degree degH P(t,x) of
P(t,x) by
deg^ P(t, x) = 2m + deg# Q(x).
By polynomials we shall of course understand linear combinations of mono-
mials. The homogeneous degree of a polynomial is therefore the maximum of the
homogeneous degrees of its monomials.
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