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.