2 GEORGIOS K. ALEXOPOULOS

ACKNOWLEDGEMENT.

I wish to thank Nikolaos Varopoulos and Noel Lohoue

for the encouragement to this research and for several inspiring discussions during

the preparation of this article.

1.1. The general strategy. Our approach is inspired by Homogenisation

theory (cf. [BLP, JKO]). We can prove for example that every left invariant

sub-Laplacian L is behaving at infinity like another, so called, homogenised sub-

Laplacian LH which lives on a nilpotent Lie group and hence it is much easier to

deal with.

The existence of the homogenised sub-Laplacian LH allows us to distinguish

those drifts whose large time effect is not important. Hence the notion of a centered

sub-Laplacian.

The main tool is a parabolic Harnack inequality which holds only for large time

(or for large balls in the elliptic case). It is proved by adaptating the method of

Krylov and Safonov (cf. [KS]).

Note that as it was shown by Varopoulos [V2, VSC], the method of Moser

[Ml, M2] can also be adapted to obtain Harnack inequalities in abstract contexts.

In our case though, Moser's approach seems dificult to adapt.

The reader can observe that the same proof also works for convolution powers

of densities on connected Lie groups as well as for convolution powers of probability

measures on discrete groups of polynomial volume growth.

Once we have a parabolic Harnack inequality, then we can prove a large time

Gaussian estimate for the heat kernel pt(x,y) by some elementary method.

We adapt, in the context of Lie groups, some ideas of Bergstrom (cf. [Be,

Saz]) who proves the central limit theorem by the so-called convolution method.

This method does not make use of the characteristic functions. It gives also simul-

taneously the speed of convergence (which according to the classical Berry-Esseen

theorem is 1/y/n). By this method, once we have a Harnack inequality, we auto-

matically have a Berry-Esseen estimate for the pt(x, y) (although not at the optimal

rate of convergence).

In order to controll the derivatives of the heat functions (i.e. the functions u

satisfying (J^ + L) u — 0) we proceed as follows. We observe that the Berry-Esseen

estimate provides with a certain compactness the space of the heat functions. Using

this compactness, we can adapt some ideas of Avellaneda and Lin [ALl, AL2] (see

also [Al, A2, ALo]) and prove a Taylor formula for the heat functions. As an

immediate consequence of this formula, we have Harnack inequalities for the space

and time derivatives of the positive heat functions.

We obtain the optimal rate of convergence for Berry-Esseen estimate for the

pt(x,y). We also obtain Berry-Esseen estimates for the time and space derivatives

of the pt(x,y).

Finally, we study the associated Riesz trasform operators by using the Berry-

Esseen estimate for the spacial derivatives of the pt(x,y).

1.2. Centered sub-Laplacians. Let G\ be the closure of [G, G] in G. Then,

G/Gi is a connected abelian Lie group and hence it can de written as a direct

product

Rn

x T

m

, where T = R/Z. Let us consider the canonical projection n :

G —• E

n

, let H =Ker(7r) and let J) be the Lie algebra of H. Then f) is an ideal of g

and [g,g] C fj.

We shall say that the sub-Laplacian L is centered it EQ G f).