20 GEORGIOS K. ALEXOPOULOS

THEOREM

1.14.9. For all Y £ g, there is a constant c 0 such that

(1.14.9) \Ypt(x,y)-Yp?(x,y)- £ (Y^(x)) XjP»(x,y)\ ct^D+2)^

1J™1

for all x,y e G and t 1.

Combining (1.8.1), (1.13.2) and (1.14.9) we have the following:

COROLLARY

1.14.10. For allY e g and all e £ (0,1) there is a constant c 0

such that

(1.14.10) \Ypt(x,y)-Yp»(x,y)- £ (Yp(x))XjP?(x,y)\

l j m

for all x,y £ G and t 1.

The above results show that the role of the correctors is indeed essential. They

also indicate how we can handle questions about the large time behavior of the

spacial derivatives of the heat kernel.

1.15. Riesz transforms. Let L = —

[E2

+ ... -f

E2)

+ E$ be a left invariant

sub-Laplacian on a connected Lie group G of polynomial volume growth and let

L

-n/2

=

_ ± _ /

t

(n/2)-le-tLdt

n £ K

T(n/2) J0

Let ei be the linear suspace of g spanned by the vector fields Ei,...,Ep and by

e2 the linear sub-space of g generated by the Lie brackets [Ei, Ej], 1 i, j p.

THEOREM

1.15.1. Let us assume that L is centered and that E0 e t\ + t2-

Then, the Riesz transform operators EiL~1/2 and L~x/2Ei, 1 i p are bounded

on U, for 1 r oo and from

L1

to

weak-L1.

The Riesz transforms are singular integral operators and so, to prove the above

results, we apply the Calderon-Zygmund theory (cf. [SI, S3, CW]). Their kernel

is, in general, singular on the diagonal, as well as at infinity. The singularity near the

diagonal can be treated by using results from the local theory of L. The geometry

and the behavior of G and L at infinity is only used for the singularity at infinity.

In order to isolate these two singularities and treat them separately we consider

the operators

L

- n / 2 , 0

=

_ l

f1

t(n/2)-le-tLdt ^ L-n/2,oo

=

_ l f°° ^ 2 ) - ! ^ ^

W 2 ) J0 T(n/2) J,

1. Local Riesz transforms. We have the following local result:

THEOREM

1.15.2. Let us assume that E0 e t\-\-t2- Then for allYi,...,Yn £ t\,

the Riesz transform operators

Rn,o = Y1...YnL-n^° and R^0 = L^^Y^Yn

are bounded on LP, for 1 p oo and from L1 to weak-L1.

2. Riesz transforms at infinity. We have the following result: