2. TH E CONTROL DISTANCE AND TH E LOCAL HARNACK INEQUALITY 21

THEOREM

1.15.3. Let us assume that L is centered. Then, for allYeg, the

Riesz transform operators

R^ =

YL-1^00,

and R*^ = L'^^Y

are bounded on

Lp,

for 1 p oo and from

L1

to

weak-L1.

Note that in general the second order Riesz transforms YiY^

- 1

'

0 0

and

i-i,ooy

1

y

2 j

Y1,Y2 G Q may be unbounded even on L? (cf. [A2]).

If G is nilpotent then there is no homogenisaton phenomena and so we can also

consider higher order Riesz transforms:

THEOREM

1.15.4. Let us assume that L is centered and that G is nilpotent.

Then, for all Yi,..., Yn G Q the Riesz transform operators

Rn,oc = yi...y

n

£"

n / 2

'°° and K,oo =

L-nl2^Yx..Yn

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

Combining the theorems 1.15.2 and 1.15.4 we have the following:

COROLLARY

1.15.5. Let L and

EQ

be as in theorem 1.15.1 and let us assume

that G is nilpotent. Then, for allY\,...,Yn E t\ the Riesz transform operators

Rn = Yx..YnL-nl2 and R*n = L-n'2Yx..Yn

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

2. The control distance and the local Harnack inequality

2.1. The control distance associated to a sub-Laplacian. To every

left invariant sub-Laplacian L — — {E\ + ... + E2) + EQ (or to every choice of left

invariant vector fields Ei,...,Ep satisfying Hormander's condition) on a connected

Lie group G we can associate a so called control distance d^.,.). This distance is

left invariant, i.e d^x^y) = diJ(e,x~1y), x,y G G and it is defined as follows:

Let us denote by C the set of all absolutely continuous paths c : [0,1] — • G, satisfying

£(£) — EiK p bi(t)Ei (c(£)), for almost all t G [0,1] and set

|c|= / (b1(t)2 + ... + bp(t)2)1/2dt.

Jo

We define

dL(x, y) = inf {|c| : c G C, c(0) = x, c(l) = y} .

We denote by B^(x) = {y G T V : d(x,y) r} the associated balls. We shall drop

the index L when there is no risk of confusion.

The behavior of the balls Br(x) as r — oo is actually a group invariant. More

precisely, let us fix a compact neighborhood U of the identity element e of G and

let \.\G be defined as in (1.1). Let us also consider a constant c 0 such that

B1/c(e)CUCBc(e).

Then

(2.1.1)

Bn/C(e) C

Un

C Bcn(e), n G N.