Proof. Take two arbitrary points in N . Because the distance is left invariant we can

choose x = 0. Because dilatations change the distances by a constant factor, we can

suppose that y ∈ B(0, 1).

A previous proposition shows that the topology generated by a homogeneous norm

and the topology generated by the Euclidean norm are the same. Therefore the space

is metrically complete. The Ball-Box Theorem implies that the ball B(0, 1) is compact.

We are in the assumptions of the abstract Hopf-Rinow theorem 1.11.

23

2 CARNOT GROUPS

An easy computation shows that

L(B(0, δµ r)) = µQ L(B(0, r))

As a consequence the Lebesgue measure is absolutely continuous with respect to the

Hausdor¬ measure HQ . Because of the invariance with respect to the group operation

it follows that the Lebesgue measure is a multiple of the mentioned Hausdor¬ measure.

The following theorem is Theorem 2. from Mitchell [23], in the particular case of

Carnot groups.

Theorem 2.11 The ball B(0, 1) has Hausdor¬ dimension Q.

Proof. We know that the volume of a ball with radius µ is cµQ .

Consider a maximal ¬lling of B(0, 1) with balls of radius µ. There are Nµ such balls

in the ¬lling; an upper bound for this number is:

Nµ ¤ 1/µQ

The set of concentric balls of radius 2µ cover B(0, 1); each of these balls has diameter

smaller than 4µ, so the Hausdor¬ ± measure of B(0, 1) is smaller than

lim Nµ (2µ)±

µ’0

which is 0 if ± > Q. Therefore the Hausdor¬ dimension is smaller than Q.

Conversely, given any covering of B(0, 1) by sets of diameter ¤ µ, there is an as-

sociated covering with balls of the same diameter; the number Mµ of this balls has a

lower bound:

Mµ ≥ 1/µQ

thus there is a lower bound

µ± ≥ µ± /µQ

cover

H± (B(0, 1))

= ∞. Therefore the Hausdor¬ dimension

which shows that if ± < Q then

of the ball is greater than Q.

We collect the important facts discovered until now: let N be a Carnot group

endowed with the left invariant distribution generated by V1 and with an Euclidean

norm on V1 .

(a) If V1 Lie-generates the whole Lie algebra of N then any two points can be joined

by a horizontal path.

(b) The metric topology of N is the same as Euclidean topology.

m

: xi ¤ r i .

(c) The ball B(0, r) looks roughly like the box x = i=1 xi

(d) the Hausdor¬ measure HQ is group invariant and the Hausdor¬ dimension of a

ball is Q.

(e) the Hopf-Rinow Theorem applies.

(f) there is a one-parameter group of dilatations, where a dilatation is an isomorphism

δµ of N which transforms the distance d in µd.

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2 CARNOT GROUPS

2.2 Pansu di¬erentiability

A Carnot group has it™s own concept of di¬erentiability, introduced by Pansu [25].

In Euclidean spaces, given f : Rn ’ Rm and a ¬xed point x ∈ Rn , one considers

the di¬erence function:

f (x + tX) ’ f (x)

X ∈ B(0, 1) ‚ Rn ’ ∈ Rm

t

The convergence of the di¬erence function as t ’ 0 in the uniform convergence gives

rise to the concept of di¬erentiability in it™s classical sense. The same convergence, but

in measure, leads to approximate di¬erentiability. Other topologies might be considered

(see Vodop™yanov [33]).

In the frame of Carnot groups the di¬erence function can be written using only

dilatations and the group operation. Indeed, for any function between Carnot groups

f : G ’ P , for any ¬xed point x ∈ G and µ > 0 the ¬nite di¬erence function is de¬ned

by the formula:

X ∈ B(1) ‚ G ’ δµ f (x)’1 f (xδµ X) ∈ P

’1

’1

In the expression of the ¬nite di¬erence function enters δµ and δµ , which are dilatations

in P , respectively G.

Pansu™s di¬erentiability is obtained from uniform convergence of the di¬erence func-

tion when µ ’ 0.

The derivative of a function f : G ’ P is linear in the sense explained further. For

simplicity we shall consider only the case G = P . In this way we don™t have to use a

heavy notation for the dilatations.

De¬nition 2.12 Let N be a Carnot group. The function F : N ’ N is linear if

(a) F is a group morphism,

(b) for any µ > 0 F —¦ δµ = δµ —¦ F .

We shall denote by HL(N ) the group of invertible linear maps of N , called the linear

group of N .

The condition (b) means that F , seen as an algebra morphism, preserves the grading

of N .

The de¬nition of Pansu di¬erentiability follows:

De¬nition 2.13 Let f : N ’ N and x ∈ N . We say that f is (Pansu) di¬erentiable

in the point x if there is a linear function Df (x) : N ’ N such that

sup {d(Fµ (y), Df (x)y) : y ∈ B(0, 1)}

converges to 0 when µ ’ 0. The functions Fµ are the ¬nite di¬erence functions, de¬ned

by

Ft (y) = δt f (x)’1 f (xδt y)

’1

25

2 CARNOT GROUPS

The de¬nition says that f is di¬erentiable at x if the sequence of ¬nite di¬erences

Ft uniformly converges to a linear map when t tends to 0.

We are interested to see how this di¬erential looks like. For any f : N ’ N ,

x, y ∈ N Df (x)y means

Df (x)y = lim δt f (x)’1 f (xδt y)

’1

t’0

provided that the limit exists.

Proposition 2.14 Let f : N ’ N , y, z ∈ N such that:

(i) Df (x)y, Df (x)z exist for any x ∈ N .

(ii) The map x ’ Df (x)z is continuous.

’1

(iii) x ’ δt f (x)’1 f (xδt z) converges uniformly to Df (x)z.

Then for any a, b > 0 and any w = δa xδb y the limit Df (x)w exists and we have:

Df (x)δa xδb y = δa Df (x)y δb Df (x)z

Proof. The proof is standard. Remark that if Df (x)y exists then for any a > 0 the

limit Df (x)δa y exists and

Df (x)δa y = δa Df (x)y

It is not restrictive therefore to suppose that w = yz, such that | y |d = 1.

We write:

δt f (x)’1 f (xδt w) = (1)(2)(3)

’1

where

’1

(1) = δt f (x)’1 f (xδt y)

(2) = δt f (xδt y)’1 f (xδt yδt z) (Df (xδt y)z)’1

’1

(3) = Df (xδt y)z

When t tends to 0 (i) implies that (1) tends to Df (x)y, (ii) implies that (3) tends to

Df (x)z and (iii) implies that (2) goes to 0. The proof is ¬nished.

If f is Lipschitz then the previous proposition holds almost everywhere. In Propo-

sition 3.2, Pansu [25] there is a more general statement:

Proposition 2.15 Let f : N ’ N have ¬nite dilatation and suppose that for almost

any x ∈ N the limits Df (x)y and Df (x)z exist. Then for almost any x and for any

w = δa y δb z the limit Df (x)w exists and we have:

Df (x)δa xδb y = δa Df (x)y δb Df (x)z

26

2 CARNOT GROUPS

Proof. The idea is to use Proposition 2.14. We shall suppose ¬rst that f is Lipschitz.

Then any ¬nite di¬erence function Ft is also Lipschitz.

Recall the general Egorov theorem:

Lemma 2.16 (Egorov Theorem) Let µ be a ¬nite measure on X and (fn )n a sequence

of measurable functions which converges µ almost everywhere (a.e.) to f . Then for

any µ > 0 there exists a measurable set Xµ such that µ(X \ Xµ ) < µ and fn converges

uniformly to f on Xµ .

Proof. It is not restrictive to suppose that fn converges pointwise to f . De¬ne, for

any q, p ∈ N, the set:

Xq,p = {x ∈ X : | fn (x) ’ f (x) | < 1/p , ∀n ≥ q}

For ¬xed p the sequence of sets Xq,p is increasing and at the limit it ¬lls the space:

Xq,p = X

q∈N

Therefore for any µ > 0 and any p there is a q(p) such that µ(X \ Xq(p)p ) < µ/2p .

De¬ne then

Xµ = Xq(p)p

p∈N

and check that it satis¬es the conclusion.

We can improve the conclusion of Egorov Theorem by claiming that if µ is a Borel

measure then each Xµ can be chosen to be open.

In our situation we can take X to be a ball in the group N and µ to be the Q

Hausdor¬ measure, which is Borel. Then we are able to apply Proposition 2.14 on Xµ .

When we tend µ to 0 we obtain the claim (for f Lipschitz).

A consequence of this proposition is Corollaire 3.3, Pansu [25].

Corollary 2.17 If f : N ’ N has ¬nite dilatation and X1 , ..., XK is a basis for the

distribution V1 such that

s ’ f (xδs Xi )

is di¬erentiable in s = 0 for almost any x ∈ N , then f is di¬erentiable almost every-

where and the di¬erential is linear, i.e. if

M

y= δti Xg(i)

i=1

then

M

Df (x)y = δti Xg(i) (f )(x)

i=1

27

2 CARNOT GROUPS

With the de¬nition of (Pansu) derivative at hand, it is natural to introduce the

class of C 1 maps from N to M .

De¬nition 2.18 Let N, M be Carnot groups. A function f : N ’ M is of class

C 1 if it is Pansu derivable everywhere and the derivative is continuous as a function

Df : N —N ’ M —M . Here N —N and M —M are endowed with the product topology.

Remark 2.19 For example left translations Lx : N ’ N , Lx (y) = xy are C 1 but

right translations Rx : N ’ N , Lx (y) = yx are not even derivable.

We can introduce the class of C 1 N manifolds.

De¬nition 2.20 Let N be a Carnot group. A C 1 N manifold is an (equivalence class

of, or a maximal) atlas over the group N such that the change of charts is locally

bi-Lipschitz.

2.3 Rademacher theorem

A very important result is theorem 2, Pansu [25], which contains the Rademacher

theorem for Carnot groups.

Theorem 2.21 Let f : M ’ N be a Lipschitz function between Carnot groups. Then

f is di¬erentiable almost everywhere.

The proof is based on the corollary 2.17 and the technique of development of a

curve, Pansu sections 4.3 - 4.6 [25].

Let c : [0, 1] ’ N be a Lipschitz curve such that c(0) = 0. To any division

Σ : 0 = t0 < .... < tn = 1 of the interval [0, 1] is a associated the element σΣ ∈ N (the

algebra) given by:

n

c(tk )’1 c(tk+1 )

σΣ =

k=0

Lemma (18) Pansu [26] implies the existence of a constant C > 0 such that

2

n

c(tk )’1 c(tk+1 )

σΣ ’ c(1) ¤C (2.3.2)

k=0

Take now a ¬ner division Σ and look at the interval [tk , tk+1 ] divided further by Σ

like this:

tk = tl < ... < tm = tk+1

The estimate (2.3.2) applied for each interval [tk , tk+1 ] lead us to the inequality:

n

d(c(tk ), c(tk+1 ))2

σΣ ’ σΣ ¤

k=0

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2 CARNOT GROUPS