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group can be joined by a geodesic.

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.
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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
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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
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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
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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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