The area and co-area formulas are fundamental tools in classical geometric measure theory. These integral formulas have sub-Riemannian analogs and indeed can be generalized to much more general metric spaces. To begin, we remind the reader of the classical area formula (see [95] Section 3.2.1):

118 Chapter 6. Geometric Measure Theory and Geometric Function Theory

**Theorem 6.1 (Euclidean area formula). Let A***⊂ R*^{n}*be a measurable set, let f :*
*A→ R*^{m}*be a Lipschitz map, and let m≥ n. Then,*

*A*

*J (f )(x) dH*^{n}*(x) =*

R^{m}

*N (f, A, y) dH*^{m}*(y)*

*where J (f ) is the Jacobian of f ,H*^{n}*is the n-dimensional Hausdorﬀ measure, and*
*N (f, A, y) = card{x ∈ A : f(x) = y}.*

To generalize Theorem 6.1 to the setting of Carnot groups, we require a notion
of horizontal Jacobian for Lipschitz maps between Carnot groups. We note that
while the co-area formula holds for arbitrary Carnot groups, the reader focused
on understanding the Heisenberg group may wish to assume thatG*i**∈ {H, R*^{n}*} in*
what follows. We begin with a deﬁnition of a metric notion of the Jacobian.

**Deﬁnition 6.2. Let**G1*,*G2 be Carnot groups equipped with sub-Riemannian
*met-rics d*_{1}*, d*_{2} *respectively. Let A* *⊂ G*1 be an *H*^{Q}*measurable set, where Q is the*
homogeneous dimension of G1*, and let f : A→ G*2 be Lipschitz with respect to
the sub-Riemannian metrics on G1 andG2*. The metric Jacobian of f at x∈ A is*
deﬁned to be

*J*_{f}^{m}*(x) = lim inf*

*r**→0*

*H**d*^{Q}_{2}*(f (B*_{A}*(x, r)))*
*H*^{Q}_{d}_{1}*(B*_{A}*(x, r))*
*where B*_{A}*(x, r) ={y ∈ A|d*1*(x, y)≤ r}.*

With this notion we easily recover the following standard integration formula:

*A*

*J*_{f}^{m}*(x)dH*_{d}^{Q}_{1}*(x) =H*^{Q}_{d}_{2}*(f (A)).* (6.1)
We note that (6.1) trivially yields a property analogous to the Sard theorem,
*namely, if J*_{f}^{m}*= 0 a.e. in A, thenH*^{Q}_{d}_{2}*(f (A)) = 0.*

A key ingredient of the proof of the sub-Riemannian area formula is an
*analysis of the set A*_{0} =*{x ∈ A : J**f*^{m}*(x) = 0}. To better understand this set, we*
introduce Pansu’s sub-Riemannian analog for the diﬀerential.

**Deﬁnition 6.3. Let (**G1*, d*_{1}) and (G2*, d*_{2}) be Carnot groups with homogeneous
*structures determined by the dilations δ*_{s}^{i}*, i = 1, 2. Suppose that f :*G1*→ G*2 is a
*measurable map. Then the Pansu diﬀerential of f at x is the map*

*D*_{0}*f (x) :*G1*→ G*2

deﬁned by

*D*0*f (x)(y) = lim*

*s**→0**δ*_{1/s}^{2} *f (x)*^{−1}*f (xδ*^{1}_{s}*y)*
whenever the limit exists.

6.1. Area and co-area formulas 119

For a Carnot group *G with sub-Riemannian metric d*_{G}, we may recognize
*the metric tangent space T*_{x}*G at a point x ∈ G as the pointed Gromov–Hausdorﬀ*
*limit of the sequence of metric spaces (x*^{−1}*G, λd*G*) as λ→ ∞. As G is equipped*
*with a family of homotheties (δ** _{s}*),

*(x*^{−1}*G, sd*G*) = (x*^{−1}*G, d*G*◦ (δ**s**× δ**s**)),*

*whence (x*^{−1}*G, d*_{G}*) and (x*^{−1}*G, sd*_{G}*) are isometric via the map δ** _{s}*. Thus the

*se-quence above is constant and T*

*G is isometric to G. From this point of view, we*

_{x}*can think of D*

_{0}

*f as a map between tangent spaces. Alternatively, by conjugating*

*with the exponential map, we can view D*0

*f as a map between the Lie algebras:*

*(D*0*f )** _{∗}*:g1

*→ g*2.

Pansu’s extension of the Rademacher diﬀerentiability theorem to Carnot groups reads as follows:

* Theorem 6.4 (Pansu–Rademacher diﬀerentiation theorem). Let*G1

*,*G2

*be Carnot*

*groups and let A*

*⊂ G*1

*be a measurable set. Let f : A*

*⊂ G*1

*→ G*2

*be Lipschitz*

*with respect to the sub-Riemannian metrics on*G1

*and*G2

*. Then, for a.e. x∈ A,*

*D*

_{0}

*f (x) exists and is a horizontal linear map (i.e., a graded homogeneous group*

*homomorphism, see Deﬁnition 2.1) between*G1

*and*G2

*.*

In Section 6.2, we prove a special case of this theorem when G1 = H and G2=R.

*Example 6.5. Let* G1 =G2 =*H. The Pansu diﬀerential D*0*f of a Lipschitz map*
*f = (f*1*, f*2*, f*3) :*H → H, acting on the Lie algebra h and expressed in terms of*
*the standard basis X*1*, X*2*, X*3, takes the form

⎛

⎝*X*_{1}*f*_{1} *X*_{1}*f*_{2} 0
*X*_{2}*f*_{1} *X*_{2}*f*_{2} 0

0 0 *X*_{1}*f*_{1}*X*_{2}*f*_{2}*− X*1*f*_{2}*X*_{2}*f*_{1}

⎞

*⎠ .*

*The next lemma establishes a link between D*_{0}*f and J*_{f}* ^{m}*.

* Lemma 6.6. Let* G1

*and*G2

*be Carnot groups with homogeneous dimensions Q =*

*Q*

_{1}

*and Q*

^{}*= Q*

_{2}

*respectively, with Q*

^{}*≥ Q, and let f : A ⊂ G*1

*→ G*2

*be Lipschitz.*

*If D*_{0}*f (x) is not injective for some x∈ A, then J**f*^{m}*(x) = 0.*

*Proof. We sketch the main ideas in the proof in the case A =* G1*. Let P =*
*D*0*f (x)(*G1) be the image of the entire group under the diﬀerential mapping. Since
*D*0*f is not injective at x, the Hausdorﬀ dimension of P is less than or equal to*
*Q− 1. Moreover, as the Pansu diﬀerential exists, we have*

*d*_{2}*(D*_{0}*f (x)(y), δ*^{2}_{1/s}*f (x)*^{−1}*f (xδ*_{s}^{1}*y)) = o(1)*

*as s→ 0. Using left invariance of the metric and the dilation property (2.20), we*
ﬁnd

*d*2*(D*0*f (x)(y), δ*_{1/s}^{2} *f (x)*^{−1}*f (xδ*^{1}_{s}*y)) =* 1

*sd*2*(f (x)δ*_{s}^{2}*D*0*f (x)(y), f (xδ*^{1}_{s}*y))*

120 Chapter 6. Geometric Measure Theory and Geometric Function Theory

so

*d*2*(f (x)δ*_{s}^{2}*D*0*f (x)(y), f (xδ*^{1}_{s}*y)) = o(s).*

*Since D*0*f (x)(δ*^{1}_{s}*y) = δ*_{s}^{2}*D*0*f (x)(y), we may rewrite this as*
*d*_{2}*(f (x)D*_{0}*f (x)(y), f (xy)) = o(d*_{1}*(x, y)).*

*Consequently, if B(x, r)* *⊂ G*1 *is the metric ball of radius r centered at x, then*
*its image under f lies in an o(r) neighborhood N of f (x)D*_{0}*f (x)(B(o, r)) where*
*o∈ G*1 *is the identity element. Since f is Lipschitz,*

*H*^{Q}*d*_{2}*(f (A))≤ (Lip f)*^{Q}*H*^{Q}*d*_{1}*(A)* (6.2)
*for all measurable A⊂ G*1, where

*Lip f (x) = lim sup*

*r**→0*

sup*{d*2*(f (x), f (y)) : d*1*(x, y)≤ r}*

*r*

*denotes the pointwise Lipschitz constant of f . Combining (6.2) with the *
*observa-tion that f (x)D*_{0}*f (x)(B(o, r)) has Hausdorﬀ dimension less than or equal to Q−1,*
it follows from a covering argument that

*H*^{Q}*d*_{2}*(f (B(x, r)))≤ H*^{Q}*d*_{2}*(N ) = o(r*^{Q}*) = o(H*^{Q}*d*_{1}*(B(x, r))).*

The result follows from the deﬁnition of the metric Jacobian. We omit the proof of the following lemma, which is an adaptation of a classical argument (see 3.2.2 in [95]). For further details, see the notes to this chapter.

**Lemma 6.7. Suppose that f : A***⊂ G*1 *→ G*2 *is a Lipschitz map and let λ > 1.*

*Let ˜A be the set of points of density of A where D*_{0}*f exists and is injective. Then*
*there exist Borel sets* *{E**i**} partitioning ˜A so that for each i,*

*• f|**E*_{i}*is injective,*

*• there exist injective horizontal linear maps L**i* *so that*
1

*λd*2*(L**i**(z), o)≤ d*2*(D*0*f (x)(z), o)≤ λd*2*(L**i**(z), o)* (6.3)
*for x∈ E**i* *and z∈ G*1*, where o∈ G*2 *is the identity, and*

*• we have*

*Lip(f|**E*_{i}*◦ L**i**|*^{−1}*C** _{i}*)

*≤ λ, Lip(L*

*i*

*|*

*E*

_{i}*◦ f|*

^{−1}*E*

*)*

_{i}*≤ λ.*(6.4) Using this setup and notion of metric Jacobian, we can prove a version of the area formula for Carnot groups.

6.1. Area and co-area formulas 121 proof of the area formula as it makes no contribution to either side of the equation.

*By Lemma 6.6, D*0*f (x) is injective for x∈ A \ A*0. Using Lemma 6.7, we ﬁnd sets
*{C**i**} partitioning A \ A*0*, with f**i**= f|**C*_{i}*injective. Let J*_{i}^{m}*(x) := J*_{f}^{m}

*i**(x). Since the*
closed balls in a Carnot group form a Vitali relation, we know that

lim

*r**→0*

*H*^{Q}*d*_{1}*(B*_{A}_{∩C}_{i}*(x, r))*

*H*^{Q}_{d}_{1}*(B**A**(x, r))* = 1 (6.5)
*for all i and all points of density x* *∈ C**i*. A quick calculation using (6.5) shows
*that J*_{f}^{m}*= J*_{i}* ^{m}* at such points:

*Using (6.1) and the fact that the f**i* are injective, we have

122 Chapter 6. Geometric Measure Theory and Geometric Function Theory

*Summing over all i yields*

*A*

*J*_{f}^{m}*(x) dH*^{Q}*d*_{1}*(x) =*

*A**\A*0

*J*_{f}^{m}*(x) dH**d*^{Q}_{1}*(x) =*

G2

*N (f, C, y) dH*^{Q}*d*_{2}*(y).*

To complete our discussion of the area formula, we note that a more geometric notion of the Jacobian is equivalent to the metric Jacobian at points of Pansu dif-ferentiability. Recall the notion of a horizontal linear map between Carnot groups (Deﬁnition 2.1).

* Deﬁnition 6.9. The horizontal Jacobian of a horizontal linear map φ :*G1

*→ G*2 is

*J*_{H}*(φ) =* *H*^{Q}*d*_{2}*(φ(A))*
*H*^{Q}_{d}_{1}*(A)* *,*

*where A is any measurable subset of*G1of positive ﬁnite measure.

We note that a simple covering argument, combined with the homogeneity
*and left invariance of the Hausdorﬀ measure, shows that the value of J*_{H}*(φ) is*
*independent of the choice of A.*

* Proposition 6.10. Let f : A⊂ G*1

*→ G*2

*be a Lipschitz map. Then J*

_{f}

^{m}*= J*

_{H}*(D*

_{0}

*f )*

*at point of Pansu diﬀerentiability in A.*

*Proof. Let x be a point of Pansu diﬀerentiability for f in A. By Lemma 6.6, we*
*may assume that D*0*f (x) is injective; else both sides are zero. Using Lemma 6.7*
*with a sequence λ**n* *→ 1, we ﬁnd sets E**n* *containing x as points of density and*
*horizontal linear maps L**n* *satisfying (6.3) and (6.4) with λ = λ**n*. Thus

*λ*^{−Q}_{n}*J**H**(L**n*) = lim

*r**→0**λ*^{−Q}_{n}*H*_{d}^{Q}_{2}*(L*_{n}*(B*_{A}*(x, r)))*
*H*_{d}^{Q}_{1}*(B**A**(x, r))*

= lim

*r**→0**λ*^{−Q}_{n}*H*_{d}^{Q}_{2}*(L*_{n}*(B*_{A}_{∩E}_{n}*(x, r)))*
*H**d*^{Q}_{1}*(B*_{A}*(x, r))* *,*
*since x is a point of density of E*_{n}*and L** _{n}* is a horizontal linear map

*≤ lim sup*

*r**→0*

*H*^{Q}_{d}_{2}*(f (B*_{A}_{∩E}_{n}*(x, r)))*
*H**d*^{Q}_{1}*(B*_{A}*(x, r))* *,*
by (6.3)

*≤ lim sup*

*r**→0*

*H*^{Q}_{d}_{2}*(f (B**A**(x, r)))*

*H**d*^{Q}_{1}*(B*_{A}*(x, r))* *= J*_{f}^{m}*(x).*