1 SOME TOPICS OF GEOMETRIC THEORY IN CARNOT GROUPS DRAFT

FRANCESCO SERRA CASSANO

Contents 1. Introduction 1 2. An introduction to Carnot Groups 2 2.1. Definition and first properties. 2 2.2. Metrics on Carnot groups 5 2.3. Hausdor↵measures in a metric space. Applications to Carnot groups. 9 2.4. Carnot groups as tangent spaces to sub-Riemannian manifolds. 17 3. Di↵erential Calculus on Carnot Groups 17 3.1. Pansu di↵erentiable maps between Carnot groups and some their characterizations 17 3.2. formula between Carnot groups 21 3.3. Whitney theorem for maps between Carnot groups 21 3.4. Sobolev spaces and Poincar´einequality. 22 3.5. Vector fields. 22 3.6. Sobolev spaces associated with vector fields. 23 3.7. Poincar´einequality. 23 3.8. BV -functions and sets of finite perimeter on Carnot groups. 26 4. Di↵erential Calculus within Carnot Groups 28 4.1. Complementary subgroups and graphs 29 4.2. Intrinsic graphs in Carnot groups 34 4.3. Regular hypersurfaces in Carnot groups. 36 4.4. Area formulas for Hausdor↵measures in metric spaces. Application to Carnot groups. 40 4.5. Regular surfaces in Carnot groups 44 4.6. Intrinsic Lipschitz Graphs in Carnot groups 48 4.7. Intrinsic di↵erentiable graphs in Carnot groups 53 4.8. H-regular surfaces, intrinsic Lipschitz graphs in Heisenberg groups vs. intrinsic di↵erentiability 57 4.9. Rectifiability in Carnot groups. 60 5. Sets of finite perimeter and minimal surfaces 64

F.S.C. is supported by MURST and INDAM, Italy and University of Trento, Italy. 1 5.1. De Giorgi’s structure theorem for sets of finite perimeter in Carnot groups of step 2 64 5.2. Minimal boundaries in Carnot groups 70 References 70

1. Introduction These notes aim at illustrating some results achieved in the in Carnot groups. First of all, I would like to thank the organizers of the ”Geome- try, Analysis and Dynamics on sub-Riemannian Manifolds” trimester, which is being held in Paris, for their kind invitation. It is also a great pleasure for me to acknowledge the help and support of several friends of mine who made this work possible: first of all, most of the results I obtained and that I presented here have been achieved jointly with Bruno Franchi and Raul Serapioni. Our long collaboration has been always an invaluable source of scientific as well as human enrichment. Without their collaboration and friendship, I would never have been able to attack this hard subject. I have to thank them also for allowing me to widely quote both our joint papers and their latest ones. I have also to thank the support of other friends and collaborators with whom I shared fruitful discussions during the last 15 years and whose work is mentioned here : A. Agrachev, G. Alberti, L. Ambrosio, Z. Balogh, S. Bianchini, F. Bigolin,U. Boscain, L. Capogna, L. Car- avenna, J.H. Cheng, G. Citti, B. Kirchheim, E. Le Donne, N. Garo- falo, P. Haj lasz, P. Koskela, B. Kleiner,G.P. Leonardi, V. Magnani, A. Malchiodi, P. Mattila, F. Montefalcone, M. Miranda jr., R. Monti, P. Pansu, S. Pauls, A. Pinamonti, S. Rigot, M. Ritor´e, J. Tyson, D. Vit- tone, P. Yang...... and many others whom I am likely to have left out, for which I apologize. These notes are not intended to provide a thorough – not even a partial – survey of the geometric measure theory in Carnot groups, since they are meant to be presented in a few lectures during this Paris trimester. Anyone interested in an exhaustive overview of this subject with a full bibliography, sharp statements, and detailed proofs, may refer to the PhD theses of V. Magnani [143], R. Monti [170] and F. Montefal- cone [165], as well as to the recent ones of F. Bigolin [35], D. Vittone [206], A. Pinamonti [185]. For more specific aspects we restrict our- selves to recommend the reader to the general monographes [71], [120], [121], [117], [116], [202], [205], [169], [49], [42], [2] and to the papers [7], [8], [10], [47], [66], [96], [102], [103][111], [122] [132], [180], [179], [181], [182], [208] and to the references therein. 2 Since these lectures especially focus on some issues of geometric mea- sure theory in Carnot groups, they of course leave out many other important topics, such as, for instance, coarea formulas (see[143]), cur- rents and Rumin’s complex (see f [8],[103],[193],[194],[18] and [107]), curvatures (see [49],[2],[127],[188] and [31]), mass transportation and optimal transport (see [11], [190],[81] and[76]).

2. An introduction to Carnot Groups 2.1. Definition and first properties. The present subsection is largely taken from [102] and [99] (see also [101]). A Carnot group G of step k (see [82], [126], [174], [121], [179], [204], [205], [42]) is a connected, simply connected whose g admits a step k strat- ification, i.e. there exist linear subspaces V1, ..., Vk such that (1) g = V ... V , [V ,V]=V ,V= 0 ,V= 0 if i>k, 1 k 1 i i+1 k 6 { } i { } where [V1,Vi] is the subspace of g generated by the commutators [X, Y ] with X V1 and Y Vi.Letmi = dim Vi,fori =1,...,k and h = m 2+ + m with2 h = 0 and, clearly, h = n. Choose a basis i 1 ··· i 0 k e1,...,en of g adapted to the stratification, that is such that

ehj 1+1,...,ehj is a base of Vj for each j =1,...,k. Let X = X1,...,Xn be the family of left invariant vector fields such that Xi(0) = ei. Given (1), the subset X1,...,Xm1 generates by com- mutations all the other vector fields; we will refer to X1,...,Xm1 as generating vector fields of the group. The exponential map is a one to one map from g onto G, i.e. any p G can be written in a unique 2 way as p = exp(p1X1 + + pnXn). Using these exponential coordi- ··· n nates, we identify p with the n-tuple (p1,...,pn) R and we identify 2 G with (Rn, ) where the explicit expression of the group operation is determined· by the Campbell-Hausdor↵formula (see [42]) and some· of its features are described in the following Proposition 2.3. If p G i mi 2 and i =1,...,k, we put p =(phi 1+1,...,phi ) R , so that we can 2 also identify p with [p1,...,pk] Rm1 Rmk = Rn. 2 ⇥···⇥ The subbundle of the tangent bundle T G that is spanned by the vector fields X1,...,Xm1 plays a particularly important role in the theory, it is called the horizontal bundle HG; the fibers of HG are

HGx = span X1(x),...,Xm1 (x) ,xG. { } 2 A subriemannian structure is defined on G, endowing each fiber of HG with a scalar product , x and with a norm x that make h· ·i |·| the basis X1(x),...,Xm1 (x) an orthonormal basis. That is if v = m1 m1 i=1 viXi(x)=(v1,...,vm1 ) and w = i=1 wiXi(x)=(w1,...,wm1 ) m1 2 are in HGx, then v, w x := vjwj and v x := v, v x. P h i j=1 P | | h i The sections of HG are called horizontal sections,avectorofHGx is P an horizontal vector while any vector in T Gx that is not horizontal is 3 a vertical vector. Each horizontal section is identified by its canoni- cal coordinates with respect to this moving frame X1(x),...,Xm1 (x). This way, an horizontal section ' is identified with a ' = n m1 ('1,...,'m1 ):R R . When dealing with two such sections ' and whose argument! is not explicitly written, we drop the index x in the scalar product writing ,' for (x),'(x) x. The same convention is adopted for the norm.h i h i Two important families of automorphism of G are the so called in- trinsic translations and the intrinsic dilations of G.Foranyx G, the 2 (left) translation ⌧x : G G is defined as ! z ⌧ z := x z. 7! x · For any >0, the dilation : G G, is defined as ! ↵1 ↵n (2) (x1, ..., xn)=( x1, ..., xn), where ↵i N is called homogeneity of the variable xi in G (see [83] Chapter 1)2 and is defined as

(3) ↵j = i whenever hi 1 +1 j hi,   hence 1 = ↵ = ... = ↵ <↵ =2 ... ↵ = k. 1 m1 m1+1   n Example 2.1 (Euclidean spaces). The Rn can be meant a (trivial) abelian step 1 Carnot group with respect to its struc- ture of . Namely the group law can be defined by the standard sum p + q, if p, q Rn, and the dilations defined p, if 2 p Rn and >0. The standard base of the Lie algebra E of Rn 2 is given by the left-invariant vector fields Xj = @j, j =1,...,n. All commutator relations in E are trivial. Example 2.2 (). The simplest example of Carnot group is provided by the Heisenberg group Hn = R2n+1. The reader can find an exhaustive introduction to this structure as well as applications n 2n+1 in [202].A point p H is denoted by p =(p1,...,p2n,p2n+1) R . 2 2 For p, q Hn, we define the group operation given by the standard sum p + q2 1 2n p q = p + q ,...,p + q ,p + q + (p q p q ) . · 1 1 2n 2n 2n+1 2n+1 2 i i+n i+n i i=1 ! X and the family of (non isotropic) dilations 2 n (p) := (p1,...,p2n, p2n+1) p H ,>0 . 8 2 The standard base of the Lie algebra, denoted hn,of Hn is given by the left invariant vector fields p X = @ n+j @ ,j=1,...,n j j 2 2n+1 p Y = @ + j @ ,j=1,...,n j n+j 2 2n+1 T = @2n+1, 4 the only non-trivial commutator relations being

[Xj,Yj]=T, j =1,...,n.

Thus the vector fields X1,...,Xn,Y1 ...,Yn satisfy H¨ormander’s rank condition, and Hn is a step 2 Carnot group, the stratification of the Lie algebra of left invariant vector fields being given by hn = V V 1 2 V = span X ,...,X ,Y ,...,Y and V = span T . 1 { 1 n 1 n} 2 { } We collect in the following proposition some more or less elementary properties of the group operation and of the canonical vector fields . Proposition 2.3. The group product has the form (4) x y = x + y + (x, y), x, y Rn · Q 8 2 n n n where =( 1,..., n):R R R and each i is a homoge- Q Q Q ⇥ ! Q neous polynomial of degree ↵i with respect to the intrinsic dilations of G defined in (2), that is

↵i i(x, y)= i(x, y), x, y G. Q Q 8 2 Moreover (i) is antisimmetric, that is Q i(p, q)= i( q, p), for all p, q G. Q Q 2 (ii) x, y G 8 2 (x, y)=... = (x, y)=0, Q1 Qm1 (5) (x, 0) = (0,y)=0 and (x, x)=Q (x, x)=0, for m

j(x, y)= j(x1,...,xhi 1 ,y1,...,yhi 1 ), if 1

Proposition 2.4. The vector fields Xj have polynomial coecients and if h` 1

(8) Xj(x)=@j + qi,j(x)@i, Xi>hl @Qi where qi,j(x)= (x, y) y=0 so that if h` 1

By (1), the rank of the Lie algebra generated by X1,...,Xm1 is n; hence X =(X1,...,Xm1 ) is a system of smooth vector fields satisfying H¨ormander’s condition. Finally, it is useful to think (9) G = G1 G2 Gk , ··· i ni th where G = exp(Vi)=R is the i layer of G and to write p G as 2 (p1,...,pk), with pi Gi. According to this (10) 2 p q = p1+q1,p2+q2+ 2(p, q),...,p+q+ (p, q) , for all p, q G. · Q Q 2 In the following we will mean by a Carnot group G a couple (Rn, ), in exponential coordinates , equipped with a basis of its · Lie algebra X1,...,Xm1 ,Xm1+1,...,Xn satisfying the previous properties. G

2.2. Metrics on Carnot groups. Definition 2.5. Given a Carnot group G =(Rn, ), a d on it is said to be invariant if · (i) d : Rn Rn [0, + ) is continuous with respect to the Eu- clidean⇥ topology;! 1 (ii) d is left-invariant with respect the family of left-translations and 1-homogeneous with respect to the one of dilations, that is (11) d(g p, g q)=d(p, q) ,d( (p), (q)) = d(p, q) · · for all p, q, g G and all >0. 2 Equivalently, a Carnot group G can be endowed by a homogeneous norm , that is a continuous function : G Rn [0, + ) (with respectk· tok the Euclidean topology), suchk· thatk ⌘ ! 1 (12) p =0 p =0; k k () 1 (13) p = p , (p) = p p G, >0; k k k k k k k k82 8 (14) p q p + q p, q G . k · k k k 6k k8 2 From a homogeneous norm , it is possible to induce a distance in G as follows k·k 1 1 (15) d(p, q)=d(q p, 0) = q p , for all p, q G. · k · k 2 Proposition 2.6 (Corollary 5.1.5,[42]). Let G be a Carnot. Then two invariant d and d˜ on G are equivalent each other, that is, there exists C>1 such that for all x, y G, 2 1 C d(x, y) d˜(x, y) Cd(x, y).   Given a Carnot group G and an invariant metric d on it, we will denote by Ud(p, r) and Bd(p, r), respectively, the open and closed ball centered at p and with radius r> 0, that is,

Ud(p, r)= q G : d(p, q)

diam(Bd(p, r)) = 2r p G,r>0 . 8 2 Moreover if µ is a on G, s- homogeneous with respect to the family of dilations for some s> 0, then µ(@B (0,r)) = 0 r> 0 . d 8 Lets us now introduce a couple of relavant invariant metrics on a Carnot group.

Definition 2.8. Let G be a Carnot group and let X1,...,Xm1 be a generating system of vector fields. An absolutely continuous curve : [0,T] G is a sub-unit ( or admissible) curve with respect to ! X1,...,Xm1 if there exist measurable real functions c1(s),...,cm1 (s), s [0,T] such that c2 1 and 2 j j  m P1 ˙ (s)= c (s)X ((s)), for a.e. s [0,T]. j j 2 j=1 X Definition 2.9. If p, q G, we define their Carnot-Carath´eodory (CC) distance as 2 d (p, q) := inf T>0 : there exists a sub-unit curve with (0) = p, (T )=q . c { } By Chow’s Theorem, the of sub-unit curves joining p and q is not empty, furthermore dc is a distance on G that induces the Euclidean topology (see [42, Theorem 19.1.3]and Proposition 2.13 below). We stress explicitly that in general Carnot–Carath´eodory distances are not Euclidean at any scale, and hence not Riemannian. A beautiful proof can be found in [199] (for a more general statement see also [150]). The of a curve : [0, 1] G is by definition ! k 1 Var() = sup dc((ti+1),(ti)). 0 t1<...

n Proposition 2.10. Let G =(R , ) be a Carnot group and let X1,...,Xm1 · be a generating system of vector fields. Then : [0, 1] G is sub-unit i↵it is 1-Lipschitz. ! It is well known that any rectifiable curve in a metric space admits an arc- length parametrization. Note also that the arc-length parametriza- tion makes the curve 1-Lipschitz (see [14, Remark 4,11] ) and hence sub-unit. A continuous rectifiable curve : [0, 1] Rn is said to be a ge- odesic, or a segment, if Var()=d((0)!,(1)). By an arclength reparametrization, a geodesic can always be reparametrized on the interval [0, Var()] in such a way that d((t),(s)) = t s for all s, t [0, Var()] (see [14, Theorem 4.2.1]). | | 2 Theorem 2.11. Let G =(Rn, ) be a Carnot group, endowed with the · CC distance dc. Then for all x, y G there exists a geodesic connecting them. 2

Several distances equivalent to dc have been used in the literature. Later on, we shall use the following one, that can also be computed explicitly 1 d (x, y)=d (y x, 0), 1 1 · where, if p =[p1,...,pk] Rm1 Rmk = Rn, then 2 ⇥···⇥ j 1/j (16) p := d (p, 0) = max "j p mj ,j =1,...,k . k k1 1 { || ||R } Here "1 = 1, and "2,..."k (0, 1] are suitable positive constants de- pending on the group structure2 (see [102, Theorem 5.1]).

Remark 2.12. Notice that for the Heisenberg group G = Hn, one can 1 2 choose "1 = "2 = 1, that is, if p =[p ,p ] = [(p1,...,p2n),p2n+1] 2n+1 1/2 2 R , then p = max (p1,...,p2n) 2n , p2n+1 . k k1 {|| ||R | | } As above, we shall denote U (p, r) and B (p, r) respectively the open and closed balls associated1 with d . 1 1 Both the CC metric dc and the metric d are invariant (see Definition 2.5).In particular they are equivalent by1 Proposition 2.6. 8 Proposition 2.13. Let G =(Rn, ) be a step k Carnot group. Let · n dc and dE denote, respectively, the CC metric on G R and the ⌘ Euclidean one on Rn. n (i) ([42, Corollary 5.15.2]) A (R ,dc) is bounded i↵it is bounded n ⇢ in (R ,dE); (ii) ([42, Corollary 5.15.1]) for each compact K Rn (with respect ⇢ to the Euclidean topology) there exists a positive constant CK such that 1 1/k C d (x, y) d (x, y) C d (x, y) x, y K ; K E  c  K E 8 2 n n (iii) The identity map id :(R ,dc) (R ,dE) is an homeomor- phism. ! Remark 2.14. By Proposition 2.6 , it follows that, given an invariant distance d on G, then the statements (i), (ii) and (iii) of Proposition 2.13 keep still true when replacing dc with d. Finally, it is useful to think G = G1 G2 G, where Gi = ni th ··· 1  exp(gi)=R is the i layer of G and to write p G as (p ,...,p ), 2 with pi Gi. According to this (17) 2 p q = p1+q1,p2+q2+ 2(p, q),...,p+q+ (p, q) , for all p, q G. · Q Q 2 The integer n k

(18) Q = ↵j = i dim Vi j=1 i=1 X X is the homogeneous of G. Proposition 2.15. Let G =(Rn, ) be a Carnot group equipped by an invariant metric d. Thenthe n-dimensional· n, is L the Haar measure of the group G. Indeed if E Rn is measurable, n n ⇢ then (⌧x(E)) = (E) for all x G. Moreover, if >0 then n( L(E)) = Q n(EL). Eventually it2 holds that L L

(19) n(B (p, r)) = n(U (p, r)) = rQ n(B (p, 1)) = rQ n(B (0, 1)). L d L d L d L d Proof. Let us simply denote the balls Bd B and Ud U. Notice n ⌘ ⌘ that, by Proposition 2.13, (Bd(p, r)) < ,foreachp G, r>0. L n1 n 2 n n Let us consider the di↵eomorphisms ⌧x : R R and : R R . Because of the group structure, it is easy to see! that ! Q det J(⌧x)=1, det J()= . where J( ) denotes the Jacobian matrix associated to a regular map : Rn Rn. Thus the first two statements easily follow. Moreover ! n n n Q n (B(p, r)) = (⌧p(B(0,r))) = (r(B(0, 1))) = r (B(0, 1)) . L L L9 L Eventually, by Propopition 2.7, it follows the equality n(B(p, r)) = n(U(p, r)). L L ⇤ 2.3. Hausdor↵measures in a metric space. Applications to Carnot groups. 2.3.1. Hausdor↵measures in metric spaces. Let us recall below three di↵erent notion of Hausdor↵measure s in a metric space. The notions of Hausdor↵and spherical Hausdor↵measures are classical and they are obtained by Carath´edory’s construction as in [80, Section 2.10.2] . Let us introduce some notation and notions. Throughout this paper (X, d) will be a separable metric space, B(x, r) := y X : d(x, y) r { 2  } are the closed ball with centre a and radius r>0. The diameter of a set E X is denoted as ⇢ diam(E) := sup d(x, y): x, y E . { 2 } If µ is an outer measure in X and A X the restriction of µ to A is denoted as ⇢ µ A(E)= µ(A E) if E X. \ ⇢ We will assume the following condition on the diameter of closed balls: there constants 0 <%0 2 and 0 > 0 such that, for all r (0, ) and x X,  2 0 2 (20) diam(B(x, r)) = %0 r. For m> 0, we denote ⇡m/2 ↵m := m ( 2 +1) being the Euler function and m (21) m := ↵m %0 .

Remark 2.16. In the case which m is an integer, the constant ↵m turns out to be equal to the m-dimensional Lebesgue measure of unit closed ball of Rm. The reason of this normalization aims to get equality (42) in the Euclidean case. We begin repeating the well known definitions of Hausdor↵measures mainly to stress their di↵erences with the less known notion of centered spherical Hausdor↵measure.

Definition 2.17. Let A X, m [0, ), (0, ), and let m be the constant (21). ⇢ 2 1 2 1 m m (i) d (A) := lim 0 d,(A), where H ! H m (A) = inf diam(E )m : A E , diam(E ) . Hd, m i ⇢ i i  ( i i ) X 10 [ m We will call d m-dimensional Hausdor↵measure. Hm m (ii) d (A) := lim 0 d,(A), where S ! S m (A) = inf diam(B(x ,r))m : A B(x ,r), Sd, m i i ⇢ i i i i n X [ diam(B(x ,r)) i i  We will call m the m-dimensional spherical Hausdoro ↵measure. (iii) S m m d (A) := sup d,0(F ) . C F A C ✓ m m m where d,0(F ) := lim 0+ d,(F ), and, in turn, d,(F ) = 0 if F = C ! C C ; and for F = 6 ; m (F ) = inf diam(B(x ,r))m : F B(x ,r), Cd, m i i ⇢ i i i i n X [ x F, diam(B(x ,r)) i 2 i i  We will call m the m-dimensional centered Hausdor↵measureo . Cd Remark 2.18. The above classical normalization, by means of constant m in the definition of Hausdor↵measures, mainly depends on the fact that, in the Eucliden case X = Rn we will get the agreement between the n- dimensional Hausdor↵measure and the Lebesegue one (see (42)). The centered Hausdor↵measures m were introduced in [195], when Cd X = Rn and d is the Euclidean measure,to estimate more eciently the Hausdor↵dimension of self-similar fractal sets. A detailed study of di↵erent types of centered Hausdor↵measures and of their properties has been carried out in [78], in the setting of a general metric space. m Notice that the 0 is not necessarily monotone (see [195, Sect. 4]). C m m It is well known that d and d are metric (outer) measures, being obtained by the classicalH Caratheodory’sS construction (see [80, 2.10.1] or [159, Theorem 4. 2]). Recall that an outer measure µ on X is said to be metric if µ(A B)= µ(A)+ µ(B) if dist(A, B) > 0 . [ m Also the measures d are metric measures in any metric space, but this fact, proved inC [78], is not as immediate as for m and m. It holds that (see [78]) H S (22) m m m 2m m . Hd Sd Cd  Hd In particular the three measures m, m and m are equivalent. Hd Sd Cd Remark 2.19. Notice that if d1 and d2 are two equivalent metric on X, then and are equivalent, as well as and , and . Hd1 Hd2 Sd1 Sd2 Cd1 Cd2 11 2.3.2. Hausdor↵(or metric) dimension in metric spaces. Definition 2.20 (Hausdor↵dimension). Let (X, d) be a (separable) metric space and let A X. We define Hausdor↵(or metric) dimen- sion of A the value ✓ Hdim(A) := inf m [0, + ): m(A)= 0 . { 2 1 Hd } m m m Remark 2.21. Since d , d and d are equivalent, the notion of Haus- dor↵dimension canH be equivalentlyS C stated with respect to the measure m m d or d . As well, the notion of metric dimension is stable with respect toS equivalentC metrics on X. Clearly the Hausdor↵dimension has the natural properties of mono- tonicity and stability with respect the countable union: (23) Hdim(A) Hdim(B) if A B X ;  ⇢ ⇢

(24) Hdim( i1=1Ai) = sup Hdim(Ai) if Ai Xi=1, 2,..., . [ i ⇢ To state the definition in other words, Hdim(A) is the unique number (it may in some metric space) for which 1 s< Hdim(A) s(A)= , )H 1 s> Hdim(A) s(A)=0. )H At the borderline case s = Hdim(A) we cannot have any general non- trivial information about the value s(A): all three cases s(A)=0, 0 < s(A) < and s(A)= Hare admissible. ButH if for some givenHA we can1 find s suchH that 0 <1 s(A) < , then s = Hdim(A) Let us now propose a simple criterionH in order1 to calculate the Haus- dor↵dimension in a metric space. Definition 2.22 (Ahlfors regularity). Let (X, d) be a metric space and let µ be a Radon measure on X. We say that the metric measure space (X, d, µ) is Ahlfors regular of dimension Q if (25) a rQ µ(B(x, r)) a rQ x X, r > 0 1   2 8 2 for suitable positive constants ai (i =1, 2).

Theorem 2.23. Let (X, d, µ) be an Ahlfors regular metric measure space of dimension Q, that is (25) holds. Then Hdim(X)= Q. For the proof of this result we will need a basic covering resul in metric space, whose the proof can be found in [159] 12 Theorem 2.24. Let (X, d) be separable metric space and let be an arbitrary family of balls (closed or open) in X such that F sup diam(B): B < . { 2F} 1 Then there exists a countable, disjoint subfamily 0 F ✓F B B B 5B [ 2F ✓[ 2F 0 where 5B denotes the ball with same center as B and radius five times the radius of B. Proof of Theorem 2.23. It suces to prove that, for a given x X 0 2 (26) 0 < Q(B(x ,r)) < r> 0 . Sd 0 18 From (26), it will follow that Hdim(B(x0,r)) = Q for each r> 0. Therefore, since X = h1=1B(x0,h), by (24), we can get the desired conclusion. [ Fix r> 0 and let 0 <

(27) Bi B(x0, 2r) i N . ⇢ 8 2 Applying Theorem 2.24 to the family of balls , we get that there exists F a countable, disjoint subfamily of , Bih :,h=1,...,N , where N can be a finite integer or infinity, suchF { that }

N (28) B(x ,r) 1 B 5B . 0 ⇢[i=1 i ⇢[h=1 ih From the definition of the spherical Hausdor↵,(28),(25) and (27), it follows that (29) N N Q Q Q (B(x0,r)) Q (5%0) r C1 µ(Bi ) Sd  ih  h  i=1 i=1 X X C µ( N B ) C µ(B(x , 2r)) C a 2Q rQ < ,  1 [i=1 ih  1 0  1 2 1 1 Q where C1 = a1 m (5%0) . Thus the right side inequality of (26) fol- lows. To get the left hand side inequality of (26), observe that, for a given covering of balls Bi : i N of B(x0,r), with Bi = B(yi,ri), { 2 } diam(Bi)=%0 ri for each i N, it follows that (30)  2 1 1 0 0 . 1  0  2 Sd, 0 8 Taking the limit as 0 in the previous inequality we get ! (31) 0 0 . 1  0  2 Sd 0 8 ⇤ Remark 2.25. Notice that, from (29) and (31), it follows that if (X, d, µ) is an Ahlfors metric measure space of dimension Q, so does the metric measure space (X, d, Q). Sd Eventually let us estimate the metric dimension of a Lipschitz curve in a metric space, which is the simplest example of 1- dimensional regular submanifold in a metric space. More precisely it holds that Theorem 2.26. Suppose that :[a, b] (X, d) is a Lipschitz curve and let = ([a, b]). Then ! 1() 1() 1() V ar() < + , Hd Sd Cd  1 and equality holds if is injective. Proof. We have only to show that (32) 1() V ar() . Cd  Indeed, from [14, Theorem 4.4.2], it follows that 1() = V ar() if is injective. Hd Thus, from (22) and (32), we get the desired conclusion. Let us prove (32). By a reparametrization of (see [14, Theorem 4.2.1], we can assume that a =0,b = V ar() with metric derivative of ˙ = 1 a.e. on [a, b]. In particular observe that is 1- Lipschitz and | |

(33) V ar(,[t, s]) = s t a t s b. 8    Given >0, choose N N such that V ar()/N < and let h := V ar()/N and J := [ih,2 (i +1)h], i =0,...,N 1. Let B := i i B(pi,h/%0) with pi = ((2i +1)h/2), i =0,...,N 1, where %0 is the constant in (20). Then, by (20), (34) diam(B )= hi=, 0,...,N 1 . i and N 1 (35) B . ✓[i=1 i Indeed, notice that, by (33), for each i =0,...,N 1, 2i +1 2i +1 h h d (t), h t h t J . 2  2  2  % 8 2 i ✓ ✓ ◆◆ 0 14 Thus (Ji) Bi for each i =0,...,N 1, and (35) follows. Therefore, from (35) and⇢ (34), we get that, for each E ✓ N 1 (36) 1 (E) 1 () diam(B )= V ar() >0 . Cd, Cd,  i 8 i=0 X Passing to the limit as 0 in (36), we get that ! 1 (E) V ar() E . Cd,0  8 ✓ Eventually, taking the supremum on all subsets E in the previous ✓ inequality, (32) follows. ⇤ 2.3.3. Applications to Carnot groups. Now we are going to apply Haus- dor↵measures and metric dimension’s issues to the case when X = G is a Carnot group equipped by an invariant distance d.

Let G be a Carnot group equipped by an invariant distance d.Let us firstly observe that (20) is satisfied with %0 = 2 by Proposition 2.7. Moreover, because the invariance of distance d, all three Hausdor↵mea- sures are left-invariant with respect to the family of left-translations ⌧x (x G), and m-homogeneous with respect to the family of 1-paramater dilations2 (> 0), that is, if µ = m,orµ = m,orµ = m, then, Hd Sd Cd for each A G, ⇢ m (37) µ(⌧x(A)) = µ(A),µ((A)) = µ(A) x G,>0 . 8 2 We denote by m the m-dimensional Hausdor↵measure obtained H n m from the Euclidean distance in R G,by c the m-dimensional ' H m Hausdor↵measure obtained from the distance dc in G, and by the H1 m-dimensional Hausdor↵measure obtained from the distance d in G. m m m m m m 1 Analogously, , , c , c , and denote the corresponding spherical andS centeredC S HausdorC ↵S.1 measures.C1 As an immediate consequence of Theorem 2.23 and (19), we get the calculation of the metric dimension of a Carnot group, proven the first time in [163].

Theorem 2.27. Let G =(Rn, ) be a Carnot group endowed with an invariant metric d. Then · Hdim(G)= Q where Q denotes the homogeneous dimension of G. (see (18)) Remark 2.28. Observe that n Q and the equality holds i↵ G is step  1 Carnot group, that is G Rn is isomorphic to the Euclidean space ⌘ Rn. In other case,namely when G is a Carnot group of step k 2, the topological dimension n is strictly less the metric dimensionQ.For instance, in the case of the Heisenberg group Hn, Hdim(Hn)= Q = dim V1 +2V2 =2n +2. 15 Proof of Theorem 2.27. From (19) it follows that the metric measure space (G,d, n) is Ahlfors regular of dimension Q. Thus, from Theorem L 2.23 we get the desired conclusion. ⇤ Q Q We are now going to compare the Hausdor↵measures d , d and Q n H S d on a given Carnot group G =(R , ) of homogeneous dimesion Q, equippedC by an invariant metric d. · We have showed that they are Radon measure on G and, from the invariance by left-translations (see (37), they are Haar measures of the group like the measure n, as well. It is well-known that, due to a gen- eral result in metric spacesL as far as uniformly distributed measures, each of them di↵ers from n only by a constant (see [159, Theorem 3.4]). Assuminig now as refenceL measure the spherical Hausor↵mea- sure Q we now compare it with Q and Q. Sd Hd Cd Theorem 2.29. Let G =(Rn, ) be a Carnot group of homogeneous dimension Q, equipped by an invariant· distance d. (i) ([190, Proposition 2.1]) Q(B)= diam(B)Q for each ball Sd Q B G; ⇢ Q Q (ii) ([190, Proposition 2.3]) d = Cd d where Cd is the isodia- metric constant, that is S H Q d (A) (38) Cd := sup S Q :0< diam(A) < + ; (Q diam(A) 1) Q Q (iii) (B)= Q diam(B) for each ball B G Cd ⇢ Proof. (iii) From (i) and the definition of Q, we get at once Cd Q(B)= diam(B)Q Q(B) diam(B)Q Sd Q Cd  Q for each ball B G. ⇢ ⇤ Corollary 2.30. Under the same assumptions of Theorem 2.29, it holds that Q = Q. Sd Cd Q Q Proof. Since both d and d are doubling measures, we can apply classical di↵erentiationS resultsC for Radon measures (see [80, Section 2.9]). Namely we can assume that Q Q d (B(x, r)) Q (39) D Q d (x) := lim SQ [0, + ) d a.e. x G ; 9 Cd S r 0 (B(x, r)) 2 1 C 2 ! Cd

Q Q Q (40) d (E)= D Q d (x) d d (x)foreachBorelsetE G . S d S C ⇢ ZE C Q By Theorem 2.29 (iii), it follows that D Q d (x)= 1foreachx G. Cd S 2 Therefore, from (39) and (40), it follows that Q Q (41) d = d on Borel sets . S C 16 Through a standard argument, by means of Borel regularity of Q and Sd Q and since they are equivalent measures, we can extend the equality Cd in (41) to all subsets of G. ⇤

The constant Cd in (38) is related to the so-called isodiametric prob- lem. Let us recall that the classical isodiametric problem in the Eu- clidean space Rn says that a ball maximize the volume among all sets with prescribed diameter, n(A) diam(A)n L  n where n is the constant defined in (21) (that is the volume of unit ball n of R ) with %0 = 2. This gives non trivial information about the ge- ometry of the Euclidean space and about the relation between the Eu- clidean metric and the Lebesgue measure. In particular a well-known consequence of the isodiametric inequality is the following agreement between the n-dimensional Lebesgue measure and the n-dimensional Hausdor↵measure defined with respect to the Euclidean distance (42) n = n . L H More generally when working on analysis on metric spaces it is a natural question to ask in what kind of settings one can have some, and what, type of isodiametric inequalities and what kind of properties can be deduced from this information. In the setting of the Carnot group the isodiametric problem has been studied in [190] and [141]. More precisely in [190] it was studied the question to know whether the sharp isodiametric inequality holds, or equivalently if balls realize the supremum in the right-hand side of (38) in Carnot groups equipped with some specific invariant distances. Observe that in this case Cd = 1 by Theorem 2.29 (i). This amounts to say that the following sharp isodiametric inequality holds. Q Q (SII) (A) Q diam(A) A G , Sd  8 ⇢ or, equivalently, from Threorem 2.29 (ii) Q = Q . Sd Hd In particular it was proven that given a non abelian Carnot group G one can always find some invariant distance on G, namely the d distance as well as some CC distances, for which the sharp isodiametric1 inequality (SII) does not hold (see [190, Theorems 3.4, 3.5 and 3.6]). Moreover an interesting link with some topic of geometric measure theory in Carnot group was established. Indeed as a consequence of those results, the purely Q-unrectifiability of Carnot groups can be obtained (see [190, Theorem 4.1] and Definition 4.82 below), whose we will discuss later. Note that it is not dicult to see that one can always find sets achiev- ing the supremum in (38), called isodiametric sets (see [190, Theorem 17 3.1]). In [141], for the case of the Heisenberg group Hn equipped by its CC distance dc, the regularity of the isodiametric sets have been stud- ied, as well as their characterization under some symmetry assumption. The characterization of a general isodiametric set in Heisenebrg group is still open. Eventually let us point out the paper [2] about the study of volume measure in sub-Riemannian manifold by means of Hausdor↵measures and [44] concerning the spherical Hausdor↵measure and the Popp mea- sure in sub-Riemannian manifold.

2.4. Carnot groups as tangent spaces to sub-Riemannian man- ifolds. An important feature of Carnot groups is that they appear as ”tangent spaces” to sub-Riemannian manifold in an appropriate sense. Tangent spaces of a sub-Riemannian manifold are themselves sub-Riemannian manifolds. They can be defined as metric spaces, us- ing Gromovs definition of tangent spaces to a metric space, and they turn out to be sub-Riemannian manifolds. Moreover, they come with an algebraic structure: nilpotent Lie groups with dilations. In the clas- sical, Riemannian case, they are indeed vector spaces, that is, Abelian groups with dilations. A general account of the many facets and con- tributions in this direction is far beyond the aim of this notes and we refer to [163, 33, 1, 169, 135] We only point out that the understanding of many of Riemannian geometric properties come from the fact that the metric tangents of a Riemannian manifold are Euclidean spaces, and the Euclidean geome- try is quite well studied. Thus a good understanding of the geometry of Carnot groups could provide a better knowledge of the geometry of sub-Riemannian structures.

3. Differential Calculus on Carnot Groups This section is taken from [102] and [143, Chapter 3]. We refer to [151] for a generalization of the content of this section.

3.1. Pansu di↵erentiable maps between Carnot groups and some their characterizations. The following definitions and results about intrinsic di↵erentiability in Carnot groups are basically due to P. Pansu ([179]), or are inspired by his ideas. The notion of P-di↵erentiability for functions acting between Carnot groups was introduced by Pansu in [179].

Definition 3.1. Let G1, G2 be Carnot groups, with homogeneous norms 1 2 1, 2 and dilations , . We say that L : G1 G2 is H-linear, ork·k is akhomogeneous·k homomorphism, if L is a group! homomorphism such that 1 2 L(g)=L(g), for all g G1 and >0. 18 2 Let us recall the following characterization of a H- linear map be- tween Carnot groups (see [143, Proposition 3.1.3 and Theorem 3.1.12]).

Theorem 3.2. Let (Gi, i) (i =1, 2) be two Carnot groups. Denote k·k by gi (i =1, 2) their Lie algebras and by g = V ... V ,,g = W ... W 1 1 k 2 1 l G1 = V1 V2 Vk, G2 = W1 W2 Wl ··· ··· the stratifications defined according to , respectively, (1) and (9). Let L : G1 G2 be a homomorphism. Then the following are equivalent: ! (i) L is H-linear; (ii) L :(G1, 1) (G2, 2) is Lipschitz and it satisfies the k·k ! k·k contact property L(Vj) Wj j =1,...,k; where, if l< kwe ✓ mean that L(Vj)= 0 for l< j k. { }  Moreover, if L is H-linear, L can be read between the Lie algebras as 1 L˜ =exp L exp : g1 g2 and L˜ is an algebra homomorphism, that is it is a linear map such! that L˜([X, Y ]) = [L˜(X), L˜(Y )] for each X, Y g . 2 1

We say that f : G1 G2 is P-di↵erentiable in g0 if there A⇢ ! 2A is a H-linear function dP fg0 : G1 G2 such that ! 1 1 1 1 1 d f (g /g) f(g ) f(g) = o g g , as g g 0 k P g0 0 · · 0 · k2 k 0 · k1 k 0 · k1 ! where o(t)/t 0ast 0+. The H-linear function df is called the ! ! g0 P-di↵erential of f in g0. The fundamental result where P-di↵erentiability applies is the so called Pansu-R¨ademacher theorem for Lipschitz function between Carnot groups . n m Theorem 3.3 (Pansu [179]). Let G1 =(R , ) and G2 =(R ,?) be two Carnot groups equipped with two CC metrics· denoted, respectively, d1 and d2. Let f : (G1,d1) (G2,d2) be a Lipschitz continuos function with openA set.⇢ Then f!is P- di↵erentiable at p for n-a.e. p . A L 2A Remark 3.4. An extension of Pansu-R¨ademecher theorem for a Lips- chitz map f : (G1,d1) (G2,d2) when is supposed to be only measurable wasA proved⇢ in [143]! and [203]. A

Given two Carnot groups (Gi, i), i =1, 2, where i is a homoge- k·k k·k neous norm on Gi, denote by H (G1, G2) the set of H-linear functions L L : G1 G2 endowed with the norm ! L H ( 1, 2) := sup L(p) 2 : p G1, p 1 1 . k kL G G {k k 2 k k  } When G1 G and G2 R, we simply denote = H (G, R) and ⌘ ⌘ LG L we will call a map L G simply G-linear. 2L 19 1 If ⌦is an in G1, we denote as CH (⌦, G2) the set of con- tinuous functions f :⌦ G2 such that dP f :⌦ H (G1, G2) is continuous. ! !L In the following we will deal with maps f :⌦ G R, that is ⇢ !1 G1 = G and G2 = R. In this case we simply denote CH (⌦, G2)as 1 C (⌦) and the Pansu di↵erential as d := dP f : G R. G G ! A map L : G R is G-linear if it is a homomorphism from G ! ⌘ (Rn, )to(R, +) and if it is positively homogeneous of degree 1 with · respect to the dilations of G, that is L(x)=Lx for >0 and x G. The R-linear set of G-linear functionals G R Given a basis 2 ! X1,...,Xn, all G-linear maps are represented as the follows. Proposition 3.5. A map L : G R is G-linear if and only if there m1 ! is a =(a1,...,am1 ) R such that, if x =(x1,...,xn) G, then m1 2 2 L(x)= i=1 aixi. Notice that, for a function f :⌦ G R, P-di↵erentiability at P ⇢ ! x0 ⌦simply means that there is a G-linear map L : G R such that2 ! 1 f(x) f(x ) L(x x) lim 0 0 · =0. x x0 ! d(x, x0) (see also [132]) Remark 3.6. The above definition is equivalent to the following one: there exists a homomorphism L from G to (R, +) such that f(⌧ ( v)) f(x ) lim x0 0 = L(v) 0+ ! uniformly with respect to v belonging to compact sets in G. In par- ticular, L is unique and we shall write L = dGf(x0). Notice that this definition of di↵erential depends only on G and not on the particu- lar choice of the canonical generating vector fields. Indeed any two Carnot–Carath´eodory distances induced by di↵erent choices of (equiv- alent) scalar products in HG are equivalent as distances. Definition 3.7. If⌦is an open set in , we shall denote by C1 (⌦,H ) G G G 1 the set of all sections ' of HG whose canonical coordinates 'j C (⌦) 2 G for j =1,...,m1. Remark 3.8. We recall that C1(⌦) C1 (⌦) and that the inclusion G may be strict, for an example see [99,⇢ Remark 5.9]. We point out that a function f C1 (⌦) may be very irregular from Euclidean point of G view (see [131])2

We say that f is di↵erentiable along Xj, j =1,...,m1,atx0 if the map f(⌧ ( e )) is di↵erentiable at = 0, where e is the j-th 7! x0 j j vector of the canonical basis of Rn.

Once a generating family of vector fields X1,...,Xm1 is fixed, we define, for any function f : G R for which the partial derivatives Xjf ! 20 exist, the horizontal of f, denoted by Gf, as the horizontal section r m1 f := (X f)X . rG i i i=1 X whose coordinates are (X1f, ..., Xm1 f). Moreover, if ' =('1,...,'m1 ) 1 is an horizontal section such that Xj'j L (G)forj =1,...,m1,we 2 loc define divG ' as the real valued function m1 m1 div (') := X⇤' = X ' G j j j j j=1 j=1 X X (see also Section 3.5). Remark 3.9. The notation we have used for the gradient in a group is partially imprecise, indeed f really depends on the choice of the rG basis X1, ..., Xm1 . If we choose a di↵erent base, say Y1, ..., Ym1 , then in general i(Xif)Xi = i(Yif)Yi. Only if the two bases are one orthonormal with respect6 to the scalar product induced by the other, we have thatP P

(Xif)Xi = (Yif)Yi. i i X X On the contrary, the notation divG used for the divergence is correct. Indeed divG is an intrinsic notion and it can be computed using the previous formula for any fixed generating family.

n Finally, if x =(x1,...,xn) R G and x0 G are given, we set 2 ⌘ 2 m1

(43) ⇡x0 (x)= xjXj(x0). j=1 X

The map x0 ⇡x0 (x) is a smooth section of HG. ! Proposition 3.10. If f :⌦ G R is Pansu-di↵erentiable at x0, ⇢ ! then it is di↵erentiable along Xj at x0 for j =1,...,m1, and (44) d f(x )(v)= f,⇡ (v) . G 0 hrG x0 ix0 For a proof see [174] Remark 3.3. The following proposition can be proved via an approximation argument as in Proposition 5.8 of [99]. Proposition 3.11. A continuous function f :⌦ R belongs to C1 (⌦) if and only if its distributional derivatives X f!are continuous G j in ⌦ for j =1,...,m1. 1 A non trivial characterization for mappings f CH (⌦, G2), similar to the one of Proposition 3.11, has been recently2 carried out in [151, Theorem 1.1]. In particular, as byproduct of this characterization we get that each continuously P-di↵erentiable function between Carnot is locally Lipschitz. 21 Theorem 3.12 (Corallary 5.8,[151]). Given two Carnot groups (Gi, i), k·k i =1, 2, where i is a homogeneous norm on Gi, let ⌦ G1 be an k·k 1 ⇢ open set and let f CH (⌦, G2). Then f :⌦ (G1, 1) G2, 2) is locally Lipschitz.2 ⇢ k·k ! k·k 3.2. Area formula between Carnot groups. The content of this section relies on [143, Chapter 4] (see also [7], [182] and [208]). An important tool in Euclidean geometric measure theory is the so- called area formula for Lipschitz maps, which turns to be true also for Lipschitz maps between Carnot groups.

Definition 3.13 (H-Jacobian). Given two Carnot groups (Gi,di), i = 1, 2, endowed with an invariant distance di, let L : G1 G2 be a ! H- linear map. Let Q denote the homogeneous dimension of G1.The jacobian of L is defined by Q d2 (L(B1)) JQ(L) := H Q (B1) Hd1 where B1 denotes the unit open ball in (G1,d1). Theorem 3.14 (Area formula, Theorem 4.3.4 [143]). Let f : A⇢ (G1,d1) (G2,d2) be a Lipschitz continuos function with measur- able set.! Then A

Q Q JQ(dP f(x))d (x)= N(f, ,y)d (y) Hd1 A Hd2 ZA ZM where N(f, ,y) denotes the multiplicity function of f, that is the A 1 number of elements of set f (y) . \A Area formula turns out to be an useful tool in order to study low- dimensional surfaces in Carnot groups (see, for instance, section 4.5 below) as well as for proving some results of unrectfiability for Carnot groups (see section 4.9 below). 3.3. Whitney theorem for maps between Carnot groups. It is well-known that an useful tool for the approximation of a function in Euclidean setting turns out to be Whitney’s extension theorem. In the setting of Carnot group we mean the possibility to extend a function ˜ f : F G1 G2 to a continuously P-di↵eriantiable map f : G1 G2, provided⇢ that!F is a closed set and a suitable notion of di↵erentiability! k on F . There is a positive answer if G2 = R . Indeed it holds Theorem 3.15. [Whitney Extention Theorem (Theorem 5.2,[102])] Let F G be a closed set, and let f : F R, k : F HG be, respectively,⇢ a continuous real function and! a continuous! horizontal section. We set 1 f(x) f(y) k(y),⇡ (y x) R(x, y) := h y · iy , d(y, x) 22 and, if K F is a compact set, ⇢ % () := sup R(x, y) : x, y K, 0

3.5. Vector fields. Consider a family X of vector fields X =(X , ..., X ) 1 m 2 Lip (Rn; Rn)m. Since we are dealing with local properties, for sake of n simplicity, we assume X1, ..., Xm are bounded in R . This assump- tion gives a simpler form to some statements below. Later on, when the vector fields will be associated with a Carnot group structure, we shall drop the boundedness assumption. This will not yield contradic- tion or lack of coherence, since the local estimates we are dealing with for groups are easily extended to the whole space by translations and dilations. As usual we shall identify vector fields and di↵erential operators. If n j Xj(x)= ci (x)@i,j=1, ..., m, i=1 X we define the m n matrix ⇥ j (45) C(x)=[ci (x)] i=1,...,n . j=1,...,m

In the case of a Carnot group G, the vector fields X1,...,Xm will agree with a system of generating vector fields X1,...,Xm1 of the group. 2 n We shall denote by Xj⇤ the operator formally adjoint to Xj in L (R ), n that is the operator which for all ', C01(R ) satisfies 2

'(x)Xj (x) dx = (x)Xj⇤'(x) dx. n n ZR ZR 1 1 m Moreover, if f Lloc is a scalar function and ' (Lloc) is a m- vector valued function,2 we define the X-gradient and2 X-divergence as the following distributions: m Xf := (X f, ..., X f), div (') := X⇤' . 1 m X j j j=1 X 23 Definition 3.16. Let ⌦be an open subset of Rn. One can define the W 1,p(⌦), 1 p , associated with the family X as X  1 the space of all the functions with finite norm u 1,p = u p + Xu p, WX 2 2 k k k k k k where Xu = Xju and the derivatives Xju are understood in the sense of| distributions.| | | The Lp-norms should be meant with respect to Lebesgue measure.P Throughout this paper, if E Rn,both E and n(E) denote its Lebesgue measure. Analogously,⇢ if µ is a measure| | in aL set X, we write µ(E)or E for the µ-measure of the set E X. | |µ ⇢ 3.6. Sobolev spaces associated with vector fields. 1,p Proposition 3.17. Endowed with its natural norm, WX (⌦), 1 p 1,2   , is a Banach space, reflexive if 1 1 !and C>0, both independent of U and f, such that f(y) (47) f(x) f C |rG | dy | U | d(x, y)Q 1 Z⌧U for x U, where ⌧U is the ball concentric with U of radius ⌧r(U). 2 We can apply now the following Hardy-Littlewood-Sobolev-type in- equality in Carnot groups ([82]) that can be stated as follows.

Theorem 3.22. Suppose 0 <↵ 0 such that q f Lp x G : I↵(x) > Cp k k for >0; |{ 2 }|  ✓ ◆ iii) if p>1, then there exists Cp > 0 such that

I f q C f p . k ↵ kL  pk kL Combining Theorems 3.21 and 3.22, we obtain pQ Proposition 3.23. Let p be fixed, 1 p1, both independent of f and U, such that i) if p>1, then 1/q 1/p (48) f(x) f q dx cr(U) f(x) p dx ; | U |  |rG | ✓ZU ◆ ✓Z⌧U ◆ ii) if p =1, then Q/(Q 1) 1 Gf L (49) x G : f(x) fU (x) > Cp kr k , |{ 2 | | }|  ✓ ◆ for >0. Inequality (48) is “almost” the Sobolev-Poincar´einequality in the group for p>1 (the non-geometric case). We say “almost” because of the presence of the dilation factor ⌧>1. But we shall get rid of ⌧ pretty soon. On the other hand, (49) is not a Sobolev-Poincar´e- type inequality in G because of the presence of the weak type norm on the 25 left. Since p = 1 is a endpoint for the weak type estimates of Theo- rem 3.22, ii), we cannot rely on Marcinckiewicz interpolation theorem to pass from a weak type estimate to the strong type estimate we are looking for. However, we can still derive from (49) a strong type in- equality, thanks to the presence on the right of the group gradient that is a local operator, and therefore enjoys a certain “stability” property under truncations. This idea was originally introduced in [139] and exploited in [198], [86], [90] and [17]. Thus, Theorem 3.23 yields the following strong type analogous.

pQ Proposition 3.24. Let p be fixed, 1 p1, both independent of f and U, such that 1/q 1/p (50) f(x) f q dx cr(U) f(x) p dx . | U |  |rG | ✓ZU ◆ ✓Z⌧U ◆ Finally, we can derive from Proposition 3.24 the geometric Sobolev- Poincar´einequality getting rid of the constant ⌧. Theorem 3.25 (Geometric Sobolev-Poincar´einequality). Let p be fixed, pQ 1 p

1/q 1/p (51) f(x) f q dx cr(U) f(x) p dx . | U |  |rG | ✓ZU ◆ ✓ZU ◆ The proof of Theorem 3.25 can be carried out as follows. We say that an open set ⌦in a quasimetric space (X, d) satisfies the Boman chain condition (⌧,M),⌧ 1,M 1, if there exists a covering W of ⌦consisting ofF open ballsU such that:

(i) U W ⌧U(x) M⌦(x) x X; 2  8 2 (ii) There is a “central” ball U1 W which can be connected to everyP ball U W by a finite chain2 of balls U ,U , ..., U = U 2 1 2 r(U) of W so that U MUj for j =1, ..., r(U). Moreover, Uj Uj 1 ⇢ \ contains a ball Rj such that Uj Uj 1 MRj for j =2, ..., r(U). [ ⇢ Theorem 3.25 is a corollary of Proposition 3.24 keewping in mind the following two results ([87]).

Theorem 3.26. Let ⌧,M 1, 1 p q< and ⌦ satisfy the Boman chain condition (⌧,M) in a quasimetric 1 space (X, d). Also, let µ and ⌫ be Borel measuresF and µ be doubling. Suppose that f and g 26 are measurable functions on ⌦ and for each ball U with ⌧U ⌦ there ⇢ exists a constant gU such that

g gU Lq (U) A f Lp (⌧U) || || dµ  || || d⌫ with A independent of U. Then there is a constant g⌦ such that

g g⌦ Lq (⌦) cA f Lp (⌦), || || dµ  || || d⌫ where c depends only on ⌧,M,q and µ. Moreover, we may choose g⌦ = gU1 where U1 is a central ball for ⌦. Theorem 3.27. Carnot-Carath´edory balls satisfy the Boman chain condition. 3.8. BV -functions and sets of finite perimeter on Carnot groups. Since with any Carnot group we can associate a H¨ormander’s family of smooth vector fields, then all our previous definitions and results still hold in this setting. In particular, within a Carnot group, we can define BV spaces in a form equivalent to that of the previous section as follows. If ⌦ Rn is open, the space of compactly supported smooth sections ✓ k of HG is denoted by C01(⌦,HG). If k N, C0(⌦,HG) is defined analogously. 2 1 The space BVG(⌦) is the set of functions f L (⌦) such that (52) 2 1 f (⌦) := sup f(x)div '(x) dx : ' C0(⌦,HG), '(x) x 1 < . || r G || G 2 | |  1 ⇢Z⌦ The space BVG,loc(⌦) is the set of functions belonging to BVG( )for each open set ⌦. U Notice the useU of⇢⇢ the intrinsic fiber norm inside the previous definition.

It is easy to see that f BVG(⌦) if and only if f BVX (⌦), where X is a family of vector fields2 that generate the horizontal2 layer. In the setting of Carnot groups, the structure theorem for BV - functions reads as follows (see [96], [47]).

Theorem 3.28. (Structure of BVG functions) If f BVG,loc(⌦) then f is a Radon measure on ⌦. Moreover, there exists2 a f - || r G || || r G || measurable horizontal section f :⌦ HG such that f (x) x =1for f -a.e. x ⌦, and ! | | || r G || 2 f(x)div '(x) dx = ', d f , G h f i || r G || Z⌦ Z⌦ 1 for all ' C0(⌦,HG). Finally the notion of gradient can be 2 rG extended from regular functions to functions f BVG defining Gf as the vector valued measure 2 r f := f =( ( ) f ,..., ( ) f ) , rG f || r G || f 1 || r G || f m1 || r G || where (f )j are the components of f with respect to the moving base Xj. 27 It is well known that the usefulness of these definitions for the Cal- culus of Variations, relies mainly in the validity of the two following theorems. In the context of subriemannian geometries they are proved respectively in [111] and [96].

Theorem 3.29. (Compactness) BVG,loc(G) is compactly embedded p Q in Lloc(G) for 1 p< Q 1 where Q, defined in (18), is the homoge-  neous dimension of G. 1 Theorem 3.30. (Lower semicontinuity) Let f,fk L (⌦), k N, be such that f f in L1(⌦); then 2 2 k ! lim inf fk (⌦) f (⌦). k G G !1 || r || ||r ||

Remark 3.31. Fine di↵erentiability properties for BVG functions have been studied in [10]. Definition 3.32. A measurable set E Rn is of locally finite G- ⇢ perimeter in ⌦(or is a G-Caccioppoli set) if the characteristic function 1 BV (⌦). In this case we call perimeter of E the measure E 2 G,loc (53) @E := 1 | |G || r G E|| and we call (generalized inward) G- to @E in ⌦the vector (54) ⌫ (x) := (x). E 1E Remark 3.33. This remark is analogous to remark 3.9. The symbol @E is somehow incorrect, indeed the value of the G-perimeter de- | |G pends on the choice of the generating vector fields X1,...,Xm1 , pre- cisely through the bound ' 1 in (52). The values of the perimeters induced by two di↵erent| families| of generating vector fields coincide only if the two families are mutually orthonormal; nevertheless the perimeters induced by di↵erent families are equivalent as measures and, as a consequence, the notion of being a G-Caccioppoli set is an intrinsic one depending only on the group G. Proposition 3.34. If E is a G-Caccioppoli set with C1 , then the G-perimeter has the following representation

n 1 @E G(⌦) = C(x)n(x) d , | | @E ⌦ | | H Z \ here n(x) is the Euclidean unit outward normal to E, C(x) is the matrix in (45) and s is the Euclidean s-dimensional Hausdor↵measure. H Remark 3.35. The G-perimeter is invariant under group translations, that is

@E (A)= @(⌧pE) (⌧pA), p G, and for any A G; | |G | |G 8 2 ⇢ indeed divG is invariant under group translations and the Jacobian determinant of ⌧p : G G equals 1. Moreover the G-perimeter is ! 28 homogeneous of degree Q 1 with respect to the dilations of the group, that is 1 Q (55) @(E) (A)= @E (A), for any Borel set A G; | |G | |G ⇢ also this fact is elementary and can be proved by changing variables in formula (52). Fundamental estimates in geometric measure theory are the so-called relative and global isoperimetric inequalities for Caccioppoli sets. They were obtained by Garofalo and Nhieu (see also [86], [90] and the monog- raphy [49]). Theorem 3.36 (Isoperimetric inequalities, Theorem 1.18,[111]). There exists a positive constant CI such that for any G- Caccippoli set E G ⇢ , for any x G, r> 0 2

(Q 1)/Q (56) min E U (x, r) , U (x, r) E C @E (U (x, r)). {| \ c | | c \ |}  I | |G c and (Q 1)/Q (57) min E , G E CI @E (G) {| | | \ |}  | |G where Uc denotes the (open) ball with respect to the CC distance dc. Isoperimetric sets have been studied in [140].

4. Differential Calculus within Carnot Groups The notion of rectifiable set is a central one in calculus of variations and in geometric measure theory. To develop a theory of rectifiable sets inside Carnot groups, and more general in metric spaces, has been the object of much research in the last 20 years (see, for instance,[74, 75, 129, 98, 99, 7, 8, 100, 102, 182, 103, 105, 146, 147, 9, 161, 114]). Rectifiable sets, in Euclidean spaces, are generalizations of C1 or of Lipschitz submanifolds. Hence, to understand the objects that, inside Carnot groups, naturally take the role of C1 or of Lipschitz submani- folds seems to be preliminary in order to develop a satisfactory theory of intrinsically rectifiable sets. In Euclidean spaces, submanifolds are locally graphs. On the other side, we stress that Carnot groups in general cannot be viewed as cartesian products of subgroups (unlike Euclidean spaces). There- fore we need a notion of intrinsic graph fitting the structure of the group G. Here we address this problem considering functions acting between complementary subgroups of a (Carnot) group G (Definition 4.2) and introducing, for these functions, the notions of intrinsic Lips- chitz continuity or of intrinsic di↵erentiability. Then, Lipschitz or C1 submanifolds will be objects that, locally, are intrinsic graphs of these functions. 29 4.1. Complementary subgroups and graphs. The content of this subsection is taken from [103], [15], [105],[95]. 4.1.1. Homogeneous subgroups. A homogeneous subgroup of a Carnot group G (see [202, 5.2.4]) is a Lie subgroup H such that g H, for all 2 g H and for all >0. Homogeneous subgroups are linear subspaces 2 of G, when G is identified with Rn with exponential coordinates. The topological dimension of a (sub)group is the dimension of its Lie algebra. The metric dimension of a subset is its Hausdor↵dimension, k with respect to the Hausdor↵measures d . The metric dimension of a homogeneous subgroup is an integer usuallyS larger than its topological dimension (see [163]).

Definition 4.1. M is a (dt,dm)-subgroup of G if M is a homogeneous subgroup of G with linear or topological dimension dt and metric di- mension d d . m t 4.1.2. Complementary subgroups. From now on G will always be a ho- mogeneous stratified group, identified with Rn with exponential coor- dinates. Definition 4.2. Let M, H be homogeneous subgroups of G.Wesay that M, H are complementary subgroups in G, if M H = e and if \ { } G = M H, · that is for each g G, there are m M and h H such that g = m h. 2 2 2 · If M, H are complementary subgroups of G and one of them is a normal subgroup then G is said to be the semi-direct product of M and H.IfbothM and H are normal subgroups then G is said to be the direct product of M and H and in this case we will also write that G = M H. By elementary⇥ facts in group theory (see e.g. [124, Lemma 2.8]) if M, H are complementary subgroups in G, so that G = M H, then it is also true that · G = H M, · that is, each g G can be written – in a unique way – as g = h¯m¯ , 2 withm ¯ M, h¯ H. Rephrased di↵erently, if M, H are complementary 2 2 subgroups in G, also H, M are complementary subgroups in G. Example 4.3. ([105, Proposition 4.1])Let G be the Heisenberg group Hn. All homogeneous subgroups of Hn are either horizontal, that is n contained in the horizontal fiber HHe , or vertical, that is containing the subgroup T. A horizontal subgroup has linear dimension and metric dimension k, with 1 k n and it is algebraically isomorphic and   isometric to Rk. A vertical subgroup can have any dimension d, with 1 d 2n + 1, its metric dimension is d + 1 and it is a normal   subgroup. All couples V, W, of complementary subgroups of Hn are of the type 30 (i) V horizontal of dimension k ,1 k n,   (ii) W vertical and normal of dimension 2n +1 k. Example 4.4. Splittings similar to the ones of Heisenberg groups exist in a general Carnot group G =(Rn, ). Indeed, choose any horizontal · homogeneous subgroup H = H1 G1 and a subgroup M = M1 ⇢ ··· M such that: H M1 = G1, and Gj = Mj for all 2 j k. Then M   and H are complementary subgroups in G and the product G = M H · is semidirect because M is a normal subgroup.

In particular, if X1,...,Xm1 ,Xm1+1,...,Xn denote a basis of Lie al- gebra , andX ,...,X denote the system of group generating vector G 1 m1 fields, chosen according to section 2.1 and let H := exp(sXj0 ):s R , { 2 } for a given 1 j0 m1. For instance choose j0 = 1. Then H is (1, 1)-   homogeneous subgroup of G, that is, according to Definition 4.1, a homogeneous subgroup with both topological an metric dimension 1. Let := span X ,...,X ,...,X and M { 2 m1 n} n 1 M := exp( )= (0,x2,...,xn): xi R,i=2,...,n R . M { 2 }⌘ Notice that is a subalgebra of of dimension n 1, since [ , ] M G M M ⇢ and also an ideal of , that is [ , ] . Thus M turns out to M G G M ⇢M be a normal subgroup of G of homogeneous dimension Q 1. Indeed n 1 it is easy to see that M = R can be considered as a Lie groups n 1 with Haar measure , satisfying (19) with Q Q 1 and closed L ⌘ n 1 with respect to the family of dilations of G. Therefore (M,d, ) is a measure metric Ahlfors regular space of dimension Q 1: asL a byprod- uct, its metric dimensionn is Q 1. In conclusion we get the splitting G = M H, with M (n 1,Q 1)-subgroup and H (1, 1)-subgroup. · 4 Example 4.5. The Engels group is E =(R , ,), were the group law is defined as ·

x1 + y1 x1 y1 x2 + y2 x2 y2 0 1 0 1 0 1 = x3 + y3 +(x1y2 x2y1)/2 x3 · y3 Bx4 + y4 + [(x1y3 x3y1)+(x2y3 x3y2)]/2C Bx4C By4C B C B C B C B +(x1 y1 + x2 y2)(x1y2 x2y1)/12 C @ A @ A B C and the family of dilation@ is A 2 3 (x1,x2,x3,x4)=(x1,x2, x3, x4).

A basis of left invariant vector fields is X1,X2,X3,X4 defined as X (x) := @ (x /2) @ + x /2 (x x + x2)/12 @ 1 x1 2 x3 3 1 2 2 x4 X (x) := @ +(x /2) @ + x /2+(x2 + x x )/12 @ 2 x2 1 x3 3 1 1 2 x4 X (x) := @ ((x + x )/2) @ 3 x3 1 2 x4 X4(x) := @x4 .

The commutation relations are [X1,X2]=X3, [X1,X3]=[X2,X3]= X4 and all the others commutators are zero. 31 Inside Engels group there are two families of complementary sub- groups. The first one is formed by 1-dimensional horizontal subgroups and 3-dimensional subgroups containing all the vertical directions. This family gives a semidirect splitting of E. The second family is formed by 2-dimensional subgroups that are not normal subgroups. All the computations can be easily done directly, better using a symbolic com- putation program. The homogeneous subgroups

M, := (t,t,0, 0) : t R , N, := (t,t,x3,x4):t, x3,x4 R , { 2 } { 2 } are complementary subgroups in E, provided that =0.More- 6 over N, is a normal subgroup, hence E is the semidirect product of M, and N,. The second family, for + = 0, is given by 6 K := (x1, x1,x3, 0) : x1,x3 R H, := (t,t,0,x4):t, x4 R . { 2 } { 2 } One can compute directly that K and H, are complementary sub- groups in E and that neither K nor H, are normal subgroups. Hence E = K H,, but the product is not a semidirect product. · 4.1.3. Components along complementary subgroups. Given M, H, com- plementary subgroups of G, the elements m M and h H such 2 2 that g = mh are unique because of M H = e and are denoted as \ { } components of g along M and H or as projections of g on M and H. Proposition 4.6. If M, H are complementary subgroups in G there is c0 = c0(M, H) > 0 such that for all g = mh

(58) c0 ( m + h ) g m + h . k k1 k k1 k k1 k k1 k k1 Proof. The right hand side is triangular inequality. For the left hand side observe that M @B(0, 1) and H @B(0, 1) are disjoint compact sets, hence have positive\ distance. The\ general statement follows by dilation. ⇤ Proposition 4.7. ([95, Corollary 2.2.13]) Let M, H be complementary subgroups of a step  group G.Ifg G let m, m¯ M and h, h¯ H be the unique elements such that 2 2 2 g = mh = h¯m.¯ Then,

1  1 1  1 1 C m m¯ + m¯  h¯  + h¯  m¯  , k k1  c0 k k1 c0 k k1 k k1 k k1 k k1 ⇣ 1  1 1  1 ⌘ 1 C h h¯ + m¯  h¯  + h¯  m¯  . k k1  c0 k k1 c0 k k1 k k1 k k1 k k1 ⇣ ⌘ Consequently, for all >0 there is c()=c(, M, H) > 0 such that for all g with g k k1  (59) m c() m¯ 1/, h c() h¯ 1/. k k1  k k1 k k1  k k1 32 From now on, we will keep the convention introduced in the proof of Corollary 4.7 and we will denote as gM and gH the components of g G. More precisely our convention is as follows, when M, H are 2 complementary subgroups in G, M will always be the first ‘factor’ and H the second one and gM M and gH H are the unique elements such that 2 2

(60) g = gMgH.

We stress that this notation is ambiguous because each component gM and gH depends on both the complementary subgroups M and H and also on the order under which they are taken. The projection maps P : G M and P : G H are defined as M ! H ! (61) PM(g) := gM, PH(g) := gH Proposition 4.8. ([95, Proposition 2.2.16]) Let M, H be complemen- tary subgroups of G, then the projection maps P : G M and M ! P : G H defined in (61) are polynomial maps. More precisely, H ! if  is the step of G, there are 2 matrices A1,...,A,B1,...,B, depending on M and H, such that (i) Aj and Bjare (n ,n )-matrices, for all 1 j , j j   and, with the notations of (4), (ii) 1 1 2 2 2 1 1 1 1    1 1  1  1 P g = A g ,A (g (A g ,B g )),...,A (g (A g ,...,B g )) ; M Q Q (iii) 1 1 2 2 2 1 1 1 1    1 1  1  1 P g = B g ,B (g (A g ,B g )),...,B (g (A g ,...,B g )) ; H Q Q (iv) Aj is the identity on Mj, and Bj is the identity on Hj, for 1 j .   Let us stress that PM and PH are not, in general, Lipschitz maps when G, M and H are endowed with the restriction of the distance d of G. Example 4.9 (Example 2.2.17,[95]). Let G be the Heisenberg group H1 =(R3, ) and let V and W be the subgroups · V = x =(x1, 0, 0) : x1 R , W = x =(0,x2,x3): x2,x3 R . { 2 } { 2 } V and W are complementary subgroups in H1, W is normal subgroup while V is not a normal subgroup. When we consider H1 = V W the · projections PV and PW are P (x ,x ,x )=(x , 0, 0), P (x ,x ,x )=(0,x ,x x x /2). V 1 2 3 1 W 1 2 3 2 3 1 2 Here P : H1 W is not Lipschitz. Indeed, let q =(1, 1, 0) and W ! p" =(1+", 1+", 0), then PWq =(0, 1, 1/2) and PWp" =(0, 1+ ", (1 + ")2/2). Hence, as " 0+, ! 1 1 2 q p = (",", 0) ", (PWq) PWp" = (0,", " " /2) p" k k1 k k1 ⇡ k 33 k1 k k1 ⇡ When we consider H1 = W V, then P and P take the form · V W PV(x1,x2,x3)=(x1, 0, 0), PW(x1,x2,x3)=(0,x2,x3 + x1x2/2)

Let q =(1, 0, 0) and p" =(1,","/2), then PWq =(0, 0, 0) and PWp" = (0,","). Hence 1 1 q p = (0,",0) ", (P q) P p" = (0,",") p" k k1 k k1 ⇡ k W W k1 k k1 ⇡ as " 0+. In this case too, P is not a Lipschitz map. ! W The example shows that both the projections either on the first factor or on the second factor can be non Lipschitz. Notice that in both cases we were considering projections on the normal factor. Indeed the projection on the complement of a normal subgroup is always metric Lipschitz continuous. Proposition 4.10. Let M, H be complementary subgroups of G. Then

(i) if H is a normal subgroup then PM is Lipschitz;

(ii) if M is a normal subgroup then PH is Lipschitz. Proof. (i) For all g = mh andg ¯ =¯mh¯, we have 1 1 1 ¯ 1 1 1 1 ¯ 1 PM(g g¯)=PM(h m m¯ h)=PM(m m¯ m¯ mh m m¯ h)=m m.¯ 1 1 Hence, for all g, g¯ G, P (g g¯)=(P g) P g¯ and, by (58), 2 M M M 1 1 1 1 1 (P g) P g¯ P (g g¯) + P (g g¯) c0 g g¯ . k M M k1  k M k1 k H k1  k k1 1 1 (ii) As before, PH(gg¯)=(PHg) PHg¯ and eventually, 1 1 1 (P g) P g¯ c0 g g¯ , for all g, g¯ G. k H H k1  k k1 2 ⇤ 4.2. Intrinsic graphs in Carnot groups. Definition 4.11. Let H be a homogeneous subgroup of G.Wesay that a set S G is a (left) H-graph (or a left graph in direction H, ⇢ or, also, an intrinsic graph) if S intersects each left coset of H in one point, at most. If A G parametrizes the left cosets of H – i.e. if A itself intersect ⇢ each left coset of H at most one time – and if S is an H-graph, then there is a unique function ' : A H such that S is the graph of ', that is E⇢ ! S = graph (') := ⇠ '(⇠):⇠ . { · 2E} Conversely, for any : A H the set graph ( ) is an H-graph. D⇢ ! One has an important special case when H admits a complementary subgroup M. Indeed, in this case, M parametrizes the left cosets of H and we have that S is a H-graph if and only if S = graph (') 34 for ' : M H. By uniqueness of the components along M and E⇢ ! H, if S = graph (') then ' is uniquely determined among all functions from M to H. From now on we will consider mainly graphs of functions acting between complementary subgroups.

Remark 4.12. Nevertheless it is relevant to mention that examples of H-graphs that are not, in general, graphs of functions acting between complementary subgroups have been considered inside the Heisenberg groups Hn = R2n+1. These sets are given as

n S = (x1, ,x2n,'(x1, ,x2n)) H . { ··· ··· }⇢ In our notation such an S is a T-graph, where T is the centre of Hn. n The left cosets of T are parametrized over the set A = HHe . We recall that the centre T has no complementary subgroup in Hn (see Example 4.3), and in general there isn’t a couple of complementary subgroups M, H of Hn and a : M H such that S = graph ( ), even locally. !

If a set S G is an intrinsic graph then it keeps being an intrinsic graph after left⇢ translations or group dilations.

Proposition 4.13. Let H be a homogeneous subgroup of G.IfS is a H-graph then, for all >0 and for all q G, S and q S are 2 · H-graphs. If, in particular, M, H are complementary subgroups in G, if S = graph (') with ' : M H, then E⇢ ! For all >0, S = graph ('), with

(62) ' : M H and E⇢ ! ' (m)= '( m), for m . 1/ 2 E

For any q G, q S = graph ('q), where 2 · 1 (63) 'q : q M H, q = m M : P (q m) and E ⇢ ! E { 2 M 2E} 1 1 1 ' (m)=(P (q m)) ' P (q m) , for all m . q H · M 2Eq Proof. If x, x0 S and x = x0 then, by definition of H-graph, x, x0 2 6 belong to di↵erent left cosets of H. Then x, x0 belong to di↵erent cosets of H, because H is a homogeneous subgroup, and also q x, q x0 · · belong to di↵erent cosets of H, by elementary properties of cosets (see e.g. [124, chapter 2, section 4]). By definition, these facts prove that both S and q S are H-graphs and that there are ' and 'q such · that S = graph (') and q S = graph ('q). To prove (62) observe that,· by uniqueness of the components, (m · '(m)) = m0 '(m0) implies that m = m0 and that ' = ' 1/. · 35 1 1 To prove (63) observe that, because p = p p , for all p , H M G 1 1 1 1 · 2 then (q m) = m q (q m) , hence · H · · · M graph (' )= m ' (m): m q { · q 2Eq} 1 1 1 = m (q m) ' (q m) : m { · · H · · M 2Eq} 1 1 1 1 = m m q (q m) ' (q m) :(q m) { · · · · M · · M · M 2E} = q graph ('). · ⇤ Remark 4.14. From (63) and the continuity of the projection maps

PM and PH it follows that the continuity of a function is preserved by translations. Precisely, given q = q q and f : M H, then the M H ! translated function fq is continuous in m M if and only if the function 2 1 f is continuous in the corresponding point (q m)M.

Remark 4.15. The algebraic expression of 'q in Proposition 4.13 can be made more explicit when G is a semi-direct product of M, H. Precisely

(i) 1 1 If M is normal in G then 'q(m)=q ' (q m) ,form q = q (q ) . H M 2E E H (ii) 1 1 1 If H is normal in G then 'q(m)=(q m) '(q m), for m q = q . H M 2E ME If both M and H are normal in G – that is if G is a direct product of M and H – then we get the well known Euclidean formula 1 (iii) ' (m)=q '(q m), for m = q . q H M 2Eq ME See also [15, Proposition 3.6]. 4.3. Regular hypersurfaces in Carnot groups. In this section we deal with the problem to introduce a notion of intrinsic regular hyper- surfaces (namely 1-codimensional topological surfaces) in the setting of a Carnot group. By that we mean submanifolds which, in the ge- ometry of the Carnot group, have the same role as C1 submanifolds have inside Euclidean spaces. Here and in what follows, intrinsic will denote properties defined only in terms of the group structure of G or, equivalently, of its Lie algebra g. This section relies on [100, 13, 103, 38, 152, 153, 106] . We define G-regular hypersurfaces in a Carnot group G, mimicking Definition 6.1 in [99], as non critical level sets of functions in C1 ( n). G R Definition 4.16. (G-regular hypersurfaces) Let G be a Carnot group. We shall say that S G is a G-regular hypersurface if for every x S there exist a neighborhood⇢ of x and a function f C1 ( ) G 2 36U 2 U such that i) S = y : f(y)=0 ;( \U { 2U } (ii) f(y) =0 fory . rG 6 2U G-regular surfaces have a unique tangent at each point. This follows from a Taylor formula for functions in C1 that is basically G proved in [179]. Proposition 4.17. If f C1 (U (p, r)), then 2 G 1 m1 (64) f(x)=f(p)+ (X f)(p)(x p )+o(d(x, p)), as x p. j j j ! j=1 X If S = x : f(x)=0 G is a G-regular hypersurface the tangent group T g S{(x)toS at x}is⇢ G m1 g (65) T S(x) := v =(v1,...,vn) G : Xjf(x)vj =0 . G { 2 } j=1 X By (2.3), T g S(x) is a proper subgroup of . We can define the tangent G G plane to S at x as (66) T S(x) := x T g S(x). G · G We stress that this is a good definition. Indeed the tangent plane does not depend on the particular function f defining the S because of point (iii) of Implicit Function Theorem below that yields g T S(x)= v G : ⌫E(x),⇡xv =0 G { 2 h ix } where ⌫E is the generalized inward unit normal defined in (54) and m ⇡x(v)= j=1 vjXj(x). Notice that the map v ⇡x(v), for x G fixed, 7! 2 P m

(67) ⇡x(v)= vjXj(x). j=1 X is a smooth section of HG. Notice also that, once more from (iii) of Theorem 4.20, it follows that ⌫E is a continuous function. 0 m 0 If v = viXi(0) HG0 we define the halfspaces S±(0,v )as i=1 2 G m m + 0 P 0 S (0,v ) := x G : xivi > 0 and S(0,v ) := x G : xivi < 0 . G { 2 } G { 2 } i=1 i=1 X X Their common boundary is the vertical plane m ⇧(0,v0) := x : x v =0 . { i i } i=1 37 X m If v = v X (y) H , S±(y, v) and ⇧(y, v) are the translated i=1 i i Gy G sets 2 P 0 0 S±(y, v) := y S±(0,v ) and ⇧(y, v)=y ⇧(0,v ) G G 0 · · where v and v have the same components vi with respect to the left invariant basis Xi. Hence m (68) S±(y, v)= x G : (xi yi)vi > 0(< 0) . G { 2 } i=1 X Clearly, TGS(x)=⇧(x, ⌫E(x)). Note also that the class of G-regular hypersurfaces is di↵erent from the class of Euclidean C1 embedded surfaces in Rn. From one side G- regular surfaces can have ’ridges’ because continuity of the derivatives of the defining functions f is required only in the horizontal directions; on the other side an Euclidean C1 surface can have so called charac- teristic points i.e. points p S where the Euclidean tangent plane T S 2 p contains the horizontal fiber HGp. Definition 4.18. If S is an Euclidean C1 hypersurface in G, we define the characteristic set of S as

(69) car(S) := x S : HGx TxS . { 2 ✓ } The points of car(S) are, under many aspects, irregular points of S. Note indeed that the tangent group does not exist in these points. It is also well known that these points are ’few’ on smooth hypersur- Q 1 faces. Precise estimates of the measure of the characteristic sets Hc of C1 surfaces in general Carnot groups groups Hn has obtained by V. Magnani [147], extending previous results of Z. Balogh [19] in the Heisenberg group, of V. Magnani [147] and of B. Franchi, R. Serapi- oni & F. Serra Cassano [102] in step 2 Carnot groups. Notice that the study of the size of the characteristic set has a long history. We refer to the contributions of M. Derridj [77], B. Franchi & R.L. Wheeden [108], D.Danielli, N.Garofalo & D.M.Nhieu [66]. Magnani’s result reads as follows. Theorem 4.19. If S is a Euclidean C1-smooth hypersurface in a Carnot group G with homogeneous dimension Q and equipped by an invariant distance d. Then Q 1 (70) (car(S)) = 0. Hd We can state now our Implicit Function Theorem, holding that a G- regular hypersurface S = f(y)=0 boundary of the set E = f(y) < { } n 1 { n 0 can be locally parameterized through a function : R R so } ! that the G-perimeter of E can be written explicitly in terms of Gf and . In view of the notion of the intrinsic graph introduced in Definitionr 4.11, we can also say that S is locally an intronsic graph in direction of a 1- dimensional horizonatal subgroup (see Corollary 4.21). 38 Theorem 4.20. (Implicit Function Theorem,[100, Theorem 2.1]) Let ⌦ be an open set in G =(Rn, ), 0 ⌦, and let f C1 (⌦) be such that · 2 2 G f(0) = 0 and X1f(0) > 0. Define E = x ⌦:f(x) < 0 ,S= x ⌦:f(x)=0 , { 2 } { 2 } and, for >0, h>0 n 1 I = ⇠ =(⇠2,...,⇠n) R , ⇠j ,Jh =[ h, h]. { 2 | | } n 1 If ⇠ =(⇠2,...,⇠n) R and t Jh, denote now by (t, ⇠) the in- 2 2 tegral curve of the vector field X1 at the time t issued from (0,⇠)= n (0,⇠2,...,⇠n) R , i.e. 2 (t, ⇠) = exp(tX1)(0,⇠). Then there exist , h > 0 such that the map (t, ⇠) (t, ⇠) is a di↵eo- ! n morphism of a neighborhood of Jh I onto an open subset of R , and, ⇥ if we denote by ⌦ the image of Int(Jh I) through this map, we have U⇢⇢ ⇥ i) E has finite G-perimeter in ;( U ii) @E = S ;( \U \U f(x) (iii) ⌫ (x)= rG for all x S , E f(x) 2 \U |rG |x where ⌫E is the generalized inner unit normal defined by (54), that can be identified with a section of HG with ⌫(x) x =1for @E -a.e. | | | |G x . In particular, ⌫E can be identified with a continuous function and2U⌫ 1. Moreover, there exists a unique function | |⌘ ' = '(⇠):I J ! h such that the following parameterization holds: if ⇠ I, put (⇠)= ('(⇠),⇠), then 2 iv) S ¯ = x ¯ : x =(⇠),⇠ I ;( \ U { 2 U 2 } (v) ' is continuous; the G-perimeter has an integral representation (area formula):

m 2 (vi) j=1 Xjf((⇠)) | | n 1 @E ( )= d (⇠). | |G U q X f((⇠)) L ZI P 1 By Dini’s theorem 4.20 and Example 4.4, it follows that, if H := (x1, 0,...,0) R, and { }⌘ n 1 M = (0,x2,...,xn): xi R,i=2,...,n R , { 2 }⌘ then (i) M and H are, respectively, (n 1,Q 1) and (1, 1)-subgroups of G, and they are complementary, that is G = M H; · (ii) S = graph (') where ' : I M H is continuous \U 39 ⇢ ! We can summarize this result in the following

Corollary 4.21. Every G-regular hypersurface is locally an intrinsic continuos graph with topological dimension n 1. Remark 4.22. Notice that, taking the Euclidean case into account, the approach of defining a regular hypersurface S G =(Rn, ) as image n 1 1 ⇢ · S = '( ) with R of a map ' CH ( , G) whose P -di↵erential U n 1 U⇢ 2 U dP '(p):R G injective for each p cannot work in a non abelian Carnot group.! Indeed, if this is the2 case,U let us consider the in- ˜ n 1 jective H-linear map L := dP '(p):R g read as homomorphism of Lie algebras. In view of Theorem 3.2,! since it is a contact map, it ˜ n 1 follows that L(R ) V1, where V1 is the first horizontal layer of the ✓ stratification of g (see (1)). Thus V1 must contain a (trivial) n 1- dimensional algebra, which implies that g must be also trivial.

Our next Theorem is a mild regularity result. Roughly speaking, it states that G regular hypersurfaces do not have cusps or spikes if they are studied with respect to the intrinsic Carnot–Carath´eodory distance, while they can be very irregular as Euclidean submanifolds. To make precise the former statement we recall the notion of essential boundary (or of measure theoretic boundary) @ F of a set F G (71) ⇤ ⇢ n(F U(x, r)) n(F c U(x, r)) @ F := x G : lim sup min L n \ , L n \ > 0 ⇤ 2 r 0+ (U(x, r)) (U(x, r)) ⇢ ! ⇢ L L Notice that the definition above makes sense in any metric measure space and that the essential boundary does not change if, in definition 71), the distance d is substituted by an equivalent distance d0.

Theorem 4.23. Let ⌦ G be a fixed open set, and E be such that ⇢ @E ⌦=S ⌦, where S is a G-regular hypersurface. Then \ \ (72) @E ⌦=@ E ⌦. \ ⇤ \ 4.4. Area formulas for Hausdor↵measures in metric spaces. Application to Carnot groups. We want now to compare the perime- ter measure, of a G-regular hypersurface S, and the intrinsic (Q 1)- Hausdor↵measure of S. Observe that it makes sense to speak of the perimeter measure of S given that S is locally the boundary of a finite G-perimeter set (as proved in Theorem 4.20). The next theorems give some explicit form of the density of the perimeter with respect to the intrinsic Hausdor↵measures concentrated on S, namely the spherical and centered Hausdor↵measures. As a consequence – as it is stated in the following corollary – G-regular hypersurfaces have coherently intrinsic Hausdor↵dimension Q 1. 40 Let us recall two area formulas for a measure µ with respect to the m-dimensional spherical and centered Hausdor↵measure, recently obtained respectively in [152] and [106]. According to Federer’s notation [80], we define a centered and a non centered density of an outer measure µ on X. Definition 4.24. Let (X, d) be a metric space and assume that (20) holds. (i) The upper and lower m-densities of µ at x X are 2 m µ(B(x, r)) ⇤ ⇥ (µ, x) := lim sup m r 0 ↵m r ! and µ(B(x, r)) ⇥m(µ, x) := lim inf . ⇤ r 0 m ! ↵m r If they agree their common value m m m ⇥ (µ, x)= ⇥⇤ (µ, x)= ⇥ (µ, x) ⇤ is called the m-density of µ at x. (ii) The m-Federer densities of µ at x X are 2 m µ(B(y, r)) ⇤ ⇥F (µ, x) := inf sup m : x B(y, r), ">0 m diam(B(y, r)) 2 n diam(B(y, r)) "  It is easy to see that o m m m m (73) ⇥⇤ (µ, x) ⇥⇤ (µ, x) 2 ⇥⇤ (µ, x) x X.  F  8 2 In a very interesting recent note [152], Magnani proved the following theorem, Theorem 4.25. Let µ be a Borel regular measure in X such that there exists a countable open covering of X whose elements have finite µ m measure; let A X be a Borel set. Suppose that d (A) < and µ A is absolutely⇢ continuous with respect to m AS. Then 1 S m m (74) µ(B)= ⇥⇤ (µ, x) d (x) F Sd ZB for each Borel set B A. ⇢ m Here ⇥F⇤ (µ, ) denotes the m-dimensional Federer density (see Def- inition 4.24 (ii)).· Moreover he showed that in a general metric space m the density ⇥F⇤ (µ, ) cannot be replaced by the ”centered” density m · ⇥⇤ (µ, ) (see Definition 4.24 (i)). Indeed· Magnani provided an example inside the Heisenberg group 1 3 X = H R , equipped with its sub-Riemannian metric d = dc. He ⌘ constructed a Radon measure µ and a set A H1 and two constants 0 t 2(A) t (k ,k ) . Sc 8 2 1 2 In particular, from what said before, it follows that the general state- ment, given A X and k> 0 ⇢ (76) ⇥m(µ, x)= k x A µ A = k m A 8 2 ) S may fail. Therefore Magnani stressed the need of an alternative proof of the assertions in [99, 102, 100, 103], that the perimeter measure @E G agrees, up to a constant, with the (Q 1)-dimensional spherical Haus-| | Q 1 dor↵measure restricted to the a G-regular boundary @E, where S1 X = G is Carnot group of Hausdor↵dimension Q and µ = @E G is the perimeter measure of a set E. Meanwhile he proved that | | .

Theorem 4.26 (Magnani [153]). Let G be a Carnot group endowed with the invariant distance d in (16). Let E G be such that its 1 ⇢ topological boundary @E is a G-regular hypersurface, then there is a constant k = k(G) > 0 for which Q 1 Q 1 ⇥ ( @E ,x)=⇥⇤ ( @E ,x)= k x @E; | |G F | |G 8 2 From Theorems 4.25 and 4.26 it follows at once that

Corollary 4.27. Let G be a Carnot group endowed with the invariant distance d . Let E G with topological boundary @E a G-regular 1 ⇢ hypersurface, then there is a constant k = k(G) > 0 such that Q 1 (77) @E = k S. | |G S1 Corollary 4.28. If S is a G-regular hypersurface then the Hausdor↵ dimension of S, with respect to the Carnot–Carath´eodory metric d or any other metric d0 comparable with it, is Q 1. Corollary 4.28 combined with Theorem 4.19 yields the following comparison result between Euclidean C1-smooth hypersurfaces and G- regular hypersurfaces. We have Theorem 4.29. If S is a Euclidean C1-smooth hypersurface in a Carnot group G with homogeneous dimension Q, then the Hausdor↵dimension of S, with respect to the Carnot–Carath´eodory metric d or any other invariant metric d0 comparable with it, is Q 1. Remark 4.30. The reverse assertion is false: there exist G-regular hy- persurfaces in G Rn that have Euclidean Hausdor↵dimension greater ⌘ than n 1: indeed in ([131]) it was showed that there exist G-regular hypersurfaces in the Heisenberg group H1 (Q =4,n = 3) with Eu- clidean Hausdor↵dimension 2.5. 42 By previous Magnani’s example it turns out that Federer densities play a privileged role when dealing with the spherical measures m.On the other hand, these densities are frequently harder to computeS than the centered ones. Therefore, in [106] we studied whether the afore- m mentioned area formula keeps being true with ”centered” densities ⇥⇤ m and measures d replaced by equivalent ones. Centered Hausdor↵ m S measures d (see Definition 2.17 (iii)) actually play this role. Indeed it holds thatC Theorem 4.31 ([106]). Let µ be a Borel regular measure in X such that there exists a countable open covering of X whose elements have µ m finite measure; let A X be a Borel set. If d (A) < and µ A is absolutely continuous⇢ with respect to m A,C then 1 Cd m ⇥⇤ (µ, ):X [0, + ] is Borel measurable · ! 1 and, for each Borel set B A, ⇢ m m (78) µ(B)= ⇥⇤ (µ, x) d (x). Cd ZB m m m Remark 4.32. Since d and d are equivalent, then d (A) < if and m C S C 1 only if d (A) < and µ A is absolutely continuous with respect to m if andS only if 1µ A is absolutely continuous with respect to m. Cd Sd Remark 4.33. According to Theorem 4.31, in the di↵erentiation results for measure stated in [100, Theorem 3.4] and [103, Theorem 4.3] the centered Hausor↵measure has to be considered in place of the spherical one. Corollary 4.34. Under the same assumptions of Theorem 4.31, if, for A X, there is a constant k>0 such that ✓ m ⇥⇤ (µ, x)= k x A 8 2 then µ A = k m A. C As an immediate consequence of Theorem 4.26 , Corollaries 4.27 and 4.34 we get Theorem 4.35. Let G be a Carnot group endowed with the invariant distance d in (16). Let S G be a G-regular hypersurface. Then 1 ⇢ Q 1 Q 1 (79) S = S S1 C1 Q 1 Q 1 Q 1 Remark 4.36. Since , and are equivalent measures, Hd Sd Cd by (79) and Radon-Nikodym’s theorem, it follows that, given a G- regular hypersurface S G, there exists a non negative density ⇣ 1 Q 1 ⇢ 2 L (S, S) such that loc Sd Q 1 Q 1 Q 1 S = ⇣ S = ⇣ S. H1 S1 43 C1 It is unknown whether ⇣ is constant. Let us recall that in the Euclidean setting n 1 n 1 n 1 S = S = S, H S C whenever S Rn is a (Euclidean) C1-regular hypersurface (see [195]). ⇢ As an other application to Carnot groups of the area formula for cen- tered Hausdor↵measure, we can provide an explicit characterization of Q 1 the centered density ⇥ ( @E G, )forsetsE whose @E is G-regular. This is corrected version of| [100,| Theorem· 3.5], which, a priori, could not apply to the spherical Hausdor↵measure for Magnani’s counterex- ample.

Theorem 4.37 ([106]). Let G be a Carnot group and d be an invariant distance. Let ⌦ G be a fixed open set and let E G such that ⇢ ⇢ @E ⌦= S ⌦, where S is a G-regular hypersurface. Then\ \ n 1 g (U (0, 1) T S(x)) Q 1 d G (80) ⇥ ( @E G,x)=L \ x S, | | ↵Q 1 8 2 where T g S(x) denotes the tangent group to S at x according to (65), G and Ud(0, 1) denotes the unit (open) ball induced by the distance d.In particular

Q 1 Q 1 (81) @E ⌦=⇥ ( @E , ) (S ⌦). | |G | |G · Cd \ Q 1 where denotes the (Q 1)-dimensional centered Hausdor↵measure Cd defined as in Definition 2.17 (iii) with X = G equipped by metric d,. Moreover, there is a constant d > 1, depending only on the distance d, such that

1 Q 1 (82) 0 < ⇥ ( @E G,x) d < x S. d  | |  182 Remark 4.38. If S = @E is G-regular, to our knowledge, the equality Q 1 Q 1 ⇥ ( @E ,x)=⇥⇤ ( @E ,x) x S | |G F | |G 8 2 holds only for particular distances on G whose the unit closed ball is convex , like d (see [153, Theorem 5.2]). In the general case, an 1 Q 1 explicit characterization of Federer density ⇥F⇤ ( @E G, ) like (80) is unknown yet. For instance, it is unknown its characterization| | · for the case of Carnot- Carath´edory distance dc unless some particular case as the Heisenberg group. 4.5. Regular surfaces in Carnot groups. A notion of intrinsic reg- ular surfaces with di↵erent topological dimension,was firstly introduced and studied in the Heisenberg groups [103] and then partially extended in Carnot groups[151]. 44 4.5.1. Regular surfaces in Heisenberg groups [103]. Let us start with some comments about possible notions of regular submanifolds of a group. It is barely worth to say that Euclidean regular submanifolds of Hn R2n+1 are not a satisfactory choice for many reasons. Indeed, Euclidean⌘ regular submanifolds need not to be group regular; this is clear for low dimensional submanifolds: the 1-dimensional, group regu- lar, objects are horizontal curves that are a small subclass of C1 lines, but, also a low codimensional Euclidean submanifold need not to be group regular due to the presence of the so-called characteristic points where no intrinsic notion of tangent space to the surface exists (see [19, 147]). On the contrary, as seen in the previous section, in Carnot groups 1-codimensional G -regular hypersurfaces, that we will call H- regular in the setting of Heisenberg groups, can be highly irregular as Euclidean objects but that enjoy intrinsic regularity properties, so that it is natural to think of them as 1-codimensional regular submanifolds of the group. What do we mean by ”intrinsic regularity” properties? We have al- ready stated how intrinsic should be meant here. We believe that the most natural requirements to be made on a subset S Hn to be con- sidered as an intrinsic regular submanifold are ⇢ (i) S has, at each point, a tangent plane and a normal plane (or better a transversal plane); (ii) tangent planes depend continuously on the point; the notion of plane has to depend only on the group structure and on the di↵erential structure as given by the horizontal bundle. Since subgroups are the natural counterpart of Euclidean subspaces, it seems accordingly natural to ask that (iii) both the tangent plane and the transversal plane are subgroups (or better cosets of sub- groups) of Hn; Hn is the direct product of them (see [103, Section 3.2]); (iv) the tangent plane to S in a point is the limit of group dilations of S centered in that point (see [103, Definition 3.4]). Notice that the requirement that the limit of a blow up procedure is a subgroup comes out naturally even in much more general settings than Hn (see [160]). Moreover, the explicit requirement of existence of both a tangent space and a transversal space is not pointless, be- cause there are subgroups in Hn, as the T axis for example, that is 2n+1 T = p =(p1,...,p2n+1) R : p1 = = p2n =0, with- { 2 ··· } out a complementary subgroup, i.e. a subgroup M Hn such that ⇢ M T = 0and Hn = M T. Finally, the distinction between normal and transversal\ planes is natural,· because not necessarily at each point a normal subgroup exists, even if a (possibly not normal but) transversal subgroup exists. Condition (iv) entails that the tangent plane has the natural geometric meaning of surface seen at infinite scale, the scale 45 however being meant with respect to intrinsic dilations. Notice thatif S is both an Euclidean smooth manifold and a group regular mani- foldthe intrinsic tangent plane is usually di↵erent from the Euclidean one. On the other hand, there are sets, bad from the Euclidean point of view, that behave as regular sets with respect to group dilations. It is natural to check if requirements (i)-(iv) are met by the classes of regular submanifolds of Hn considered in the literature. C1 horizontal curves: they are Euclidean C1 curves; their (Eu- • clidean) tangent space in a point is a 1-dimensional ane sub- space contained in the horizontal fiber through the point, hence it is also a coset of a 1-dimensional subgroup of Hn. The normal space is the complementary subspace of the tangent space, and it is again a subgroup. Clearly both of them depend continu- ously on the point. It can also be shown (see [103, Theorem 3.5]) that the Euclidean tangent lines are also limits of group dilation of the curve, so that they are also tangent in the group sense. Legendre submanifolds: they are n-dimensional, hence maximal • dimensional, integral manifolds of the horizontal distribution (see [16]). The tangent spaces are n-dimensional ane sub- spaces of the horizontal fiber that are also cosets of subgroups of Hn. The comple-mentary ane subspaces are the normal subgroups. As before the tangent spaces are limit of intrinsic dilations of the surface (see [103, Theorem 3.5]). H-regular hypersurfaces : we recall that, locally,they are given • as level sets of C1 ( n) functions from n to , with P- H H H R di↵erential of maximal rank (see Definition 4.16). We showed before that H-regular hypersurfaces have a natural normal space (i.e. the span of the horizontal normal vector) at each point, hence it is a coset of a 1-dimensional subgroup contained in the horizontal fibre; that the natural tangent space is a subgroup obtained as limit of intrinsic dilations of the surface; and finally, notwithstanding that these surfaces can be highly irregular as Euclidean surfaces, the intrinsic normal subgroup and the in- trinsic tangent subgroup depend continuously on the point. In conclusion, all the surfaces in these examples are intrinsically regular submanifolds in the sense that they satisfy requirements (i)-(iv). Notice that C1 horizontal curves have topological dimension 1, Legendre sub- manifolds have topological dimension n, and H- regular hypersufarces have topological dimension 2n (the systematic specification topologi- cal is not pointless, because, as already noticed before, other di↵erent play a role in the geometry of Carnot groups). Our aim is now to fill the picture, finding other classes of intrinsi- cally regular submanifolds of arbitrary topological dimension. Notice that, from the analytical point of view, horizontal curves and Legendre surfaces are 46 given locally as images in Hn respectively of intervals I R or of open ⇢ sets in Rn through P-di↵erentiable maps with injective di↵erentials. On the contrary, 1-codimensional H-regular surfaces are given locally as level sets of P-di↵erentiable functions with surjective di↵erentials. The first idea coming to the mind, and the one we take here, is to generalize both these approaches. Notice that, even if in the Euclidean setting they are fully equivalent, this is no longer true in Heisenberg groups. Thus we define

Definition 4.39. Let k be an integer, 1 k n.   (i) A subset S Hn is a k-dimensional H-regular surface if for any ⇢ p S there are open sets Hn, with p and a function ' 2: such that p U, '⇢is injective, '2isU continuously P- V!U 2U k n di↵erentiable with dP '(x): R H injective for each x , and ! 2V S = '( ). \U V (ii) A subset S Hn is a k-codimensional (or (2n+1 k)-dimensional) ⇢ H-regular surface if for any p S there are open sets Hn, 2 k 1 kU⇢ with p and a function f : R , f CH ( , R ) , with 2Un k U! 2 U dP f(p): H R surjective for each p , such that ! 2U S = p : f(p)= 0 . \U { 2U } Remark 4.40. For k = 1, Definition 4.39 (i) defines horizontal, con- tinuously di↵erentiable, curves. On the other hand, Definition 4.39 (i) cannot be extended to the case k> n. Indeed, if this is the case, let ˜ k n us consider the injective H-linear map L := dP '(p):R h read as homomorphism of Lie algebras. In view of Theorem 3.2,! since it is a ˜ k contact map, it follows that L(R ) V1, where V1 is the first horizon- n ✓ tal layer of the stratification of h (see (1)). Thus V1 must contain a (trivial) k- dimensional algebra with k> n. On the other hand it is well known that V1 can contain at most a (trivial) algebra of dimension n. In turn Definition 4.39 (ii), for k = 1, gives the notion of H-regular hypersurface introduced in previous section. Definition 4.39 (ii)-unlike the previous one-could be formally extended to k> n, but we restrict ourselves to 1 k n because only in this situation it is possible   1 to prove (see Remark 4.70) that a CH surface of codimension k is locally a graph in a consistent suitable sense. Nevertheless we point out that level sets of P-di↵erentiable maps ' : H1 R2 with surjective P-di↵erential have been studied (see [138] and [133]).!

Let us introduce the definition of Heisenberg tangent cone to a set A at a point p, which extend the classical Euclidean one in [80, 3.1.21]. 47 Definition 4.41. Let A Hn. The intrinsic (Heisenberg) tangent cone to A at 0 is the set ⇢

Tan (A, 0) := v = lim r (xh): forsome(rh)h (0, + ),xh A H h + h ⇢ 1 2 n ! 1 with lim xh =0 . h + ! 1 o and the cone at a point p is given as Tan (A, p) := ⌧pTan (⌧ p(A), 0). H H Theorem 4.42 (Theorem 3.5,[103]). If S is a k-dimensional H-regular surface, 1 k 1,n, then:   (i) S is an Euclidean k-dimensional submanifold of R2n+1 of class C1. (ii) Let TanS denote the Euclidean tangent bundle of S. Then Tan(S, p)=Tan (A, p) p S. H 8 2 (iii) k S is comparable with k S. S1 H Remark 4.43. It is easy to check that low dimensional H-regular sur- faces are graphs because they are Euclidean C1 submanifolds and be- cause low dimensional intrinsic graphs in Hn turn out to be Euclidean graphs.

Definition 4.44. Let S = x : '(x)= 0 be a k- codimensional H- 1{ k } regular surface with ' CH ( , R ) and 1 k n andl let p S. The tangent group to S 2at p, denotedU T g (S, p), is the subgroup defined2 H as T g (S, p) := ker d '(p) H P Theorem 4.45. Let S be a k- codimensional H-regular surface with 1 k n.   (i) ([103, Theorem 3.27])S is locally an intrinsic graph. More pre- cisely, for each p S there exist: 2 n (i1) two complementary subgroups W and V of H , that is Hn = W V, W and V are, respectively, a (2n+1 k, 2n+2 k)- normal subgroup· and a (k, k)-subgroup; n (i2) an open set H with p , a relatively open set U⇢ 2U W and a continuos function ' : W V V⇢such that V⇢ ! S = (⇠) := ⇠ '(⇠): ⇠ . \U { · 2V} In particular the topological dimension of S turns out to be 2n+ 1 k. (ii) ([103, Proposition 3.29]) Tan (S, p)= ⌧ (T g (S, p)) for each H p H p S. (iii) ([103,2 Theorem 4.1]) By previous claim (i) and with the notation therein, there exists a positive continuous function ⇣ C0( ), 48 2 U depending only f which can be explicited represented (see [103, formula (48)]), such that

2n+2 k 2n+1 k (83) (S )= ⇣ d . C1 \U H ZV In particular the metric dimension of S turns out to be 2n+2 k. Claim (i) of Theorem 4.45 follows by a suitable Implicit Function Theorem which holds when Hn can be splitted in complementary sub- groups (see [103, Proposition 3.13]). From claim (ii) it follows that the notion of tangent group is inde- pendent of the defining function f of S. Area formula (83) was erroneously stated with respect to the spher- 2n+2 k ical Hausdor↵measure in [103, formula (48)]. Indeed, taking into account previous Magnani’sS1 example and according to Remark 2n+2 k 4.33, the centerd Hausdor↵measure has to be used in place of the spherical one. C1 4.5.2. Regular surfaces in a general Carnot group. A notion of regu- lar surfaces in a general Carnot group has been recently introduced by Magnani in [151]. More precisely, as a byproduct of an Implicit Function Theorem ([151, Theorem 1.3]) and a Rank Theorem ([151, Theorem 1.4]) in Carnot groups, and in the same spirit of the no- tion of H-regular surface, two types of (G, M) -regular sets have been introduced ([151, Definitions 10.3 and 10.4]): those contained in G, which are suitable level sets and those in M, which are suitable images. In particular it can be proved they are locally intrinsic graphs ([151, Corollary 1.5]). 4.6. Intrinsic Lipschitz Graphs in Carnot groups. The notion of intrinsic Lipschitz continuity, for functions acting between complemen- tary subgroups M and H of G, was originally suggested by Corollary 3.17 of [103] together with the fact that a H-regular surface keeps being H-regular after a (left) translation, (see the Definition given in [104]). We propose here a geometric definition. We say that f : M H, is intrinsic Lipschitz continuous if, at each p graph (f), there! is an 2 (intrinsic) closed cone with vertex p, axis H and fixed opening, inter- secting graph (f) only in p. The equivalence of this definition and other ones, more algebraic, is the content of Proposition 4.49. Notice also that M and H are metric spaces, being subsets of G, hence it makes sense to speak also of metric Lipschitz continuous functions from M to H. As usual, f : M H is said (metric) Lipschitz if there is L>0 ! such that, for all m, m0 M, (84) 2 1 1 f(m) f(m0) = d (f(m),f(m0)) Ld (m, m0)=L m m0 k · k1 1  1 k · k1 The notions of intrinsic Lipschitz continuity and of Lipschitz continuity are di↵erent ones (see Example 4.51) and we will try to convince the 49 reader, that intrinsic Lipschitz continuity seems more useful in the context of functions acting between subgroups of a Carnot group. 4.6.1. Definition and first properties of intrinsic Lipschitz functions. . Let us come to the basic definitions. By intrinsic (closed) cone we mean Definition 4.46. If M, H are complementary subgroups in G, q G 2 and 0, the cones CM,H(q, ), with base M, axis H,vertexq, opening are defined as

C , (q, )=q C , (e, ), where C , (e, )= p : p p . M H · M H M H { k Mk1  k Hk1} If 0 <1 <2, CM,H(q, 1) CM,H(q, 2) and CM,H(e, 0) = H. Moreover for all t> 0 ⇢

t (CM,H(e, )) = CM,H(e, ) . Definition 4.47. We say that f : M H is intrinsic L-Lipschitz in if there is L>0 such that E⇢ ! E (85) C (p, 1/L) graph (f)= p , for all p graph (f). M,H \ { } 2 The Lipschitz constant of f in is the infimum of the L>0 such that (85) holds. An intrinsic LipschitzE (continuous) function, with Lipschitz constant noe exceeding L> 0, is called a L-Lipschitz function. We will call a set S G an intrinsic Lipschitz graph, if there exists ⇢ an intrinsic Lipschitz function f : M H such that S = graph (f) E⇢ ! for suitable complementary subgroups M and H. We will call the topo- logical dimension of M and H, respectively, dimension and codimension of graph (f). Because of Proposition 4.13 and Definition 4.46 left translations of intrinsic Lipschitz H-graphs, or of intrinsic L-Lipschitz functions, keeps being intrinsic Lipschitz H-graphs, or intrinsic L-Lipschitz functions. We state these facts in the following theorem. Theorem 4.48. Let G be a Carnot group, then for all q G, if f : 2 M H is intrinsic L-Lipschitz if and only if fq : q M H is E⇢ ! E ⇢ ! intrinsic L-Lipschitz,where fq is the function defined in (63). The geometric definition of intrinsic Lipschitz graphs has equivalent algebraic forms (see also [15], [104],[95]). Proposition 4.49. Let M, H be complementary subgroups in G, f : M H and L>0. Then (i) to (iv) are equivalent. E⇢ ! (i) f is intrinsic L-Lipschitz in . E 1 1 (ii) P (¯q q) L P (¯q q) , for all q, q¯ graph (f). k H k1  k M k1 2 (iii) fq¯ 1 (m) L m , for all q¯ =¯mf(¯m) graph (f) and m q¯ 1 . k k1  k k1 2 2E (iv) 1 1 f(m1) m2 f(m1m2) L f(m1) m2 , for all m1,m2 . k H · k1  k Mk1 2E 50 Moreover, f : M H is intrinsic Lipschitz if and only if the distance of twoE points⇢ q,!q¯ graph (f), is bounded by the norm of their 2 projection on the domain M. Precisely (v) 1 1 1 1 q¯ q c0(1 + L) P (¯q q) = P (¯q q) L P (¯q q) , k k1  k M k1 )k H k1  k M k1

where c0 < 1 is the constant in Proposition 4.6, and conversely (vi) 1 1 1 1 P (¯q q) L P (¯q q) = q¯ q (1 + L) P (¯q q) , k H k1  k M k1 )k k1  k M k1 for all q, q¯ graph (f). 2 Proof. The equivalence between (i) and (ii) follows from definition (4.46), observing that ifq ¯ graph (f), then C (¯q, 1/L) graph (f)= 2 M,H \ 1 q¯ is equivalent with CM,H(e, 1/L) graph (fq¯ )= e . The{ } equivalence of (ii) and (iii) follows\ once more{ from} the defini- tion of cone and left invariance of the definition. Indeed we recall that 1 1 fq¯ (m)= (f(¯m)m)H f m¯ (f(¯m)m)M , then (¯qm)H =(¯mf(¯m)m)H = (f(¯m)m) and (¯qm) =(¯mf(¯m)m) =¯m(f(¯m)m) . H M M M (iv) follows from (iii) changing variables. Let m2 := (f(¯m)m)M that is 1 m =(f(¯m) m2)M then 1 1 1 f(¯m)m f m¯ (f(¯m)m) = f(¯m)(f(¯m) m2) f mm¯ 2 H · M M H · 1 1 1 1 = f(¯m)f(¯m) m2 f(¯m) m2 f mm¯ 2 H H · 1 = f(¯m) m2 f mm¯ 2 . H · Then conclude with m ¯ = m1. 1 1 For the last two inequalities observe that, if q¯ q c0(1 + L) q¯ q M then k k1  k k1 1 1 1 1 1 q¯ q q¯ q q¯ q L q¯ q . k Hk1  c0 k k1 k Mk1  k Mk1 Conversely 1 1 1 1 q¯ q q¯ q + q¯ q (1 + L) q¯ q . k k1 k Mk1 k Hk1  k Mk1 ⇤ Remark 4.50. If G is the semi-direct product of M and H, (ii), (iii) and (iv) of Proposition 4.49 take a more explicit form. Indeed, recalling Remark 4.15, we get (i) If M is normal in G then f is intrinsic L-Lipschitz if and only if 1 1 1 f(¯m) f(m) L f(¯m) m¯ mf(¯m) , for all m, m¯ . k k1  k k1 2E (ii) If H is normal in G then 1 1 1 ((¯qm) )Hf (¯qm)M = m f(¯m) mf mm¯ , 51 hence (iii) of Proposition 4.49 becomes 1 1 m f(¯m) mf mm¯ L m , for all m, m¯ . k k1  k k1 2E Hence, changing variables we get that f is intrinsic L-Lipschitz if and only if 1 1 1 1 m0 mf¯ (¯m) m¯ m0f m0 L m¯ m0 , for all m0, m¯ . k k1  k k1 2E (iii) If G is a direct product of M and H, that is G = M H,weget the well known expression for Lipschitz functions ⇥ 1 1 f(¯m) f(m) L m¯ m , for all m, m¯ . k k1  k k1 2E Hence in this case intrinsic Lipschitz functions are the same as the usual metric Lipschitz functions from (M,d )to(H,d ). 1 1 Example 4.51. ([15, Example 3.21]) Here we show that condition (84) is not invariant under left translations of the graph. It follows that nei- ther intrinsic Lipschitz continuity implies (metric) Lipschitz continuity 1 nor the opposite. Let G = H = W V where V := v =(v1, 0, 0) · { 1/2 } and W = w =(0,w2,w3) . Notice that w = max w2 , w3 , if { } k k1 {| | | | } w W and v = v1 if v V. 1 2 k k | | 2 1/2 (i) ' : W V, defined as '(w) := (1 + w3 , 0, 0) , satisfies (84) with L!= 1, hence ' is Lipschitz.| On| the contrary ' is not intrinsic Lipschitz. Indeed, let p := (1, 0, 0) graph (()'), from 2 1/2 Proposition 4.13 we have ' 1 (w)=(w + w , 0, 0) . For p | 2 3| 1 'p , Proposition 4.49 (iii) does not hold. This shows also that condition ((84) is not invariant under graph translations. 1/2 (ii) : W V, defined as (w) := (1 + w3 w2 , 0, 0) , is intrinsic! Lipschitz; indeed, with p =(1|, 0, 0) and| '(w) := 1/2 ( w3 , 0, 0) we have (w)='p(w), so that is intrinsic Lips- chitz| | because ' is intrinsic Lipschitz. On the contrary is not Lipschitz, in the sense of (84), as can be easily observed. Analogously (iii) the constant function ' : V W defined as '(v) := (0, 1, 0), ! for all v V, is Lipschitz but it is not intrinsic Lipschitz; 2 (iv) : V W defined as (v):=(0, 1, v1) for all v V, is intrinsic! Lipschitz continuous but it is not Lipschitz. 2 Remark 4.52. It is a natural question to ask if intrinsic Lipschitz func- tions are metric Lipschitz functions provided that appropriate choices of the metrics in the domain or in the target spaces are made. The answer is almost always negative. Nevertheless something relevant can be stated (see [95, Remark 3.1.8]). We conclude this section stressing that that intrinsic Lipschitz func- tions, even if non metric Lipschitz, nevertheless are metric Holder con- tinuous . 52 Proposition 4.53 (Proposition 3.1.10, [95]). Let M, H be complemen- tary subgroups in a step  group G. Let L>0 and f : M H be an intrinsic L-Lipschitz function. Then E⇢ ! (i) f is bounded on bounded subsets of . Precisely, for all R>0 there E is C1 = C1(M, H,f,R) > 0 such that

(86) f(m) C1, for all m such that m R. k k1  2E k k1  (ii) f is 1 -Holder continuous on bounded subset of . Precisely, for all  E R>0, there is C2 = C2(G, M, H,C1,L,R) > 0 such that (87) 1 1 1/ f(¯m) f(m) C2 m¯ m , for all m, m¯ with m , m¯ R. k k1  k k1 2E k k1 k k1  4.6.2. Surface measure of intrinsic Lipschitz graphs. If M, H are com- plementary subgroups in G and f : E M H is intrinsic Lipschitz then the metric dimension of graph (f⇢) is the! same as the metric di- mension of the domain E. That is, if s is the metric dimension of E then s (graph (f) ) < , S1 \U 1 for any bounded G. A non trivial corollary of this estimate is that 1-codimensionalU intrinsic⇢ Lipschitz graphs are boundaries of sets of locally finite G-perimeter (see [Theorem 4.2.9][92]). Upper and lower bounds on the Hausdor↵measure of a Lipschitz graph are trivially true in Euclidean spaces. Indeed if f : Rk Rn k is Lipschitz then k n ! the map f : R R defined as f (x) := (x, f(x)) is a Lipschitz parametrization of! the Euclidean graph of f ; this gives the upper bound on the dimension of the graph. On the other side, the projection n k n k k R R R R is Lipschitz continuous, with Lipschitz constant 1, yielding⌘ ⇥ the lower! bound. Such a proof cannot work here. From one side the projection PM or PH are not Lipschitz continuous (see Example 4.9), on the other side, the natural parametrization : M ! G,f (m) := m f(m), is almost never a Lipschitz continuous map · between the two metric spaces M and G (see[95, Remark 3.1.8]). Notice that the following result contains a stronger statement: it states that a Lipschitz graph parametrized on a homogeneous subgroup of dimension dm is (locally) Alhfors dm-regular.

Theorem 4.54. ([95, Theorem 3.2.1])Let M, H be complementary sub- groups in G. Let dm denote the metric dimension of M.Iff : M H ! isintrinsic L-Lipschitz in M then there is c = c(M, H) > 0 such that, c dm 0 rdm dm (graph (f) B(p, r)) c(1 + L)dm rdm 1+L S1 \  ✓ ◆ for all p graph (f) and r> 0, where c is the constant in Proposition 2 0 4.6. In particular, graph (f) has metric dimension dm. 53 4.7. Intrinsic di↵erentiable graphs in Carnot groups. A func- tion f : M H, acting between complementary subgroups of G, is ! intrinsic di↵erentiable in a point m M if can be approximated by suitable intrinsic linear functions. Intrinsic2 linear functions, acting be- tween complementary subgroups, are those functions whose graphs are homogeneous subgroups. Definition 4.55. Let M and H be complementary subgroups in G. Then L : M H is an intrinsic linear function if L is defined on all of ! M and if graph (L)= m L(m):m M is an homogeneous subgroup { · 2 } of G. Before using intrinsic linear functions in the definition of intrinsic di↵erentiability of functions we begin collecting some of their proper- ties. Proposition 4.56. ([92, Proposition 3.1.5]) Let M and H be comple- mentary subgroups in G. (i) If L : M H is intrinsic linear then the homogeneous subgroups ! graph (L) and H are complementary subgroups and G = graph (L) H. · (ii) If N is an homogeneous subgroup such that N and H are complemen- tary in G then there is a unique intrinsic linear function L : M H ! such that N = graph (L). Intrinsic linear functions are not necessarily group homomorphism between their domains and codomains, as the following example shows. Let V, W be the complementary subgroups of H1 defined as V = v = { (v1, 0, 0) and W = w =(0,w2,w3) . For any fixed a R,the } { } 2 function L : V W defined as ! L(v)=(0, av , av2/2) 1 1 is intrinsic linear because graph (L)= (t, at, 0) : t R is a 1- { 2 } dimensional homogeneous subgroup of H1. This L is not a group ho- momorphism from V to W. Intrinsic linear functions can be algebraically characterized as fol- lows. Proposition 4.57. (([92, Proposition 3.1.3])) Let M and H be com- plementary subgroups in G. Then L : M H is an intrinsic linear function if and only if !

(i) L(m)=(L(m)), for all m M and R; 2 2 (ii) 1 1 1 L(m1m2)= L(m1) m2 L L(m1) m2 , for all m1,m2 M. H M 2 Intrinsic linear functions are intrinsic Lipschitz . Proposition 4.58. (([92, Proposition 3.1.6])) Let M and H be com- plementary subgroups in G and L : M H be intrinsic linear, then 54 ! (i) L is a polynomial function; (ii) L is intrinsic C-Lipschitz continuous with C := sup L(m) : m =1 {k k1 k k1 } Let us point out that, in the particular case of Heisenberg group Hn, an explicit characterization of intrinsic linear can be provided, in term of H-linear functional. Indeed

Proposition 4.59 (Proposition 3.23,[15]). Let Hn = W V as in Ex- ample 4.3. Then · n (i) L : V W is intrinsic linear i↵ L : V H , defined as (v)=!v L(v), is H- linear. ! L · (ii) L : W V is intrinsic linear i↵ L is H-linear. ! We use intrinsic linear functions to define intrinsic di↵erentiability in a way that is formally completely similar to the usual definition of P -di↵erentiability. First if f(0)=0 we say that f : M H is intrinsic di↵erentiable in ! 0 if there exist an intrinsic linear map L : M H such that, for all ! m M, 2 1 (88) L(m) f(m) = o( m ), as m 0 , k · k1 k k1 k k1 ! where o(t)/t 0ast 0+. Up to this point the definition of intrinsic di↵erentiability! is the same! as the definition of P - di↵erentiability. The di↵erences appear (see Definition 4.60 below) when we extend the pre- vious notion to any pointm ¯ M using (88) in a translaten invariant 2 way. That is, givenm ¯ M we considerp ¯ =¯m f(¯m) and the trans- 2 · 1 1 lated function fp¯ , that, by definition, satisfies fp¯ (0) = 0 (see (63)). 1 Now we say that f is intrinsic di↵erentiable inm ¯ i↵ fp¯ satisfies (88). We give also an uniform version of Definition 4.60 in Definition 4.66.

Definition 4.60. Let M and H be complementary subgroups in G and f : M H with relatively open in M.For¯m and A⇢ ! A 2A 1 1 p¯ :=m ¯ f(¯m) graph (f) let fp¯ : p¯ M H be the translated function· in(63).2 We say that f is intrinsicA ⇢ di↵!erentiable in m¯ if 2A 1 fp¯ is intrinsic di↵erentiable in e that is if there is an intrinsic linear map df = df m¯ : M H such that, for all m p¯ 1 , ! 2A 1 df m¯ (m) fp¯ 1 (m) (i) lim k · k1 =0. m 0 m k k1! k k1 The intrinsic linear map df m¯ is called the intrinsic di↵erential of f at m¯ .

1 Writing explicitly the expression of fp¯ in (i), we can equivalently state that f is intrinsic di↵erentiable in m¯ if 2A (ii) 1 1 df m¯ (m) (f(¯m)m) f (¯m(f(¯m)m) ) = o( m ), as m 0, k · H · M k1 k k1 k k1 ! 55 or, changing variables, if (iii) 1 1 1 1 df m¯ ((f(¯m) m) ) (f(¯m) m) f (¯mm) = o( (f(¯m) m) ), k M · H · k1 k Mk1 1 as (f(¯m) m) 0, or finally if k Mk1 ! (iv) 1 1 1 1 1 1 1 df m¯ (f(¯m) m¯ m) (f(¯m) m¯ m) f (m) = o( (f(¯m) m¯ m) ), k M · H · k1 k Mk1 1 1 as (f(¯m) m¯ m) 0. Here o(t) is such that limt 0+ o(t)/t =0. k Mk1 ! ! Remark 4.61. From Proposition 4.56 (ii), we immediately get that the intrinsic di↵erential is unique, if it exists. Remark 4.62. If a function is intrinsic di↵erentiable it keeps being in- trinsic di↵erentiable after a left translation of the graph. Precisely, let q = m f(m )andq = m f(m ) graph (f), then f is intrinsic di↵er- 1 1 1 2 2 2 2 1 1 entiable in m1 if and only if fq2 q (fq ) is intrinsic di↵erentiable · 1 ⌘ 1 q2 in m2. In particular, f is intrinsic di↵erentiable in m1 if and only if f 1 is intrinsic di↵erentiable in e. q1 Proposition 4.63. ([92, Proposition 3.2.3]) Let M, H be complemen- tary subgroups in G and f : M H with relatively open in M. If f is intrinsic di↵erentiableA in⇢m ! , then fAis continuous in m. 2A Remark 4.64. Once more we write explicitly the form taken by (i) and (ii) of Definition 4.60 when G is the semidirect product of M and H. If M is a normal subgroup then f : M H is di↵erentiable inm ¯ M if ! 2 1 1 1 df m¯ (m) f(¯m) f mf¯ (¯m)mf(¯m) = o( m ), k · · k1 k k1 as m 0 or, changing variables, if k k1 ! 1 1 1 1 1 1 df m¯ f(¯m) m¯ mf(¯m) f(¯m) f (m) = o( f(¯m) m¯ mf(¯m) ), k · · k1 k k1 1 1 as f(¯m) m¯ mf(¯m) 0. k k1 ! If H is a normal subgroup then f : M H is di↵erentiable inm ¯ M if ! 2 1 1 1 df m¯ (m) m f(¯m) m f (¯mm) = o( m ), as m 0, k · · · · k1 k k1 k k1 ! or, changing variables, if

1 1 1 1 1 1 1 df m¯ m¯ m (¯m m) f(¯m) (¯m m) f (m) = o( m¯ m ), k · · · · k1 k k1 1 as m¯ m 0. k k1 ! If both M and H are normal subgroups then f : M H is di↵erentiable ! inm ¯ M if 2 1 1 df m¯ (m) f(¯m) f (¯mm) = o( m ), as m 0. k · · k1 k k1 k k1 ! 56 Remark 4.65. Let us recall Pansu’s di↵erentiability (see Definition 3.1): a function f, acting between two nilpotent groups G1 and G2, is P- di↵erentiable ing ¯ G1 if there is an homogeneous homomorphism 2 L : G1 G2 such that ! 1 1 1 1 1 L(¯g g) f(¯g) f(g) = o( g¯ g ), as g¯ g 0. k · · · kG2 k · kG1 k · k! We remark here that P- di↵erentiability and intrinsic di↵erentiability, when both of them make sense, are in general di↵erent notions. Indeed, let V, W be the complementary subgroups of H1 defined as V = v =(v1, 0, 0) and W = w =(0,w2,w3) . As observed before, { } { } an intrinsic linear function L : V W is of the form ! 2 L(v)=(0, av1, av1/2), for any fixed a R. 2 An homogeneous homomorhism h : V W is of the form ! h(v)=(0, av1, 0), for any fixed a R. 2 Obviously, L is intrinsic di↵erentiable in v = 0 while h is P-di↵erentiable in v = 0. On the other side, it is easy to check that neither L is P- di↵erentiable nor h is intrinsic di↵erentiable in v =0. We remark also that P-di↵erentiability is not conserved after graph translations. Indeed, consider once more the function h : V W, then 2 !1 graph (h)= (t, at, at /2) : t R .Letp := (0,p2, 0) H , p =0. { 2 }2 2 6 Then p graph (h)= (t, at + p2, (at p2t)/2) : t R is the graph of · { 2 } the function hp : V W defined as hp(v)=(0, av1 + p2, p2v1). It is ! easy to check that hp is not Pansu di↵erentiable in v =0. Finally, if G is the direct product of M and H it is easy to convince oneself that f : M H is Pansu di↵erentiable f is intrinsic di↵erentiable. ! () Definition 4.66. ([15, Definition 3.16]) Let M and H be complemen- tary subgroups in G and f : M H with relatively open in A⇢ ! A M. We say that f is uniformly intrinsic di↵erentiable in if A (i) f is intrinsic di↵erentiable at eachm ¯ M; (ii) df depends continuously onm ¯ , that2 is, for each compact m¯ K⇢ , there is ⌘ = ⌘ : [0, + ) [0, + ), with limt 0+ ⌘(t)=0 suchA that K 1 ! 1 ! 1 1 sup df m¯ 1 (m) df m¯ 2 (m) ⌘ m¯ 1 m¯ 2 ; m k · k1  k · k1 2K (iii) for each compact , there is " = " , : [0, + ) K⇢A A K 1 ! + 1 [0, + ), with limt 0 "(t) = 0 and such that for all m p¯ , and for1 allm ¯ ! 2K 2K 1 df m¯ (m) fp¯ 1 (m) "( m ) m k · k1  k k1 k k1 wherep ¯ =¯m f(¯m). · 57 The uniform intrinsic di↵erentiability implies intrinsic Lipschitz con- tinuity.

Proposition 4.67. Let M and H be complementary subgroups in G and f : M H be uniformly intrinsic di↵erentiable in , with A⇢ ! A A relatively open in M. Then for all m there is r> 0 such that f is intrinsic Lipschitz in B(m, r). 2A A\ Proof. Letp ¯ =¯m f(¯m) graph (f). Since is relatively open there · 2 A exists r> 0 such that := M B(¯m, r) . Observe now that K \ ⇢A (89) p¯ 1 = m¯ 1 = M B(e, r) , K K \ and, by Propositions 4.57 (i) and 4.58 (ii), there exists a positive con- stant C1 > 0 such that

(90) df m¯ (m) C1 m m M B(e, r) . k k1  k k1 8 2 \ Since 1 f 1 (m)= df (m) df (m) f 1 (m) , p¯ m¯ · m¯ · p¯ by the triangle inequality,(90), (89), Definition 4.66 (iii) and Proposi- tion 4.49 (iii), we get the desired conclusion. ⇤

4.8. H-regular surfaces, intrinsic Lipschitz graphs in Heisen- berg groups vs. intrinsic di↵erentiability. In this section we deal with some relationships among intrinsic regular surfaces, intrinsic Lip- schitz graphs and intrinsic di↵erentiability in the framework of the Heisenberg group. More precisely we will present: a characterization of a H-regular surface as locally uniformly • intrinsic di↵erentiable graph (Theorem 4.68); a R¨ademacher type- theorem for 1-codimensional intrinsic Lip- • schitz graphs (Theorem 4.72); a characterization of a function whose the graph is H-regular as • continuous distributional solution of a suitable first nonlinear PDE, namely a balance law (see Theorem 4.75); a characterization of an intrinsic Lipschitz function as continu- • ous distributional solution of a suitable balance law (see Theo- rem 4.77).

Theorem 4.68 (Theorem 4.2,[15]). The following are equivalent (i) S Hn is a H-regular surface (see Definition 4.39); (ii) For⇢ all p S there is an open set such that p and S is the (intrinsic)2 graph of a uniformlyU intrinsic2 diU↵erentiable\U function ' acting between complementary subgroups of Hn. 58 More precisely, with 1 k n, if S is k- dimensional H-regular then ' is defined on a subset of (k, k)- dimensional horizontal subgroup; if if S is k- codimensional H-regular then ' is defined on a subset of (2n +1 k, 2n +2 k)-normal subgroup. Remark 4.69. Theorem 4.68 is an extension of previous result [13, The- orem 4.1] for 1-codimensional intrinsic graphs in Heisenberg groups (see also [60]). Remark 4.70. Describing H-regular submanifolds as (intrinsic di↵eren- tiable) graphs is more general and flexible than using parametrizations or level sets. Indeed, di↵erently from Rn-where d-dimensional C1 em- bedded submanifolds are equivalently defined as non-critical level sets n n d of di↵erentiable functions R R or as images of injective di↵eren- d n ! 1 d n d n tiable maps R R (or as graphs of C functions R R ) - in H , ! ! low dimensional H-regular surfaces cannot be seen as non critical level sets and low codimensional ones cannot be seen as (bilipschitz) images of open sets. The reasons for this are rooted in the algebraic structure of Hn. From one side, low dimensional horizontal subgroups of Hn are not normal subgroups, hence they are not kernels of homogeneous homomorphisms, while they are tangent spaces to low dimensional sub- manifolds; on the other side, injective homogeneous homomorphisms Rd Hn do not exist, if d n + 1(see Remark 4.40). ! From Proposition 4.67 and Theorem 4.68 it follows that Corollary 4.71. Each H- regular surface S Hn is locally a graph of an intrinsic Lipschitz function. ⇢ An interesting result as far as the calculus within Carnot groups is a R¨adamacher type-theorem for 1- codimensional Lipschitz graph within the Heisenberg group. More precisely Theorem 4.72 (Theorem 4.29, [105]). Let Hn = W VHn with V · (1, 1)-dimensional subgroup. If W is relatively open, and f : U⇢ U! V is intrinsic Lipschitz, then f is intrinsic di↵erentiable in , a.e. with U respect to the Haar measure of W. Remark 4.73. It is a (dicult?) open problem whether a R¨adamacher- type theorem holds for higher codimensional Lipschitz graphs in Heisen- berg groups. For instance, assume that H2 R5 = W V with W := ⌘ · exp(span Y1,Y2,T )= (0, 0,w3,w4,w5) and V :== exp(span X1,X2 )= (v ,v , 0{, 0, 0 complementary} { subgroups,} according to the notation{ of} { 1 2 } Example 2.2. Note that subgroup W is isomorphic to the first Heisen- berg group H1 R3 and V to R2. Given an intrinsic Lipschitz map ⌘ f =(f1,f2, 0, 0, 0) : W V is unknown whether it is intrinsic di↵er- ! entiable a.e. with respect to the Haar measure of W. Remark 4.74. Franchi, Marchi and Serapioni recently extended Theo- rem 4.72 to 1-codimensional intrinsic Lipschitz within a more general 59 class Carnot groups called of type ?, which contains step 2 Carnot groups (see Definition 5.11 and [92]). For sake of simplicity, we will restrict to the first Heisenberg group H1 in order to introduce the two remaining characterizations. Moreover we 1 1 will considered so-called X1 graph in H , that is we will consider H 3 ⌘ R = W V with W := exp(span Y1,T )= (0,w2,w3) and V := exp(span ·X )= (v , 0, 0 complementary{ } subgroups,{ according} to { 1} { 1 } the notation of Example 2.2. Note that subgroup W is isomorphic to the (abelian) group R2 and V to R. Moreover each function ' : A⇢ W V can be meant as a function '(0,w2,w3)= (f(w2,w3), 0, 0) ! with f : R2 R. A⇢ ! Theorem 4.75 (Theorem 5.1,[13] and Theorem 1.2,[38]). Let H1 = 2 W V, with W = (0,w2,w3) R and V = (v1, 0, 0) R and let · { }⌘ { }⌘ ' =(f,0, 0) : W V, where is a relatively open set in W. Then the followingA⇢ are equivalent:! A (i) S = graph (')= w '(w): w is H-regular hypersurface; { · 2A} (ii) there exists a continuous function g : R, such that f is a distributional continuous solution of theA balance! law 1 (91) @ f + @ f 2 = g in , w2 2 w3 A that is 1 f@ + f 2@ d 2 = g d 2 C1( ) . w2 2 w3 L L 8 2 c A ZA ✓ ◆ ZA Remark 4.76. Notice that a distributional continuous solution f of (91) with continuous source g can be very irregular. For instance, it may occur that f/BV ( ) (see [131]). 2 loc A Theorem 4.77 ([61] and Theorem 1.1, [36]). Let H1 = W V, with 2 · W = (0,w2,w3) R and V = (v1, 0, 0) R and and let ' = { }⌘ { }⌘ (f,0, 0) : W V, where is a relatively open set in W. Then the followingA⇢ are equivalent:! A (i) ' is (locally) intrinsic Lipschitz; (ii) there exists a measurable (locally) bounded function g : R, such that f is a distributional continuous solution of theA balance! law (91) Example 4.78. In H1, with the notations of Theorem 4.75, fix 1/2 < ↵ ↵< 1 and consider ' : W V, '(0,w2,w3)= (w3 , 0, 0). Then, ↵ ! | | since f(w2,w3)= w3 is a distributional continuos solution of (91) | | 2↵ 2 with g(w2,w3)= 2↵ w3 w3, then S = graph (') is a H regular hy- persurface. Moreover,| one| can easily check that S is not an Euclidean graph in any neighborhood of the origin (see [103, Example 3.8]). Ar- guing in the same way, if ↵ =1/2 , by Theorem 4.77, S turns out to be an intrinsic (locally) Lipschitz graph. 60 Remark 4.79. The approximation, by means regular functions, and a Poincar´einequality for 1-codimensional intrinsic Lipschitz graphs in Heisenberg groups have been obtained, respectively, in [61] and [62]. 4.9. Rectifiability in Carnot groups. Rectifiable sets are basic con- cepts of geometric measure theory. They were introduced in the 1920s in the plane by Besicovitch and in 1947 in general dimensions in Eu- clidean spaces by Federer. A systematic study of rectifiable sets in general metric spaces was made by Ambrosio and Kirchheim in [7] in 2000. However, the definitions they used are not always appropriate in Heisenberg groups, and many other Carnot groups, when equipped with their natural CC metric; indeed, often there are only trivial rec- tifiable sets of measure zero. Let us recall that, according to Federer [80, 3.2.14]

k Definition 4.80. AsetE (X, d) is said to be countably d- rectifi- ⇢ kH able if there a sequence of Lipschitz function fi : Ai (R , Rk ) (X, d)(i =1, 2,...,) such that ⇢ k·k ! k (E 1 f (A )) = 0 . Hd \[i=1 i i Remark 4.81. It is also well-known that in the Euclidean setting that is when X = Rn, by Whitney extension theorem, countable k-rectifiability can be also expressed in terms of k- dimensional surfaces. Namely it holds that E is countably k- rectifiable if and only if there is a se- H1 n k quence of k-dimensional C surfaces (Si)i R such that (E ⇢ H \ 1 S )= 0 [i= i Definition 4.82. A metric space (X, d) is said to be purely k- unrec- k tifiable if for each Lipschitz function f : A (R , k ) (X, d), ⇢ k·kR ! k(f(A)) = 0 . Hd Ambrosio and Kirchheim proved the following result.

Theorem 4.83. ([7, Theorem 7.2]) The first Heisenberg group (H1,d) is purely k- unrectifiable for k =2, 3, 4, for each invariant distance d. A general criterion for unrectifiability was proved by Magnani by means of the area formula for Lipschitz maps between Carnot groups (see Theorem 3.14).

Theorem 4.84. ([146, Theorem 1.1]) Let G be a Carnot group and let V1 denote the first horizontal layer in its stratification (see (1)). Then (G,d) is purely k-unrectifiable i↵ V1 does not contain a (trivial) Lie subalgebra of dimension k. Therefore, taking the previous unrectifiability results into account, a new suitable notion of rectifiability needs in Carnot groups fitting better the new geometry. 61 4.9.1. (Q 1)-rectifiable sets in Carnot groups. A suitable notion of rectifiability in Carnot group for sets of metric dimension (Q 1) has firstly been introoduced in the setting of Heisenberg group [99] and then step 2 Carnot groups for studying the structure of sets of finite G-perimeter [102] . We remind that De Giorgi’s celebrated structure theorem in Eu- clidean spaces ([72], [73]) states that if E Rn is a set of locally finite perimeter, then the associated perimeter measure⇢ @E is concentrated on a portion of the topological boundary @E, the| so-called| reduced n 1 boundary @⇤E @E. In addition, @⇤E is -rectifiable, i.e. @⇤E, up to a set of (n⇢ 1)-Hausdor↵measure zero,H is a countable union of compact subsets of C1 submanifolds and the perimeter measure is the (n 1)-Hausdor↵measure of the reduced boundary. Roughly speak- ing, this says that the perimeter measure in supported on a portion of the topological boundary @E, that can be expressed – after removing a negligible set of “bad points” – as the countable union of compact subsets of “good hypersurfaces”. If in the spirit of De Giorgi’s theorem we want to describe the struc- ture of sets of finite intrinsic perimeter in a Carnot group G, we need a natural notion of rectifiable subsets, and in this perspective, the correct definition of “good hypersurfaces”, i.e. of intrinsic C1-regular submanifold of G given in Definition 4.16 a key tool. Keeping in mind this notion, the following definition is the natural counterpart of the corresponding Euclidean definition (see [100] and [92]). Definition 4.85. (i) E G is said to be 1-codimensional (Q 1)- ⇢ dimensional G-rectifiable if there exists a sequence of G-regular hypersurfaces (Si)i such that

1 Q 1 (92) E Si =0. S1 \ i=1 ! [ (ii) E G is said to be 1- codimensional (Q 1)-dimensional GL- rectifiable⇢ if there exists a sequence of 1- codimensional intrinsic Lipschitz graphs (Si)i such that

1 Q 1 (93) E Si =0. S1 \ i=1 ! [ Remark 4.86. Franchi, Marchi and Serapioni recently proved that, for Carnot group of type ? (see Definition 5.11) 1-codimensional (Q 1)- dimensional G-rectifiability and 1-codimensional (Q 1)-dimensional G-rectifiability are equivalent (see [92, Proposition 4.4.4]. Before we enter the study of the rectifiability of the reduced boundary (whatever this means, as we shall see below), let us point out the relationships between our definition in Carnot groups and the standard Euclidean notion. The following result proved in [102] yields that “ 62 negligible” subsets of codimension 1 in a Carnot group with respect to the Euclidean distance are “ negligible” subsets of codimension 1 with respect to Carnot–Carath´edory distance. Proposition 4.87. ([102, Proposition 4.4]) Let G =(Rn, ) be a Carnot group. For any ↵ 0 and R>0 there is a constant c(↵·,R) > 0 such that for any set E G U (0,R) ⇢ \ 1 ↵+Q n ↵ (94) (E) c(↵,R) (E),↵0. H1  H In particular, for all E G, ⇢ ↵ ↵+Q n (95) (E)=0 = (E)=0,↵0. H )H1 Proposition 4.87 combined with Theorem 4.19 yields Theorem 4.88. Let G = Rn be a Carnot group. Then, if S is a (n 1)-dimensional Euclidean rectifiable set of Rn then S is also (Q 1)- dimensional G-rectifiable. On the other hand, there are (Q 1)-dimensional G-rectifiable sets in a Carnot groups G =(Rn, ) that are not (n 1)-dimensional Euclidean · rectifiable. Indeed, in [26] a set N R3 is constructed, such that for an appropriate ">0, ⇢ 3 (N) = 0 and 2+"(N) > 0. H1 H Hence N is (trivially) 3 = (Q 1)-dimensional H1-rectifiable, but it is not 2-dimensional Euclidean rectifiable because its Euclidean Hausdor↵ dimension is strictly larger than 2. As we mentioned above, a sharper result in this direction is contained in [131]: there exist G-regular hy- persurfaces in the Heisenberg group H1 (Q =4,n = 3) with Euclidean Hausdor↵dimension 2.5. We recall here that relationships between Euclidean and intrinsic Hausdor↵measure in Heisenberg groups have been deeply investigated in [26], where also sharp results were obtained. This study was extend more recently to a general Carnot group in [28]. Eventually let us stress that each H- regular t- graph in Hn R2n+1, ⌘ that is graph with respect to the subgroup T = (0,...,0,p2n+1) , keeps the same Euclidean metric dimension. Indeed{ } Theorem 4.89. ([200, Corollary 1.2]) Let S Hn be an H-regular hypersurface that is not (Euclidean) countably ⇢ 2n-rectifiable; then S H is not a t-graph.In particular each H regular t-graph has Euclidean metric dimension 2n.

4.9.2. Rectifiability in Heisenberg groups. In the setting of Heisenberg groups, a more complete notion of rectifiability can be introduced. With the intrinsic notion of regular submanifolds proposed in Defi- nition 4.39 and the one of intrinsic Lipschiz graph in Definition 4.46, it is possible, following the two usual approachs , to give an (intrinsic) notion of rectifiable sets in Hn. 63 Definition 4.90. (i) E Hn is a (k, H)-rectifiable set if there is ⇢ a sequence of k-dimensional H- regular surfaces Si such that

km (E i1=1Si)= 0 S1 \[ where km = k, if 1 n ,and km = k + 1 ,if n +1 k 2n. n    (ii) E H is a (k, HL)-rectifiable set if : ⇢ (ii1)for1 k n, there exists a sequence of Lipschitz  k n function fi : Ai R H such that ⇢ ! k (E i1=1fi(Ai)) = 0 : S1 \[ (ii2)forn+1 k 2n, there exists a sequence of intrinsic Lipschitz graph S= graph (f ) of dimension k, with f : A i i i i ⇢ Wi Vi, such that ! k+1 (E i1=1Si)= 0, S1 \[ where Wi and Vi are , respectively, (k, k + 1)- and (2n +1 k, 2n +1 k)- subgroups, which are complementary. Remark 4.91. We recall that, by a combination of Proposition 4.87 with Theorem 4.19, for n +1 k 2n,ak-dimensional Euclidean   rectifiable set E R2n+1 Hn always is a (k, H)-rectifiable set, while ⇢ ⌘ the converse is false. On the contrary, for 1 k n,a(k, H)- rectifiable set is k-dimensional Euclidean rectifiable while the opposite is false: for instance, the vertical T axis in H1 R3 gives the simplest example of an Euclidean 1-dimensional rectifiable⌘ set that is not 1- dimensional H-rectifiable. Remark 4.92. Definition 4.90 (i) is non trivial only if 1 k n,   because, for n +1 k and f : A Rk Hn Lipschitz, always k (f(A)) = 0 (see Theorem 4.84), hence⇢ we! will consider this notion ofS1 rectifiability only if 1 k n.   Remark 4.93. We explicitly observe that we do not know if (k, HL)- rectifiability and (k, H)- rectifiability are equivalent if 1 k n.In Euclidean spaces these two definitions are equivalent and in fact the type of definition below was the original definition of Federer and it has also often been used in general metric spaces. We dont know if they are equivalent in Hn. The problem is that we dont have a suitable Whitney extension theorem for maps f : F Rk Hn when F is closed. Eventually, by Remark 4.86, they are equivalent⇢ ! for k =2n. In the paper [161] we approached the problem of characterizing in- trinsic k-rectifiable sets in Hn through almost everywhere existence and uniqueness of generalized tangent subgroups or tangent measures, ac- cording to the notions introduced, respectively, by Mattila [159] and Preiss [187] in the Eucldiean setting. About generalized tangent groups the result reads as follows. 64 Theorem 4.94. Let E Hn be a set which has locally finite km measure. ⇢ S1

(i) ([161, Theorem 3.14]) Let 1 k n. Then E is (k, HL)-rectifiable   k if and only if it has an approximate tangent subgroup Tp, a.e. The latter means that, for k a.e.p E, there exists a homogeneousS1 S1 2 subgroup Tp, of topological and metric dimension k , such that k 1 (E B(p, r) q : d (p q, Tp) >sd (p, q) ) lim S1 \ \{ 1 · 1 } =0, r 0 k ! r for each s> 0. (ii) ([161, Theorem 3.15])Let n +1 k 2n. Then E is (k, H)-   rectifiable if and only if it has an approximate tangent subgroup Tp, k a.e. The latter means that, for k a.e.p E, there exists a homo- S1 S1 2 geneous subgroup Tp, of topological dimension k and metric dimension k +1, such that k+1 1 (E B(p, r) q : d (p q, Tp) >sd (p, q) ) lim S1 \ \{ 1 · 1 } =0, r 0 k+1 ! r for each s> 0, and k+1 (E B(p, r)) (96) lim inf S1 \ > 0 . r 0 k+1 ! r Remark 4.95. We point out that some other notions of rectifiability in Carnot groups were introduced by Pauls [182] and, recently, by Ghezzi- Jean [114]. However, to our knowledge, there is no comparison among these notions of rectifiablity.

5. Sets of finite perimeter and minimal surfaces 5.1. De Giorgi’s structure theorem for sets of finite perimeter in Carnot groups of step 2. Thus we are left with the notion of reduced boundary for subsets of a Carnot group. The definition we give is a simple translation of the Euclidean one, as follows. Definition 5.1. (Reduced boundary) Let E be a G-Caccioppoli set; we say that x @⇤ E if 2 G i) @E (U(x, r)) > 0foranyr>0;( | |G

ii) there exists lim ⌫E d @E ;( r 0 | |G ! ZU(x,r)

(iii) lim ⌫E d @E =1. r 0 G U(x,r) | | m ! Z R 1 The limits in Definition 5.1 should be understood as a convergence of the averages of the coordinates of ⌫E with respect to the chosen moving base of the fibers. 65 Definition 5.1 is a straightforward extention of its Euclidean coun- terpart, but its utility is not obvious. Indeed, in the Euclidean setting, it is immediate to show that the perimeter measure is concentrated on the reduced boundary, since, by Lebesgue–Besicovitch Di↵erentiation Lemma, given a Radon measure µ,foranyf L1 (dµ) 2 loc

lim f(y) dµE f(x) r 0 y x

lim ⌫E d @E = ⌫E(x), for @E -a.e. x, r 0 G G ! U (x,r) | | | | Z 1 that is @E -a.e. x G belongs to the reduced boundary @⇤ E. | |G 2 G From now on the group G will be a step 2 Carnot group. Indeed, the keystep for the main result of this paper, i.e. the so-called Blow-up Theorem stated below, fails to be true for general groups of step greater than 2, as we can see from the Example 5.4. Specializing our notations, in step 2 Carnot groups, we have g = V V , [V ,V ]=V , [V ,V ]= 0 , 1 2 1 1 2 1 2 { } and Q = m +2(n m ). 1 1 We can prove now the following results. i) At each point of the reduced boundary of a G-Caccioppoli set there is a (generalized) tangent group; ii) Both the reduced boundary and the measure-theoretic boundary are (Q 1)-dimensional G-rectifiable sets; Q 1 iii) @E G = c @⇤E, i.e. the perimeter measure equals a constant times| the| sphericalS1 (Q 1)-dimensional Hausdor↵measure restricted to the reduced boundary. iv) An intrinsic holds for G-Caccioppoli sets. 66 The precise meaning of statement i) is the content of the following Blow-up Theorem. It is precisely point i) that can be false in a general Carnot group. Indeed we provide an example of a G-regular hypersur- face S = @E in a step 3 group (the so-called Engel group, see e.g. [116], [169]) such that 0 @ E but E has not generalized tangent group at G⇤ that point. 2 Statement iii) fits in the general problem of comparing di↵erent geo- metric measures in Carnot groups. A good reference for this prob- lem, in Euclidean spaces, is Matilla’s book [159]. In the setting of the Heisenberg group, in [66] it is proved that the perimeter of an Euclidean C1,1-hypersurface is equivalent to its (Q 1)-dimensional intrinsic Hausdor↵measure, whereas in [99] it is proved that on the boundary of sets of finite intrinsic perimeter the (Q 1)-dimensional intrinsic spherical Hausdor↵measure coincide – after a suitable nor- malization – with the perimeter measure. In the setting of general Carnot groups the problem is essentially open. The equivalence of the intrinsic perimeter and of the (Q 1)-dimensional intrinsic Hausdor↵ measure for C1 -hypersurfaces in a general Carnot groups has been G proved in the previous subsection. In addition, the perimeter measure of a smooth set in general subriemannian spaces equals the intrinsic Minkowski content, as it is proved in [174]. In Ahlfors-regular metric spaces, a general representation theorem of the perimeter measure of sets of finite perimeter in terms of the Hausdor↵measure is proved in [3] (see also the refined result for subriemannian manifolds in [4]), showing that the intrinsic perimeter admits a density # with respect to the Hausdor↵measure that is locally summable and bounded away from zero. Statement iii) says precisely that, thanks to i) and ii), in step 2 Carnot groups the function # is constant.

To state our result, let us fix few notations. For any set E G, ⇢ x0 G and r>0 we consider the translated and dilated sets Er,x0 defined2 as

1 1 Er,x0 = x : x0 r(x) E = ⌧x E. { · 2 } r 0 If x0 is fixed and there is no ambiguity, we shall write simply Er, and in addition we set Ex0 = E1,x0 . Moreover if v HGx0 we define the + 2 halfspaces S (v) and S(v)as G G + S (v) := x : ⇡x x, v x 0 (97) G { h 0 i 0 } S(v) := x : ⇡ x, v 0 . G { h x0 ix0  } where ⇡x0 is the map defined in (43). g + The common topological boundary T (v)ofS (v) and of S(v) is G G G the subgroup of G T g (v) := x : ⇡ x, v =0 . G { h x0 ix0 } 67 Theorem 5.3. (Blow-up Theorem) If E is a G-Caccioppoli set, x @ E and ⌫ (x ) H is the inward normal as defined in (54) 0 G⇤ E 0 Gx0 then2 2 1 (98) lim 1 = 1 + in L ( ) Er,x S (⌫ (x0)) loc G r 0 0 G E ! and for all R>0 + (99) lim @Er,x0 (U (0,R)) = @S (⌫E(x0)) (U (0,R)). r 0 G 1 G G 1 ! | | | | Notice that, by Proposition 3.34, + n 1 g @S (⌫E(x0)) (U (0,R)) = (T (⌫E(0)) U (0,R)). | G |G 1 H G \ 1 As we have already pointed out, Theorem 5.3 fails to hold in general Carnot groups of step k>2. In fact, the core of the following example consists in showing that in Carnot groups of step greater than 2 can exists cones (i.e. dilation-invariant sets) that are not flat (they are not of the form S±(v) for some horizontal vector v) but nevertheless with G vertex belonging to the reduced boundary. The following counterexample was inspired by Martin Reimann, and then Roberto Monti found a preliminary form of the counterexample itself. Example 5.4. Let us recall the definition of Engel algebra and group. 4 Let E =(R , ) be the Carnot group whose Lie algebra is g = V1 V2 V3 · with V1 = span X1,X2 , V2 = span X3 , and V3 = span X4 , the only non zero commutation{ } relations being{ } { } [X ,X ]= X , [X ,X ]= X . 1 2 3 1 3 4 In exponential coordinates the group law takes the form 4 4 x y = H( x X , y X ), · i i i i i=1 i=1 X X where H is given by the Campbell-Hausdor↵formula 1 1 (100) H(X, Y )=X + Y + [X, Y ]+ ([X, [X, Y ]] [Y,[X, Y ]]) . 2 12 In exponential coordinates an explicit representation of the vector fields is x x x x x x2 X = @ + 2 @ +( 3 1 2 )@ ,X= @ 1 @ + 1 @ 1 1 2 3 2 12 4 2 2 2 3 12 4 x X = @ 1 @ ,X= @ . 3 3 2 4 4 4 Let E = x R4 : f(x) 0 , where { 2 } 1 1 f(x)= x (x2 + x2) x x + x . 6 2 1 2 2 1 3 4 68 Since @E = x R4 : f(x)=0 is a smooth Euclidean manifold, then { 2 } E is a G-Caccioppoli set (see Proposition 3.34). Moreover, 1 f(x)= 0, (x2 + x2) , rE 2 1 2 and, by the Implicit Function Theorem (Theorem 4.20), f(x) ⌫ (x)= rE =(0, 1) E f(x) |rE | for all x @E N, where N = x E : x1 = x2 =0. Since 2 \ { 2 } @E E(N) = 0, then the origin belongs to the reduced boundary of E. | | 3 On the other hand, since f(x)= f(x)for>0, it follows that E,0 = E = E, so that (98) fails to be true since E is not a vertical halfspace. Even if we do not enter into the details of the proof of Theorem 5.3, we want to stress the technical point where the assumption on the step of G is used. In the Euclidean setting an elementary statement says @f @f that = = = 0 implies f = f(x1). In Carnot groups the @x2 ··· @xn corresponding statement should be that the vanishing of X2f to Xm1 f yields that f is a function of just one variable. But this is false as examples in the Heisenberg group H1 show (see example 5.6 below).

What is possible to prove in step 2 groups is that if Y1,...,Ym1 are left invariant smooth orthonormal (horizontal) sections, if Y f = = 2 ··· Ym1 f = 0 and if Y1f is positive, then f is an increasing function of one variable. Example 5.4 shows that in groups of step 3 or larger, even this last weaker statement is false. Lemma 5.5. ([102, Lemma 3.6]) Let G be a step 2 group and let

Y1,...,Ym1 be left invariant smooth orthonormal sections of HG. As- sume that g : G R satisfies ! (101) Y g 0 and Y (g)=0 if j =2,...,m . 1 j 1 Then the level lines of g are ”vertical hyperplanes orthogonal to Y1” that is sets that are group translations of S(Y ) := p : ⇡ p, Y (0) =0 . 1 { h 0 1 i } Example 5.6. ([9, Remark 5.4])The following simple example shows that the sign condition is essential for the validity of the conclusions of Lemma 5.5, even in the first Heisenberg group H1.Letp =(x, y, t) 1 y x 2 H , and the vector fields X1 := @x @t and Y1 := @y + @t, the 2 2 function 1 g(x, y, t) := t xy 2 ✓ ◆ 1 (with : R R smooth) satisfies X1g(x, y, t)= y0 t xy and ! 2 ✓ ◆ Y1g = 0. Therefore the sets Et := g

X11Et 0orX11Et 0 (depending on the sign of 0 and y), but are not locally halfspaces.

We can state now our main structure theorem for G-Caccioppoli sets. Theorem 5.7. (Structure of G-Caccioppoli sets)Let G be a step 2 Carnot group , endowed with the invariant distance d . Let E G be a -Caccioppoli set and let @ E denote its reduced boundary.1 Then⇢ G G⇤ (i) @E = @E @⇤ E; | |G | |G G (ii) there exists a constant #0 = #0(G) > 0 s.t. Q 1 ⇥ ( @E ,x)= # x @⇤ E; | |G 0 8 2 G Q 1 where the (Q 1)-dimensional density ⇥ ( @E , ) must be | |G · understood according to Definition 4.24 (i) with X = G and d = d ; (iii) @ E is1 (Q 1)-dimensional -rectifiable; G⇤ G (iv) @E = # Q 1 @ E, G 0 G⇤ | | Q 1 S where denotes the (Q 1)-dimensional spherical Hausdor↵ S measure defined according to Definition 2.17 (ii) with X = G and d = d . 1 Finally, the following divergence theorem is then an easy consequence of Theorem 5.7, but we stress the fact that the measure theoretic boundary of E (see (71)), @ ,GE, appears in the identity (ii). As in the Euclidean space, the corresponding⇤ statement for the reduced bound- ary holds straightforwardly. However, the interest of the statement for the measure theoretic boundary comes not only from the fact that – as in the Euclidean setting – the last one is sometimes easier to deal with, but mainly from the fact that the measure theoretic boundary – unlike the reduced boundary – is independent of the choice of the metric.

Theorem 5.8. (Divergence Theorem) Let E be a G-Caccioppoli set, then there exists a positive constant #0 = #0(G) such that Q 1 (i) @E G = #0 @ ,GE, | | S1 ⇤ and the following version of the divergence theorem holds (ii) n Q 1 1 div G'd = #0 ⌫E,' d , ' C0(G,HG). 1 E L @ , Eh i S 8 2 Z Z ⇤ G Remark 5.9. Ambrosio, Kleiner and Le Donne partially extended The- orem 5.3 in a general Carnot group [9]. More precisely they proved that ,at each point of the reduced boundary, a vertical half-spaces belongs to the family of tangent spaces at that point, but, a priori, the family could be not a singleton. 70 Remark 5.10. Marchi fully extended Theorem 5.7 to a special class of Carnot group, called Carnot group of type ? (see [158]).

Definition 5.11. We say that a stratified Lie algebra g = V1 Vk is of type ? if there exists a basis X ,...,X of V such that··· { 1 m1 } 1 [Xj, [Xj,Xi]] = 0, for all i, j =1,...,m1. A Carnot group G is said to be a group of type ? if its Lie algebra g is of type ?. Obviously each step 2 Carnot group is of type ?, but it can be proved that there are Carnot groups . Remark 5.12. Ambrosio, Ghezzi and Magnani exented Theorem 5.3 for sets of finite perimeter in a sub-Riemannian manifolds, whose the tangent group is isomorphic to a step 2 Carnot group [6]. 5.2. Minimal boundaries in Carnot groups. TO BE ADDED

5.2.1. Existence of minimal boundaries `ala De Giorgi and calibrations.

5.2.2. Existence of non parametric minimal surfaces in Heisenberg groups: minimal t- and 1- codimensional intrinsic graphs.

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Francesco Serra Cassano: Dipartimento di Matematica, Universita` di Trento, Via Sommarive 14, 38050, Povo (Trento) Italy E-mail address: [email protected]

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