Intrinsically Lipschitz functions with normal target in Carnot groups
II N T R I N S I C A L L Y L I P S C H I T ZF U N C T I O N S W I T H N O R M A L T A R G E T I NC A R N O T G R O U P S g i oac c h i n o a n t o n e l l i * a n d a n d r e a m e r l o ** abstract We provide a Rademacher theorem for intrinsically Lipschitz functions φ : U ⊆ W → L , where U is a Borel set, W and L are complementary subgroups of a Carnot group, where we require that W is a Carnotsubgroup and L is a normal subgroup. Our hypotheses are satisfied for example when W is a horizontal subgroup.Moreover, we provide an area formula for this class of intrinsically Lipschitz functions. keywords Carnot groups, intrinsically Lipschitz functions, Rademacher theorem, area formula. msc ( ) C , E , A , Q , A . In the last years there has been an increasing interest towards Geometric Measure Theory in the non-smooth set-ting, and in particular in the setting of sub-Riemannian Carnot groups for what concers the notion of rectifiability.This line of research was initiated by the result in [ ] in which the authors proved that the first Heisenberg group H is not k -rectifiable, according to Federer’s definition (see [ ]), for k ≥ ] for a wide study of the notion of intrinsically Lipschitzfunction and to [ ] for some recent developments. The notion of intrinsically differentiable function has beenintroduced in [ ] and then widely studied in the last years: we refer the reader to [ , , , , ] for somedevelopments.One of the main open questions in this area of research is whether a Rademacher type theorem holds. Namely,is it true that every intrinsically Lipschitz function between complementary subgroups of a Carnot group is in-trinsically differentiable almost everywhere? Some first answers have been given in [ , ] in the setting of Carnotgroups G of type (cid:63) , i.e., a class larger than Carnot groups of step , and for maps φ : U ⊆ W → L , where U is open and W and L are complementary subgroups of G , with L horizontal and one-dimensional . Up to now it isnot known either if the results in [ , ] hold in arbitrary Carnot groups or if, in some special case, one can prove aRademacher type theorem with L horizontal not necessarily one-dimensional.In this note we prove that a Rademacher type theorem with normal target is true in arbitrary Carnot groups,provided that the function is defined on a subset of a Carnot subgroup . We provide also an area formula for mapsthat satisfy our hypotheses.
Theorem . . Let W and L be two complementary subgroups of a Carnot group G . Let us assume that L is normal and letU ⊆ W be a Borel set. Let φ : U ⊆ W → L be an intrinsically Lipschitz function. Then, if we call Φ : U → G the graphmap of φ , we have that Φ is Lipschitz. * Scuola Normale Superiore, Piazza dei Cavalieri, , Pisa, Italy, ** Universitá di Pisa, Largo Bruno Pontecorvo, , Pisa, Italy. a r X i v : . [ m a t h . M G ] J un reliminaries 2 If in addition W is a Carnot subgroup of G , then φ is intrinsically differentiable H k (cid:120) W -almost everywhere on U, wherek : = dim H ( W ) ; the map Φ is Pansu differentiable H k (cid:120) W -almost everywhere on U, and the following formula holds: H k ( Φ ( V )) = (cid:90) V J ( d Φ x ) d H k ( x ) , for every Borel set V ⊆ U , ( ) where d Φ x is the Pansu differential of Φ at x ∈ V and J ( d Φ x ) is its Jacobian, see ( ) . We stress that the hypotheses of Theorem . are satisfied for example when W is horizontal. Thus our theoremholds for intriniscally Lipschitz horizontal surfaces in arbitrary Carnot groups. We stress also that an area formulafor maps that parametrizes C H -submanifolds, that is a more restrictive condition than being intrinsically Lipschitz,has been recently obtained in [ ].The proof of Theorem . is considerably simpler than the proof of a low-codimensional Rademacher theorembecause in that case it may not be true, as in our case, that the intrinsic Lipschitz property of φ reads as thegraph map Φ being Lipschitz (see Remark . ). In fact we apply Pansu-Rademacher theorem and Magnani’s areaformula to deduce that the graph map Φ is Pansu differentiable almost everywhere and the area formula holds.Thus in order to complete the proof of Theorem . , we are left to relate the Pansu differentiability of the graphmap Φ with the intrinsic differentiability of the map φ , and this is done in Proposition . . Acknowledgments : The authors are grateful to Enrico Le Donne for several suggestions that led to an improve-ment of this note. They are also grateful to Sebastiano Don for precious comments.
For the basic terminology on Carnot groups and graded groups we refer the reader to [ ]. We recall somenotation. Every graded group has a one-parameter family of dilations that we denote by { δ λ : λ > } . We willindicate with δ λ both the dilation of factor λ on the group and its differential.Given a graded group G , we fix a homogeneous norm (cid:107) · (cid:107) on G , which is unique up to bi-Lipschitz equivalence.Moreover, by [ , Theorem . ] on every Carnot group there exists a slight variation of the anisotropic homoge-neous norm that induces a left-invariant homogeneous distance d . We eventually fix this particular homogeneousnorm from now on . We also recall that with homogeneous homomorphism we mean a homomorphism that commuteswith δ λ , for every λ >
0. We now deal with complementary subgroups and maps between them.
Definition . (Complementary subgroups) . Given a Carnot group G with identity e , we say that two subgroups W and L are complementary subgroups in G if they are homogeneous , i.e., closed under the action of δ λ for every λ > G = W · L , and W ∩ L = { e } .Given two complementary subgroups W and L in a Carnot group G , we denote the projection maps from G onto W and onto L by π W and π L , respectively. Defining g W : = π W g and g L : = π L g for any g ∈ G , one has g = ( π W g ) · ( π L g ) = g W · g L .We now describe what is the parametrizing function of some translation of the intrinsic graph of a function. Unlessanything different is declared, we eventually fix from now on two arbitrary complementary subgroups W and L of a Carnot group G . Definition . (Intrinsic graph) . Given a function φ : U ⊆ W → L , we define the intrinsic graph of φ as follows:graph ( φ ) : = { Φ ( w ) : = w · φ ( w ) : w ∈ U } = : Φ ( U ) . Definition . (Intrinsic translation of a function) . Given a function φ : U ⊆ W → L , we define, for every q ∈ G , U q : = { a ∈ W : π W ( q − · a ) ∈ U } ,and φ q : U q ⊆ W → L by setting φ q ( a ) : = π L ( q − · a ) − · φ (cid:0) π W ( q − · a ) (cid:1) . ( ) reliminaries 3 Proposition . . Given a function φ : U ⊆ W → L , and q ∈ G , the following facts hold:(a) graph ( φ q ) = q · graph ( φ ) ;(b) if L is normal one gets that, for all a ∈ U, π W ( q − · a ) = q − W · a , π L ( q − · a ) = a − · q W · q − L · q − W · a , ( ) and thus φ q ( a ) = a − · q W · q L · q − W · a · φ ( q − W · a ) , U q = q W · U ; ( ) Proof.
The proof of (a) directly follows from ( ), which yields a · φ q ( a ) = q · π W ( q − · a ) · φ ( π W ( q − · a )) , ∀ a ∈ U q . ( )Let us prove (b). Since L is normal, the following holds: a − · q W · q − L · q − W · a = a − · q W · q − L · ( a − · q W ) − ∈ L .Moreover it holds that q − W · a ∈ W , and since q − W · a · a − · q W · q − L · q − W · a = q − · a ,the equation ( ) holds and then ( ) is a consequence of ( ) and ( ).For the notion of intrinsically Lipschitz function we refer the reader to [ ]. We explicitly recall here, for thereader’s benefit, only the notions of intrinsically linear and intrinsically differentiable functions. For these defini-tions, and the study of some properties related, we refer the reader to [ , , , , , ]. Eventually, in order to fixthe notation, we recall the notion of Pansu differentiability, see also [ ]. Definition . (Intrinsically linear function) . The map (cid:96) : W → L is said to be intrinsically linear if graph ( (cid:96) ) is ahomogeneous subgroup of G . Definition . (Intrinsically differentiable function) . Let φ : U ⊆ W → L be a function with U Borel in W .Fix a density point a ∈ D ( U ) of U , let p : = φ ( a ) − · a − and denote by φ p : U p ⊆ W → L the shiftedfunction introduced in Definition . . We say that φ is intrinsically differentiable if there is an intrinsically linear map d φ φ a : W → L such that lim b → e , b ∈ U p (cid:107) d φ φ a [ b ] − · φ p ( b ) (cid:107)(cid:107) b (cid:107) =
0. ( )The function d φ φ a is called the intrinsic differential of φ at a . Notice that one can take the limit in ( ) because, if a ∈ D ( U ) , then e ∈ D ( U p ) . This last claim comes from the invariant properties of Proposition . . Remark . (Intrinsic differentiability and tangent subgroups) . Whenever the intrinsic differential introduced inDefinition . exists, it is unique: see [ , Theorem . . ]. Let us recall the notion of tangent subgroup to the graphof a function. If we fix φ : U ⊆ W → L , we say that a homogeneous subgroup T of G is a tangent subgroup tograph ( φ ) at a · φ ( a ) if the following facts hold:(a) T is a complementary subgroup of L ;(b) The limit lim λ → ∞ δ λ (cid:16) ( a · φ ( a )) − · graph ( φ ) (cid:17) = T ,holds in the sense of Hausdorff convergence on compact subsets of G . If k : = dim H ( W ) is the Hausdorff dimension of W , we say that a ∈ U is a density point of U , and we write a ∈ D ( U ) , if H k (cid:120) W ( U ∩ B ( a , r )) / H k (cid:120) W ( B ( a , r )) → r goes to 0. roof of the theorem 4 In [ ] the authors state the following: a function φ : U ⊆ W → L , with U open , is intrinsically differentiable at a if and only if graph ( φ ) has a tangent subgroup T at a · φ ( a ) and moreover T = graph ( d φ φ a ) . The proof of thislatter claim follows from [ , Theorem . . ], that shows one part of the statement, and generalizing [ , Theorem . ]. Definition . (Pansu differentiability) . Let W and G be two arbitrary graded groups endowed with two homoge-neous left-invariant distances d W and d G , respectively. Given f : U ⊆ W → G with U Borel and a density point a ∈ D ( U ) , we say that f is Pansu differentiable at a if there exists a homogeneous homomorphism d f a : W → G ,that we call Pansu differential at a , such thatlim a ∈ U , a → a d G ( f ( a ) − · f ( a ) , d f a [ a − · a ]) d W ( a , a ) = In what follows we prove that, in case L is normal , the intrinsic differentiability of a function φ can be read asthe Pansu differentiability of the graph map Φ . We start with a lemma. Lemma . . Let W and L be two complementary subgroups of a Carnot group G , with L normal . The map (cid:96) : W → L is intrinsically linear if and only if the graph map of (cid:96) , i.e., L : W → G defined as L ( w ) : = w · (cid:96) ( w ) , is a homogeneoushomomorphism.Proof. The proof is contained in [ , Part (i) of Proposition . ]. In the reference it is stated and proved forHeisenberg groups, but the proof of Part (i) holds verbatim in our context, because it merely uses the fact that L is normal.The forthcoming proposition is inspired by [ , Proposition . (i)]. We give a detailed proof in our context, sincethe last part of the argument, i.e., the part in which we invoke the forthcoming Lemma . , is different with respectto the reference. Proposition . . Let W and L be two complementary subgroups of a Carnot group G , with L normal . Let φ : U ⊆ W → L be a function with U Borel . Given a density point a ∈ D ( U ) , we have that φ is intrinsically differentiable at a if andonly if the graph map Φ : U ⊆ W → G is Pansu differentiable at a . Moreover if any of the previous two holds we have thefollowing formula: d Φ a [ w ] = w · d φ φ a [ w ] , ∀ w ∈ W . Proof.
Let us first notice that p : = φ ( a ) − · a − = a − · a · φ ( a ) − · a − , and thus, from the fact that L is normal, ( p ) W = a − , ( p ) L = a · φ ( a ) − · a − . ( )Let us assume that φ is intrinsically differentiable at a . We are going to prove that the graph map Φ is Pansudifferentiable at a . From the intrinsic differentiability of φ , we know that there exists the intrinsic differential d φ φ a as in Definition . , which is an intrinsically linear map by definition. We define its graph map d Φ a [ w ] : = w · d φ φ a [ w ] , ∀ w ∈ W . ( )From Lemma . , it follows that the map d Φ a is a homogeneous homomorphism. We show that it is the Pansudifferential of the graph map Φ . Indeed, let us take w ∈ U and compute d Φ a [ a − · w ] − · Φ ( a ) − · Φ ( w ) = d φ φ a [ a − · w ] − · w − · a · φ ( a ) − · a − · w · φ ( w ) == d φ φ a [ a − · w ] − · φ p ( a − · w ) , ( )where in the first equality we used the definition ( ) and in the second one we used ( ) and the explicit expressionin the first equality of ( ). Notice that, from the second equality of ( ) and the first equality of ( ), we get that U p = a − · U . Thus ( ), jointly with the intrinsic differentiability of φ , see ( ), tells us thatlim w → a , w ∈ U (cid:107) d Φ a [ a − · w ] − · Φ ( a ) − · Φ ( w ) (cid:107)(cid:107) a − · w (cid:107) = roof of the theorem 5 that implies the Pansu differentiability of Φ at a with differential d Φ a .Vice versa, let us assume that the graph map Φ is Pansu differentiable at a . Thus there exists a homogeneoushomomorphism d Φ a : W → G such thatlim w → a , w ∈ U (cid:107) d Φ a [ a − · w ] − · Φ ( a ) − · Φ ( w ) (cid:107)(cid:107) a − · w (cid:107) =
0. ( )By using the fact that L is normal, and by simple computations, we get that π W (cid:0) d Φ a [ a − · w ] − · Φ ( a ) − · Φ ( w ) (cid:1) = (cid:0) ( d Φ a [ a − · w ]) W (cid:1) − · a − · w , ∀ w ∈ U .Now notice that since there exists a geometric constant C > (cid:107) g W (cid:107) ≤ C (cid:107) g (cid:107) for every g ∈ G (see [ ,Proposition . ]), from the previous equality and ( ) we deducelim w → a , w ∈ U (cid:13)(cid:13)(cid:13)(cid:0) ( d Φ a [ a − · w ]) W (cid:1) − · a − · w (cid:13)(cid:13)(cid:13) (cid:107) a − · w (cid:107) =
0. ( )Since d Φ a is a homogeneous homomorphism, the limit in ( ) allows us to use the forthcoming Lemma . with U e : = a − · U . This leads to the equality π W ◦ ( d Φ a ) = ( id ) | W .Thus, by using the splitting, for every w ∈ W it holds d Φ a [ w ] = : w · d φ φ a [ w ] for some map d φ φ a : W → L .Now we are in a position to apply Lemma . to deduce that d φ φ a is intrinsically linear, because its graph isa homogeneous homomorphism being a Pansu differential. Now the intrinsic differentiability of φ at a , withintrinsic differential d φ φ a , follows by joining the computations in ( ) with ( ). Lemma . . Let W and L be two complementary subgroups of a Carnot group G and let U e ⊆ W be a Borel set containingthe identity e and for which e is a density point . Let us assume F : W → G is a continuous homogeneous map, i.e., itcommutes with δ λ for every λ > . Let us further assume that lim w → e , w ∈ U e (cid:13)(cid:13) ( F ( w ) W ) − · w (cid:13)(cid:13) (cid:107) w (cid:107) = Then F ( w ) W = w for every w ∈ W .Proof. Notice first that the map w → ( F ( w ) W ) − is a homogeneous map. Indeed, from the homogeneity of F , thehomogeneity of the projection onto W and the homogeneity of the inverse, respectively, we get (( F ( δ λ w )) W ) − = (( δ λ F ( w )) W ) − = ( δ λ ( F ( w ) W )) − = δ λ (( F ( w ) W )) − , ∀ λ >
0. ( )Set S W : = { w ∈ W : (cid:107) w (cid:107) = } . Since e is a density point of U e ⊆ W , the following holds: there exists D ⊆ S W ,dense in S W , such that, for every w ∈ D , there exists an infinitesimal sequence { t j } j ∈ N such that δ t j w ∈ U e for all j ∈ N . Fix w ∈ D . We claim that F ( w ) W = w so that by density and the continuity of F the thesis follows. Indeed, letus fix (cid:101) >
0. By hypothesis, and since there exists an infinitesimal sequence { t j } j ∈ N such that δ t j w ∈ U e , we getthat there is t j > (cid:107) ( F ( δ t j w ) W ) − · δ t j w (cid:107) ≤ (cid:101) (cid:107) δ t j w (cid:107) .By the homogeneity in ( ), the homogeneity of the norm and the fact that (cid:107) w (cid:107) =
1, since w ∈ D ⊆ S W , we get (cid:107) ( F ( w ) W ) − · w (cid:107) ≤ (cid:101) .Thus from the fact that (cid:101) > Indeed if U ⊆ W is Borel and for H k (cid:120) W -almost every point in U there exist a sequence h j ( x ) converging to e and 0 < λ ( x ) < B ( x · h j ( x ) , λ ( x ) (cid:107) h j ( x ) (cid:107) ) ∩ E = ∅ for all j ∈ N , then H k (cid:120) W ( E ) = eferences We are now in a position to prove our Theorem . . Let us recall the definition of the Jacobian of a homogeneousmap. Take a homogeneous map F : W → G between graded groups W and G , equipped with homogeneousleft-invariant distances d W and d G , respectively. Denote by k : = dim H ( W ) the Hausdorff dimension of W . The Jacobian of F is J ( F ) : = H k ( F ( B ( e , 1 ))) H k ( B ( e , 1 )) , ( )where B ( e , 1 ) is the ball centered at the identity e , and of radius . Proof of Theorem . . Since L is normal , we can use [ , Proposition . ] and thus, from the fact that φ is intrinsicallyLipschitz, we deduce that the graph map Φ : U ⊆ W → G is Lipschitz. If in addition W is a Carnot subgroup ,we are in a position to apply Rademacher theorem (see [ ] and [ , Theorem . ]) to the graph map Φ : U ⊆ W → G , in order to conclude that it is H k (cid:120) W -almost everywhere Pansu differentiable on U . Eventually, we applyProposition . to conclude that every point of Pansu differentiability of Φ is a point of intrinsic differentiabilityof φ . Finally, the area formula ( ) is a direct consequence of the area formula in [ , Theorem . . ] applied to thegraph map Φ , after having noticed that Φ : U ⊆ W → G is injective. Remark . . By joining the conclusions of Theorem . and Remark . we can conclude that in our hypotheses theintrinsically Lipschitz property guarantees the existence, at ( Φ ) ∗ ( H k (cid:120) U ) -almost every point on the graph of φ , ofa tangent subgroup. Moreover, from Proposition . we get that, whenever the Pansu differential d Φ x exists, then φ is intrinsically differentiable at x and d Φ x [ w ] = w · d φ φ x [ w ] for all w ∈ W . Taking into account this equality, thedefinition of the Jacobian ( ) and Remark . we stress that the area element J ( d Φ x ) in the area formula ( ) onlydepends on the geometry of the tangent subgroup of graph ( φ ) at Φ ( x ) , which is graph ( d φ φ x ) . Remark . . The hypotheses of Theorem . are satisfied whenever we take an intrinsically Lipschitz function φ : U ⊆ W → L , with W horizontal , and U Borel . Thus our result applies in particular to intrinsically Lipschitzhorizontal surfaces in arbitrary Carnot groups.
Remark . . If we do not assume L to be normal in the hypotheses of Theorem . , the graph map Φ : U ⊆ W → L may not be Lipschitz when φ is intrinsically Lipschitz. The forthcoming example is also found in [ ].Indeed, let us take the second Heisenberg group H , with a basis of the Lie algebra given by ( X , X , X , X , X ) ,where the only nontrivial relations are [ X , X ] = [ X , X ] = X . Let us identify H with R by means of exponen-tial coordinates of the first kind. Set, in exponential coordinates, W : = { ( x , x , x , x ) : x , x , x , x ∈ R } and L : = { ( x , 0, 0, 0, 0 ) : x ∈ R } . We notice that W is a Carnot subgroup and L is not normal .It is easily seen that the map φ : W → L defined as φ ( x , x , x , x ) : = (
1, 0, 0, 0, 0 ) for every ( x , x , x , x ) ∈ R is intrinsically Lipschitz. Moreover, if we fix (cid:101) > Φ (
0, 0, (cid:101) , 0, 0 ) = (
1, 0, (cid:101) , 0, − (cid:101) /2 ) and Φ (
0, 0, 0, 0, 0 ) = (
1, 0, 0, 0, 0 ) . Thus (cid:107) Φ (
0, 0, 0, 0, 0 ) − · Φ (
0, 0, (cid:101) , 0, 0 ) (cid:107) = (cid:107) (
0, 0, (cid:101) , 0, − (cid:101) ) (cid:107) ∼ (cid:101) → (cid:101) ,and then Φ cannot be Lipschitz with respect to the induced metric, since (cid:107) (
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