aa r X i v : . [ m a t h . M G ] M a y A COAREA-TYPE INEQUALITY IN CARNOT GROUPS
FRANCESCA CORNI
Abstract.
We prove a coarea-type inequality for continuously Pansu differentiable func-tions, with everywhere surjective Pansu differential and uniformly lower Ahlfors regularlevel sets, acting between two Carnot groups endowed with homogeneous distances.
Contents
1. Introduction 12. Preliminar definitions and results 33. Coarea-type Inequality 94. Applications 19Acknowledgement 22References 221.
Introduction
Geometric measure theory in non-Euclidean metric spaces has been relevantly developedduring the last decades. One of the first goals in this line of research is the study of Carnotgroups, that are connected, simply connected, nilpotent, stratified Lie groups. These arethe simplest models of sub-Riemannian manifolds. One can canonically associate to eachCarnot group a family of non-isotropic dilations defined according to the stratification ofthe Lie algebra of the group. We study Carnot groups endowed with a distance that ishomogeneous with respect to these dilations. Within the study of these metric spaces, along-standing open problem is the validity of the coarea formula for Lipschitz maps actingbetween two Carnot groups. Up to now, for these mappings only a coarea-type inequality isavailable [11]. Some stronger results have been proved for specific situations. For instance,one can refer to [3],[13], [20] for Lipschitz real-valued maps acting on a generic Carnot group,to [9],[15], [19], [21] for continuously Pansu differentiable mappings from a Heisenberggroup H n to R k (where, depending on k , higher regularity on the Pansu differential maybe required) and to [8], [12], [14] for Euclidean regular maps from a Carnot group to R k .Moreover, a very general result has recently been proved in [7]. The authors consider twoCarnot groups G and M , endowed with homogeneous distances, an open set Ω ⊂ G anda map f : Ω → M , with Pansu differential Df ( x ) continuous on Ω. Then, the coareaformula holds for f if, at every point x ∈ Ω, either Df ( x ) is surjective and ker( Df ( x )) canbe complemented with a homogeneous subgroup (Definition 2) or Df ( x ) is not surjective. Date : May 6, 2020.2010
Mathematics Subject Classification.
Primary 28A75; Secondary 28A78, 22E30.
Key words and phrases.
Carnot groups, coarea formula, spherical measure, packing measure.
A key step in the proof of the coarea formula [7, Theorem 1.3] is a suitable implicit functiontheorem (Theorem 2.6). Fix a value m ∈ M and consider a point x ∈ f − ( m ) such that Df ( x ) is surjective. Assume that there exists a homogeneous subgroup V complementaryto ker( Df ( x )) and choose any homogeneous subgroup W complementary to V . Then thereexist an open neighbourhood Ω ⊂ G of x , an open set U ⊂ W and a map φ : U ⊂ W → V such that Ω ∩ f − ( m ) is the intrinsic graph of φ (Definition 3). It is not clear how to provethe existence of an analogous parametrization if we assume the Pansu differential Df ( x )only to be surjective. In this work we bypass this lack and we prove a weaker coarea-type result, that permits, under a further regularity condition, to deal with more generalsituations. More precisely we prove the following result. Theorem 1.1.
Let ( G , d ) , ( M , d ) be two Carnot groups endowed with homogeneousdistances, of metric dimension Q, P and topological dimension q, p , respectively. Let f ∈ C G ( G , M ) be a function and assume that Df ( x ) is surjective at every point x ∈ G . Let Ω be a closed bounded subset of G . Assume that there exist two constants ˜ r, C > suchthat for any m ∈ M , the level set f − ( m ) is ˜ r -locally C -lower Ahlfors ( Q − P ) -regular withrespect to the measure S Q − P . Then there exists a constant L = L ( C, G , p ) such that Z Ω C P ( Df ( x )) d S Q ( x ) ≤ L Z M S Q − P ( f − ( m ) ∩ Ω) d S P ( m ) . The factor C P ( Df ( x )) is the coarea factor of the Pansu differential Df ( x ) (Definition9) and S α denotes the α -dimensional spherical Hausdorff measure built with respect tohomogeneous distance. Refer to Definition 11 for the notion of locally lower Ahlfors regularset.It is immediate to extend Theorem 1.1, to the case when Ω is a measurable subset of G (Theorem 3.3). As an example of its generality, notice that Theorem 1.1 can be applied toany continuously Pansu differentiable functions f : H → R satisfying the requirements.The proof of Theorem 1.1 is inspired to an abstract procedure presented in [23], whereit is used to prove a coarea-type inequality for functions from a metric space to a measurespace, for packing-type measures. Analogous argument involving suitable packing measuresis adapted here to prove Claim 1 of Theorem 3.2.By applying Theorem 1.1, we deduce new results about slicing of measurable functionson the level sets of f (Corollaries 4.2 and 4.3).We conclude our introduction with a brief discussion about the hypotheses of Theorem1.1. In particular we compare them with assumptions of available results in literature.The hypothesis of Theorem 1.1 about Ahlfors regularity of the level sets of the map f is not redundant: if we consider f ∈ C G ( G , M ), with Pansu differential everywheresurjective on G , the lower Ahlfors regularity of level sets is not guaranteed, even locally.One can refer to [9, Corollary 6.2.4], where the author presents some explicit examplesof this phenomenon. In addition, in [9], a particular situation is considered: a class ofhigher regular mappings from the Heisenberg group H n to the Euclidean R n is studied.Let us consider a homogeneous distance d on H n . The author considers functions f ∈ C ,α G ( H n , R n ) with α > a, b ∈ H n d L ( H n , R n ) ( Df ( a ) , Df ( b )) . d ( a, b ) α . COAREA-TYPE INEQUALITY IN CARNOT GROUPS 3
By [9, Corollary 5.5.6], if we assume that the Pansu differential of f is everywhere surjective,the level sets of f are uniformly locally Ahlfors 2-regular with respect to S . Therefore,the validity of the inequality of Theorem 1.1 for this class of regular functions is ensuredby our result. This confirms the coarea-type equality proved in [9, Theorem 6.2.5]. Tosummarize, in this setting, we can weaken the hypothesis adopted in [9, Theorem 6.2.5]about the required regularity of the considered map, passing from C ,α G -regular maps, with α >
0, to continuously Pansu differentiable functions. We compensate the lower regularityof the map with the more geometrical hypothesis about the Ahlfors regularity of its levelsets. We need to remark that these considerations are limited, up to now, to maps fromthe Heisenberg group H n to R n , about which more results are available.We stress that in Theorem 1.1 the assumption about the uniform local lower Ahlforsregularity of the level sets of the map f can be read also as a substitute of the existence ofa suitable splitting of G . In fact, this condition is automatically verified if one assumes theexistence of a p -dimensional homogeneous subgroup V ⊂ G complementary to ker( Df ( x ))for every point x ∈ G (Corollary 4.6). The proof of this observation uses tools of thetheory of intrinsic Lipschitz graphs ([6]). We stress that Corollary 4.6 is just an exampleof application of Theorem 1.1. In fact, as we discussed above, it can be derived also by thecoarea formula in [7, Theorem 1.3].2. Preliminar definitions and results
In this section we introduce some definitions and notations.When we write a . b , we mean that there exists some positive constant C such that a ≤ Cb . If C depends on some parameter d , it will be specified with a subscript. Forinstance, a . d b means that there exists a constant C depending on d such that a ≤ Cb .Analogous notations are assumed for & . Definition 1. A Carnot group G is a connected, simply connected, nilpotent Lie groupsuch that its Lie algebra Lie( G ) is stratified i.e. there exist linear subspaces V , V , . . . , V k such that Lie( G ) = V ⊕ · · · ⊕ V k and [ V , V i ] = V i +1 V k = { } V i = { } if i > k, where [ V , V i ] = span { [ X, Y ] : X ∈ V , Y ∈ V i } .The number k is called the step of G .The topological dimension of G is q = P ki =1 dim( V i ); the number Q = P ki =1 ( i dim( V i ))is called homogeneous dimension of G .We denote the left translation associated to an element x ∈ G by τ x : G → G , τ x ( y ) = xy. We can naturally introduce on Lie( G ) a family of non-isotropic linear dilations δ t ( v ) = k X i =1 t i v i if v = k X i =1 v i with v i ∈ V i . Since G is simply connected, the exponential map exp : Lie( G ) → G is a global diffeomor-phism, then we can identify G with Lie( G ) and any dilation δ t can be identified with thefunction exp ◦ δ t ◦ exp − : G → G , we denote this map again by δ t . FRANCESCA CORNI
A Lie subgroup W ⊆ G is called homogeneous if it is closed with respect to the familyof anisotropic dilations, hence if for every t > δ t ( W ) ⊆ W .Through the exponential map and according to the Baker-Campbell-Hausdorff formulawe can move the group product of G to an isomorphic polynomial group product on Lie( G ):for X, Y ∈ Lie( G ), we call it BCH ( X, Y ). In particular Lie( G ) endowed with BCH ( · , · )is isomorphic to G itself ([24, Theorem 4.2]), so we identify G and Lie( G ) as Lie groups.We fix a basis of G , ( v , . . . , v q ) and we identify G with R q through the chosen basis asfollows(1) ϕ : G → R q , ϕ ( p ) = ( x , . . . , x q ) if p = q X i =1 x i v i . The product on G can be moved to a polynomial group product on R q (see [1, Proposition2.2.22]). By the identification of G with Lie( G ) and R q , G can be seen as R q endowed atthe same time with the structure of Lie group, with a polynomial group product, and thestructure of Lie algebra, and hence of linear space. The inverse of an element with respectto the group product is ( x , . . . , x q ) − = ( − x , . . . , − x q ) while the identity element is thenull vector of R q and we denote it by 0.We assume that G is a Carnot group endowed with a homogeneous distance d , that isa distance such that d ( zx, zy ) = d ( x, y ) for every x, y, z ∈ G and d ( δ t ( x ) , δ t ( y )) = td ( x, y )for every t > x, y ∈ G .We set k x k := d ( x,
0) for every x ∈ G ; the metric closed ball centered at x of radius r is denoted by B ( x, r ) := { y ∈ G : d ( x, y ) ≤ r } and for every set S ⊂ G , we calldiam( S ) := sup { d ( x, y ) : x, y ∈ S } . Notice that diam( B ( x, r )) = 2 r for all x ∈ G and r >
0, for any fixed homogeneous distance d . When nothing more is specified, by ”ball”we will mean ”closed ball”. For a ball B , we denote the radius of B by r ( B ).We fix on G a scalar product with respect to which ( v , . . . , v q ) is an orthonormal basis;extending it by left invariance, we obtain a Riemannian metric g on G . The norm arisingfrom the fixed scalar product turns out to be identified through ϕ with the Euclideanmetric on R q . We will denote this norm on G by | · | . Proposition 2.1. [1, Proposition 5.15.1]
Let G be a Carnot group of step k endowed witha homogeneous distance d . For every compact subset K ⊂ G there exists a constant C K such that for any x ∈ K C K | x | ≤ k x k ≤ C K | x | k . We recall some definitions and results about splitting a Carnot groups into the productof complementary subgroups and about regularity for maps acting between homogeneoussubgroups (for more details, please refer to [25, Section 4]).
Definition 2.
Let G be a Carnot group. Two homogeneous subgroups W , V are said complementary subgroups of G if W ∩ V = { } and for every g ∈ G , there exist w ∈ W , v ∈ V such that: g = wv , i.e. G = WV . We denote by π W : G → W and π V : G → V the group projections on the subgroups: if g = wv with w ∈ W , v ∈ V , π W ( g ) = w and π V ( g ) = v .If r > w ∈ W , we denote by B W ( w, r ) := B ( w, r ) ∩ W and if v ∈ V , B V ( v, r ) := B ( v, r ) ∩ V . COAREA-TYPE INEQUALITY IN CARNOT GROUPS 5
Proposition 2.2. [6, Proposition 2.12] If G = WV is the product of two complementarysubgroups, there exists c = c ( W , V ) > such that (2) c ( k w k + k v k ) ≤ k wv k ≤ k w k + k v k for all w ∈ W , v ∈ V . Definition 3.
Let G = WV be the product of two complementary subgroups and let U ⊂ W be a set. If we consider a map φ : U → V , we define its intrinsic graph as the setgraph( φ ) = { wφ ( w ) : w ∈ U } . The map Φ : U → graph( φ ) , Φ( w ) := wφ ( w ) is called the graph map of φ . Definition 4.
Let G = WV be the product of two complementary subgroups and L bea constant. Let U ⊂ W be open, we say that a function φ : U → V is called intrinsic L -Lipschitz if(3) k π V (Φ( w ′ ) − Φ( w )) k ≤ L k π W (Φ( w ′ ) − Φ( w )) k for every w, w ′ ∈ U . Definition 5 (Carath´eodory’s construction) . Let
F ⊂ P ( G ) be a non-empty family ofclosed subsets of a Carnot group G , equipped with a homogeneous distance d . Let ζ : F → R + be a function such that 0 ≤ ζ ( S ) < ∞ for any S ∈ F . If δ >
0, and A ⊂ G , we define(4) φ δ,ζ ( A ) = inf ( ∞ X j =0 ζ ( B j ) : A ⊂ ∞ [ j =0 B j , diam( B j ) ≤ δ, B j ∈ F ) , If F coincides with the family of closed balls, F b , with respect to the distance d and ζ ( B ( x, r )) = r α we call S α ( A ) := sup δ> φ δ,ζ ( A )the α -spherical Hausdorff measure of A .If G is a Carnot group of topological dimension q and homogeneous dimension Q , then Q is the metric dimension of G with respect to any homogeneous distance d . The sphericalmeasures S m are invariant by left translation, hence, for any positive m , S m ( τ x ( A )) = S m ( A ) for every x ∈ G and A ⊂ G . By uniqueness of the Haar measure, S Q coincidesup to a constant with the Lebesgue measure L q (for more details please refer to [25,Propositions 2.19, 2,32]). Moreover, for any positive m , S m ( δ t ( A )) = t m S m ( A ) for every t > A ⊂ G . Definition 6 (Packing) . Let N , ℓ be two natural numbers, with ℓ ≥
1. Let X be a metricspace. A ℓ -packing is a countable collection of closed balls { B j } such that concentric balls ℓB j are pairwise disjoint. An ( N, ℓ )-packing is a collection of balls { B j } which is the unionof at most N ℓ -packings
Remark 1.
In a doubling metric space it is not restrictive to assume that once fixed anumber ℓ ≥
1, there exists a natural number N , only depending on ℓ , such that there existsa ( N, ℓ )-packing that covers the whole space. For instance, in [23, Remark 4] it is provedthat if a metric space X is doubling at small scales, fine covering of ( N, ℓ )-packings existwith N depending only on ℓ . FRANCESCA CORNI
Definition 7 (Packing premeasure) . Let ℓ ≥ N be natural numbers. Let G be aCarnot group endowed with a homogeneous distance d and let be δ > α >
0; let E ⊂ G ,we introduce P ∆ αd,δ,N,ℓ ( E ) = sup { X i r ( B i ) α : { B i } ( N, ℓ )-packing of
E,E ⊆ ∪ i B i , B i centered on E, r ( B i ) ≤ δ } and define P ∆ αd,N,ℓ ( E ) := inf δ> P ∆ αd,δ,N,ℓ ( E ) . We define also a modified packing object. In particular, in this case we do not requirethe packings to cover the set,˜ P ∆ αd,δ,N,ℓ ( E ) = sup (X i r ( B i ) α : { B i } ( N, ℓ )-packing of
E, B i centered on E, r ( B i ) ≤ δ ) , and ˜ P ∆ αd,N,ℓ ( E ) := inf δ> ˜ P ∆ αd,δ,N,ℓ ( E ) . We denote by H αE the Hausdorff measure on G i.e. the measure obtained by Carath´eodory’sconstruction assuming F is the family of all closed sets and ζ ( B ) = L α ( { y ∈ R α : | y | ≤ } )2 α diam( B ) α . We will denote the closed Euclidean ball of center x and radius r > B E ( x, r ) = { x ∈ G : | x | ≤ r } . From now on, we consider two Carnot groups endowed with homogeneous distances( G , d ), ( M , d ), of metric dimension Q and P and topological dimension q and p , respec-tively. The Lie algebras of G and M are stratified and we identify as above Lie( G ) with G and Lie( M ) with M so the groups can be seen as direct sum of linear subspaces G = V ⊕ · · · ⊕ V k M = W ⊕ · · · ⊕ W M . We denote by δ t and δ t the natural anisotropic dilations of parameter t > G and M ,respectively.A map L : G → M is a h-homomorphism if it is a group homomorphism such that L ( δ t ( x )) = δ t ( L ( x )) for any x ∈ G and t >
0. In this case we say L ∈ L ( G , M ).Given two h-homomorphisms L, T ∈ L ( G , M ), we define the distance d L ( G , M ) ( L, T ) :=sup q ∈ B (0 , d ( L ( q ) , T ( q )) and we denote by k L k L ( G , M ) := d L ( G , M ) ( L, I ), where I : G → M denotes the map that associates to any point of G the unit element of M .If we identify G with R q and M with R p through two fixed bases as in (1), L is a linearmap from R q to R p . Then we can consider its Jacobian | L | = p det( LL ∗ ). Observe that | L | is the Euclidean algebraic Jacobian of L from R q to R p , or, equivalently, it is the Jacobianof L between the two Lie algebras G and M with respect to the fixed scalar products. Formore details about h-homomorphisms, please refer to [10, Section 3.1].An invertible h-homomorphism is called a h-isomorphism .If L : G → M is a surjective h-homomorphism and N is its kernel, we call L a h-epimorphism if there exists a homogeneous subgroup of G , H , complementary to N . In this COAREA-TYPE INEQUALITY IN CARNOT GROUPS 7 case the restriction L | H is a h-isomorphism (for more details please refer to [14, Definition2.2, Proposition 7.14]).We denote by k x k := d ( x,
0) for every x ∈ G and by k x k := d ( x,
0) for every x ∈ M .Let Ω be an open set in G and f : Ω → M be a continuous function. Fix a point x ∈ Ω.If there exists a h-homomorphism L : G → M that satisfies k L ( x − y ) − f ( x ) − f ( y ) k = o ( k x − y k ) as k x − y k → ,f is said Pansu differentiable at x . If such a map L exists, it is unique and it is called the Pansu differential of f at x ; we denote it by Df ( x ). This definition has been introducedin [22]. We say that f ∈ C G (Ω , M ) or that f is continuously Pansu differentiable on Ω ifthe function Df : Ω → L ( G , M ) is continuous. Proposition 2.3.
Let G and M be two Carnot groups and let Ω ⊂ G be an open set. If f ∈ C G (Ω , M ) , the function k Df k L ( G , M ) : Ω → R , x → k Df ( x ) k L ( G , M ) is continuous. Definition 8. [16, Definition 4.5] Let f : K → Y be a vector-valued continuous functionfrom a compact metric space ( K, d ) to a vector-valued metric space ( Y, d ). Then wedefine the modulus of continuity of f on K as ω K,f ( t ) = max x,y ∈ Kd ( x,y ) ≤ t d ( f ( x ) , f ( y )) . If we consider an open set Ω ⊂ G and a map f : Ω ⊂ G → M ; for j = 1 , . . . , M , we call F j := π j ◦ f , where π j : M → W j is the orthogonal projections onto the j -th layers of M .If f is Pansu differentiable at x ∈ Ω, by [16, Theorem 4.12], F is Pansu differentiable at x and DF ( x ) = π ◦ Df ( x ) (notice that DF : Ω → W ). Theorem 2.4. [16, Theorem 1.2]
Let ( G , d ) and ( M , d ) be two Carnot groups endowedwith homogeneous distances. Let k be the step of G and Ω ⊂ G be an open subset. Let usconsider a map f ∈ C G (Ω , M ) . Let Ω , Ω ⊂ G be two open subsets of G such that { x ∈ G : d ( x, Ω ) ≤ cH diam(Ω ) } ⊂ Ω with Ω compactly contained in Ω , and c = c ( G , d ) and H = H ( G , d ) geometric con-stants, only depending on ( G , d ) . Then there exists a constant C , only depending on G , max x ∈ Ω k DF ( x ) k L ( G ,W ) and on the modulus of continuity ω Ω ,DF such that d ( f ( x ) − f ( y ) , Df ( x )( x − y )) d ( x, y ) ≤ C [ ω Ω ,DF ( cHd ( x, y ))] /k . for every x, y ∈ Ω with x = y . Remark 2.
By [16, Theorem 4.12], if f is continuously Pansu differentiable, then x → DF ( x ) is a continuous map from Ω to L ( G , W ), and so by Proposition 2.3, the modulusof continuity ω Ω ,DF ( s ) goes to zero as s goes to zero. Definition 9 (Coarea factor) . Let L : G → M be a h-homomorphism and let be Q ≥ P .We call coarea factor of L , C P ( L ), the unique constant such that S Q ( B (0 , C P ( L ) = Z M S Q − P ( L − ( ξ ) ∩ B (0 , d S P ( ξ ) . By [11, Proposition 1.12], and so in particular by the left invariance of the involvedspherical Hausdorff measures, by uniqueness of the Haar measure on Carnot groups and
FRANCESCA CORNI by the Euclidean coarea formula, C P ( L ) is well defined, and it is not zero if and only if L is surjective and in this case it can be computed as follows. By B G (0 ,
1) and B M (0 ,
1) herewe denote the closed unitary balls in G and M , respectively, then C P ( L ) = S Q − P (ker( L ) ∩ B G (0 , H q − pE (ker( L ) ∩ B G (0 , S P ( B M (0 , L p ( B M (0 , L q ( B G (0 , S Q ( B G (0 , | L | = Z S Q − P (ker( L ) ∩ B G (0 , H q − pE (ker( L ) ∩ B G (0 , | L | = Z S Q − P (ker( L ) ∩ B (0 , H q − pE (ker( L ) ∩ B (0 , | L | , (5)where Z = S P ( B M (0 , L p ( B M (0 , L q ( B G (0 , S Q ( B G (0 , . Observe that Z is a geometrical constant not dependingon L . Notice that in the last line of (5) we have dropped the subscript G to denote theball B G (0 , Definition 10.
Let G be a Carnot group endowed with a homogeneous distance d . Let F b be the family of closed balls with positive radius in G . Let α > x ∈ G and µ be aBorel regular measure on G . We call spherical α -Federer density of µ at x the real number θ α ( µ, x ) := inf ǫ> sup (cid:26) µ ( B ) r ( B ) α : x ∈ B ∈ F b , diam( B ) < ǫ (cid:27) . This density is the right object to represent the abstract way to differentiate any Borelregular measure absolutely continuous with respect to the α -spherical one in a metric spacesatisfying general hypothesis. Theorem 2.5 ([18, Theorem 7.2]) . Let α > and let µ be a Borel regular measure on G such that there exists an open numerable covering of G whose elements have µ -finitemeasure. Let d be a homogeneous distance. If B ⊂ A ⊂ G are Borel sets, then θ α ( µ, · ) isa Borel function on A . In addition, if S α ( A ) < ∞ and µ x A is absolutely continuous withrespect to S α x A , then we have µ ( B ) = Z B θ α ( µ, x ) d S α ( x ) . Definition 11.
Let (
X, d, µ ) be a metric measure space, consider a subset E ⊂ X and twopositive numbers α, C >
0. We say that E is locally C -lower Ahlfors α -regular with respectto µ if there is ¯ r > x ∈ E , 0 < r < ¯ r , µ ( B ( x, r ) ∩ E ) ≥ Cr α . If we need to stress the value of ˜ r , we say that E is ˜ r -locally C -lower Ahlfors α -regularwith respect to µ . If ˜ r = ∞ , we say that E is C -lower Ahlfors α -regular with respect to µ .Now we report a result contained in [7, Lemma 2.10], that is, up to now, the most generalavailable implicit function theorem into this setting. Similar statements are proved in [16,Theorem 1.4] and [4, Theorem 3.27]. Theorem 2.6.
Let G and M be two Carnot groups and let Ω ⊂ G be open. Let f ∈ C G (Ω , M ) be a function and fix x ∈ Ω . Assume that Df ( x ) is a surjective h-epimorphism COAREA-TYPE INEQUALITY IN CARNOT GROUPS 9 and consider V a subgroup complementary to ker( Df ( x )) . Fix a homogeneous subgroup W complementary to V . Write x = w v with respect to the splitting WV .Then there exist an open set A ⊂ W , with w ∈ A , and a continuous map φ : A → V such that f ( aφ ( a )) = f ( x ) for every a ∈ A . Remark 3.
By [7, Corollary 2.16], the parametrization φ given by Theorem 2.6 is L -Lipschitz for some positive constant L . Proposition 2.7. [6, Theorem 3.9]
Let G = WV be the product of two complementarysubgroups endowed with a homogeneous distance d . Let be N be the metric dimension of W . If φ : W → V is intrinsic L -Lipschitz in W , then (cid:18) c ( W , V )1 + L (cid:19) N r N ≤ S N (graph( φ ) ∩ B ( x, r )) for all x ∈ graph( φ ) and r > . Remark 4.
The proof of Proposition 2.7 is based on local arguments. Let us now assumethat φ is defined from an open set A ⊂ W to V and w ∈ A . Assume that, for a positiveconstant ˜ r > π W ( B (Φ( w ) , ˜ r )) ⊂ A , then for any 0 < r < ˜ r we have that S N (graph( φ ) ∩ B (Φ( w ) , r )) ≥ (cid:16) c ( W , V )1+ L (cid:17) N r N . In [11], the author proved a coarea-type inequality. We recall it here, adapting it to ourcontext.
Theorem 2.8. [11, Theorem 2.6]
Let A ⊆ G be a measurable set and f : A → M be aLipschitz map, then (6) Z M S Q − P ( f − ( m ) ∩ A ) d S P ( m ) ≤ Z A C P ( Df ( x )) d S Q ( x ) . Coarea-type Inequality
We will need the following simple proposition in order to prove the main theorem.
Proposition 3.1.
Let G be a Carnot group endowed with a homogeneous distance d . Let W ⊂ G be a homogeneous subgroup of topological dimension n and metric dimension N .Then for every Borel set B ⊂ W we have that L n ( B ) = sup w ∈ B (0 , H nE ( B ( w, ∩ W ) S N ( B ) Proof.
Let us consider µ W ( A ) := H nE x W ( A ) = L n ( W ∩ A ) for any set A ⊂ G . Since both µ W and S N x W are left invariant on W , they coincide up to a constant, so we can applyTheorem 2.5, hence we know that µ W ( A ) = R A θ N ( µ W , x ) S N ( x ). Let us then compute forany x ∈ W θ N ( µ W , x ) = inf r> sup { z : x ∈ B ( z,t ) }
Remark 5.
It is immediate to observe that any continuously Pansu differentiable functionis locally metric Lipschitz, hence by [15, Theorem 2.1] we have that for every measurable set A ⊂ G the function m → S Q − P ( A ∩ f − ( m )) is S P -measurable. In particular this resultfollows by [15, Theorem 2.1] combined with Monotone Convergence Theorem applied,for every m ∈ M , to the sequence of characteristic functions Ω n ∩ f − ( m ) ր A ∩ f − ( m ) ,where we have considered an exhaustion of compact sets for A , Ω n ր A . Notice that themeasurability of the map x → C P ( Df ( x )) follows by [11, Theorem 2.6].Theorem 1.1 is a direct consequence of the following result. In fact, it is analogous toTheorem 3.2, where hypotheses have been slightly weakened even if they are definitelymore technical. Theorem 3.2.
Let ( G , d ) , ( M , d ) be two Carnot groups, endowed with homogeneousdistances, of metric dimension Q, P and topological dimension q, p , respectively. Let Ω ′ be an open subset of G . Let f ∈ C G (Ω ′ , M ) be a function and assume Df ( x ) is surjectiveat every x ∈ Ω ′ .Let Ω ⋐ Ω ′ be a closed bounded set such that there exists an open set Ω ′′ and a positivenumber s > such that Ω ′′ is compactly contained in Ω and if we call Ω s := { x ∈ G : d ( x, Ω) < s } and we call R := cH diam(Ω s ) , (7) Ω sR := { x ∈ G : d ( x, Ω s ) ≤ R } ⊂ Ω ′′ where H = H ( G , d ) , c = c ( G , d ) are geometric constants that depend on G and d as inTheorem 2.4. Assume that there exist two constants ˜ r, C > such that for any m ∈ M , thelevel set f − ( m ) is ˜ r -locally C -lower Ahlfors ( Q − P ) -regular with respect to the measure S Q − P . Then there exists a constant L = L ( C, G , p ) such that Z Ω C P ( Df ( x )) d S Q ( x ) ≤ L Z M S Q − P ( f − ( m ) ∩ Ω) d S P ( m ) . Proof.
In the proof we will denote by B ( x, r ) the closed metric ball of G with respect of d with center x and radius r . Let us preliminarily introduce for δ >
0, and E ⊂ G S f d ,δ,ζ ( E ) = inf nX ζ ( B i ) : B i closed balls , E ⊂ ∪ i B i , r ( B i ) ≤ δ o COAREA-TYPE INEQUALITY IN CARNOT GROUPS 11 with ζ ( B ( x, r )) = r Q − P S P ( f ( B ( x, r )). Define the Carath´eodory’s measure S f d ,ζ ( E ) := sup δ> S f d ,δ,ζ ( E ) . Let be ℓ = 1 and fix N = N (2 , G ) the minimum natural number such that there exists a( N, G that covers G itself. We define P f d ,δ,ζ ,N,ℓ ( E ) = sup { X ζ ( B i ) : { B i } ( N, ℓ )-packing of
E,B i centered on E, E ⊆ ∪ i B i , r ( B i ) ≤ δ } . and P f d ,ζ ,N,ℓ ( E ) := inf δ> P f d,δ,ζ ,N,ℓ ( E ) . Claim 1 S f d ,ζ (Ω) ≤ P f d ,ζ ,N, (Ω) ≤ Z M P ∆ Q − Pd ,ζ ,N, ( f − ( m ) ∩ Ω) d S P ( m ) Proof.
We follow the scheme of [23, Proposition 15]. Let { B i } be a ( N, r ( B i ) ≤ δ and with B i centered on Ω. Consider any ball B i and choose m ∈ f ( B i ). Pick x i ∈ B i ∩ f − ( m ) ∩ Ω and call B i,m the smallest ball centered at x i thatcontains B i . Observe that B i,m ⊂ B i , so that, for each m ∈ M , the collection { B i,m } is a( N, f − ( m ) ∩ Ω consisting of balls with radius less or equal than 2 δ centeredon f − ( m ) ∩ Ω, that covers f − ( m ) ∩ Ω. Then X i r ( B i ) Q − P S P ( f ( B i )) = X i ( Z M f ( B i ) ( m ) d S P ( m ))( r ( B i )) Q − P = Z M X i f ( B i ) ( m )( r ( B i )) Q − P d S P ( m )= Z M X { i : m ∈ f ( B i ) } ( r ( B i )) Q − P d S P ( m ) ≤ Z M X { i : ∃ B i,m } ( r ( B i,m )) Q − P d S P ( m ) ≤ Z M P ∆ Q − Pd , δ,N, ( f − ( m ) ∩ Ω) d S P ( m ) . Hence, by Monotone Convergence Theorem P f d ,ζ ,N, (Ω) ≤ Z M P ∆ Q − Pd ,N, ( f − ( m ) ∩ Ω) d S P ( m ) . Let us now observe that S f d ,ζ (Ω) ≤ P f d ,ζ ,N, (Ω) . Any ( N, δ isa covering of Ω of balls of radius smaller than δ so, surely, for any δ S f d ,δ,ζ (Ω) ≤ P f d ,δ,ζ ,N, (Ω)so letting δ go to zero, we get the thesis. (cid:3) Claim 2
There exists a constant T = T ( C, G ) such that for every m ∈ M P ∆ Q − Pd ,N, ( f − ( m ) ∩ Ω) ≤ T S Q − P ( f − ( m ) ∩ Ω) . Proof.
Of course, for any δ > δ < ˜ r ) and m ∈ M we have that P ∆ Q − Pd ,δ,N, ( f − ( m ) ∩ Ω) ≤ N ˜ P ∆ Q − Pd ,δ, , ( f − ( m ) ∩ Ω) . Let us consider a (1 , { B i } of balls of radius smaller than δ , centered on f − ( m ) ∩ Ω such that X i r ( B i ) Q − P ≥
12 ˜ P ∆ Q − Pd ,δ, , ( f − ( m ) ∩ Ω) . Then P ∆ Q − Pd ,δ,N, ( f − ( m ) ∩ Ω) ≤ N ˜ P ∆ Q − Pd ,δ, , ( f − ( m ) ∩ Ω) ≤ N X i r ( B i ) Q − P ≤ NC S Q − P ( f − ( m ) ∩ B i )= 2 NC S Q − P ( f − ( m ) ∩ B i ∩ Ω δ ) ≤ NC S Q − P ( f − ( m ) ∩ Ω δ ) , where Ω δ := { x ∈ G : d ( x, Ω) < δ } .Now we let δ go to zero. Since Ω is closed, for every m ∈ M , f − ( m ) ∩ Ω δ ց f − ( m ) ∩ Ωas δ →
0, hence by Monotone convergence theorem, for every m ∈ M we get P ∆ Q − Pd ,N, ( f − ( m ) ∩ Ω) ≤ T S Q − P ( f − ( m ) ∩ Ω) , where T = NC = N (2 , G ) C , hence T = T ( C, G ). (cid:3) By combining Claim 1. and Claim 2. we obtain the existence of a constant T = T ( C, G )such that S f d ,ζ (Ω) ≤ P f d ,ζ ,N, (Ω) ≤ T Z M S Q − P ( f − ( m ) ∩ Ω) d S P ( m ) . From now on, we denote by δ it the intrinsic dilations by t >
0, on G for i = 1 and on M for i = 2, respectively. Claim 3 If k is the step of G , S f d ,ζ (Ω) & k,p Z Ω | Df ( x ) | d S Q ( x ) . Proof.
The proof is composed of two main steps.First, we can observe that Df ( x ) is a continuous function on Ω ′ , so we can consider thefollowing measure on Ω ′ : for any A ⊂ Ω ′ , µ ( A ) := R A | Df ( x ) | d S Q ( x ). We want to compare µ with the Carath´eodory’s measure built with coverings of closed balls measured by thefunction ζ ( B ( x, r )) := | Df ( x ) | r Q . COAREA-TYPE INEQUALITY IN CARNOT GROUPS 13
We denote this measure by S f d ,ζ = sup δ> φ δ,ζ .We want to prove that there exists ¯ r > < r ≤ ¯ r and for every x ∈ Ω, µ ( B ( x, r )) . ζ ( B ( x, r )). By Federer 2.10.17 (1), this implies that µ ( A ) . S f d ,ζ ( A )for any A ⊆ Ω (so also for A = Ω).Since Ω is closed and bounded, it is compact. The function | Df ( · ) | : Ω s → R , x →| Df ( x ) | is a continuous function on a compact set. Let us fix ǫ = min x ∈ Ω s | Df ( x ) | >
0; itis positive since Df ( x ) is everywhere surjective by hypothesis. Moreover, the map | Df ( · ) | is uniformly continuous, then there exists r ′ > | | Df ( x ) | − | Df ( y ) | | ≤ ǫ if | x − y | ≤ r ′ , x, y ∈ Ω s .Let us fix ¯ r < min { r ′ , s } . Let us fix any x ∈ Ω and 0 < r ≤ ¯ r and let us study µ ( B ( x, r )) | Df ( x ) | r Q = 1 r Q Z B ( x,r ) | Df ( y ) || Df ( x ) | d S Q ( y ) ≤ r Q Z B ( x,r ) | | Df ( y ) | − | Df ( x ) | || Df ( x ) | d S Q ( y ) + 1 r Q Z B ( x,r ) | Df ( x ) || Df ( x ) | d S Q ( y ) ≤ r Q Z B ( x,r ) ǫ min x ∈ Ω s | Df ( x ) | S Q ( y ) + S Q ( B (0 , S Q ( B (0 , b ;where 0 < b = S Q ( B (0 , < ∞ , hence for any x ∈ Ω and 0 < r ≤ ¯ r , we have that µ ( B ( x, r )) | Df ( x ) | r Q ≤ b and then, as we said above, µ (Ω) ≤ b S f d ,ζ (Ω), so µ (Ω) . S f d ,ζ (Ω) . Second part of the proof, we want to compare S f d ,ζ (Ω) with S f d ,ζ (Ω), and, in partic-ular, we want to prove that S f d ,ζ (Ω) . k,p S f d ,ζ (Ω) . As we observed, the measure S f d ,ζ is defined as a Carath´eodory’s measure defined withcoverings of closed balls measured by the function ζ ( B ( x, r )) = r Q − P S P ( f ( B ( x, r )). Thestrategy will rely on the comparison between ζ and ζ . In particular we fix h > h < s . We want to prove that there exists ¯ r > < r ≤ ¯ r forevery x ∈ Ω h := { y ∈ G : dist( y, Ω) < h } ,(8) ζ ( B ( x, r )) & k,p ζ ( B ( x, r )) , this would give the desired thesis.Let us first define for x ∈ Ω h and r > A x,r := δ r ( f ( x ) − f ( B ( x, r )) , A x := Df ( x )( B (0 , . The proof will be composed of various steps, and it will be useful to give name to thefollowing conditions:(a) lim r → sup x ∈ Ω h (cid:12)(cid:12) A x,r ( m ) − A x ( m ) (cid:12)(cid:12) = 0 for any m ∈ M ;(b) lim r → sup x ∈ Ω h (cid:12)(cid:12) S P ( A x,r ) − S P ( A x ) (cid:12)(cid:12) . We will first prove that ( b ) implies the thesis (8). Let x ∈ Ω h and denote by V ( x ) := (ker( Df ( x )) ⊥ . For every r small enough ζ ( B ( x, r )) = r Q − P S P ( B ( x, r )) = r Q S P ( f ( B ( x, r )) r P = r Q S P ( f ( x ) − f ( B ( x, r )) r P = r Q S P ( δ /r ( f ( x ) − f ( B ( x, r ))) , hence ζ ( B ( x, r )) ζ ( B ( x, r )) = S P ( δ /r ( f ( x ) − f ( B ( x, r )) | Df ( x ) | . Observe that the map Df ( x ) | V ( x ) : V ( x ) → M is injective and surjective and that | Df ( x ) | = | Df ( x ) | V ( x ) | , by the choice of V ( x ). If π V ( x ) is the orthogonal projection on V ( x ), for some geometric constant G we have that S P ( Df ( x )( B (0 , G L p ( Df ( x )( B (0 , G | Df ( x ) | L p ( π V ( x ) ( B (0 , y ∈ G as y = m y ( π V ( x ) ( y )), with m y ∈ ker( Df ( x ))). Observe that for every x ∈ Ω h , V ( x ) = (ker( Df ( x )) ⊥ is a linear subspace of constant topological dimension p > Df ( x ) is surjective at any point x ∈ Ω h . We notice that the factor L p ( V ( x ) ∩ B E (0 , x . Remember now that by Proposition 2.1 applied to K = B (0 , C B (0 , such that1 C B (0 , | x | ≤ k x k ≤ C B (0 , | x | k , where k is the step of G . Hence L p ( π V ( x ) ( B (0 , ≥ L p ( V ( x ) ∩ B (0 , ≥ L p (cid:18) V ( x ) ∩ B E (cid:18) , C B (0 , ) k (cid:19)(cid:19) = 1( C B (0 , ) kp L p ( V ( x ) ∩ B E (0 , D ( k, p ) > . Hence for every x ∈ Ω h , we have that S P ( A x ) | Df ( x ) | ≥ GD ( k, p ) := D ′ ( k, p ) = D ′ > . If we now assume ( b ) to be true, and we fix ǫ = D ′ min x ∈ Ω h | Df ( x ) | >
0, there exists0 < ¯ r ≤ s − h such that for every 0 < r ≤ ¯ r and for every x ∈ Ω h , (cid:12)(cid:12) S P ( A x,r ) − S P ( A x ) (cid:12)(cid:12) ≤ sup x ∈ Ω h (cid:12)(cid:12) S P ( A x,r ) − S P ( A x ) (cid:12)(cid:12) ≤ ǫ, so that for every 0 < r ≤ ¯ r and for every x ∈ Ω h , S P ( A x,r ) ≥ S P ( A x ) − ǫ and so for every x ∈ Ω h and 0 < r ≤ ¯ rζ ( B ( x, r )) ζ ( B ( x, r )) = S P ( A x,r ) | Df ( x ) | ≥ S P ( A x ) | Df ( x ) | − ǫ | Df ( x ) | ≥ D ′ − ǫ | Df ( x ) | ≥ D ′ − ǫ min x ∈ Ω h | Df ( x ) | = D ′ > COAREA-TYPE INEQUALITY IN CARNOT GROUPS 15 by the choice of ǫ . This gives the thesis.Second point, we prove that ( a ) implies ( b ). Surely, we have thatlim r → sup x ∈ Ω h (cid:12)(cid:12) S P ( A x,r ) − S P ( A x ) (cid:12)(cid:12) ≤ lim r → sup x ∈ Ω h (cid:12)(cid:12)(cid:12)(cid:12)Z M A x,r ( m ) − A x ( m ) d S P ( m ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim r → Z M sup x ∈ Ω h (cid:12)(cid:12) A x,r ( m ) − A x ( m ) (cid:12)(cid:12) d S P ( m ) . (9)We want now to apply the dominated Lebesgue convergence theorem using ( a ). In orderto do this, we prove that for r ≤ s − h , for any m ∈ M (10) sup x ∈ Ω h (cid:12)(cid:12) A x,r ( m ) − A x ( m ) (cid:12)(cid:12) ≤ B (0 ,W ) ( m )for some constant W >
0. Notice that 2 B (0 ,W ) ∈ L S P ( M ) . First, consider that sup x ∈ Ω h (cid:12)(cid:12) A x,r ( m ) − A x ( m ) (cid:12)(cid:12) ≤ sup x ∈ Ω h (cid:12)(cid:12) A x,r ( m ) (cid:12)(cid:12) +sup x ∈ Ω h | A x ( m ) | .For any x ∈ Ω h and m ∈ M , if we assume A x ( m ) = 1, it implies that m = Df ( x )( η ) forsome η ∈ B (0 , k m k = k Df ( x )( η ) k ≤ k Df ( x ) k L ( G , M ) ≤ max x ∈ Ω h k Df ( x ) k L ( G , M ) =: k Df k Ω h .For any x ∈ Ω h , r ≤ s − h and m ∈ M , if A x,r ( m ) = 1, m = δ /r ( f ( x ) − f ( q r )) for some q r ∈ B ( x, r ) ⊆ Ω s , hence k m k = k δ /r ( f ( x ) − f ( q r )) k = k Df ( x )( δ /r ( x − q r )) δ /r ( Df ( x )( x − q r )) − f ( x ) − f ( q r )) k ≤ k Df ( x )( δ /r ( x − q r )) k + k δ /r ( Df ( x )( x − q r )) − f ( x ) − f ( q r ) k ≤ k Df ( x ) k L ( G , M ) + K ( ω Ω ′′ ,DF ( Hc ( s − h ))) k ≤ k Df k Ω h + K ( ω Ω ′′ ,DF ( Hc ( s − h ))) k , where ω Ω ′′ ,DF is the modulus of continuity of x → DF ( x ) defined in Definition 8 and K is a constant that plays the role of C of Theorem 2.4.Hencesup x ∈ Ω h (cid:12)(cid:12) A x,r ( m ) − A x ( m ) (cid:12)(cid:12) ≤ sup x ∈ Ω h A x,r ( m ) + sup x ∈ Ω h A x ( m ) ≤ B (0 , k Df k Ω h + K ( ω Ω ′′ ,DF ( Hc ( s − h ))) k ) ( m ) + B (0 , k Df k Ω h ) ( m ) ≤ B (0 , k Df k Ω h + K ( ω Ω ′′ ,DF ( Hc ( s − h ))) k ) ( m ) . and this implies that (10) is true, with W = k Df k Ω h + ( ω Ω ′′ ,DF ( Hc ( s − h ))) k ).We can then apply the dominated Lebesgue convergence Theorem to (9), and since wehave assumed ( a ) to be true, we obtain ( b ).It remains to prove ( a ). By contradiction, we assume ( a ) to be false. Then, there exists at least one element m ∈ M such that the limit lim r → sup x ∈ Ω h (cid:12)(cid:12) A x,r ( m ) − A x ( m ) (cid:12)(cid:12) does not exist or lim r → sup x ∈ Ω h (cid:12)(cid:12) A x,r ( m ) − A x ( m ) (cid:12)(cid:12) > . In both cases, since all the considered elements are positive, there exists at least a positiveinfinitesimal sequence r n such thatlim n →∞ sup x ∈ Ω h (cid:12)(cid:12) A x,rn ( m ) − A x ( m ) (cid:12)(cid:12) > . This implies that there exists ˜ n > n ≥ ˜ n ,sup x ∈ Ω h (cid:12)(cid:12) A x,rn ( m ) − A x ( m ) (cid:12)(cid:12) > r n ≤ s − h. Hence for every n ≥ ˜ n there exists at least an element x n ∈ Ω h ⊆ Ω h such that (cid:12)(cid:12) A xn,rn ( m ) − A xn ( m ) (cid:12)(cid:12) > (cid:12)(cid:12) A xn,rn ( m ) − A xn ( m ) (cid:12)(cid:12) = 1 . Since Ω h is a compact set, the sequence x n converges up to a subsequence to some ¯ x ∈ Ω h .Let us first prove that there exists some ¯ n such that for every n ≥ ¯ n (12) (cid:12)(cid:12) A xn,rn ( m ) − A ¯ x ( m ) (cid:12)(cid:12) = 1 . Let us then assume by contradiction that there exists a subsequence x n k such that(13) lim k →∞ (cid:12)(cid:12)(cid:12) A xnk ,rnk ( m ) − A ¯ x ( m ) (cid:12)(cid:12)(cid:12) = 0then, on this subsequence, we have that(14) (cid:12)(cid:12)(cid:12) A xnk ,rnk ( m ) − A xnk ( m ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) A xnk ,rnk ( m ) − A ¯ x ( m ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) A xnk ( m ) − A ¯ x ( m ) (cid:12)(cid:12)(cid:12) . Let us prove that (14) goes to zero as k → ∞ and this would give a contradiction with(11). The fact that (14) goes to zero follows by the assumption (13) and by the fact that(15) (cid:12)(cid:12)(cid:12) A xnk ( m ) − A ¯ x ( m ) (cid:12)(cid:12)(cid:12) → k → ∞ . Let us prove (15): assume by contradiction that on a subsequence of x n k , x n kp , for p sufficiently large,(16) (cid:12)(cid:12)(cid:12) A xnkp ( m ) − A ¯ x ( m ) (cid:12)(cid:12)(cid:12) = 1 . Let us assume m / ∈ A ¯ x , then m ∈ A x nkp so that m = Df ( x n kp )( η n kp ) for η n kp ∈ B (0 , η n kpq → ¯ η ∈ B (0 ,
1) and letting q go to infinite, by the continuity of the differential, we know that m = Df (¯ x )(¯ η ) ∈ A ¯ x ,but this is not possible. COAREA-TYPE INEQUALITY IN CARNOT GROUPS 17
So it must be true that m ∈ A ¯ x . Hence, by the definition of A ¯ x and the continu-ity of the Pansu differential, it is true that for some η ∈ B (0 , m = Df (¯ x )( η ) =lim p →∞ Df ( x n kp )( η ) = lim p →∞ X p where for every p , X p := Df ( x n kp )( η ) ∈ A x nkp , hence m ∈ lim sup p →∞ A x nkp , and then lim sup p →∞ A xnkp ( m ) = lim sup p →∞ A xnkp ( m ) = 1 butat the same time, by (16), m / ∈ A x nkp for p sufficiently large and this implies thatlim p →∞ A xnkp ( m ) = 0 = lim sup p →∞ A xnkp ( m ) and this gives a contradiction.Let us then continue from (12). We need to prove that (11) is not possible. Since m isfixed, there are only two possibilities:(a1) m ∈ A ¯ x ;(a2) m / ∈ A ¯ x .We show that neither (a1) nor (a2) can be true. Assume that (a1) is true, then m = Df (¯ x )( η ) for some η ∈ B (0 , m / ∈ A x n ,r n for n ≥ ¯ n and so clearly thereexists the following limit(17) lim n →∞ A xn,rn ( m ) = 0 . Let us define for any n , q n := x n δ r n ( η ) ∈ B ( x n , r n ) ⊆ Ω s . Consider for any n ≥ ¯ nδ /r n ( f ( x n ) − f ( q n )) = Df (¯ x )( η ) δ /r n ( Df (¯ x )( x − n q n ) − Df ( x n )( x − n q n )) δ /r n ( Df ( x n )( x − n q n ) − f ( x n ) − f ( q n ))and observe that by Theorem 2.4, since x n , q n ∈ Ω s and (7) holds k δ /r n ( Df ( x n )( x − n q n ) − f ( x n ) − f ( q n )) k ≤ K ( ω Ω ′′ ,DF ( cHr n )) k → n → ∞ and k ( Df (¯ x )( η )) − Df ( x n )( η ) k ≤ d L ( G , M ) ( Df ( x n ) , Df (¯ x )) → n → ∞ by the continuity of Df ( x ). Hence lim n →∞ δ /r n ( f ( x n ) − f ( q n )) = m . Thispermits to conclude that m ∈ lim sup n →∞ A x n ,r n and so thatlim sup n →∞ A xn,rn ( m ) = lim sup n →∞ A xn,rn ( m ) = 1 . At the same time, (17) implies that there exists the limitlim sup n →∞ A xn,rn ( m ) = lim n →∞ A xn,rn ( m ) = 0 , so we reach a contradiction.Assume now (a2), then m ∈ A x n ,r n for every n ≥ ¯ n and then by (12), m / ∈ A ¯ x . For every n ≥ ¯ n there exists q n ∈ B ( x n , r n ) ⊆ Ω s such that m = δ /r n ( f ( x n ) − f ( q n )) = Df (¯ x )( δ /r n ( x − n q n )) Df (¯ x )( δ /r n ( x − n q n )) − Df ( x n )( δ /r n ( x − n q n )) δ /r n ( Df ( x n )( x − n q n ) − f ( x n ) − f ( q n ))and again by Theorem 2.4 and by continuity of Df we obtain that m = lim n →∞ δ /r n ( f ( x n ) − f ( q n )) = lim n →∞ Df (¯ x )( δ /r n ( x − n q n )) that up to a subsequence is equal to Df (¯ x )( η ) for some η ∈ B (0 , m ∈ A ¯ x thatis a contradiction with (11). Hence, finally (a) is proved (cid:3) Claim 4
For every x ∈ Ω ′ , C P ( Df ( x )) . q,p,k | Df ( x ) | . Proof.
Since we have assumed that | Df ( x ) | is surjective at every x , by (5), we have that C P ( Df ( x )) = Z S Q − P (ker( Df ( x )) ∩ B (0 , H q − pE (ker( Df ( x )) ∩ B (0 , | Df ( x ) | . By Proposition 3.1, for any x ∈ Ω ′ and any Borel set B ⊂ ker( Df ( x )) S Q − P ( B ) = 1sup w ∈ B (0 , H q − pE (ker( Df ( x )) ∩ B ( w, H q − pE ( B ) , hence, by taking into account Proposition 2.1, we have that S Q − P (ker( Df ( x )) ∩ B (0 , H q − pE (ker( Df ( x )) ∩ B (0 , w ∈ B (0 , H q − pE ( B ( w, ∩ ker( Df ( x )) ≤ H q − pE ( B (0 , ∩ ker( Df ( x )) ≤ H q − pE ( B E (0 , C B (0 , ) k )) ∩ ker( Df ( x ))= 1 L q − p ( B E (0 , C B (0 , ) k ) ∩ ker( Df ( x )) =: D ′′ ( q, p, k ) > . In the last passage we considered that ker( Df ( x )) is a linear subspace of constant topo-logical dimension q − p . By combining all claims the proof is achieved. (cid:3)(cid:3) It is easy to extend Theorem 1.1 to the case in which Ω is not necessarily compact butit is any measurable set.
Theorem 3.3.
Let ( G , d ) , ( M , d ) be two Carnot groups, endowed with homogeneousdistances, of metric dimension Q, P and topological dimension q, p , respectively. Let f ∈ C G ( G , M ) be a function and assume Df ( x ) to be surjective at any point x ∈ G . Let A ⊂ G be a measurable set. Assume that there exist two constants ˜ r, C > such that forany m ∈ M , the level set f − ( m ) is ˜ r -locally C -lower Ahlfors ( Q − P ) -regular with respectto the measure S Q − P . Then there exists a constant L = L ( C, G , p ) such that Z A C P ( Df ( x )) d S Q ( x ) ≤ L Z M S Q − P ( f − ( m ) ∩ A ) d S P ( m ) . Proof.
Let us consider an increasing sequence of compact sets in Ω n ⊆ A such that Ω n ր A .Hence by Theorem 1.1, there exists L = L ( C, G , p ) such that for every n ∈ N Z Ω n C P ( Df ( x )) d S Q ( x ) ≤ L Z M S Q − P ( f − ( m ) ∩ Ω n ) d S P ( m ) ≤ L Z M S Q − P ( f − ( m ) ∩ A ) d S P ( m ) , COAREA-TYPE INEQUALITY IN CARNOT GROUPS 19 so if we let n go to ∞ , by Monotone Convergence Theorem we get the thesis. (cid:3) Applications
Corollary 4.1.
In the hypothesis of Theorem 3.3, let u : A → R be a non-negative mea-surable function, then there exists a constant L = L ( C, G , p ) such that Z A u ( x ) C P ( Df ( x )) d S Q ( x ) ≤ L Z M Z f − ( m ) ∩ A u ( x ) d S Q − P ( x ) d S P ( m ) . Proof.
We can write u = P ∞ k =1 1 k A k with A k measurable sets (see [2, Theorem 7]). ByMonotone Convergence Theorem we have that Z A u ( x ) C P ( Df ( x )) d S Q ( x ) = ∞ X k =1 k Z A ∩ A k C P ( Df ( x )) d S Q ( x ) ≤ ∞ X k =1 k L Z M S Q − P ( f − ( m ) ∩ A ∩ A k ) d S P ( m ) ≤ ∞ X k =1 k L Z M Z f − ( m ) ∩ A A k ( x ) d S Q − P ( x ) d S P ( m )= L Z M Z f − ( m ) ∩ A ∞ X k =1 k A k ( x ) d S Q − P ( x ) d S P ( m )= L Z M Z f − ( m ) ∩ A u ( x ) d S Q − P ( x ) d S P ( m ) . (18) (cid:3) Corollary 4.2.
In the hypothesis of Theorem 3.3, let u : A → R be a measurable function.If we assume • u is S Q − P -summable on f − ( m ) ∩ A for S P -a.e. m ∈ M • R M R f − ( m ) ∩ A | u ( x ) | d S Q − P ( x ) d S P ( m ) < ∞ then u is summable on A .Proof. We can write u = u + − u − . By applying passages analogous to (18), consideringTheorem 2.8 instead of Theorem 1.1, we then obtain that(19) − Z A u − ( x ) C P ( Df ( x )) d S Q ( x ) ≤ − Z M Z f − ( m ) ∩ A u − ( x ) d S Q − P ( x ) d S P ( m ) . Hence by (18) applied to u + , (19), and our hypothesis, we have that Z A u ( x ) C P ( Df ( x )) d S Q ( x ) = Z A u + ( x ) C P ( Df ( x )) d S Q ( x ) − Z A u − ( x ) C P ( Df ( x )) d S Q ( x ) ≤ L Z M Z f − ( m ) ∩ A u + ( x ) d S Q − P ( x ) d S P ( m ) − Z M Z f − ( m ) ∩ A u − ( x ) d S Q − P ( x ) d S P ( m ) ≤ L Z M Z f − ( m ) ∩ A | u ( x ) | d S Q − P ( x ) d S P ( m ) < ∞ . Now, it is enough to prove that C P ( Df ( x )) >
0, for every x ∈ A . This follows from thefacts that | Df ( x ) | > x ∈ A and that, taking into consideration Proposition 2.1,for every x we have the following S Q − P (ker( Df ( x )) ∩ B (0 , H q − pE (ker( Df ( x )) ∩ B (0 , w ∈ B (0 , H q − pE ( B ( w, ∩ ker( Df ( x ))) ≥ H q − pE ( B (0 , ∩ ker( Df ( x )))= 1 L q − p ( B (0 , ∩ ker( Df ( x ))) ≥ L q − p ( B E (0 , C B (0 , ) ∩ ker( Df ( x )))= 1(2 C B (0 , ) q − p > . (20) (cid:3) Corollary 4.3.
In the hypothesis of Theorem 3.3, if A ( x ) = 0 for S Q − P -a.e. x ∈ f − ( m ) ,for S P -a.e m ∈ M , then A ( x ) = 0 for S Q -a.e x ∈ G .Proof. It follows by Theorem 3.3 and (20). (cid:3)
Now we see how to apply Theorem 1.1 to the particular geometrical case in which thereexists a p -dimensional homogeneous subgroup V complementary to ker( Df ( x )) for anypoint x of a neighbourhood of a fixed compact set Ω. We fix then again ( G , d ), ( M , d )two Carnot groups, endowed with homogeneous distances, of metric dimension Q, P andtopological dimension q, p , respectively. For any set Ω ⊂ G , and any real number D > D = { y ∈ G : d ( y, Ω) < D } . By modifying the proof of [7, Lemma 2.9], combining it with an easy compactnessargument and Theorem 2.4, the following immediately follows.
Proposition 4.4.
Let us consider a map f ∈ C G ( G , M ) and a compact set Ω ⊂ G . Assumethat there exists a p -dimensional homogeneous subgroup V such that Df ( x ) | V : V → M isa h-isomorphism for every x ∈ Ω . Then there exists a constant R > such that for every x ∈ Ω , for every y ∈ B ( x, R ) and v ∈ V such that yv ∈ B ( x, R ) d ( f ( y ) , f ( yv )) ≥ R k v k . Notice that our hypothesis implies that V is complementary to ker( Df ( x )) for every x ∈ Ω.Indeed, any continuously Pansu differentiable map is locally metric Lipschitz. Hence,combining Proposition 4.4 with the proof of [7, Corollary 2.16] we get the following.
Proposition 4.5.
Let us consider a map f ∈ C G ( G , M ) and a compact set Ω ⊂ G . Letus assume that there exists a p -dimensional homogeneous subgroup V such that Df ( x ) | V : V → M is a h-isomorphism for every x ∈ Ω D for some D > . Then there exists a constant L such that for every m ∈ M , x ∈ f − ( m ) ∩ Ω , the set f − ( m ) ∩ B ( x, R ) is an intrinsicLipschitz graph with constant L , where R is the constant of Proposition 4.4 applied to Ω . COAREA-TYPE INEQUALITY IN CARNOT GROUPS 21
Proof.
By hypothesis, at any point x ∈ Ω D , ker Df ( x ) is a normal homogeneous subgroupcomplementary to V , hence Df ( x ) is a surjective h-epimorphism. Assume R is smallerthan D . Let us fix a homogeneous subgroup W complementary to V . For every m ∈ M and x ∈ f − ( m ) ∩ Ω, the set f − ( m ) ∩ B ( x, R ) is contained into the intrinsic graph of afunction φ m,x : U m,x ⊂ W → V , for some open set U m,x ⊂ W . The map φ m,x is given byTheorem 2.6, repeatedly applied to different points of f − ( m ) ∩ B ( x, R ), if necessary.Now we need to observe that the notion of intrinsic Lipschitz function introduced in [7]is equivalent to our notion (it is immediate to compare Definition in [7] with [6, Definition9, Definition 10, Proposition 3.1]). Then by [7, Corollary 2.16], f − ( m ) ∩ B ( x, R ) is theintrinsic Lipschitz graph of an intrinsic L -Lipschitz function φ m,x for some constant L depending on R , and on the Lipschitz constant of f | B ( x,R ) , that can be uniformly boundedby the sup x ∈ Ω Lip( f | B ( x,R ) ) ≤ Lip( f | Ω D ) < ∞ . As a consequence, the sets f − ( m ) ∩ B ( x, R )are intrinsic L -Lipschitz for some positive L independent of x ∈ Ω and m ∈ M . (cid:3) Corollary 4.6.
Let f ∈ C G ( G , M ) be a function with Df ( x ) surjective at every x ∈ G and let Ω ⊂ G be a compact set. Assume that there exists a p -dimensional subgroup V of G such that Df ( x ) | V is an h-isomorphism for every x ∈ Ω D for some D > . Set λ = sup x ∈ Ω Lip( f | B ( x,R ) ) , where R is the constant given by Proposition 4.4 applied to Ω .Then there exists a constant ≤ T ( G , λ, R, p ) < ∞ , such that Z Ω C P ( Df ( x )) d S Q ( x ) ≤ T Z M S Q − P ( f − ( m ) ∩ Ω) d S P ( m ) . Proof.
We can assume
R < D . Set W any homogeneous subgroup complementary to V .By Propositions 4.5 and 2.7, there exists a constant K > m ∈ M and x ∈ f − ( m ) ∩ Ω, for every 0 < r < R , S Q − P ( f − ( m ) ∩ B ( x, r )) ≥ Kr Q − P , where K is a constant depending on c ( W , V ) > φ m,x : U m,x ⊂ W → V of { f − ( m ) ∩ B ( x, r ) } { m ∈ M ,x ∈ f − ( m ) ∩ Ω } .Moreover observe that by Proposition 4.5, φ m,x are intrinsic L -Lipschitz, for some constant L independent of m and x . Now notice that our observation can take the place of thehypothesis that level sets f − ( m ) are uniformly locally lower Ahlfors ( Q − P )-regular withrespect to S Q − P in Theorem 1.1 (more precisely in Claim 2 on Theorem 3.2), hence we canapply our result to this situation, and we directly get the thesis. (cid:3) Remark 6.
We have seen, in the proof of Corollary 4.6, that the existence of a p -dimensional homogeneous subgroup V complementary to ker( Df ( x )) for every point x ∈ G ,implies that the level sets of f are R -locally C -lower Ahlfors ( Q − P )-regular with respectto S Q − P , for some positive constants C and R , locally independent of the choice of the levelset. We want to highlight that the opposite may be false. In fact, there exist continuouslyPansu differentiable maps between Carnot groups, with everywhere surjective differential,such that their level sets are lower Ahlfors regular, but at the same time ker( Df ( x )) doesnot admit any complementary subgroup.We present a simple example related to the first Heisenberg group H , that is the sim-plest non-commutative Carnot group. It can be represented as a direct sum of two linearsubspaces H = H ⊕ H , where H = span( e , e ), H = span( e ) with unique non trivialrelation [ e , e ] = e . For every p, q ∈ H , pq = p + q + [ p, q ]. Let us consider the map f : H → R , f ( x, y, z ) = ( ax + by, cx + dy ) , with det (cid:20) a bc d (cid:21) = 0 . Observe that f ∈ L ( H , R ), then the Pansu differential of f is constant on H : for every¯ x ∈ H , Df (¯ x )( x, y, z ) = f ( x, y, z ) , hence, ker( Df (¯ x )) = span( e ) for every ¯ x ∈ H . Notice that span( e ) is a normal homoge-neous subgroup of metric dimension 2 that does not admit any complementary subgroup(see for instance [5, Proposition 4.1]). Let us now focus on the level sets of f . If we fix v ∈ R , we have that f − ( v ) = w span( e ) for some w = w ( v ) ∈ H , hence any level setis a coset of span( e ). Then, by left invariance and homogeneity of the distance, the levelsets f − ( v ) are C -lower Ahlfors 2-regular with respect to S , for some positive constant C ,independent of the choice of v . Acknowledgement.
We would like to thank Pierre Pansu for directing us to the study ofthe problem during my stay in Orsay. We would also like to express our gratitude to BrunoFranchi and Francesco Serra Cassano, for many fruitful conversations and suggestions.
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