C^{1,α}-rectifiability in low codimension in Heisenberg groups
aa r X i v : . [ m a t h . M G ] F e b C ,α -RECTIFIABILITY IN LOW CODIMENSION IN HEISENBERGGROUPS KENNEDY OBINNA IDU AND FRANCESCO PAOLO MAIALE
Abstract.
A natural notion of higher order rectifiability is introduced for sub-sets of Heisenberg groups H n in terms of covering a set almost everywhere bya countable union of ( C ,αH , H )-regular surfaces, for some 0 < α ≤
1. We provethat a sufficient condition for C ,α -rectifiability of low-codimensional subsets inHeisenberg groups is the almost everywhere existence of suitable approximatetangent paraboloids. MSC (2010):
Keywords: rectifiability, approximate tangent paraboloids, Heisenberg group .1.
Introduction
Rectifiable sets are focal to studies in geometric measure theory and admit variousapplications in several branches of mathematical analysis. Interests in such sets arisemainly for their geometric, measure-theoretic, and analytic properties which includea notion of (approximate) tangent spaces defined almost everywhere, a version ofthe area and coarea formulas ([1],[16]), and a framework for studying boundednessof a class of singular integral operators (see e.g., [4, 5, 3]).In metric spaces, particularly Carnot groups, the definition of rectifiability di-verges along several, not necessarily equivalent, directions (see e.g., [12, 19, 2, 10]).The original definition by Federer [8, Section 3.2.14] is in terms of composing a setwith countably many Lipschitz images of subsets of the Euclidean space R n . Thisis adopted in [1] and shown to be inappropriate in general metric spaces consideringeven the basic setting of the Heisenberg group, H . Mattila et al. in [18] defined rec-tifiability in the Heisenberg group H n considering a countable union of C H -regularsurfaces. This is related to the approach of using notions of regular surfaces in thesense of Franchi, Serapioni and Serra Cassano (see e.g., [11], [13], [14]). Severalresults can be found on characterizations and basic properties of rectifiable sets inEuclidean spaces and general metric spaces (see e.g., [1, 8, 9, 6, 17, 18]). A wellknown characterization in the Heisenberg group H n is in terms of the a.e. existenceof the approximate tangent spaces (see e.g., [18]).A missing piece in the study of rectifiability in metric spaces is the natural notionof higher order rectifiability which, in the Heisenberg group, can be defined in terms Date : February 11, 2021. ,α -RECTIFIABILITY IN LOW CODIMENSION IN HEISENBERG GROUPS 2 of composing a set with countably many ( C m,αH , H )-regular surfaces, for some m ∈ N and 0 < α ≤
1. Our goal in this article is to initiate progress along this line.1.1.
Main results.
We prove a characterization of C ,α -rectifiability in low-codimensionin terms of the H k m -a.e. existence of an approximate tangent paraboloid (as in Def-inition 2.15).Throughout this paper, we denote by k and k m the dimension and metric dimen-sion respectively in the sense that V ∈ V ( H n , k ) with k ∈ { n + 1 , . . . , n } means V is a k -dimensional plane in the vertical Grassmannian and k m = k + 1. Theorem 1.1.
Fix < α ≤ and n +1 ≤ k ≤ n . Let E ⊂ H n be a H k m -measurableset with H k m ( E ) < ∞ such that for H k m -a.e. p ∈ E there are V p ∈ V ( H n , k ) and λ > such that lim sup r → + r k m H k m ( E ∩ B ( p, r ) \ Q α ( p, V p , λ )) = 0 , and assume also that for H k m -a.e. p ∈ E there holds Θ k m ∗ ( E, p ) > . (1) Then E is C ,α -rectifiable in the sense of Definition 2.5. Next, we prove that a slightly stronger opposite implication is also true.
Proposition 1.2. If E ⊂ H n is a C ,α -rectifiable set with H k m ( E ) < ∞ , then for H k m -a.e. p ∈ E there exist V p ∈ V ( H n , k ) and λ > such that lim r → + r k m H k m ( E ∩ B ( p, r ) \ Q α ( p, V p , λ )) = 0 . In the case α = 1 we can take any λ > arbitrarily small. Preliminaries
The Heisenberg group H n is the simplest Carnot group whose Lie algebra h n hasa step two stratification: h n = h ⊕ h where h = span { X , . . . , X n , Y , . . . , Y n } and h = span { T } with commutators[ X i , Y j ] = δ ij T and [ X i , X j ] = [ Y i , Y j ] = 0 . The vector fields X , . . . , X n , Y , . . . , Y n define a vector subbundle of the tangentvector bundle T H n , the so-called horizontal vector bundle H H n . Via exponentialcoordinates H n can be identified with R n +1 , and we may express the group law bythe Baker-Campbell-Hausdorff formula as p · q := p ′ + q ′ , p n +1 + q n +1 − n X i =1 ( p i q i + n − p i + n q i ) ! , where p ′ := ( p , · · · , p n ). The inverse of p is p − := ( − p ′ , − p n +1 ) and e = 0 is theidentity of H n . The centre of H n is the subgroup T := { p = (0 , . . . , , p n +1 ) } . For ,α -RECTIFIABILITY IN LOW CODIMENSION IN HEISENBERG GROUPS 3 any q ∈ H n and r > , we denote as τ q : H n → H n the left translation p q · p = τ q ( p )and as δ r : H n → H n the dilation p (cid:0) rp ′ , r p n +1 (cid:1) =: δ r p. We denote by k · k the homogeneous norm (with respect to dilations) and d themetric given, respectively, by k p k := d ( p, e ) := max n(cid:13)(cid:13) p ′ (cid:13)(cid:13) R n , | p n +1 | / o and d ( p, q ) = d (cid:0) q − · p, e (cid:1) = (cid:13)(cid:13) q − · p (cid:13)(cid:13) , for all p, q ∈ H n , where k · k R n stands for the Euclidean norm in R n .Let Ω be an open subset of H n ≡ R n +1 and k ≥ C m (Ω) the space of real-valued functions whichare m times continuously differentiable in the Euclidean sense. We further denote by C m (Ω , H H n ) the set of all C m -sections of H H n construed in the sense of regularitybetween smooth manifolds.Let f ∈ C (Ω). We define the horizontal gradient of f as ∇ H f := ( X f, . . . , X n f, Y f, . . . , Y n f )or, equivalently as a section of the horizontal bundle H H n , as ∇ H f := n X j =1 ( X j f ) X j + ( Y j f ) Y j with canonical coordinates ( X f, . . . , X n f, Y f, . . . , Y n f ). Definition 2.1.
Let
U ⊂ H n and f : U → R a continuous function. We say f ∈ C H ( U ) if ∇ H f exists and is continuous in U . Furthermore, if ∇ H f is α -H¨oldercontinuous for some < α ≤ , then we say f ∈ C ,αH ( U ) .We write (cid:2) C H ( U ) (cid:3) k as the set of k -tuples f = ( f , · · · , f k ) such that f i ∈ C H ( U ) for each ≤ i ≤ k . We define h C ,αH ( U ) i k analogously. We remark that the inclusion C ( U ) ⊂ C H ( U ) is strict (see e.g., [11, Remark5.9]).2.1. C ,α -rectifiability in low codimension. In Proposition 2.20 we state theresult which gives that the metric dimension in H n is given by k m = k + 1 if n + 1 ≤ k ≤ n. This tells us that the notion of rectifiability via Lipschitz maps is only interesting inlow dimension ( k ≤ n ). Indeed, any Lipschitz function f : A ⊂ R k → H n satisfies H k m ( f ( A )) = 0when dimension and metric dimension are not equal (i.e., the case k > n ). Therefore,we need to find a different notion of rectifiability.The idea, looking at the Euclidean case, is to first introduce a notion of regularsurfaces which is more fitting in our setting. ,α -RECTIFIABILITY IN LOW CODIMENSION IN HEISENBERG GROUPS 4 Definition 2.2.
Let k ∈ { n + 1 , . . . , n } . A set S ⊂ G is a k -dimensional ( C ,αH , H ) -regular surface if for any p ∈ S there are U ⊆ H n open and f ∈ [ C ,αH ( U )] k satisfying(a) d H f q surjective at all q ∈ U ;(b) S ∩ U = { q ∈ U : f ( q ) = 0 } . The operator d H is the usual Pansu differential and it is represented by the hori-zontal gradient ∇ H f mentioned above. This definition (with C ,αH replaced by C H )was already given in [18] so we refer the reader to that paper for more details. Definition 2.3.
Let S be a k -dimensional ( C ,αH , H ) -regular surface and let f be asabove. The tangent group to S at p ∈ S , denoted as T G S ( p ) , is T G S ( p ) := { p ∈ G : d H f p ( p ) = 0 } . The following characterization of H -regular surfaces is an immediate consequenceof the definition: Proposition 2.4.
A set S is a k -dimensional ( C ,αH , H ) -regular surface if and only if S is locally the intersection of (2 n +1 − k ) 1 -codimensional ( C ,αH , H ) -regular surfaceswith linearly independent normal vectors. We recall that, for any open set Ω ⊂ H n , the Taylor’s expansion of a function f ∈ C αH (Ω) based at the point x ∈ Ω, is given by f ( x ) = f ( x ) + d H f x ( x − x ) + o ( d ( x , x ) α ) . (2)To conclude this introductory section, we can finally give the formal definition of C ,α -rectifiability for a subset of a homogeneous group. Definition 2.5.
A measurable set E ⊂ H n is C ,α -rectifiable if there exist k -dimensional ( C ,αH , H ) -regular surfaces S i , with i ∈ N , such that H k m E \ [ i ∈ N S i ! = 0 . Whitney’s extension theorem.
The following Whitney-type extension the-orem was proved in [20, Theorem 4] and holds for all Heisenberg groups.
Theorem 2.6 ( C ,α -extension) . Let F be a closed subset of a Carnot group G . Let α ∈ (0 , and f : F → R , g : F → H G satisfying the following: there exists M > such that(i) | f ( x ) | , | g ( x ) | ≤ M , on every compact subset of F ;(ii) | f ( x ) − f ( y ) − h g ( x ) , π ( y − x ) i| ≤ M d ( x, y ) α for every x, y ∈ F ;(iii) | g ( x ) − g ( y ) | ≤ M d ( x, y ) α for every x, y ∈ F ;where h· , ·i is the inner product in H G (identified as an Euclidean space). Thenthere exists an extension ˜ f : G → R , ˜ f ∈ C ,α G ( G ) such that g ( x ) = ∇ G ˜ f ( x ) for all x ∈ F . ,α -RECTIFIABILITY IN LOW CODIMENSION IN HEISENBERG GROUPS 5 The intrinsic Grassmannian.
A subgroup S ⊂ H n is a homogeneous sub-group if δ r ( S ) ⊆ S for all r >
0, where δ r is the intrinsic dilation defined by δ r ( p ) = ( rp , . . . , rp n , r p n +1 ) . A homogeneous subgroup S is either horizontal , i.e. contained in exp( h ), or vertical ,i.e. it contains the center of H n . Horizontal subgroups are commutative whilevertical subgroups are non-commutative and normal in H n .In the sequel, we often use the distance d ( p, S ) := inf s ∈ S d ( p, s ), where d ( p, s ) := k p − s k . Definition 2.7.
Two homogeneous subgroups S and T of H n are complementarysubgroups in H n if S ∩ T = { } and H n = T · S . If, in addition, T is normal wesay that H n is the semidirect product of S and T and write H n = T ⋊ S . If H n is the semidirect product of homogeneous subgroups S and T , then we candefine unique projections π S : H n → S and π T : H n → T in such a way thatid H n = π T · π S . Furthermore, if T is normal in H n , then the following algebraic equalities hold: π T ( p − ) = π − S ( p ) · π − T ( p ) · π S ( p ) , π S ( p − ) = π − S ( p ) ,π T ( δ λ p ) = δ λ π T ( p ) , π S ( δ λ p ) = δ λ π S ( p ) ,π T ( p · q ) = π T ( p ) · π S ( p ) · π T ( q ) · π − S ( p ) , π S ( p · q ) = π S ( p ) · π S ( q ) . Proposition 2.8. If H n = T ⋊ S as above, then the projections π S and π T arecontinuous, π S is a h -homomorphism and there is c ( S, T ) := c > such that c k π S ( p ) k ≤ d ( p, T ) ≤ k π S ( p ) k ,c k π − S ( p ) · π T ( p ) · π S ( p ) k ≤ d ( p, S ) ≤ k π − S ( p ) · π T ( p ) · π S ( p ) k (3) holds for all p ∈ H n . This result was proved in [18] for the Heisenberg group and generalized in [15] toall homogeneous groups.
Remark 2.9.
In [15], we also proved that the constant c does not depend on S and T if we consider for 1 ≤ k ≤ n a k -homogeneous subgroup S and write H n = S ⊥ ⋊ S where S ⊥ is the vertical subgroup defined as follows. If S = h f , . . . , f k i , we take S ⊥ = h f , . . . , f k i ⊥ H ⊕ h e n +1 i , where ⊥ H denotes the orthogonal in the horizontal layer of H n with respect to thefixed scalar product. In this case, we denote by c G the universal constant.We are now ready to introduce the notion of intrinsic Grassmannian as in [18]. ,α -RECTIFIABILITY IN LOW CODIMENSION IN HEISENBERG GROUPS 6 Definition 2.10. A k -homogeneous subgroup S belongs to the k -Grassmannian G ( H n , k ) if there is a (2 n + 1 − k ) -subgroup T such that H n = T · S . Moreover,the union G ( H n ) = n +1 [ k =0 G ( H n , k ) is often referred to as the intrinsic Grassmannian of H n . Proposition 2.11.
The trivial subgroups { e } and H n are the unique elements of G ( H n , and G ( H n , n + 1) respectively and(i) for ≤ k ≤ n , G ( H n , k ) coincides with the set of all horizontal k -homogeneoussubgroups;(ii) for n + 1 ≤ k ≤ n , G ( H n , k ) coincides with the set of all vertical k -homogeneous subgroups.Furthermore, any vertical subgroup T with linear dimension in { , . . . , n } is not anelement of the intrinsic Grassmannian of H n . This result was proved in [18, Proposition 2.17]. Notice that in the Carnot settingthis result is no longer true since the equality G ( G , k ) = { vertical k -homogeneous subgroups } for any k does not hold if we do not put additional assumptions on G or to thepossible values of k . Remark 2.12. If S is a ( C ,αH , H )-regular surface, then T H S ( p ) ∈ G ( H n ). Remark 2.13.
The Grassmannian G ( H n ) is a subset of the Euclidean counterpart(in R n +1 ) and is endowed with the same topology. Moreover, G ( H n , k ) is a compactmetric space with respect to the distance ρ ( S , S ) = max k x k =1 d ( π S ( x ) , π S ( x )) . Intersection lemma.
The main objects we deal with in this paper are α -paraboloids and cylinders, so we first recall the definitions in this setting: Definition 2.14.
Fix α ∈ (0 , , λ > and S ∈ G ( H n ) . The α -paraboloid of aperture λ is defined as Q α ( x, S, λ ) = (cid:8) y ∈ H n : d ( x − y, S ) ≤ λd ( x, y ) α (cid:9) , while the α -cylinder of aperture λ is given by C α ( x, S, λ ) = B ( S, λr α ) = (cid:8) y ∈ H n : d ( x − y, S ) ≤ λr α (cid:9) . Definition 2.15.
Let E ⊂ H n be H k m -measurable and α ∈ (0 , . We say that ahomogeneous subgroup V p , of dimension k and metric dimension k m , is an approx-imate tangent paraboloid to E at p if Θ ∗ k m ( E, p ) > and lim r → r − k m H k m ( E ∩ B ( p, r ) \ Q α ( p, V p , λ )) = 0 for all λ > . We write apPar k m H ( E, p ) for the set of all approximate tangent paraboloids to E at p and, if there is only one, we denote it by V p . ,α -RECTIFIABILITY IN LOW CODIMENSION IN HEISENBERG GROUPS 7 The following result gives the relation between cylinders and paraboloids, whichexplains why to prove our main result we can equivalently use cylinders.
Lemma 2.16 ([7], Lemma 2.3) . For S ∈ G ( H n , k ) and r > fixed, suppose thatfor every r < r H k ( E ∩ B ( x, r ) \ C α ( x, S, λ )) ≤ ǫr k . Then for all r < r we have H k (cid:0) E ∩ B ( x, r ) \ Q α ( x, S, λ ′ ) (cid:1) ≤ ǫ − − k r k , where λ ′ := 4 α λ . Using the Taylor’s expansion (2), it is not difficult to prove the following (repeatingsimilar arguments as in [18, Lemma 2.28]).
Lemma 2.17.
Fix n < k ≤ n . Let S ⊂ H n be a k -dimensional ( C ,αH , H ) -regularsurface and x ∈ S . Then for all λ > there exists r > such that S ∩ B ( x, r ) ⊂ Q α ( x, T H S ( x ) , λ ) . We now prove that vertical subgroups in the Grassmannian have horizontal com-plements that can be chosen in a continuous way.
Lemma 2.18.
Given T ∈ G ( H n , k ) with n < k ≤ n , we can always find unit vectors ν , . . . , ν n +1 − k ∈ h , continuously depending on T , such that the subgroup S := exp (span { ν , . . . , ν n +1 − k } ) is horizontal complement of T . Furthermore T = ∩ kj =1 N ( ν j ) and for all p ∈ G andall α ∈ (0 , there holds the inclusion Q α ( p, T, λ ) ⊆ k \ j =1 Q α ( p, N ( ν j ) , λ ) . The proof of the first part can be found in [13, Lemma 3.26] (the continuity followsfrom the construction). In [18, Lemma 2.32] they proved the inclusion X ( p, V, s ) ⊆ k \ j =1 X ( p, N ( ν j ) , s ) , but it is relatively easy to see that we can replace cones with paraboloids.The following is a basic result with proof which closely follows an Euclideanversion that can be found in [7, Lemma 2.1]. Lemma 2.19.
Let
S, T ∈ G ( H n , k ) with n < k ≤ n and set ϑ := ρ ( S, T ) . Thenthere is Z ∈ G ( H n , k − and ℓ > such that for any positive number η we have B ( S, η ) ∩ B ( T, η ) ⊆ B (cid:18) Z, nηℓϑ (cid:19) . ,α -RECTIFIABILITY IN LOW CODIMENSION IN HEISENBERG GROUPS 8 Proof.
First, we claim that there is e ∈ T with k e k = 1 such that there holds k π S ⊥ ( e ) k = ℓϑ . Indeed, an application of the triangular inequality gives ϑ ≥ ρ ( S ⊥ , T ⊥ ) − C T − C S ≥ sup t ∈ T ⊥ k t k =1 k π S ⊥ ( t ) k − C T − C S , where C T = max k x k =1 d ( π T ( x ) , π ⊥ T ( x )) and C S = max k x k =1 d ( π S ( x ) , π ⊥ S ( x )) . It follows that ϑ + C T + C S ≥ sup t ∈ T ⊥ k t k =1 k π S ⊥ ( t ) k > L , L > L ϑ ≥ sup t ∈ T ⊥ k t k =1 k π S ⊥ ( t ) k ≥ L ϑ. By compactness, we can find e ∈ T ⊥ and ℓ ∈ [ L , L ] such that k π S ⊥ ( e ) k = ℓϑ, and this concludes the proof of the claim. Now consider an orthonormal basis e k +1 , . . . , e n +1 of S and define Z := span { e, e k +1 , . . . , e n +1 } ⊥ . It is easy to verify that dim( Z ) = k − x ∈ B ( S, η ) ∩ B ( T, η ) we have ( k x − · e i k ≤ η for i = k + 1 , . . . , n + 1 , k x − · e k ≤ η. . We now set e ′ := π S ⊥ ( e ) k π S ⊥ ( e ) k and consider the resulting orthonormal basis { e ′ , e k +1 , . . . , e n +1 } of Z ⊥ . Then for x ∈ B ( V, η ) ∩ B ( W, η ), using the triangle inequality, we have k x − · e ′ k = 1 k π S ⊥ ( e ) k k x − · π S ⊥ ( e ) k = 1 ℓϑ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) x − · e − n +1 X i = k +1 ( e − · e i ) e i !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ℓϑ (2 n + 1 − k + 1) η. ,α -RECTIFIABILITY IN LOW CODIMENSION IN HEISENBERG GROUPS 9 We finally infer that k π Z ⊥ ( x ) k ≤ k x − · e ′ k + n +1 X i = k +1 k x − · e i k≤ k x − · e ′ k + n +1 X i = k +1 k x − · e i k≤ ℓϑ (2(2 n + 1) − k + 1) η ≤ nℓϑ η, and this concludes the proof. (cid:3) Other technical results.
The first statement we need for the proof of ourmain result was proved in [18, Proposition 2.20]:
Proposition 2.20.
Let S ∈ G ( H n , k ) with n < k ≤ n . Then H k euc is a left-invariantHaar measure on S and there results H k euc x S = H k m x S (4) where k m is the metric dimension of S and it is equal to k m = k + 1 . Lemma 2.21.
Let S ∈ G ( H n ) , α ∈ (0 , and ¯ λ, M > . Then for all λ ≤ ¯ λ , givenany p ∈ H n and q ∈ H n \ Q α ( p, S, λ ) with d ( p, q ) ≤ M , we have B (cid:18) q, λ d ( p, q ) α (cid:19) ⊂ H n \ Q α (cid:18) p, S, λC (cid:19) , where C := C ( M, ¯ λ ) > .Proof. Let r := d ( p, q ) and z ∈ B ( q, λr α / d ( p, z ) α ≤ [ d ( p, q ) + d ( q, z )] α ≤ (cid:20) r + λ r α (cid:21) α . To prove that z does not belong to the paraboloid above, we notice that d ( p − z, S ) ≥ d ( p − q, S ) − d ( p − z, p − q ) > ≥ λd ( p, q ) α − d ( z, q ) ≥≥ λr α − λ r α = λ r α , where the last inequality follows from the fact that q ∈ H n \ Q α ( p, S, λ ). We nowhave to study the sign of the following function: T r,α,λ ( t ) := t λ r α − (cid:20) r + λ r α (cid:21) α . The idea is to consider λ a variable and find ¯ t in such a way that the functions isincreasing with respect to λ ∈ (0 , ¯ λ ) withlim λ → T r,α (¯ t, λ ) ≥ . This is easily achieved by computing the derivative. Notice that, since we fix ¯ λ > t for which the above holds. (cid:3) ,α -RECTIFIABILITY IN LOW CODIMENSION IN HEISENBERG GROUPS 10 Proofs of main results
The goal of this section is to prove Theorem 1.1 and Proposition 1.2. We haveadopted similar techniques used in [7, Section 3] and [18, Section 3] for some of thekey results.
Proof of Proposition 1.2.
Let E ⊂ H n be C ,α -rectifiable and let { Γ i } i ∈ N be thefamily of ( C ,αH , H )-regular surfaces such that H k E \ [ i ∈ N Γ i ! = 0 . In particular E is H k -rectifiable so (see [18, Theorem 3.15]) for H k -a.e. x ∈ E thereexists an approximate tangent subgroup T x ∈ G ( H n , k ) and Θ k ∗ ( E, x ) >
0. For each i ∈ N denote by E i the set E ∩ Γ i ; by standard density properties (e.g., [18, Lemma3.6]) for H k -a.e. x ∈ E i we have thatΘ k ( E \ E i , x ) = 0 . (5)By Lemma 2.17, for any λ > E i ∩ B ( x, r ) \ Q α ( x, T x , λ ) = ∅ for every x ∈ E i and r small enough. The conclusion follows by the latter fact and (5). (cid:3) For the sake of simplicity, in the next technical result we will use the followingnotation for cylinders: C rα ( x ) := C α ( x, V x , r ) . Lemma 3.1.
Let E ⊂ H n , take n < k ≤ n and fix M, λ, δ, r > . Suppose that forevery z ∈ E and for every < ρ ≤ r we have H k ( E ∩ B ( z, ρ )) ≤ M ρ k . (6) Consider any two points x, y such that d ( x, y ) ≤ r and V x , V y ∈ G ( H n , k ) satisfying ( H k ( E ∩ B ( x, r )) ≥ δr k H k ( E ∩ B ( y, r )) ≥ δr k (7) and ( H k ( E ∩ B ( x, r ) \ C rα ( x )) ≤ εr k H k ( E ∩ B ( y, r ) \ C rα ( y )) ≤ εr k (8) where ε ≤ δ . Then there exists a positive constant C := C ( n, δ, M, λ ) such that ρ ( V x , V y ) ≤ Cr α . The proof of this result is similar to the Euclidean version (see [7, Lemma 3.5] formore details) thanks to Lemma 2.19.
Remark 3.2.
Using Lemma 2.16, it is easy to see that the cylinders can be replacedwith α -paraboloids in Lemma 3.1.We now prove, thanks to this result, that Lemma 2.18 can be improved, namelywe can get the α -H¨older continuity of the dependence. ,α -RECTIFIABILITY IN LOW CODIMENSION IN HEISENBERG GROUPS 11 Lemma 3.3.
Let n < k ≤ n and let U be an open subset of H n . Suppose that wehave a mapping U ∋ p V p ∈ G ( H , k ) that satisfies the following property: for all p, q ∈ U , V p and V q satisfy the assump-tions of Lemma 3.1. If we denote by W p = span (exp { ν ( p ) , . . . , ν n +1 − k ( p ) } ) the horizontal complement given in Lemma 2.18, then the mappings p ν j ( p ) , j = 1 , . . . , n + 1 − k are α -H¨older continuous in some U ′ ⊂ U , with α given in Lemma 3.1.Proof. Up to replacing U with U ′ ⊂ U , we can always assume that d ( p, q ) is suffi-ciently small so an application of Lemma 3.1 gives ρ ( V p , V q ) ≤ cd ( p, q ) α . On the other hand, in Lemma 2.18 we proved that V p ν j ( p ) , j = 1 , . . . , n + 1 − k are continuous maps for p ∈ U ′ , which means that d ( ν j ( p ) , ν j ( q )) ≤ c ′ ρ ( V p , V q )holds for all p, q ∈ U ′ . If we put together these two inequalities, we get d ( ν j ( p ) , ν j ( q )) ≤ c ′ ρ ( V p , V q ) ≤ ˜ cd ( p, q ) α , where ˜ c := c · c ′ , which means that the α -H¨older continuity is proved. (cid:3) Proof of Theorem 1.1.
Following [18], we first prove uniqueness almost ev-erywhere of approximate tangent paraboloids.
Proposition 3.4.
Let E ⊂ H n be H k m -measurable with H k m ( E ) < ∞ , and let A be the set of points of E for which there is an approximate tangent parabolid ofdimension k and metric dimension k m . Then the following holds:(a) A is H k m -measurable;(b) E has an unique approximate tangent paraboloid V p at H k m -a.e. p ∈ A ;(c) the mapping A ∋ p V p ∈ G ( H n , k ) is measurable. The proof of this result follows the same strategy of [18, Proposition 3.9] so werefer the reader to that paper for more details.We are now ready to prove our main result. We follow closely the strategy in [18,Theorem 3.15] and point out the main differences in our case.
Proof of Theorem 1.1.
Suppose that for H k m -a.e. p ∈ E there are r ( p ) , ℓ ( p ) > V p = apPar k m H ( E, p ) such that, for 0 < r < r ( p ), there holds H k m ( E ∩ B ( p, r )) > ℓ ( p ) r k m , (9)and, for all λ >
0, we also havelim r → H k m x E ( B ( p, r ) \ Q α ( p, V p , λ )) r k m = 0 . (10) ,α -RECTIFIABILITY IN LOW CODIMENSION IN HEISENBERG GROUPS 12 For i ≥ E i as E i := (cid:26) p ∈ E : min { r ( p ) , ℓ ( p ) } > i (cid:27) , and notice that the difference between E and E ∗ := ∪ i ≥ E i is H k m -negligible so wecan work in E ∗ without loss of generality.Now recall that, given any p ∈ E ∗ , we can find k horizontal unit vectors ν h ( p ) inthe horizontal bundle H H np transversal to V p in such a way that T p := exp (span { ν ( p ) , . . . , ν n +1 − k ( p ) } )is a horizontal subgroup of H n with H n = V p · T p . Moreover, using Proposition 3.4and the continuity part of Lemma 2.18, we also find that E ∗ can be written as E ∗ = [ j ≥ F j with H k m ( F j ) < ∞ and ν h (cid:12)(cid:12) F j is H k m -measurable. As a consequence ν h : E ∗ → H H n is a measurable sections of H H n for each 1 ≤ h ≤ n + 1 − k . Define for appropriateindices and for p ∈ E ∗ the function ρ i,h,j ( p ) := sup (cid:26) |h ν h ( p ) , π ( p − q ) i| d ( p, q ) α : q ∈ E i , < d ( p, q ) α < j (cid:27) , where π : H n → h is the projection onto the first layer given by π ( q ) := n X j =1 q j X j . We claim that for all i ≥ ≤ h ≤ n + 1 − k it turns out thatlim j →∞ ρ i,h,j ( p ) = 0 . (11)We argue by contradiction. More precisely, if (11) fails, then there is ¯ λ > τ ∈ (0 , i − ) - there is q ∈ E i such that |h ν h ( p ) , π ( p − q ) i| > ¯ λd ( p, q ) α with r := d ( p, q ) α < τ . Then q ∈ B ( p, r ) \ Q α ( p, N ( ν h ( p )) , λ ) and, applying Lemma 2.21, we get the inclusion B (cid:18) q, ¯ λr (cid:19) ⊂ B ( p, r ) \ Q α (cid:18) p, N ( ν h ( p )) , ¯ λC (cid:19) . (12)Finally, putting Lemma 2.18 together with (12) and (9), yields H k m (cid:18) E ∩ B ( p, r ) \ Q α (cid:18) p, V p , ¯ λC (cid:19)(cid:19) ≥ H k m (cid:18) E ∩ B ( p, r ) \ Q α (cid:18) p, N ( ν h ( p )) , ¯ λC (cid:19)(cid:19) ≥ H k m (cid:18) E ∩ B (cid:18) q, ¯ λr (cid:19)(cid:19) ≥ i α (cid:18) ¯ λr (cid:19) k +1 , ,α -RECTIFIABILITY IN LOW CODIMENSION IN HEISENBERG GROUPS 13 and this is a contradiction with (10): the claim (11) holds true. Apply Lusin theoremto each ν h and Egoroff theorem to the sequence ( ρ i,h,j ) j ; we can write E i = E i, ∪ ( ∪ β ≥ K i,β )with E i, H k m -negligible, K i,β compact, ν h (cid:12)(cid:12) K i,β continuous and ρ i,h,j goes to zerouniformly in K i,β with respect to j . So we can apply Lemma 3.3 (the assumptionsof Lemma 3.1 are satisfied by the lower-density inequality (1)) and obtain that ν h (cid:12)(cid:12) K i,β is actually α -H¨older continuous so we can now apply Whitney theorem (see Theorem2.6) in K i,β and obtain functions f i,β,h ∈ C ,αH ( H n )with f i,β,h (cid:12)(cid:12) K i,β = 0, ∇ H f i,β,h (cid:12)(cid:12) K i,β = ν h and |∇ H f i,β,h | 6 = 0. Consider the set S i,β,h := { p ∈ H n : f i,β,h ( p ) = 0 } and notice that it is a 1-codimensional ( C ,αH , H )-regular surface containing K i,β .Finally, we can consider the intersection S i,β := k \ h =1 S i,β,h . By Proposition 2.4 we have that S i,β is a k -codimensional ( C ,αH , H )-regular surfacethat contains the set K i,β . Moreover, we have E ⊂ E ∪ ( ∪ i ≥ ∪ β ≥ S i,β ) , which means that E is C ,α -rectifiable, concluding the proof. (cid:3) References [1]
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Email address : [email protected] Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy.