Marstrand-Mattila rectifiability criterion for 1 -codimensional measures in Carnot Groups
MM A R S T R A N D - M A T T I L AR E C T I F I A B I L I T Y C R I T E R I O N F O R1 - C O D I M E N S I O N A L M E A S U R E S I NC A R N O T G R O U P S a n d r e a m e r l o * abstract This paper is devoted to show that the flatness of tangents of 1-codimensional measures in CarnotGroups implies C G -rectifiability. As applications we prove a criterion for intrinsic Lipschitz rectifiability of finiteperimeter sets in general Carnot groups and we show that measures with ( n + ) -density in the Heisenberggroups H n are C H n -rectifiable, providing the first non-Euclidean extension of Preiss’s rectifiability theorem. keywords Marstrand-Mattila rectifiability criterion, Preiss’s rectifiability Theorem, Carnot groups. msc ( ) A , A , C . introduction In Euclidean spaces the following rectifiability criterion is available, known as Marstrand-Mattila rectifiabilitytheorem. It was first proved by J. M. Marstrand in [ ] for m = n =
3, later extended by P. Mattila to every m ≤ n in [ ] and eventually strengthened by D. Preiss in [ ]. Theorem . Suppose φ is a Radon measure on R n and let m ∈ {
1, . . . , n − } . Then, the following are equivalent:(i) φ is absolutely continuous with respect to the m-dimensional Hausdorff measure H m and R n can be covered φ -almostall with countably many m-dimensional Lipschitz surfaces,(ii) φ satisfies the following two conditions for φ -almost every x ∈ R n :(a) < Θ m ∗ ( φ , x ) ≤ Θ m , ∗ ( φ , x ) < ∞ ,(b) Tan m ( φ , x ) ⊆ { λ H m (cid:120) V : λ > V ∈ Gr ( n , m ) } , where the set of tangent measures Tan m ( φ , x ) is introduced inDefinition . . The rectifiability of a measure, namely that (i) of Theorem holds, is a global property and as such it isusually very difficult to verify in applications. Rectifiability criteria serve the purpose of characterizing such globalproperty with local ones, that are usually conditions on the density and on the tangents of the measure. Most of themore basic criteria impose condition (ii a) and the existence of an affine plane V ( x ) , depending only on the point x , on which at small scales the support of measure is squeezed on around x . The difference between these variouselementary criteria relies on how one defines “squeezed on”, for an example see Theorem . of [ ]. However,the existence of just one plane approximating the measure at small scales may be still too difficult to prove inmany applications and this is where Theorem comes into play. The Marstrand-Mattila rectifiability criterion saysthat even if we allow a priori the approximating plane to rotate at different scales, the density hypothesis (ii a)guarantees a posteriori this cannot happen almost everywhere. * Universitá di Pisa, Largo Bruno Pontecorvo, , Pisa, Italy a r X i v : . [ m a t h . M G ] A ug It is well known that if a Carnot group G has Hausdorff dimension Q , then it is ( Q − ) -purely unrectifiable inthe sense of Federer, see for instance Theorem . of [ ]. Despite this geometric irregularity, in the fundationalpaper [ ] B. Franchi, F. Serra Cassano and R. Serapioni introduced the new notion of C G -rectifiability in Carnotgroups, see Definition . . This definition allowed them to establish De Giorgi’s rectifiability theorem for finiteperimeter sets in the Heisenberg groups H n : Theorem (Corollary . , [ ]) . Suppose Ω ⊆ H n is a finite perimeter set. Then its reduced boundary ∂ ∗ H Ω is C G -rectifiable. It is not hard to see that an open set with smooth boundary is of finite perimeter in H n , but there are finiteperimeter sets in H whose boundary is a fractal from an Euclidean perspective, see for instance [ ]. This meansthat the Euclidean and C G -rectifiability are not equivalent.The main goal of this paper is to establish a 1-codimensional analogue of Theorem in Carnot groups: Theorem . Suppose φ is a Radon measure on G . The following are equivalent:(i) φ is absolutely continuous with respect to the ( Q − ) -dimensional Hausdorff measure H Q− and φ -almost all G canbe covered by countably many C G -surfaces,(ii) φ satisfies the following two conditions for φ -almost every x ∈ G :(a) < Θ Q− ∗ ( φ , x ) ≤ Θ Q− ∗ ( φ , x ) < ∞ ,(b) Tan Q− ( φ , x ) is contained in M , the family of non-null Haar measures of the elements of Gr ( Q − ) , the -codimensional homogeneous subgroups of G . While the fact that (i) implies (ii) follows for instance from Lemma . and Corollary . of [ ], the viceversa isthe subject of investigation of this work. Besides the already mentioned importance for the applications, Theorem is also relevant because it establishes that C G -rectifiability is characterized in the same way as the Euclidean one,and this is the main motivation behind the definition of P -rectifiable measures, given in Definition . . Our mainapplication of Theorem , is the proof of the first extension of Preiss’s rectifability theorem outside the Euclideanspaces: Theorem . Suppose φ is a Radon measure on the Heisenberg group H n such that for φ -almost every x ∈ H n , we have: < Θ n + ( φ , x ) : = lim r → φ ( B ( x , r )) r n + < ∞ , where B ( x , r ) are the metric balls relative to the Koranyi metric. Then φ is absolutely continuous with respect to H n + and H n can be covered φ -almost all with countably many C H n -regular surfaces. Finally, an easy adaptation of the arguments used to prove Theorem also provides the following rectifiabilitycriterion for finite perimeter sets in arbitrary Carnot groups. Theorem asserts that if the tangent measures to theboundary of a finite perimeter set are sufficiently close to vertical hyperplanes, then the boundary can be coveredby countably many intrinsic Lipschitz graphs. Theorem . There exists an ε G > such that if Ω ⊆ G is a finite perimeter set for which: lim sup r → d x , r ( | ∂ Ω | G , M ) : = lim sup r → inf ν ∈ M W ( | ∂ Ω | G (cid:120) B ( x , r ) , ν (cid:120) B ( x , r )) r Q ≤ ε G , for | ∂ Ω | G -almost every x ∈ G where W is the -Wasserstein distance, then G can be covered | ∂ Ω | G -almost all with countablymany intrinsic Lipschitz graphs. We present here a survey on the strategy of the proof of our main result, Theorem . For the sake of discussion,let us put ourselves in a simplified situation. Assume E is a compact subset of a Carnot group G = ( R n , ∗ ) suchthat: ( α ) there exists an η ∈ N such that η − r Q− ≤ H Q− ( E ∩ B ( x , r )) ≤ η r Q− for any 0 < r < diam ( E ) and any x ∈ E ,( β ) the functions x (cid:55)→ d x , r ( H Q− (cid:120) E , M ) converge uniformly to 0 on E as r goes to 0.The cryptic condition ( β ) can be reformulated, thanks to Propositions . and . in the following more geometricway. For any (cid:101) > r ( (cid:101) ) > H Q− -almost any x ∈ E and any 0 < ρ < r ( (cid:101) ) there is a plane V ( x , ρ ) ∈ Gr ( Q − ) , depending on both the point x and the scale ρ , for which: E ∩ B ( x , ρ ) ⊆ { y ∈ G : dist ( y , x ∗ V ( x , ρ )) ≤ (cid:101)ρ } , ( ) B ( y , (cid:101)ρ ) ∩ E (cid:54) = ∅ for any y ∈ B ( x , ρ /2 ) ∩ x ∗ V ( x , ρ ) . ( )In Euclidean spaces if a Borel set E satisfies ( ) and ( ) it is said weakly linear approximable , see for instance Chapter of [ ]. The condition ( ) says that at small scales E is squeezed on the plane x ∗ V ( x , ρ ) , while ( ) implies thatinside B ( x , ρ ) any point of x ∗ V ( x , ρ ) is very close to E , see the picture below:Figure : On the left we see that ( ) implies that at the scale ρ the set E , in yellow, is contained in a narrow stripof size 2 (cid:101)ρ around x ∗ V ( x , ρ ) . On the right we see that ( ) implies that any ball centred on the plane x ∗ V ( x , ρ ) inside B ( x , ρ /2 ) and of radius (cid:101)ρ must meet E .The first step towards the proof of our main result is the following technical proposition that is a suitablyreformulated version of Theorem . of [ ] and of Lemma . of [ ]. Proposition shows that if at some point x the set E has also big projections on some plane W , i.e. ( ) holds, then around x the set E is almost a W -intrinsicLipschitz surface. Proposition . Let k > η and ω > . Suppose further that x ∈ E and ρ > are such that:(i) d x , k ρ ( H Q− (cid:120) E , M ) ≤ ω ,(ii) there exists a plane W ∈ Gr ( Q − ) such that: ( ρ / k ) Q− ≤ H Q− (cid:120) W ( P W ( B ( x , ρ ) ∩ E )) , ( ) where P W is the splitting projection on W, see Definition . .If k is chosen suitably large and ω are suitably small, there exists an α = α ( η , k , ω ) > with the following property. Forany z ∈ E ∩ B ( x , ρ ) and any y ∈ B ( x , k ρ /8 ) ∩ E for which ωρ ≤ d ( z , y ) ≤ k ρ /2 , we have that y is contained in the conezC W ( α ) , that is introduced in Definition . . We remark that thanks to our assumption ( β ) on E , hypothesis (i) of the above proposition is satisfied almosteverywhere on E whenever ρ < ˜ r ( ω ) , where ˜ r ( ω ) is suitably small and depends only on ω . Let us explain someof the ideas of the proof of Proposition . If the plane W is almost orthogonal to V ( x , ρ ) , the element of Gr ( Q − ) for which ( ) and ( ) are satisfied by E at x at scale ρ , we would have that the projection of E on W would be toosmall and in contradiction with ( ), see Figure .Figure : The weak linear approximability of E implies that E ∩ B ( x , ρ ) is contained inside V ωρ , an ωρ -neighbourhood of the plane V ( x , ρ ) . If V ( x , ρ ) and W are almost orthogonal, i.e. the Euclidean scalarproduct of their normals is very small, it can be shown that that the projection P W on W of V ωρ ∩ B ( x , ρ ) has H Q− -measure smaller than ( ωρ ) Q− .It is possible to push the argument and prove that if the constants k and ω are chosen suitably, the planes V ( x , ρ ) and W must be at a very small angle, and in particular inside B ( x , ρ ) the plane x ∗ V ( x , ρ ) must be very close to x ∗ W . So close in fact that it can be proved that E ∩ B ( x , ρ ) is contained in a 2 ωρ -neighbourhood W ωρ of W . Thisimplies that z , y ∈ W ωρ , and since W and V ( x , ρ ) are at a small angle, it is possible to show that dist ( y , zW ) ≤ ωρ .Furthermore, by assumption on y , z we have d ( z , y ) > ωρ and thus we infer that dist ( y , zW ) ≤ d ( y , z ) . Thisimplies in particular that y ∈ zC W ( ) .The second step towards the proof of the main result is to show that at any point x of E and for any ρ > W x , ρ ∈ Gr ( Q − ) on which E has big projectons. Theorem . There is a η ∈ N , such that for H Q− -almost every x ∈ E and ρ > sufficiently small there is a planeW x , ρ ∈ Gr ( Q − ) for which: H Q− (cid:0) P W x , ρ ( E ∩ B ( x , ρ )) (cid:1) ≥ η − ρ Q− . ( )We now briefly explain the ideas behind the proof of Theorem , that are borrowed from Chapter , § . and§ . of [ ]. Fix two parameters η ∈ N and ω > ω < η Q ( Q + ) and for which: B + : = B ( δ η − ( v ) , η − ) ⊆{ y ∈ B (
0, 1 ) : (cid:104) y , v (cid:105) > ω } , B − : = B + ∗ δ η − ( v − ) ⊆{ y ∈ B (
0, 1 ) : (cid:104) y , v (cid:105) < − ω } ,where δ λ are the intrinsic dilations introduced in ( ) and v ∈ V is an arbitrary vector with unitary Euclidean norm.Thanks to the assumption ( ) on E , for any 0 < ρ < r ( ω ) we have that: E ∩ B ( x , ρ ) ⊆ { y ∈ B ( x , ρ ) : dist ( y , x ∗ V ( x , ρ )) ≤ ωρ } .In particular thanks to the assumptions on η and ω we infer that E ∩ x δ ρ B + = ∅ = E ∩ x δ ρ B − . Let W x , ρ : = V ( x , ρ ) and for any z ∈ x δ ρ B + we define the curve: γ z ( t ) : = z δ η − t ( n ( W x , ρ ) − ) , as t varies in [
0, 1 ] and where n ( W x , ρ ) is the unit Euclidean normal to W x , ρ . The curve γ z must intersect W x , ρ at thepoint P W x , ρ ( z ) since γ z ( ) ∈ x δ ρ B − and as a consequence we have the inclusion γ z ([
0, 1 ]) ⊆ P − W x , ρ ( P W x , ρ ( z )) . Sinceconditions ( ) and ( ) heuristically say that E almost coincides with the plane x ∗ W x , ρ inside B ( x , ρ ) and it has veryfew holes, most of the curves γ z should intersect the set E too.More precisely, we prove that if some γ z does not intersect E , there is a small ball U z centred at some q ∈ E suchthat γ z ∩ U z (cid:54) = ∅ . It is clear that defined the set: F : = E ∪ (cid:91) z ∈ x δ r B + , γ z ∩ E = ∅ U z ,we have P W x , r ( x δ r B + ) ⊆ P W x , r ( F ) . So, intuitively speaking adding these balls U z allows us to close the holes of E .An easy computation proves that H Q− ( P W x , r ( x δ r B + )) ≥ r Q− / η Q− and thus in order to be able to conclude theproof of ( ) we should have some control over the size of the projection of the balls U z . This control is achievablethanks to ( ), see Proposition . and Theorem . , and in particular we are able to show that: H Q− (cid:18) P W x , r (cid:16) (cid:91) z ∈ x δ r B + γ z ∩ E = ∅ U z (cid:17)(cid:19) ≤ ω r Q− .This implies that E satisfies the big projection properties, i.e. ( ) holds with η : = η Q− . This part of the argumentis rather delicate and technical. For the details we refer to the proof of Theorem . .The third step towards the proof of Theorem is achieved in Subsection . , where we prove the following: Theorem . There exists a intrinsic Lipschitz graph Γ such that H Q− ( E ∩ Γ ) > . The strategy we employ to prove the above theorem is the following. We know that at H Q− -almost every pointof x ∈ E there exists a plane W x , r such that H Q− ( P W x , r ( E ∩ B ( x , r ))) ≥ η − r Q− . For any x ∈ E at which theprevious inequality holds, we let B the points y ∈ B ( x , r ) for which there is a scale s ∈ ( r ) for which W y , s isalmost orthogonal to W x , r . Choosing the angle between W y , s and W x , r sufficiently big it is possible to prove thatthe projection of B on W x , r is smaller than η − r Q− /2. This follows from the intuitive idea that if y ∈ B , the set E ∩ B ( y , s ) is contained in a narrow strip that is almost orthogonal to W x , r inside B ( y , s ) and thus its projection on W x , r has very small H Q− -measure. On the other hand, Proposition . tells us that S Q− (cid:120) V ( P W x , r ( E ∩ B ( x , r ) \ B )) ≤ c ( V ) S Q− ( E ∩ B ( x , r ) \ B ) , and this allows us to infer that there are many points z ∈ B ( x , r ) ∩ E for which W z , s is contained in a (potentially large) fixed cone with axis W x , r for any 0 < s < r . This uniformity on the scalesallows us to infer thanks to Proposition that E ∩ B ( x , r ) \ B is an intrinsic Lipschitz graph.Since properties ( α ) and ( β ) are stable for the restriction-to-a-subset operation, Theorem implies by means of aclassical argument that E can be covered H Q− -almost all with intrinsic Lipschitz graphs.Therefore, we are reduced to see how we can improve the regularity of the surfaces Γ i covering E from intrinsicLipschitz to C G . Since the blowups of H Q− (cid:120) E are almost everywhere flat, the locality of the tangents, i.e. Proposi-tion . , implies that the blowups of the measures H Q− (cid:120) Γ i are flat as well, where we recall that a measure is saidflat if it is the Haar measure of a 1-codimensional homogeneous subgroup of G . Furthermore, since intrinsic Lips-chitz graphs can be extended to boundaries of sets of finite perimeter, see Theorem . , they have an associatednormal vector field n i . Therefore, for H Q− -almost every x ∈ Γ i the elements of Tan Q− ( H Q− (cid:120) Γ i , x ) are also theperimeter measures of sets with constant horizontal normal n i ( x ) , see Propositions B. , B. , and B. . The aboveargument shows that on the one hand Tan Q− ( H Q− (cid:120) Γ i , x ) are flat measures and on the other if seen as bound-ary of finite perimeter sets, they must have constant horizontal normal coinciding with n i ( x ) almost everywhere.Therefore, for H Q− -almost every x ∈ E ∩ Γ i the set Tan Q− ( H Q− (cid:120) Γ i , x ) must be contained in the family of Haarmeasures of the 1-codimensional subgroup orthogonal to n i ( x ) . The fact that E ∩ Γ i is covered with countablymany C G -surfaces follows by means of the rigidity of the tangents discussed above and a Whitney-type theorem,that is obtained in Appendix B with an adaptation of the arguments of [ ]. structure of the paper In Section we recall some well known facts about Carnot groups and Radon measures. Section is divided infour parts. The main results of Subsection . are Propositions . and . , that allow us to interpret the flatnessof tangents in a more geometric way. Subsection . is devoted to the proof of Proposition . , that is roughlyTheorem . Subsection . is the technical core of this work and the main result proved in it is Theorem . ,that codifies the fact that the flatness of tangents implies big projections on planes. Finally, in Subsection . weput together the results of the previous three subsections to prove Theorem . that asserts that for any Radonmeasure satisfying the hypothesis of Theorem , there is an intrinsic Lipschitz graph of positive φ -measure. InSection we prove Theorem . that is the main result of the paper and its consequences. In Appendix A weconstruct the dyadic cubes that are needed in Section and in Appendix B we recall some well known facts aboutfinite perimeter sets in Carnot groups and intrinsic Lipschitz graphs whose surface measures has flat tangents. notation We add below a list of frequently used notations, together with the page of their first appearance: |·|
Euclidean norm, (cid:107)·(cid:107) smooth-box norm, (cid:104)· , ·(cid:105) scalar product in the Euclidean spaces, V i layers of the stratification of the Lie algebra of G , n i dimension of the i -th layer of the Lie algebra of G , π i ( · ) projections of R n onto V i , h i the topological dimension of the vector space V ⊕ . . . ⊕ V i , Q homogeneous dimension of the group G , Q i coefficients of the coordinate representation of the group operation τ x left translation by x , δ λ intrinsic dilations, U i ( x , r ) open Euclidean ball in V i of radius r > x , B ( x , r ) open ball of radius r > x , B ( x , r ) closed ball of radius r > x , T x , r φ dilated of a factor r > φ at the point x ∈ H n , Tan m ( φ , x ) set of m -dimensional tangent measures to the measure φ at x , (cid:42) weak convergence of measures, Gr ( m ) the m -dimensional Grassmanian, M ( m ) the set of the Haar measures of the elements of Gr ( m ) , n ( V ) the normal of the plane V ∈ Gr ( Q − ) , N ( V ) the 1-dimensional homogeneous subgroup generated by n ( V ) π N ( V ) the orthogonal projection of V onto V : = V ∩ V where V ∈ Gr ( Q − ) , π V the orthogonal projection of V onto N ( V ) , P V splitting projection on the plane V ∈ Gr ( Q − ) , P N ( V ) splitting projection on the 1-dimensional subgroup N ( V ) , C V ( α ) cone of amplitude α with axis V , Λ ( α ) upper bound on | π ( w ) | / (cid:107) w (cid:107) for any x ∈ C V ( α ) , Lip + ( K ) non-negative 1-Lipschitz functions with support contained in the compact set K . S α α -dimensional spherical Hausdorff measure relative to the metric (cid:107)·(cid:107) , C α α -dimensional centred spherical Hausdorff measure relative to the metric (cid:107)·(cid:107) , H keu Euclidean k -dimensional Hausdorff measure, Θ m ∗ ( φ , x ) m -dimensional lower density of the measure φ at x , Θ m , ∗ ( φ , x ) m -dimensional upper density of the measure φ at x , c ( Q ) centre of the cube Q reliminaries 7 α ( Q ) measure of the distance of the cube Q from the planes V ∈ Gr ( Q − ) d x , r ( · , M ) distance of the Radon measure φ inside the ball B ( x , r ) from flat measures, d H ( · , · ) Hausdorff distance of sets ∆ dyadic cubes, Since dyadic cubes are used extensively throughout the paper we introduce some nomenclature for the relation-ships of two cubes. For any couple of dyadic cubes Q , Q ∈ ∆ :(i) if Q ⊆ Q , then Q is said to be an ancestor of Q and Q a sub-cube of Q ,(ii) if Q is the smallest cube for which Q (cid:40) Q , then Q is said to be the parent of Q and Q the child of Q .Finally, throughout the paper we should adopt the convention that the set of natural numbers N does notinclude the number 0. The set of natural numbers including 0 will be denoted by N . This preliminary section is divided into four subsections. In Subsections . and . we introduce the setting, fixnotations and prove some basic facts on splitting projections and intrinsic cones. In Subsection . we recall somewell known facts on Radon measures and their blowups and finally in Subsection . we introduce the two mainnotions of 1-codimensional rectifiable sets available in Carnot groups. . Carnot groups
In this subsection we briefly introduce some notations on Carnot groups that we will extensively use throughoutthe paper. For a detailed account on Carnot groups and sub-Riemannian geometry we refer to [ ].A Carnot group G of step s is a connected and simply connected Lie group whose Lie algebra g admits astratification g = V ⊕ V ⊕ · · · ⊕ V s . The stratification has the further property that the entire Lie algebra g isgenerated by its first layer V , the so called horizontal layer . We denote by n the topological dimension of g , by n j the dimension of V j and by h j the number ∑ ji = n i .Furthermore, we let π i : G → V i be the projection maps on the i -th layer of the Lie algebra V i and denote by U i ( a , r ) the Euclidean ball of radius r and centre a inside the i -th layer V i . We shall remark that more often thannot, we will shorten the notation to v i : = π i v .The exponential map exp : g → G is a global diffeomorphism from g to G . Hence, if we choose a basis { X , . . . , X n } of g , any p ∈ G can be written in a unique way as p = exp ( p X + · · · + p n X n ) . This means thatwe can identify p ∈ G with the n -tuple ( p , . . . , p n ) ∈ R n and the group G itself with R n endowed with ∗ , theoperation determined by the Campbell-Hausdorff formula. From now on, we will always assume that G = ( R n , ∗ ) and as a consequence, that the exponential map exp acts as the identity.The stratificaton of g carries with it a natural family of dilations δ λ : g → g , that are Lie algebra automorphismsof g and are defined by: δ λ ( v , . . . , v s ) = ( λ v , λ v , . . . , λ s v s ) , ( )where v i ∈ V i . The stratification of the Lie algebra g naturally induces a stratification on each of its Lie sub-algebras h , that is: h = V ∩ h ⊕ . . . ⊕ V s ∩ h . ( )Furthermore, note that since the exponential map acts as the identity, the Lie algebra automorphisms δ λ are alsogroup automorphisms of G . Definition . (Homogeneous subgroups) . A subgroup V of G is said to be homogeneous if it is a Lie subgroup of G that is invariant under the dilations δ λ for any λ > reliminaries 8 Thanks to Lie’s theorem and the fact that exp acts as the identity map, homogeneous Lie subgroups of G arein bijective correspondence through exp with the Lie sub-algebras of g that are invariant under the dilations δ λ .Therefore, homogeneous subgroups in G are identified with the Lie sub-algebras of g (that in particular are vectorsub-spaces of R n ) that are invariant under the intrinsic dilations δ λ .For any nilpotent, Lie algebra h with stratification W ⊕ . . . ⊕ W s , we define its homogeneous dimension as:dim hom ( h ) : = s ∑ i = i · dim ( W i ) .Thanks to ( ) we infer that, if h is a Lie sub-algebra of g , we have dim hom ( h ) : = ∑ si = i · dim ( h ∩ V i ) . It is aclassical fact that the Hausdorff dimension (for a definition of Hausdorff dimension see for instance Definition . in [ ]) of a nilpotent, connected and simply connected Lie group coincides with the homogeneous dimensiondim hom ( h ) of its Lie algebra. Therefore, the above discussion implies that if h is a vector space of R n which is alsoan α -dimesional homogeneuous subgroup of G , we have: α = s ∑ i = i · dim ( h ∩ V i ) = dim hom ( h ) . ( ) Definition . . Let Q : = dim hom ( g ) and for any m ∈ {
1, . . . ,
Q − } we define the m -dimensional Grassmanian of G , denoted by Gr ( m ) , as the family of all homogeneous subgroups V of G of Hausdorff dimension m .Furthermore, thanks to ( ) and some easy algebraic considerations that we omit, one deduces that for theelements of Gr ( Q − ) the following identities hold:dim ( V ∩ V ) = n − ( V ∩ V i ) = dim ( V i ) for any i =
2, . . . , s . ( )Thanks to ( ), we infer that for any V ∈ Gr ( Q − ) there exists a n ( V ) ∈ V such that: V = V ⊕ V ⊕ . . . ⊕ V s ,where V : = { w ∈ V : (cid:104) n ( V ) , w (cid:105) = } . In the following we will denote with N ( V ) the 1-dimensional homogeneoussubgroup generated by the horizontal vector n ( V ) . We shall remark that the above discussion implies that theelements of Gr ( Q − ) are hyperplanes in R n whose normal lies in V . It is not hard to see that the viceversa holdstoo and that the elements of Gr ( Q − ) are normal subgroups of G .For any p ∈ G , we define the left translation τ p : G → G as: q (cid:55)→ τ p q : = p ∗ q .As already remarked above, we assume without loss of generality that the group operation ∗ is determined by theCampbell-Hausdorff formula, and therefore it has the form: p ∗ q = p + q + Q ( p , q ) for all p , q ∈ R n ,where Q = ( Q , . . . , Q s ) : R n × R n → V ⊕ . . . ⊕ V s , and the Q i s have the following properties. For any i =
1, . . . s and any p , q ∈ G we have:(i) Q i ( δ λ p , δ λ q ) = λ i Q i ( p , q ) ,(ii) Q i ( p , q ) = − Q i ( − q , − p ) ,(iii) Q = Q i ( p , q ) = Q i ( p , . . . , p i − , q , . . . , q i − ) .Thus, we can represent the product ∗ more precisely as: p ∗ q = ( p + q , p + q + Q ( p , q ) , . . . , p s + q s + Q s ( p , . . . , p s − , q , . . . , q s − )) . ( ) Definition . . A metric d : G × G → R is said to be homogeneous and left invariant if for any x , y ∈ G we have: reliminaries 9 (i) d ( δ λ x , δ λ y ) = λ d ( x , y ) for any λ > d ( τ z x , τ z y ) = d ( x , y ) for any z ∈ G .Throughout the paper we will always endow, if not otherwise stated, the group G with the following homoge-neous and left invariant metric: Definition . . For any g ∈ G , we let: (cid:107) g (cid:107) : = max { (cid:101) | g | , (cid:101) | g | , . . . , (cid:101) s | g s | s } ,where (cid:101) = (cid:101) , . . . (cid:101) s are suitably small parameters depending only on the group G . For the proof that (cid:107)·(cid:107) is a left invariant, homogeneous norm on G for a suitable choice of (cid:101) , . . . , (cid:101) s , we refer to Section of [ ].Furthermore, we define: d ( x , y ) : = (cid:107) x − ∗ y (cid:107) ,and let B ( x , r ) : = { z ∈ G : d ( x , z ) < r } be the open metric ball relative to the distance d centred at x at radius r > Remark . . Fix an othonormal basis E : = { e , . . . , e n } of R n such that: e j ∈ V i , whenever h i ≤ j < h i + . ( )From the definition of the metric d , it immediately follows that the ball B ( r ) is contained in the box:Box E ( r ) : = (cid:8) p ∈ R n : for any i =
1, . . . , s whenever |(cid:104) p , e j (cid:105)| ≤ r i / (cid:101) i for any h i ≤ j < h j + (cid:9) . Definition . . Let A ⊆ H n be a Borel set. For any 0 ≤ α ≤ n + δ >
0, define: C αδ ( A ) : = inf (cid:40) ∞ ∑ j = r α j : A ⊆ ∞ (cid:91) j = B r j ( x j ) , r j ≤ δ and x j ∈ A (cid:41) , S αδ ( A ) : = inf (cid:40) ∞ ∑ j = r α j : A ⊆ ∞ (cid:91) j = B r j ( x j ) , r j ≤ δ (cid:41) ,and S αδ , E ( ∅ ) : = = : C αδ ( ∅ ) . Eventually, we let: C α ( A ) : = sup B ⊆ A sup δ > C αδ ( B ) be the centred spherical Hausdorff measure, S α ( A ) : = sup δ > S αδ ( A ) be the spherical Hausdorff measure.Both C α and S α are Borel regular measures, see [ ] and Section . of [ ] respectively.In the following definition, we introduce a family of measures that will be of great relevance throughout thepaper. Definition . (Flat measures) . For any m ∈ {
1, . . . ,
Q − } we define the family of m -dimensional flat measures M ( m ) as: M ( m ) : = { λ S m (cid:120) V : for some λ > V ∈ Gr ( m ) } . ( )In order to simplify notation in the following we let M : = M ( Q − ) .The following proposition gives a representation of ( Q − ) -flat measures, that will come in handy later on: Proposition . . For any V ∈ Gr ( Q − ) we have: S Q− (cid:120) V = β − H n − eu (cid:120) V , where β : = H n − eu ( B (
0, 1 ) ∩ V ) and β does not dopend on V. reliminaries 10 Proof.
Let E : = { z ∈ G : (cid:104) z , n ( V ) (cid:105) < } and let ∂ E be the perimeter measure of E , see Definition B. . Thanks toidentity ( . ) in [ ], it can be proven that: ∂ E = n ( V ) H n − eu (cid:120) V .On the other hand, since the reduced boundary ∂ ∗ E = V of E is a C G -surface, see Definition . , thanks toTheorem . of [ ] we conclude that: β ( (cid:107)·(cid:107) , n ( V )) S Q− (cid:120) V = | ∂ E | G = H n − eu (cid:120) V ,where β ( (cid:107)·(cid:107) , n ( V )) : = max z ∈ B ( ) H n − eu ( B ( z , 1 ) ∩ V ) . Furthermore since B (
0, 1 ) is convex, Theorem . of [ ]implies that: β ( (cid:107)·(cid:107) , n ( V )) = H n − eu ( B (
0, 1 ) ∩ V ) .Finally note that the right hand side of the above identity does not depend on V since B (
0, 1 ) is invariant underrotations of the first layer V .Proposition . has the following useful consequence: Proposition . . A function ϕ : G → R is said to be radially symmetric if there is a profile function g : [ ∞ ) → R suchthat ϕ ( x ) = g ( (cid:107) x (cid:107) ) . For any V ∈ Gr ( Q − ) and any radially symmetric, positive function ϕ we have: ˆ ϕ d S Q− (cid:120) V = ( Q − ) ˆ s Q− g ( s ) ds . Proof.
It suffices to prove the proposition for positive simple functions, since the general result follows by BeppoLevi’s convergence theorem. Thus, let ϕ ( z ) : = ∑ Ni = a i χ B ( r i ) ( z ) and note that for any V ∈ Gr ( Q − ) we have: ˆ ϕ ( z ) d S Q− (cid:120) V = N ∑ i = a i S Q− (cid:120) V ( B ( r i )) = β − N ∑ i = a i H n − eu (cid:120) V ( B ( r i )) = β − H n − eu (cid:120) V ( B (
0, 1 )) N ∑ i = a i r Q− i =( Q − ) N ∑ i = a i ˆ r i s Q− ds = ( Q − ) ˆ N ∑ i = a i s Q− χ [ r i ] ( s ) ds = ( Q − ) ˆ s Q− g ( s ) ds . . Cones and splitting projections
For any V ∈ Gr ( Q − ) , the group G can be written as a semi-direct product of V and N ( V ) , i.e.: G = V (cid:111) N ( V ) . ( )In this subsection we specialize some of the results on projections of Section . of [ ] to the case in which splittingof G is given by ( ). Definition . (Splitting projections) . For any g ∈ G , there are two unique elements P V g ∈ V and P N ( V ) g ∈ N ( V ) such that: g = P V g ∗ P N ( V ) g .The following result is a particular case of Proposition . . of [ ]. Proposition . . For any V ∈ Gr ( Q − ) , we let:A g : = g − Q (cid:0) π V g , π n ( V ) g (cid:1) , A i g i : = g i − Q i (cid:0) π V g , A g , . . . , A i − g i − , π n ( V ) g , 0, . . . , 0 (cid:1) for any i =
3, . . . , s , where π n ( V ) g : = (cid:104) g , n ( V ) (cid:105) n ( V ) and π V g = g − π n ( V ) g . With these definitions, the projections P V and P N ( V ) havethe following expressions in coordinates:P V g = (cid:16) π V g , A g , . . . , A s g s (cid:17) , and P N ( V ) g = (cid:16) π n ( V ) g , 0, . . . , 0 (cid:17) . Furthermore, for any x , y ∈ G the above representations and the fact that V ∈ Gr ( Q − ) is normal imply: reliminaries 11 (i) P V ( x ∗ y ) = x ∗ P V y ∗ P N ( V ) x − ,(ii) P N ( V ) ( x ∗ y ) = P N ( V ) ( x ) ∗ P N ( V ) ( y ) = P N ( V ) ( x ) + P N ( V ) ( y ) ,where here the symbol + has to be intended as the sum of vectors.Remark . . Throughout the paper the reader should always keep in mind that the projections P V are not Lipschitzmaps and that this is the major source of the technical problems we have to overcome in order to prove our mainresult, Theorem . .The splitting projections allows us to give the following intrinsic notion of cone. Definition . . For any α > V ∈ Gr ( Q − ) , we define the cone C V ( α ) as: C V ( α ) : = { w ∈ G : (cid:107) P N ( V ) ( w ) (cid:107) ≤ α (cid:107) P V ( w ) (cid:107)} .The following result is very useful, since one of the major difficulties when dealing with geometric problems inCarnot groups is that d ( x , y ) ≈ | x − y | s if x and y are not suitably chosen. However, Proposition . shows thatif y (cid:54)∈ xC V ( α ) , then d ( x , y ) is bi-Lipschitz equivalent to the Euclidean distance | x − y | . Proposition . . Suppose x , y ∈ G are such that x − y (cid:54)∈ C V ( α ) for some α > and V ∈ Gr ( Q − ) . Then, there is adecreasing function α (cid:55)→ Λ ( α ) for which: d ( x , y ) ≤ Λ ( α ) | π ( x − y ) | . Proof.
For the sake of notation, we define v : = x − y and claim that: | v i | ≤ Λ i | v | i , for any i =
2, . . . , s , ( )for some constant Λ i depending only on α and i . Before proving ( ) by induction, note that thanks to Proposition . , the definition of the distance d and the fact that v (cid:54)∈ C V ( α ) we have: (cid:101) i | A i v i | i ≤ α − |(cid:104) v , n ( V ) (cid:105)| , for any i =
2, . . . , s . ( )If i = Q ( · , · ) we have: | Q (cid:0) π V v , π n ( W ) v (cid:1) | = | v | | Q (cid:0) δ | v | − ( π W v ) , δ | v | − ( π n ( W ) v ) (cid:1) | = | v | (cid:12)(cid:12)(cid:12) Q (cid:18) π W v | v | , π n ( W ) v | v | (cid:19)(cid:12)(cid:12)(cid:12) ≤ | v | sup a , b ∈ U ( ) Q ( a , b ) = : λ | v | . ( )Putting ( ) and ( ) together we have: (cid:101) ( | v | − λ | v | ) ≤ (cid:101) | v − Q (cid:0) π W v , π n ( W ) v (cid:1) | = (cid:101) | A v | ≤ α − |(cid:104) v , n ( V ) (cid:105)| ≤ α − | v | .Therefore, for i =
2, inequality ( ) holds with the choice Λ ( α ) : = λ + ( α(cid:101) ) . Note that the function Λ isdecreasing in α .Suppose now that for any i =
2, . . . , k inequality ( ) holds and that the functions Λ i ( α ) are decreasing in α .Then, thanks to the homogeneity of Q k + , we have: | Q k + (cid:0) π W v , A v , . . . , A k v k , π n ( W ) v , 0, . . . , 0 (cid:1) | = | v | k + | Q k + (cid:0) δ | v | − ( π W v ) , δ | v | − ( A v ) , . . . , δ | v | − ( A k v k ) , δ | v | − ( π n ( W ) v ) , 0, . . . , 0 (cid:1) |≤| v | k + sup a , b ∈ U ( ) a i ∈ U i ( Λ i ) , i = k . | Q i ( a , . . . , a k , b , 0, . . . , 0 ) | = : λ k + | v | k + . ( )Note that α (cid:55)→ λ k + is decreasing in α and that the bounds ( ) and ( ) imply: (cid:101) k + k + ( | v k + | − λ k + | v | k + ) ≤ (cid:101) k + k + ( | v k + | − | Q k + (cid:0) π W v , A v , . . . , A k v k , π n ( W ) v , 0, . . . , 0 (cid:1) | ) ≤ (cid:101) k + k + | A k + v k + | ≤ α − ( k + ) | v | k + . reliminaries 12 Defined Λ k + ( α ) : = λ k + + ( α(cid:101) k + ) k + , inequality ( ) is proved. Finally we infer that: d ( x , y ) = max {| v | , (cid:101) | v | , . . . , (cid:101) k + | v s | s } ≤ s ∑ i = Λ i ( α ) i | v | = : Λ ( α ) | v | = Λ ( α ) | π ( x − y ) | .The fact that Λ ( α ) is a decreasing function, is easily seen from the fact that the Λ i ( α ) s are decreasing.The following proposition allows us to exactly estimate the distance of a point g ∈ G from a plane V ∈ Gr ( Q − ) and gives us a bound on the Lipschitz constant of P V at the origin. Proposition . . For any V ∈ Gr ( Q − ) and any g ∈ G we have: dist ( P N ( V ) g , V ) = | π n ( V ) g | , and in particular dist ( g , V ) = | π n ( V ) g | . Furthermore, there is a constant C > depending only on G such that: (cid:107) P V ( g ) (cid:107) ≤ C (cid:107) g (cid:107) . Proof.
For any v ∈ V and g ∈ G , thanks to Proposition . and identity ( ), we have: P N ( V ) ( g ) − ∗ v = (cid:0) v − π n ( V ) g , v + Q (cid:0) − π n ( V ) g , v (cid:1) , . . . , v s + Q s (cid:0) − π n ( V ) g , 0, . . . , 0, v . . . , v s (cid:1)(cid:1) .Since by assumption (cid:104) v , n ( V ) (cid:105) =
0, in order to minimize (cid:107) P N ( V ) ( g ) − ∗ v (cid:107) as v varies in V , we choose: v ∗ : = v ∗ : = − Q (cid:0) − π n ( V ) g , 0 (cid:1) , v ∗ : = − Q (cid:0) − π n ( V ) g , 0, 0, v ∗ (cid:1) ,... v ∗ s : = − Q s (cid:0) − π n ( V ) g , 0, . . . , 0, 0, v ∗ , . . . , v ∗ s − (cid:1) . ( )Let us prove that v ∗ is an element of V and that it is of minimal distance for P N ( V ) ( g ) on V . In order to show this,let us note that thanks to the definition of d , we have:dist ( P N ( V ) ( g ) , V ) = inf v ∈ V (cid:107) P N ( V ) ( g ) − ∗ v (cid:107) ≥ inf v ∈ V |− π n ( V ) g + v | = | π n ( V ) g | , ( )Furthermore, thanks to the definition of the operation and few algebraic computations that we omit, we have: P N ( V ) ( g ) − ∗ v ∗ = ( − π n ( V ) g , 0, . . . , 0 ) . ( )Therefore, since v ∗ ∈ V thanks to ( ) and ( ) the first part of the proposition is proved. The second part followssince: dist ( g , V ) = inf v ∈ V d ( g , v ) = inf v ∈ V d ( P V g ∗ P N ( V ) g , v ) = inf v ∈ V d ( P N ( V ) g , P V g − ∗ v ) = dist ( P N ( V ) g , V ) ,where the last identity comes from the fact that the translation by P V g − is surjective on V .We proceed with the proof of the last claim of the proposition, estimating the norm of each component of P V ( g ) .By definition of π V and π n ( V ) we have: | π n ( V ) ( g ) | ≤ | g | ≤ (cid:107) g (cid:107) and | π V ( g ) | ≤ | g | ≤ (cid:107) g (cid:107) .In order to estimate the norm of π ( P V g ) , . . . , π s ( P V g ) we proceed by induction and make use of their representa-tions yielded by Proposition . . The base case is the estimate of the norm of π ( P V g ) : | π ( P V g ) | = | A g | ≤ | g | + | Q (cid:0) π V g , π n ( V ) g (cid:1) | ≤ (cid:107) g (cid:107) (cid:16) (cid:101) − + (cid:12)(cid:12)(cid:12) Q (cid:16) π V g (cid:107) g (cid:107) , π n ( V ) g (cid:107) g (cid:107) (cid:17)(cid:12)(cid:12)(cid:12)(cid:17) ≤ ( (cid:101) − + λ ) (cid:107) g (cid:107) ,where λ is the constant introduced in ( ). Note that the constant c : = (cid:101) − + λ depends only on G . reliminaries 13 Suppose now that for any i =
2, . . . , k , there is a constant c i , depending only on G , such that | π i ( P V g ) | ≤ c i (cid:107) g (cid:107) i .This implies that: | π k + ( P V g ) | = | A k + g k + | ≤ | g k + | + | Q k ( π V g , A g , . . . , A k g k π n ( V ) g , 0, . . . , 0 ) |≤(cid:107) g (cid:107) k + (cid:16) (cid:101) − k + + (cid:12)(cid:12)(cid:12) Q k (cid:16) π V g (cid:107) g (cid:107) , A g (cid:107) g (cid:107) , . . . , A k g k (cid:107) g (cid:107) k , π n ( V ) g (cid:107) g (cid:107) , 0, . . . , 0 (cid:17)(cid:12)(cid:12)(cid:12)(cid:17) ≤ sup a , b ∈ U ( ) a i ∈ U i ( c i ) , i = k . | Q i ( a , . . . , a k , b , 0, . . . , 0 ) | · (cid:107) g (cid:107) k + = : c k + (cid:107) g (cid:107) k + .Note that since the c i depended only on G , so does c k + . Thanks to the definition of (cid:107)·(cid:107) , we eventually deducethat: (cid:107) P V g (cid:107) ≤ s ∑ i = (cid:101) i c ii (cid:107) g (cid:107) = : C (cid:107) g (cid:107) ,which concludes the proof of the proposition.The following result is the analogue of Proposition . . of [ ], where M : = V and H : = N ( V ) . Proposition . . For any V ∈ Gr ( Q − ) and any g ∈ G , defined C : = ( C + ) − we have:C ( (cid:107) P N ( V ) g (cid:107) + (cid:107) P V g (cid:107) ) ≤ (cid:107) g (cid:107) ≤ (cid:107) P N ( V ) g (cid:107) + (cid:107) P V g (cid:107) . ( ) Proof.
The right hand side of ( ) follows directly from the triangular inequality. Furthermore, thanks to Proposi-tion . we deduce that (cid:107) P N ( V ) ( g ) (cid:107) = | π n ( V ) ( g ) | ≤ (cid:107) g (cid:107) and (cid:107) P N ( V ) g (cid:107) + (cid:107) P V g (cid:107) ≤ ( C + ) (cid:107) g (cid:107) .The following proposition allows us to estimate the distance of parallel 1-codimensional planes. Proposition . . Let x , y ∈ G and V ∈ Gr ( Q − ) . Defined: dist ( xV , yV ) : = max (cid:26) sup v ∈ V dist ( xv , yV ) , sup v ∈ V dist ( yv , xV ) (cid:27) , we have:(i) dist ( xV , yV ) = dist ( x , yV ) = dist ( y , xV ) = | π n ( V ) ( π ( x − y )) | ,(ii) dist ( u , xV ) ≤ dist ( u , yV ) + dist ( xV , yV ) , for any u ∈ G .Proof. For any v ∈ V we have:dist ( xv , yV ) = inf w ∈ V dist ( xv , yw ) = inf w ∈ V d ( x , y ( y − xv − x − y ) w ) = inf w ∈ V d ( x , yw ) = dist ( x , yV ) ,where the second last identity comes from the fact that v ∗ : = y − xv − x − y ∈ V and the transitivity of thetranslation by v ∗ on V . Therefore, we have sup v ∈ V dist ( xv , yV ) = d ( x , yV ) and thus by Proposition . we infer:dist ( xV , yV ) = max (cid:8) dist ( x , yV ) , dist ( y , xV ) (cid:9) = max {| π n ( V ) ( π ( y − x )) | , | π n ( V ) ( π ( x − y )) |} = | π n ( V ) ( π ( x − y )) | = dist ( x , yV ) = dist ( y , xV ) ,where the last identity comes from the arbitrariness of x and y . In order to prove (ii), let w ∗ be the element of V for which dist ( u , yV ) = d ( u , yw ∗ ) and note that:dist ( u , xV ) = inf v ∈ V d ( u , xv ) ≤ d ( u , yw ∗ ) + inf v ∈ V d ( yw ∗ , xv ) = dist ( u , yV ) + inf v ∈ V d ( yw ∗ , xv )= dist ( u , yV ) + d ( xw ∗ , yV ) ≤ dist ( u , yV ) + dist ( xV , yV ) .This concludes the proof of the proposition. reliminaries 14 The following result is a direct consequence of Proposition . . of [ ]. The bound ( ) is obtained with thesame argument used by V. Chousionis, K. Fässler and T. Orponen to prove Lemma . of [ ]. Proposition . . For any V ∈ Gr ( Q − ) there is a constant ≤ c ( V ) ≤ S Q− ( B ( C ) ∩ V ) = : C such that for anyp ∈ G and any r > we have: S Q− (cid:120) V (cid:0) P V ( B ( p , r )) (cid:1) = c ( V ) r Q− . Furthermore, for any Borel set A ⊆ G for which S Q− ( A ) < ∞ , we have: S Q− (cid:120) V ( P V ( A )) ≤ c ( V ) S Q− ( A ) . ( ) Proof.
The existence of the constant c ( V ) is yielded by Proposition . . of [ ]. Furthermore Propositions . , . and the fact that B (
0, 1 ) ∩ V ⊆ P V ( B (
0, 1 )) for any V ∈ Gr ( Q − ) , imply that:1 = β − H n − eu ( B (
0, 1 ) ∩ V ) = S Q− (cid:120) V ( B (
0, 1 ) ∩ V ) ≤ S Q− (cid:120) V (cid:0) P V ( B (
0, 1 )) (cid:1) = c ( V ) ,and: S Q− (cid:120) V ( P V ( B (
0, 1 ))) ≤ S Q− ( B ( C ) ∩ V ) = C ,where the constant C does not depend on V thanks to Proposition . and since the ball B ( C ) is invariantunder rotations of V .Suppose now A is a Borel set with finite S h -measure and let { B ( x i , r i ) } i ∈ N be a sequence of balls covering A such that ∑ i ∈ N r Q− i ≤ S Q− ( A ) . Then: S Q− ( P V ( A )) ≤ S Q− (cid:16) P V (cid:16) (cid:91) i ∈ N B ( x i , r i ) (cid:17)(cid:17) ≤ c ( V ) ∑ i ∈ N r Q− i ≤ c ( V ) S Q− ( A ) .This concludes the proof. . Densities and tangents of Radon measures
In this subsection we briefly recall some facts and notations about Radon measures on Carnot groups and theirblowups.
Definition . . If φ is a Radon measure on G , we define: Θ m ∗ ( φ , x ) : = lim inf r → φ ( B ( x , r )) r m and Θ m , ∗ ( φ , x ) : = lim sup r → φ ( B ( x , r )) r m ,and say that Θ m ∗ ( φ , x ) and Θ m , ∗ ( φ , x ) are respectively the lower and upper m -density of φ at the point x ∈ G . Definition . (weak convergence of measures) . A sequence of Radon measures { µ i } i ∈ N is said to be weaklyconverging in the sense of measures to some Radon measure ν , if for any continuous functions with compactsupport f ∈ C c , we have: ˆ f d µ i → ˆ f d ν .Throughout the paper, we denote such convergence with µ i (cid:42) ν . Definition . . For any couple of Radon measures φ and ψ and any compact set K ⊆ G we let: F K ( φ , ψ ) : = sup (cid:110)(cid:12)(cid:12)(cid:12) ˆ f d φ − ˆ f d ψ (cid:12)(cid:12)(cid:12) : f ∈ Lip + ( K ) (cid:111) , ( )where Lip + ( K ) is the set of non-negative 1-Lipschitz functions whose support is contained in K . Furthermore, if K = B ( x , r ) we shorten the notation to F x , r ( φ , ψ ) : = F B ( x , r ) ( φ , ψ ) .The next lemma is an elementary fact on Radon measures. We omit its proof. reliminaries 15 Lemma . . If φ is a Radon measure, for any x ∈ G there are at most countably many radii R > for which: φ ( ∂ B ( x , R )) > F K . Proposition . . Assume that { µ i } i ∈ N is a sequece of Radon measures such that lim sup i → ∞ µ i ( B ( R )) < ∞ for everyR > and let µ be a Radon measure on G . The following are equivalent:(i) µ i (cid:42) µ ,(ii) lim i → ∞ F K ( µ i , µ ) = for any compact set K ⊆ G .Proof. As a first step, we prove that (i) implies (ii). If (i) holds, for any non-negative Lipschitz function f withcompact support we have: lim i → ∞ ˆ f d µ i = ˆ f d µ .Fix a compact subset K of G contained in B ( R ) for some R >
0. It is immediate to see that Lip + ( K ) is a compactmetric space when endowed with the supremum distance (cid:107)·(cid:107) ∞ and thus it is totally bounded. Therefore, forany (cid:101) >
0, there is a finite set S ⊆ Lip + ( K ) such that whenever f ∈ Lip + ( K ) , we can find a g ∈ S such that (cid:107) f − g (cid:107) ∞ < (cid:101) . This implies that: (cid:12)(cid:12)(cid:12) ˆ f d µ i − ˆ f d µ (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ˆ f d µ i − ˆ gd µ i (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˆ gd µ i − ˆ gd µ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˆ gd µ − ˆ f d µ (cid:12)(cid:12)(cid:12) ≤ (cid:101) ( µ i ( K ) + µ ( K )) + (cid:12)(cid:12)(cid:12) ˆ gd µ i − ˆ gd µ (cid:12)(cid:12)(cid:12) .Thanks to the arbitrariness of f , we infer that: F K ( µ i , µ ) ≤ (cid:101) ( µ i ( K ) + µ ( K )) + sup g ∈ S (cid:12)(cid:12)(cid:12) ˆ gd µ i − ˆ gd µ (cid:12)(cid:12)(cid:12) . ( )Taking the limit as i goes at infinity on both sides of ( ), thanks to (i) we have:lim sup i → ∞ F K (cid:0) µ i , µ ) ≤ (cid:101) ( µ ( K ) + lim sup i → ∞ µ i ( K )) ≤ (cid:101) ( µ ( K ) + lim sup i → ∞ µ i ( B ( R )) (cid:1) .The arbitrariness of (cid:101) eventually concludes the proof.We are left to prove that (ii) implies (i). Let g be a non-negative Lipschitz function with compact support. Sincewe assumed that (ii) holds, we infer that:lim i → ∞ (cid:12)(cid:12)(cid:12) ˆ gd µ i − ˆ gd µ (cid:12)(cid:12)(cid:12) = Lip ( g ) lim i → ∞ (cid:12)(cid:12)(cid:12) ˆ g Lip ( g ) d µ i − ˆ g Lip ( g ) d µ (cid:12)(cid:12)(cid:12) ≤ Lip ( g ) lim i → ∞ F supp ( g ) ( µ i , µ ) = K be a compact set in G . By the continuity of the measure from above, for any (cid:101) > s > µ ( B ( K , s )) ≤ µ ( K ) + (cid:101) , where B ( K , s ) : = { x ∈ G : dist ( x , K ) ≤ s } . Defined f ( z ) : = min {
1, dist ( z , B ( K , s ) c ) / s } ,we conclude that: µ ( K ) + (cid:101) ≥ ˆ f d µ = lim i → ∞ ˆ f d µ i ≥ lim sup i → ∞ µ i ( K ) .The arbitrariness of (cid:101) finally implies that lim sup i → ∞ µ i ( K ) ≤ µ ( K ) for any compact set K ⊆ G . Assume now g isa continuous function whose support is contained in a ball B ( R ) with µ -null boundary. This choice can be donewithout loss of generality thanks to Lemma . . Since g is continuous, it can be easily proved that for any D ⊆ R we have: ∂ g − ( D ) ⊆ g − ( ∂ D ) . ( )Let (cid:101) > A : = { t ∈ R : µ ( g − ( { t } )) > } is the set of atoms of the measure push forward g ( µ ) .Since g is bounded, there is an N ∈ N and a finite sequence { t i } i = N such that: reliminaries 16 (a) t ≤ −(cid:107) g (cid:107) ∞ < t < . . . < t N − < (cid:107) g (cid:107) ∞ ≤ t N ,(b) t n ∈ R \ A and | t n + − t n | < (cid:101) for any n =
1, . . . , N − E n : = B ( R ) ∩ g − ([ t n , t n + ]) for any n =
1, . . . , N − ) we have: µ ( ∂ E n ) ≤ µ ( ∂ B ( R ) ∪ g − ( { t n } ∪ { t n + } )) ≤ µ ( ∂ B ( R )) + µ ( g − ( { t n } )) + µ ( g − ( { t n + } )) =
0. ( )Thanks to the definition of the E i s it is possible to show that E i ∩ E j ⊆ ∂ E i ∪ ∂ E j and thus by ( ) we infer that ∑ N − n = µ ( E n ) ≤ µ ( B ( R )) . This implies that:lim sup i → ∞ ˆ gd µ ≤ lim sup i → ∞ N − ∑ n = t n + µ i ( E n ) ≤ N − ∑ n = t n + µ ( E n ) ≤ N − ∑ n = ( t n + − t n ) µ ( E n ) + N − ∑ n = t n µ ( E n ) ≤ (cid:101) N − ∑ n µ ( E n ) + ˆ gd µ ≤ (cid:101)µ ( B ( R )) + ˆ gd µ .The arbitrariness of (cid:101) implies that lim sup i → ∞ ´ gd µ i ≤ ´ gd µ . Repeating the argument for − g we deduce thatlim sup i → ∞ ´ − gd µ i ≤ ´ − gd µ , concluding the proof of the proposition. Definition . (Tangent measures) . Let φ be a Radon measure on G . For any x ∈ G and any r >
0, we define T x , r φ to be the Radon measure for which: T x , r φ ( B ) = φ ( x δ r ( B )) for any Borel set B ⊆ G .We define Tan m ( φ , x ) , the set of the m -dimensional tangent measures to φ at x , as the collection of Radon measures ν for which there is an infinitesimal sequence { r i } i ∈ N such that: r − mi T x , r φ (cid:42) ν . Proposition . . Let φ be a Radon measure and ν ∈ Tan m ( φ , x ) , i.e., r − mi T x , r i φ (cid:42) ν for some r i → . If y ∈ supp ( ν ) ,there exists a sequence { z i } i ∈ N ⊆ supp ( φ ) such that δ r i ( x − z i ) → y.Proof. A simple argument by contradiction yields the claim, the proof follows verbatim its Euclidean analogue,Proposition . in [ ]. Proposition . . Suppose φ is a Radon measure on G such that: < Θ m ∗ ( φ , x ) ≤ Θ m , ∗ ( φ , x ) < ∞ , for φ -almost every x ∈ G . Then
Tan m ( φ , x ) (cid:54) = ∅ for φ -almost every x ∈ G .Proof. This is an immediate consequence of the local uniform boundness of the rescaled measures T x , r φ . The prooffollows verbatim the Euclidean one, see for instance Lemma . of [ ].The following result is the analogue of Proposition . of [ ] which establishes the locality of tangents inthe Euclidean space. We give here a detailed proof, since we need to avoid to use the Besicovitch coveringtheorem, which may not hold for the distance d . This proposition is of capital importance since it will ensure usthat restricting and multiplying by a density a measure with flat tangents will yield a measure still having flattangents. Proposition . (Locality of the tangents) . Let φ be a Radon measure such that: < Θ m ∗ ( φ , x ) ≤ Θ m , ∗ ( φ , x ) < ∞ , for φ -almost every x ∈ G . ( ) Then, for any ρ ∈ L ( φ ) we have Tan m ( ρφ , x ) = ρ ( x ) Tan m ( φ , x ) for φ -almost every x ∈ G . reliminaries 17 Proof.
First of all, let us prove that φ is asymptotically doubling:lim sup r → φ ( B ( x , 2 r )) φ ( B ( x , r )) ≤ lim sup r → φ ( B ( x , 2 r ))( r ) m m r m φ ( B ( x , r )) ≤ m Θ m , ∗ ( φ , x ) Θ m ∗ ( φ , x ) < ∞ , ( )for φ -almost every x ∈ G . Thanks to Theorem . . and the Lebesgue differentiation Theorem in [ ], we have: φ ( B ) : = φ (cid:16)(cid:110) x ∈ G : lim sup r → B ( x , r ) | ρ ( y ) − ρ ( x ) | d φ ( y ) > (cid:111)(cid:17) =
0. ( )Let x ∈ G \ B , suppose ν ∈ Tan m ( φ , x ) , and let r i → ν i : = T x , r i φ r mi (cid:42) ν .Defined ν (cid:48) i : = r − mi T x , r i ( ρφ ) for every ball B ( l ) , we have: | ρ ( x ) ν i − ν (cid:48) i | ( B ( l )) ≤ r mi ˆ B ( x , lr i ) | ρ ( y ) − ρ ( x ) | d φ ( y ) = φ ( B ρ r i ( x )) r mi · B ( x , lr i ) | ρ ( y ) − ρ ( x ) | d φ ( y ) = : ( I ) i · ( I I ) i ,where | ρ ( x ) ν i − ν (cid:48) i | is the total variation of the measure ρ ( x ) ν i − ν (cid:48) i . For φ -almost every x ∈ B thanks to ( ) wehave Θ m , ∗ ( φ , x ) < ∞ and lim i → ∞ ( I I ) i =
0. This implies that:lim i → ∞ | ρ ( x ) ν i − ν (cid:48) i | ( B ( l )) = l >
0. If in addition to this, we also assume that ν ( ∂ B ( l )) =
0, since ν i (cid:42) ν , then by Proposition . of[ ] we also have that: lim i → ∞ | ρ ( x ) ν − ν (cid:48) i | ( B ( l )) =
0. ( )Finally, since by Proposition . we have ν ( ∂ B ( l )) = l >
0, thanks to ( ) we conclude that ν (cid:48) i (cid:42) ρ ( x ) ν . Proposition . . Assume φ is a Radon measure supported on a compact set K such that for φ -almost every x ∈ G we have: < Θ Q− ∗ ( φ , x ) ≤ Θ Q− ∗ ( φ , x ) < ∞ . Then, for any ϑ , γ ∈ N the set E φ ( ϑ , γ ) : = (cid:8) x ∈ K : ϑ − r Q− ≤ φ ( B ( x , r )) ≤ ϑ r Q− for any < r < γ (cid:9) is compact.Proof. Since K is compact, in order to verify that E φ ( ϑ , γ ) is compact, it suffices to prove that it is closed. If E φ ( ϑ , γ ) is empty or finite, there is nothing to prove. So, suppose there is a sequence { x i } i ∈ N ⊆ E φ ( ϑ , γ ) converging tosome x ∈ K . Fix an 0 < r < γ and assume that δ > r + δ < γ . Therefore, if d ( x , x i ) < δ and r − d ( x , x i ) >
0, we have: ϑ − (cid:0) r − d ( x , x i ) (cid:1) Q− ≤ φ (cid:0) B ( x i , r − d ( x , x i )) (cid:1) ≤ φ ( B ( x , r )) ≤ φ (cid:0) B ( x i , r + d ( x , x i )) (cid:1) ≤ ϑ ( r + d ( x , x i )) Q− ,Taking the limit as i goes to ∞ , we see that x ∈ E ( ϑ , γ ) . Proposition . . Assume φ is a Radon measure supported on a compact set K such that for φ -almost every x ∈ G we have: < Θ Q− ∗ ( φ , x ) ≤ Θ Q− ∗ ( φ , x ) < ∞ . Then, for any ϑ , γ , µ , ν ∈ N the set: E φϑ , γ ( µ , ν ) = { x ∈ E φ ( ϑ , γ ) : ( − µ ) φ ( B ( x , r )) ≤ φ ( B ( x , r ) ∩ E φ ( ϑ , γ )) for any < r < ν } , is compact. reliminaries 18 Proof. If E φϑ , γ ( µ , ν ) is empty or finite, there is nothing to prove. Furthermore, thanks to Proposition . the sets E φ ( ϑ , γ ) are compact and thus to prove our claim it is sufficient to show that E φϑ , γ ( µ , ν ) is closed in E φ ( ϑ , γ ) . Takea sequence { y i } i ∈ N ⊆ E φϑ , γ ( µ , ν ) converging to some y ∈ E φ ( ϑ , γ ) . Fix an 0 < r < ν and a δ ∈ (
0, 1/4 ) and let i ( δ ) ∈ N be such that for any i ≥ i ( δ ) we have d ( y , y i ) < δ r . These choices imply: ( − µ ) φ ( B ( y i , r − d ( y , y i ))) ≤ φ ( B ( y i , r − d ( y , y i )) ∩ E φ ( ϑ , γ )) ≤ φ ( B ( y , r ) ∩ E φ ( ϑ , γ )) .Note that the sequence of functions f i ( z ) : = χ B ( y i , r − d ( y , y i )) ( z ) converges pointwise φ -almost everywhere to χ B ( y , r ) ( z ) . This is due to the fact that for any i ≥ i ( δ ) on the one hand we have supp ( f i ) (cid:98) B ( y , r ) and onthe other the functions f i are equal to 1 on B ( y , r ( − δ )) . Thus, the dominated convergence theorem implies: ( − µ ) φ ( B ( y , r )) = lim i → ∞ ( − µ ) φ ( B ( y i , r − d ( y , y i ))) ≤ φ ( B ( y , r ) ∩ E ( ϑ , γ )) .Since r was aribtrarily chosen in (
0, 1/ ν ) , this shows that y ∈ E ϑ , γ ( µ , ν ) concluding the proof. Proposition . . Assume φ is a Radon measure supported on a compact set K such that for φ -almost every x ∈ G we have: < Θ Q− ∗ ( φ , x ) ≤ Θ Q− ∗ ( φ , x ) < ∞ . Then, for any < (cid:101) < there are ϑ , γ ∈ N such that for any ϑ ≥ ϑ , γ ≥ γ and µ ∈ N there is a ν = ν ( ϑ , γ , µ ) ∈ N such that: φ ( K / E φϑ , γ ( µ , ν )) ≤ (cid:101)φ ( K ) . ( ) Proof.
Assume at first that ϑ and γ are fixed. It is easy to check that: E φϑ , γ ( µ , ν ) ⊆ E φϑ , γ ( µ , ν ) , whenever µ ≤ µ , ( ) E φϑ , γ ( µ , ν ) ⊆ E φϑ , γ ( µ , ν ) , whenever ν ≤ ν . ( )Fix some µ ∈ N and let x ∈ E φ ( ϑ , γ ) \ (cid:83) ν ∈ N E φϑ , γ ( µ , ν ) . For such an x , we can find a sequence r ν such that r ν ≤ ν and: φ ( B ( x , r ν ) ∩ E φ ( ϑ , γ )) < ( − µ ) φ ( B ( x , r ν )) .This would imply that lim inf r → φ ( B ( x , r ) ∩ E φ ( ϑ , γ )) φ ( B ( x , r )) ≤ ( − µ ) , showing thanks to Lebescue differentiability Theorem of [ ] that: φ (cid:18) E φ ( ϑ , γ ) \ (cid:91) ν ∈ N E φϑ , γ ( µ , ν ) (cid:19) =
0. ( )Thanks to inclusion ( ) and identity ( ), for any µ ∈ N we can find some ν ∈ N such that: φ ( E φ ( ϑ , γ ) \ E φϑ , γ ( µ , ν )) ≤ (cid:101)φ ( E φ ( ϑ , γ )) /2.We finally prove that there are ϑ , γ ∈ N such that φ ( K \ E φ ( ϑ , γ )) ≤ (cid:101)φ ( K ) /2. For any x ∈ K \ (cid:83) ϑ , γ ∈ N E φ ( ϑ , γ ) , itis immediate to see that Θ Q− ∗ ( φ , x ) = Θ Q− ∗ ( φ , x ) = ∞ , which thanks to the choice of φ , implies: φ (cid:18) K \ (cid:91) ϑ , γ ∈ N E φ ( ϑ , γ ) (cid:19) =
0. ( )Thanks to the fact that E φ ( ϑ , γ ) ⊆ E φ ( ϑ (cid:48) , γ (cid:48) ) whenever γ ≤ γ (cid:48) and ϑ ≤ ϑ (cid:48) and the continuity from above of themeasure, this proves the proposition.The following result allows us to compare the measure φ when restricted to E φ ( ϑ , γ ) with the spherical Haus-dorff measure. reliminaries 19 Proposition . . Let ψ be a Radon measure supported on a Borel set E. Suppose further that there are < δ ≤ δ suchthat: δ ≤ Θ m ∗ ( ψ , x ) ≤ Θ m , ∗ ( ψ , x ) ≤ δ , for ψ -almost every x ∈ G . Then, ψ = Θ m , ∗ ( ψ , x ) C m (cid:120) E where C m is the centred spherical Hausdorff measure introduced in Definition . and inparticular: δ C m (cid:120) E ≤ ψ (cid:120) E ≤ δ C m (cid:120) E , and δ S m (cid:120) E ≤ ψ (cid:120) E ≤ δ m S m (cid:120) E . Proof.
Thanks to
Lebesgue differentiability theorem of [ ], for any Borel set A ⊆ G and ψ -almost all x ∈ A ∩ E wehave: lim sup r → ψ ( B ( x , r ) ∩ E ∩ A ) r m ≤ lim sup r → ψ ( B ( x , r ) ∩ E ∩ A ) ψ ( B ( x , r )) ψ ( B ( x , r )) r m ≤ δ .Thanks to Proposition . . of [ ], we conclude that ψ ≤ δ S m (cid:120) E and in particular ψ is absolutely continuouswith respect to S m (cid:120) E . Furthermore, thanks to Theorem . of [ ], we infer that: ψ = Θ m , ∗ ( ψ , x ) C m (cid:120) E . ( )Finally, from ( ) we deduce that: δ S m (cid:120) E ≤ δ C m (cid:120) E ≤ ψ ≤ δ C m (cid:120) E ≤ δ m S m (cid:120) E ,where the first and last inequality follow from the fact that the measures C m and S m are equivalent and satisfy thebounds S m ≤ C m ≤ m S m , for a reference see [ ].An immediate consequence of the above proposition is the following: Corollary . . For any ϑ , γ ∈ N we have ϑ − S Q− (cid:120) E φ ( ϑ , γ ) ≤ φ (cid:120) E φ ( ϑ , γ ) ≤ ϑ Q− S Q− (cid:120) E φ ( ϑ , γ ) . The following result will be used in the proof of the very important Proposition . . It establishes the naturalrequest that if a sequence of planes V i in Gr ( Q − ) convergences in the Grassmanian to some plane V ∈ Gr ( Q − ) (i.e. the normals converge as vectors in V ), then the surface measures on the V i s converge weakly to the surfacemeasure on V . Proposition . . Suppose that { V i } i ∈ N is a sequence of planes in Gr ( Q − ) such that n ( V i ) → n for some n ∈ V . Then,there exists a V ∈ Gr ( Q − ) such that n ( V ) = n and: S Q− (cid:120) V i (cid:42) S Q− (cid:120) V . Proof.
For any continuous function of compact support f ∈ C c we have:lim i → ∞ ˆ f d S Q− (cid:120) V i − ˆ f d S Q− (cid:120) V = lim i → ∞ β − (cid:18) ˆ f d H n − eu (cid:120) V i − ˆ f d H n − eu (cid:120) V (cid:19) =
0, ( )where the last identity comes from the fact that H n − eu (cid:120) V i (cid:42) H n − eu (cid:120) V . . Rectifiabile sets in Carnot groups
In this subsection we recall the two main notions of rectifiability in Carnot groups that will be extensivelyused throughout the paper. First of all, let us recall the definition of horizontal vector fields and of horizontaldistribution.
Definition . . For any i =
1, . . . , n and any x ∈ G we let X i ( x ) : = ∂ t ( x ∗ δ t ( e i )) | t = , and say that the map X i : G ∼ = R n → R n so defined is the i -th horizontal vector field . Furthermore, we define the horizontal distribution of G to be the following n -dimensional distribution of planes of R n : H G ( x ) : = span { X ( x ) , . . . , X n ( x ) } .Finally, for any open set Ω in G we denote by C ( Ω , H G ) the sections of H G of class C with support contained in Ω . reliminaries 20 The definition of regular surface we are about to give is reminiscent of the characterisation of smooth surfacesin the Euclidean spaces through the local inversion theorem. Heuristically speaking, a C G -surface is a set that istransverse to H G and whose sections with H G are C -surfaces. Definition . ( C G -surfaces) . We say that a closed set C ⊆ G is a C G - surface if there exists a continuous function f : G → R such that C = f − ( ) and whose horizontal distributional gradient ∇ G f : = ( X f , . . . , X n f ) can berepresented by a continuous, never-vanishing section of H G . Remark . . Thanks to Corollary . of [ ], if C is a C G -regular surface, then S Q− (cid:120) C is σ -finite.The second notion of regular surface we give in this subsection is inspired by the characterisation of Lipschitzgraphs through cones. Definition . (Intrinsic Lipschitz graphs) . Let V ∈ Gr ( Q − ) and E be a Borel subset of V . A function f : E → N ( V ) is said to be intrinsic Lipschitz if there exists an α > v ∈ E we have: gr ( f ) : = { w f ( w ) : w ∈ E } ⊆ v f ( v ) C V ( α ) .A Borel set A ⊆ G is said to be an intrinsic Lipschitz graph if there is an intrinsic Lipschitz function f : E ⊆ V → N ( V ) such that A = f ( E ) .The following extension theorem is of capital importance for us: Theorem . (Theorem . , [ ]) . Suppose V ∈ Gr ( Q − ) and let f : E → N ( V ) be an intrinsic Lipschitz function.Then there is an intrinsic Lipschitz function ˜ f : V → N ( V ) such that f ( v ) = ˜ f ( v ) for any v ∈ E. The following result is an immediate consequence of Theorem . : Proposition . . If f : E ⊆ V → N ( V ) is an intrinsic Lipschitz function, then S Q− (cid:120) f ( E ) is σ -finite.Proof. Theorem . together with Theorem . . of [ ] immediately implies that S Q− ( gr ( f ) ∩ B ( R )) < ∞ forany R > C G -surface and of intrinsic Lipschitz surface rise the two following definitions of rectifiabil-ity: Definition . . A Borel set A ⊆ G of finite S Q− -measure is said to be:(i) C G - rectifiable if there are countably many C G -surfaces Γ i such that S Q− ( A \ (cid:83) i ∈ N Γ i ) = intrinsic rectifiable if there are countably many intrinsic Lipschitz graphs Γ i such that S Q− ( A \ (cid:83) i ∈ N Γ i ) = . of [ ]. Proposition . . (Decomposition Theorem) Suppose F is a family of Borel sets in G for which S Q− (cid:120) C is σ -finite for anyC ∈ F . Then, for any Borel set E ⊆ G such that S Q− ( E ) < ∞ , there are two Borel sets E u , E r ⊆ E such that:(i) E u ∪ E r = E,(ii) E r is contained in countable union of elements of F ,(iii) S Q− ( E u ∩ C ) = for any C ∈ F .Such decomposition is unique up to S Q− -null sets, i.e. if F u and F r are Borel sets satisfing the three properties listed above,we have: S Q− ( E r (cid:52) F r ) = S Q− ( E u (cid:52) F u ) = he support of codimensional measures with flat tangents is intrinsic rectifiable 21 Proof.
Define R ( E ) : = { E (cid:48) ⊆ E : there are { Γ i } i ∈ N ⊆ F such that E (cid:48) ⊆ (cid:83) i ∈ N Γ i } and: α : = sup E (cid:48) ∈R ( E ) S Q− ( E (cid:48) ) .Suppose { E i : i ∈ N } ⊆ R ( E ) is a maximizing sequence, i.e. lim i → ∞ S Q− ( E i ) = α . Then we let E r : = (cid:83) i ∈ N E i andnote that E r is covered countably many sets in F and S Q− ( E r ) = α . The set E \ E r is unrectifiable with respect to F . Indeed, if there was an intrinsic Lipschitz graph Γ such that S Q− ( G \ E ∩ Γ ) >
0, we would infer that: S Q− ( E r ∪ ( G \ E r ∩ Γ )) > α .Since E r ∪ ( G \ E r ∩ Γ ) ∈ R ( E ) , this would contradict the maximality of α . If F r and F u are as in the statement, wehave: S Q− ( E r ∩ E u ) = S Q− ( E r ∩ F u ) = S Q− ( F r ∩ E u ) = S Q− ( F r ∩ F u ) = E r ∪ E u = E = F r ∪ F u and the above chain of identities proves the uniqueness of the decomposition. Corollary . . For any Borel set E ⊆ G such that S Q− ( E ) < ∞ , there are two Borel sets E u , E r ⊆ E such that:(i) E u ∪ E r = E,(ii) there are countably many intrinsic Lipschitz functions f i : V i → N ( V i ) , where V i ∈ Gr ( Q − ) , whose graphs cover S Q− -almost all E r ,(iii) S Q− ( E u ∩ C ) = for any C intrinsic Lipschitz graph.Proof. Thanks to Proposition . we know that every intrinsic Lipschitz graph is S Q− - σ -finite. If we choose F in the statement of Proposition . to be the family of all intrinsic Lipschitz graphs of G , we get two sets E u and E r whose union is the whole E , such that E u has S h -null intersection with every intrinsic Lipschitz graph and E r can be covered by countably many graphs of intrinsic Lipschitz functions f i : E i ⊆ V i → N ( V i ) . The conclusionfollows from Theorem . . codimensional measures with flat tangents is intrinsic rectifiable Throughout this section we assume φ to be a fixed Radon measure on G whose support is a compact set K andsuch that for φ -almost every x ∈ G we have:(i) 0 < Θ Q− ∗ ( φ , x ) ≤ Θ Q− ∗ ( φ , x ) < ∞ ,(ii) Tan Q− ( φ , x ) ⊆ M , where M is the family of 1-codimensional flat measures introduced in Definition . .The main goal of this section is to prove the following: Theorem . . There is an intrinsic Lipschitz graph Γ such that φ ( Γ ) > . The strategy we employ in order to prove Theorem . is divided in four parts. First of all in Subsection . we show that the hypothesis (ii) on φ implies that for φ -almost any x ∈ K and r > V x , r for which K as a set is very close in the Hausdorff distance to V x , r . In Subsection . we prove that if K ∩ B ( x , r ) has big projection on some plane W , then W is very close to V x , r and there exists an α > y , z ∈ B ( x , r ) for which d H ( y , z ) ≥ dist ( W , V x , r ) r , we have z ∈ yC W ( α ) . Subsection . is the technical core ofthis section, and its main result Theorem . , shows that for φ -almost any x ∈ K we have that the set B ( x , r ) ∩ K has a big projection on V x , r . Finally, in Subsection . making use of the results of the previous subsections, weconstruct the wanted φ -positive intrinsic Lipschitz graph. he support of codimensional measures with flat tangents is intrinsic rectifiable 22 . Geometric implications of flat tangents
In this subsection we reformulate the hypothesis (ii) on φ in more geometric terms. Furthermore, in the followingDefinition . , we introduce two functionals on Radon measures that will be used in the following to estimate atsmall scales the distance of supp ( φ ) from the planes in Gr ( Q − ) . These functionals can be considered the Carnotanalogue of the bilateral beta numbers of Chapter of [ ] or of the functional d ( · , M ) of Section of [ ]. Definition . . For any x ∈ G and any r > d x , r ( φ , M ) : = inf Θ > V ∈ Gr ( Q− ) F x , r ( φ , Θ S Q− (cid:120) xV ) r Q , and ˜ d x , r ( φ , M ) : = inf Θ > z ∈ G , V ∈ Gr ( Q− ) F x , r ( φ , Θ S Q− (cid:120) zV ) r Q ,where F x , r was introduced in ( ).In the following proposition we summarize some useful properties of the functionals introduced above. Proposition . . The functionals d x , r ( · , M ) and ˜ d x , r ( · , M ) satisfy the following properties.(i) for any x ∈ G , k > and r > we have d x , kr ( φ , M ) = d k ( r − ( Q− ) T x , r φ , M ) ,(ii) for any r > the function x (cid:55)→ d x , r ( φ , M ) is continuous,(iii) for any x , y ∈ G and r , s > for which B ( y , s ) ⊆ B ( x , r ) , we have ( s / r ) Q ˜ d y , s ( φ , M ) ≤ ˜ d x , r ( φ , M ) ,(iv) for any x ∈ G and any s ≤ r, we have ( s / r ) Q d x , s ( φ , M ) ≤ d x , r ( φ , M ) .Proof. It is immediate to see that f belongs to Lip + ( B ( x , kr )) if and only if there is a g ∈ Lip + ( B ( k )) such that f ( z ) = rg ( δ r ( x − z )) . This implies that:1 ( kr ) Q (cid:18) ˆ f d φ − Θ ˆ f d S Q− (cid:120) xV (cid:19) = k Q r Q− (cid:18) ˆ g ( δ r ( x − z )) d φ ( z ) − Θ ˆ g ( δ r ( x − z )) d S Q− (cid:120) xV (cid:19) = k Q (cid:18) ˆ g ( z ) d T x , r φ r Q − ( z ) − Θ ˆ g ( z ) d S Q− (cid:120) V (cid:19) ,and this proves (i). To show that the map x (cid:55)→ d x , r ( φ , M ) is continuous, we prove the following stronger fact. Forany x , y ∈ G we have: | d x , r ( φ , M ) − δ y , r ( φ , M ) | ≤ d ( x , y ) r Q φ ( B ( x , r + d ( x , y ))) . ( )In order to prove ( ), for any (cid:101) > Θ ∗ > V ∗ ∈ Gr ( Q − ) be such that: (cid:12)(cid:12)(cid:12) ˆ f d T y , r φ r Q− − Θ ∗ ˆ f d S Q− (cid:120) V ∗ (cid:12)(cid:12)(cid:12) ≤ d y , r ( φ , M ) + (cid:101) , for any f ∈ Lip + ( B (
0, 1 )) ,Furthermore, by definition of d y , r we can find an f ∗ ∈ Lip + ( B (
0, 1 )) such that: d x , r ( φ , M ) − (cid:101) ≤ (cid:12)(cid:12)(cid:12) ˆ f ∗ d T x , r φ r Q− − Θ ∗ ˆ f ∗ d S Q− (cid:120) V ∗ (cid:12)(cid:12)(cid:12) .This choice of f ∗ , Θ ∗ and V ∗ implies: d x , r ( φ , M ) − d y , r ( φ , M ) ≤ (cid:12)(cid:12)(cid:12) ˆ f ∗ d T x , r φ r Q− − Θ ∗ ˆ f ∗ d S Q− (cid:120) V ∗ (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ˆ f ∗ d T y , r φ r Q− − Θ ∗ ˆ f ∗ d S Q− (cid:120) V ∗ (cid:12)(cid:12)(cid:12) + (cid:101) ≤ (cid:12)(cid:12)(cid:12) ˆ f ∗ d T x , r φ r Q− − ˆ f ∗ d T y , r φ r Q− (cid:12)(cid:12)(cid:12) + (cid:101) ≤ r − ( Q− ) ˆ | f ∗ ( δ r ( x − w )) − f ∗ ( δ r ( y − w )) | d φ ( w ) + (cid:101) ≤ d ( x , y ) r Q φ ( B ( x , r + d ( x , y ))) + (cid:101) .Interchanging x and y , the bound ( ) is proved thanks to the arbitrariness of (cid:101) . Finally the statements (iii) and(iv) follow directly by the definitions. he support of codimensional measures with flat tangents is intrinsic rectifiable 23 The following proposition allows us to rephrase the rather geometric condition on φ that is the flatness of thetangents, into a more malleable functional-analytic condition that is the φ -almost everywhere convergence of thefunctions x (cid:55)→ d x , kr ( φ , M ) to 0. Proposition . . The two following conditions are equivalent:(i) lim r → d x , kr ( φ , M ) = for φ -almost every x ∈ G and any k > ,(ii) Tan ( φ , x ) ⊆ M for φ -almost every x ∈ G .Proof. Let us prove that (i) implies (ii). Fix k > { r i } i ∈ N is an infinitesimal sequence such that: r − ( Q− ) i T x , r i φ (cid:42) ν ∈ Tan Q− ( φ , x ) . ( )By definition of d x , kr ( · , M ) for any i ∈ N we can find Θ i ( k ) > V i ( k ) ∈ Gr ( Q − ) such that whenever f ∈ Lip + ( B ( k )) , we have: (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d T x , r i φ r Q− i − Θ i ( k ) ˆ f d S Q− (cid:120) V i ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ F k (cid:16) r − ( Q− ) i T x , r i φ , Θ i ( k ) S Q− (cid:120) xV i ( k ) (cid:17) ≤ k Q d k (cid:16) r − ( Q− ) i T x , r i φ , M (cid:17) = k Q d x , kr i ( φ , M ) , ( )where the last identity comes from Proposition . (i). Putting ( ) and ( ) together with the fact that we areassuming that (i) holds, we infer:lim i → ∞ (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d ν − Θ i ( k ) ˆ f d S Q− (cid:120) V i ( k ) (cid:12)(cid:12)(cid:12)(cid:12) =
0, for any f ∈ Lip + ( B ( k )) . ( )Defined ϕ ( z ) : = max { k − d ( z ) , 0 } , we deduce that:0 = lim sup i → ∞ (cid:12)(cid:12)(cid:12)(cid:12) ˆ ϕ d ν − Θ i ( k ) ˆ ϕ d S Q− (cid:120) V i ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ k Q Q lim sup i → ∞ Θ i ( k ) − ˆ ϕ d ν , ( )where the last identity comes Proposition . and the fact that ϕ is radial. The bound ( ) implies in particularthat: 0 ≤ lim sup i → ∞ Θ i ( k ) ≤ Q k Q ˆ ϕ d ν .Therefore, there are a Θ ( k ) ≥
0, an n ( k ) ∈ V and (not relabeled) a subsequence for which:lim i → ∞ Θ i ( k ) = Θ ( k ) and lim i → ∞ n ( V i ( k )) = n ( k ) ,where n ( k ) = n ( V ( k )) for some V ( k ) ∈ Gr ( Q − ) . This implies that for any f ∈ Lip + ( B ( k )) we have: (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d ν − Θ ( k ) ˆ f d S Q− (cid:120) V ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d ν − Θ i ( k ) ˆ f d S Q− (cid:120) V i ( k ) (cid:12)(cid:12)(cid:12)(cid:12) + Θ i ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d S Q− (cid:120) V − ˆ f d S Q− (cid:120) V i ( k ) (cid:12)(cid:12)(cid:12)(cid:12) + | Θ ( k ) − Θ i ( k ) | ˆ f d S Q− (cid:120) V ( k ) . ( )Since the sequence { Θ i ( k ) } i ∈ N is bounded, thanks to Proposition . and the fact that inequality ( ) holds forany i ∈ N , we infer that: (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d ν − Θ ( k ) ˆ f d S Q− (cid:120) V ( k ) (cid:12)(cid:12)(cid:12)(cid:12) =
0, for any f ∈ Lip + ( B ( k )) . ( )Since Lipschitz functions with support contained in B ( k ) are dense in L ( B ( k )) , thanks to ( ) we concludethat ν (cid:120) B ( k ) = Θ ( k ) S Q− (cid:120) V ( k ) ∩ B ( k ) . Let k ≤ k and note that: Θ ( k ) S Q− (cid:120) V ( k ) ∩ B ( k ) = ν (cid:120) B ( k ) = Θ ( k ) S Q− (cid:120) V ( k ) ∩ B ( k ) . he support of codimensional measures with flat tangents is intrinsic rectifiable 24 The above identity yields Θ ( k ) = Θ ( k ) , V ( k ) = V ( k ) and in particular ν = Θ ( ) S Q− (cid:120) V ( ) .We are left to prove the viceversa. Assume by contradiction that (i) does not hold. This implies that we can finda k >
0, an (cid:101) > { r i } i ∈ N such that:lim inf r i → d x , kr i ( φ , M ) > (cid:101) . ( )Thanks to the weak- ∗ pre-compactness of the sequence r − ( Q− ) i T x , r i φ and the fact that we are assuming that (ii)holds, we can find a (non-relabeled) sequence { r i } i ∈ N a Θ > V ∈ Gr ( Q − ) such that r − ( Q− ) i T x , r i φ (cid:42) Θ S Q− (cid:120) V . For any i ∈ N let f ∗ i ∈ Lip ( B ( k )) be such that: d k ( r − ( Q− ) i T x , r i φ , M ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ f ∗ i d T x , r i φ r Q− i − Θ ˆ f ∗ i d S Q− (cid:120) V (cid:12)(cid:12)(cid:12)(cid:12) ,and note that since Lip + ( B ( k )) is compact when endowed with the supremum distance, we can assume withoutloss of generality that f ∗ i is uniformly converging to some f ∗ ∈ Lip + ( B ( k )) . Thanks to Proposition . thisimplies that:lim inf r j → d x , kr i ( φ , M ) = lim inf r j → d k ( r − ( Q− ) i T x , r i φ , M ) ≤ r j → (cid:12)(cid:12)(cid:12)(cid:12) ˆ f ∗ i d T x , r i φ r Q− i − Θ ˆ f ∗ i d S Q− (cid:120) V (cid:12)(cid:12)(cid:12)(cid:12) = r − ( Q− ) i T x , r i φ (cid:42) Θ S Q− (cid:120) V , contradicting ( ) Proposition . . Let δ ∈ N and assume φ is a Radon measure such that for φ -almost any x ∈ G we have:(i) δ − ≤ Θ Q− ∗ ( φ , x ) ≤ Θ Q− ∗ ( φ , x ) ≤ δ ,(ii) lim sup r → d x , kr ( φ , M ) < ( Q δ ) − .Then for any φ -almost all x ∈ G and any k > we have: lim sup r → d x , kr ( φ , M ) = sup { d k ( ν , M ) : ν ∈ Tan Q− ( φ , x ) } . Proof.
Since φ is locally finite, we can assume without loss of generality that it is supported on a compact set K .For any (cid:101) >
0, we let C (cid:101) be a compact subset of K such that: . φ ( K \ C (cid:101) ) < (cid:101)φ ( K ) , . for any x ∈ C (cid:101) we have δ − ≤ Θ Q− ∗ ( φ , x ) ≤ Θ Q− ∗ ( φ , x ) ≤ δ .Fix a point x ∈ C (cid:101) such that Tan Q− ( φ (cid:120) C (cid:101) , x ) = Tan Q− ( φ , x ) (cid:54) = ∅ and recall that such a choice can be donewithout loss of generality thanks to Proposition . .Suppose { r i } i ∈ N is an infinitesimal sequence such that lim i → ∞ d x , kr i ( φ , x ) = lim sup r → d x , kr ( φ , x ) and assumewithout loss of generality ν is an element of Tan Q− ( φ , x ) such that: r − ( Q− ) i T x , r i φ (cid:42) ν .Let us prove that lim sup r → d x , kr ( φ , M ) ≤ d k ( ν , M ) . For any 0 < η < Θ S Q− (cid:120) V be an element of M suchthat F k ( ν , Θ S Q− (cid:120) V ) / k Q ≤ d k ( ν , M ) + η . With this choice, thanks to the triangular inequality we infer that:lim sup i → ∞ d k ( r − ( Q− ) i T x , r i φ , M ) ≤ lim sup i → ∞ F k ( r − ( Q− ) i T x , r i φ , Θ S Q− (cid:120) V ) k Q ≤ lim sup i → ∞ F k ( r − ( Q− ) i T x , r i φ , ν ) + F k ( ν , Θ S Q− (cid:120) V ) k Q ≤ d k ( ν , M ) + η , ( ) he support of codimensional measures with flat tangents is intrinsic rectifiable 25 where the last inequality comes from the choice of Θ and V and Proposition . . The arbitrariness of η concludesthe proof of the first claim.As a second and final step of the proof, let us fix a ν ∈ Tan Q− ( φ , x ) and show that lim sup r → d x , kr ( φ , M ) ≥ d k ( ν , M ) . Since ν ∈ Tan Q− ( φ , x ) , we can find an infinitesimal sequence { r i } i ∈ N such that: r − ( Q− ) i T x , r i φ (cid:42) ν .Furthermore, for any 0 < η < i ∈ N there exists a Θ i > V i ∈ Gr ( Q − ) such that: F k ( r − ( Q− ) i T x , r i φ , Θ i S Q− (cid:120) V i ) k Q ≤ d k ( r − ( Q− ) i T x , r i φ , M ) + η = d x , kr ( φ , M ) + η ,where the last identity above comes from Proposition . (i). In addition to this, thanks to assumption (ii) and sincethe sequence { r i } i ∈ N is infinitesimal, there exists a i ∈ N such that for any i ≥ i we have d x , kr i ( φ , M ) ≤ ( Q δ ) − .This implies for any i ≥ i that: (cid:12)(cid:12)(cid:12)(cid:12) ˆ g ( w ) d T x , r i φ ( w ) r Q− i − Θ i ˆ g ( w ) d S Q− (cid:120) V i ( w ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ F k ( r − ( Q− ) i T x , r i φ , Θ i S Q− (cid:120) V i ) ≤ ( Q δ ) − k Q + η k Q , ( )where g ( x ) : = min { dist ( x , B ( k ) c ) , ( − η ) k } . Thanks to the definition of g and to ( ) we infer: Θ i (( − η ) k ) Q − ( − η ) k φ ( B ( x , kr i )) r Q− i ≤ Θ i ˆ B ( ( − η ) k ) g ( w ) d S Q− (cid:120) V i ( w ) − ˆ B ( k ) ( − η ) k d T x , r i φ ( w ) r Q− i ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ g ( w ) d T x , r i φ ( w ) r Q− i − Θ i ˆ g ( w ) d S Q− (cid:120) V i ( w ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( Q δ ) − k Q + η k Q , ( )With a similar argument, one can also prove that: ( − η ) k φ ( B ( x , ( − η ) kr i )) r Q− i − Θ i ( − η ) k Q ≤ ( Q δ ) − k Q + η k Q . ( )Rearranging inequality ( ) and dividing both sides by k Q , thanks to the choice of x and to the arbitrariness of i ,we have: lim sup i → ∞ ( − η ) Q Θ i ≤ lim sup i → ∞ ( − η ) φ ( B ( x , kr i ))( kr i ) Q− + ( Q δ ) − + η ≤ δ + ( Q δ ) − + η .Thanks to the arbitrariness of 0 < η < i → ∞ Θ i ≤ δ + ( Q δ ) − .Similarly, rearranging inequality ( ) and dividing both sides by k Q , thanks to the arbitrariness of i we infer that: ( − η ) Q− Θ Q− ∗ ( φ , x ) − ( Q δ ) − − η = lim inf i → ∞ ( − η ) Q− φ ( B ( x , ( − η ) r i ))( − η ) Q− ( kr i ) Q− − ( Q δ ) − − η ≤ ( − η ) lim inf i → ∞ Θ i .The arbitrariness of η and the choice of x conclude that lim inf i → ∞ Θ i ≥ ( − −Q ) δ − .Thus, up to subsequences we can assume that Θ i converge to some Θ ∈ [( − −Q ) δ − , δ + ( Q δ ) − ] and thatthere exists a V ∈ Gr ( Q − ) such that n ( V i ) → n ( V ) . By Proposition . this implies that Θ i S Q− (cid:120) V i (cid:42) Θ S Q− (cid:120) V . Therefore, thanks to the triangular inequality, this implies for any i ∈ N that: d k ( ν , M ) ≤ F k ( ν , r − ( Q− ) i T x , r i φ ) + F k ( r − ( Q− ) i T x , r i φ , Θ i S Q− (cid:120) V i ) + F k ( Θ i S Q− (cid:120) V i , Θ S Q− (cid:120) V ) k Q ≤ F k ( ν , r − ( Q− ) i T x , r i φ ) k Q + d k ( r − ( Q− ) i T x , r i φ , M ) + η + F k ( Θ i S Q− (cid:120) V i , Θ S Q− (cid:120) V ) k Q .Finally, thanks to the arbitrariness of i and of η and to Proposition . , we conclude that: d k ( ν , M ) ≤ lim sup i → ∞ d k ( r − ( Q− ) i T x , r i φ , M ) . he support of codimensional measures with flat tangents is intrinsic rectifiable 26 Notation . . Throughout Section we let 0 < ε < . yields two naturalnumbers ϑ , γ ∈ N , that from now on we consider fixed, such that: φ ( K \ E ( ϑ , γ )) ≤ ε φ ( K ) .These ϑ and γ have the further property, again thanks to Proposition . , that for any µ ≥ ϑ there is a ν ∈ N forwhich: φ ( K \ E ϑ , γ ( µ , ν )) ≤ ε φ ( K ) .Furthermore, we define η : = Q and let: δ G : = min (cid:40) ( Q + ) ϑ , η Q + ( − η ) Q − ( ϑ ) Q + (cid:41) .Eventually, if ˜ d x , r ( φ , M ) ≤ δ for some 0 < δ < δ G , we define Π ( x , r ) to be the subset of planes V ∈ Gr ( Q − ) forwhich there exists a Θ > z ∈ G such that: F x , r ( φ , Θ S Q− (cid:120) zV ) ≤ δ r Q .The following two propositions are the main results of this subsection. They are so relevant since they give amore geometric interpretation of the condition “flatness of the tangents” and In particular tell us that E ( ϑ , γ ) is aweakly linearly approximable set. For a discussion on how this will play a role in the proof of the main result ofthis work, we refer to the Introduction. Proposition . . Let x ∈ E φ ( ϑ , γ ) be such that ˜ d x , r ( φ , M ) ≤ δ for some δ < δ G and < r < γ . Then for everyV ∈ Π ( x , r ) we have: sup w ∈ E φ ( ϑ , γ ) ∩ B ( x , r /4 ) dist (cid:0) w , xV (cid:1) r ≤ + Q ϑ Q δ Q = : C δ Q . Proof.
Let V be any element of Π ( x , r ) and suppose z ∈ G , Θ > (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d φ − Θ ˆ f d S Q− (cid:120) zV (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ r Q , for any f ∈ Lip + ( B ( x , r )) .Since the function g ( w ) : = min { dist ( w , B ( x , r ) c ) , dist ( w , zV ) } belongs to Lip + ( B ( x , r )) , we deduce that:2 δ r Q ≥ ˆ g ( w ) d φ ( w ) − Θ ˆ g ( w ) d S Q− (cid:120) zV = ˆ g ( w ) d φ ( w ) ≥ ˆ B ( x , r /2 ) min { r /2, dist ( w , zV ) } d φ ( w ) .Suppose that y is a point in B ( x , r /4 ) ∩ E φ ( ϑ , γ ) furthest from zV and let D = dist ( y , zV ) . If D ≥ r /8, this wouldimply that:2 δ r Q ≥ ˆ B ( x , r /2 ) min { r /2, dist ( w , zV ) } d φ ( w ) ≥ ˆ B ( y , r /16 ) min { r /2, dist ( w , zV ) } d φ ( w ) ≥ r φ ( B ( y , r /16 )) ≥ r Q ϑ Q ,which is not possible thanks to the choice of δ . This implies that D ≤ r /8 and as a consequence, we have:2 δ r Q ≥ ˆ B ( x , r /2 ) min { r /2, dist ( w , zV ) } d φ ( w ) ≥ ˆ B ( y , D /2 ) min { r /2, dist ( w , zV ) } d φ ( w ) ≥ D φ ( B ( y , D /2 )) ≥ ϑ − (cid:18) D (cid:19) Q , ( )where the second inequality comes from the fact that B ( y , D /2 ) ⊆ B ( x , r /2 ) . This implies thanks to ( ), that:sup w ∈ E ( ϑ , γ ) ∩ B ( x , r /4 ) dist ( w , zV ) r ≤ Dr ≤ + Q ϑ Q δ Q = C δ Q /2. he support of codimensional measures with flat tangents is intrinsic rectifiable 27 In particular we infer that dist ( x , zV ) / r ≤ C δ Q /2. Therefore, thanks to Proposition . , we deduce:sup w ∈ E ( ϑ , γ ) ∩ B ( x , r /4 ) dist ( w , xV ) r ≤ sup w ∈ E ( ϑ , γ ) ∩ B ( x , r /4 ) dist ( w , zV ) + dist ( xV , zV ) r ≤ C δ Q . Proposition . . Let x ∈ E φ ( ϑ , γ ) and < r < γ be such that for some < δ < δ G we have:d x , r ( φ , M ) + d x , r ( φ (cid:120) E φ ( ϑ , γ ) , M ) ≤ δ . ( ) Then for any V ∈ Π ( x , r ) and any w ∈ B ( x , r /2 ) ∩ xV we have E φ ( ϑ , γ ) ∩ B ( w , δ Q + r ) (cid:54) = ∅ .Remark . . The set Π ( x , r ) in the above statement is non-empty since Proposition . insures that ( ) implies˜ d x , r ( φ , M ) ≤ δ . Proof.
Fix some V ∈ Π ( x , r ) and suppose that Θ > F x , r ( φ , Θ S Q (cid:120) xV ) + F x , r ( φ (cid:120) E φ ( ϑ , γ ) , Θ S Q (cid:120) xV ) r Q ≤ δ .Defined g ( x ) : = min { dist ( x , B (
0, 1 ) c ) , η } , we deduce that: ϑ − ( − η ) Q− η r Q − Θ η r Q ≤ η r φ (cid:0) B ( x , ( − η ) r ) (cid:1) − η r Θ S Q− (cid:120) xV ( B ( x , r )) ≤ ˆ rg ( δ r ( x − z )) d φ ( z ) − Θ ˆ rg ( δ r ( x − z )) d S Q− (cid:120) xV ≤ δ r Q ,since rg ( δ r ( x − · )) ∈ Lip + ( B ( x , r )) . Simplifying and rearranging the above chain of inequalities, we infer that: Θ ≥ ϑ − ( − η ) Q− − δ / η ≥ ( ϑ ) − ( − η ) Q− = ( ϑ ) − ( − Q ) Q− ,where the first inequality comes from the choice of δ and the last equality from the definition of η , see Definition . . Since the function Q (cid:55)→ ( − Q ) Q− is decreasing and lim Q→ ∞ ( − Q ) Q− = e , we deduce that Θ ≥ ϑ e . Suppose that δ ( Q + ) < λ < w ∈ xV ∩ B ( x , r /2 ) such that φ (cid:0) B ( w , λ r ) ∩ E ( ϑ , γ ) (cid:1) =
0. This would imply that: Θ η ( − η ) Q− λ Q r Q = Θ ηλ r S Q− (cid:120) xV (cid:0) B ( w , ( − η ) λ r ) (cid:1) ≤ Θ ˆ λ rg ( δ λ r ( w − z )) d S Q− (cid:120) xV ( z )= Θ ˆ λ rg ( δ λ r ( w − z )) d S Q− (cid:120) xV ( z ) − ˆ λ rg ( δ λ r ( w − z )) d φ ( z ) ≤ δ r Q , ( )where the last inequality comes from the choice of Θ , V and the fact that λ rg ( δ λ r ( w − · )) ∈ Lip + ( B ( x , r )) . Thanksto ( ), the choice of λ and the fact that 1/4 e ϑ < Θ , we have that: δ QQ + e ϑ η ( − η ) Q− < Θ λ Q η ( − η ) Q− ≤ δ .However, few algebraic computations that we omit show that the above inequality chain is in contradiction withthe choice of δ < δ G . he support of codimensional measures with flat tangents is intrinsic rectifiable 28 . Construction of cones complementing supp ( φ ) in case it has big projections on planes This subsection is devoted to the proof of Proposition . that tells us that if the measure φ is well approximatedinside a ball B ( x , r ) by some plane V and if there exists some other plane W on which the S Q− -measure of theprojection P W ( supp ( φ ) ∩ B ( x , r )) is comparable with r Q− , then at scales comparable with r the set supp ( φ ) is a W -intrinsic Lipschitz surface. In other words, we can find an α > y ∈ zC W ( α ) whenever y , z ∈ B ( x , r ) and d ( z , y ) (cid:38) r .Before proceeding with the statement and the proof of Proposition . , we fix some notation that will be exten-sively used throughout the rest of the paper. Notation . . Throughout this paragraph we assume σ ∈ N to be a fixed positive natural number. First of all, letus define the following two numbers: ζ ( σ ) : = − Q σ − and N ( σ ) : = (cid:98)− ( ζ ( σ )) (cid:99) + C ( σ ) : = ( n − ) C ( σ ) , C ( σ ) : = Q σ , C ( σ ) : = C ( ζ ( σ ) − ) m , C ( σ ) : = C m + N ζ ( σ ) − .Finally, we introduce six further new constants that depend only on σ . Although we could avoid giving an explicitexpression for such constants, we choose nonetheless to make them explicit. This is due to a couple of reasons. Firstof all, having their values helps keeping under control their interactions in proofs, getting more precise statements.Secondly, fixing these constants once and for all, we avoid the practise of choosing them “large enough” whennecessary. In doing so we hope to help the reader not to get distracted with the problem of whether these choiceswere legitimate or not.For the sake of readibility, we choose not make the dependence on σ of the numbers N , ζ and the constants C , . . . , C explicit in the following. We let:(i) A ( σ ) : = (cid:26) C , − ζ N log 2 − (cid:27) ,(ii) k ( σ ) : = N + C ζ − A ( + e NA ) and 0 < R < − ( N + ) ζ k ,(iii) ε G ( σ ) : = Q− n − β ∏ sj = (cid:101) nii ( A k ) Q− C Q − C where β is the constant introduced in Proposition . ,(iv) ε ( σ ) : = min (cid:26) δ G , ε Q G ( C C A k ) Q + (cid:0) + kR − Λ ( )( C + ) (cid:1) Q , (cid:16) k − k (cid:17) Q + , ( A C + A kC C e NA ) Q ( Q + ) (cid:27) ,(v) ε ( σ ) : = Q C ( A C ) Q− ,Since in the rest of Section we make an extensive use of the dyadic cubes constructed in Appendix A, we recallhere some of the notation. For any Radon measure ψ such that for ψ -almost every x ∈ G we have:0 < Θ Q− ∗ ( ψ , x ) ≤ Θ Q− ∗ ( ψ , x ) < ∞ ,and any ξ , τ ∈ N we denote by ∆ ψ ( ξ , τ ) the family of dyadic cubes relative to ψ and to the parameters ξ and τ yielded by Theorem A. . Furthermore, for any compact subset κ of E ψ ( ξ , τ ) and l ∈ N we let: ∆ ψ ( κ ; ξ , τ , l ) : = { Q ∈ ∆ ψ ( ξ , τ ) : Q ∩ κ (cid:54) = ∅ and Q ∈ ∆ φ j ( ξ , τ ) for some j ≥ l } ,where ∆ φ j ( ξ , τ ) is the j -th layer of cubes, see Theorem A. . Finally, for any Q ∈ ∆ ψ ( E ψ ( ξ , τ ) ; ξ , τ , 1 ) , we define: α ( Q ) : = ˜ d c ( Q ) ,2 k diam Q ( ψ , M ) + ˜ d c ( Q ) ,2 k diam Q ( ψ (cid:120) E ψ ( ξ , τ ) , M ) ,where c ( Q ) ∈ Q is the centre of the cube Q , see Theorem A. .Eventually, we recall for the reader’s sake some nomenclature on dyadic cubes that was already introduced inthe section Notation at the beginning of the paper. For any couple of dyadic cubes Q , Q ∈ ∆ ψ ( ξ , τ ) : he support of codimensional measures with flat tangents is intrinsic rectifiable 29 (i) if Q ⊆ Q , then Q is said to be an ancestor of Q and Q a sub-cube of Q ,(ii) if Q is the smallest cube for which Q (cid:40) Q , then Q is said to be the parent of Q and Q the child of Q . Notation . . If not otherwise stated, in order to simplify notation throughout Section we will always denote by ∆ : = ∆ φ ( ϑ , γ ) the family of dyadic cubes constructed in Theorem A. relative to the the measure φ that was fixedat the beginning of this Section and to the parameters ϑ , γ , fixed in Notation . . Furthermore, we let: E ( ϑ , γ ) : = E φ ( ϑ , γ ) , E ( µ , ν ) : = E φϑ , γ ( µ , ν ) and ∆ ( κ , l ) : = ∆ φ ( κ ; ϑ , γ , l ) .Finally, if the dependence on σ of the constants introduced above is not specified, we will always assume that σ = ϑ , where once again ϑ is the one natural number fixed in Notation . . Remark . . For any compact set κ of E ( ϑ , γ ) , we let M ( κ , l ) be the set of maximal cubes of ∆ ( κ , l ) ordered byinclusion. The elements of M ( κ , l ) are pairwise disjoint and enjoy the following properties:(i) for any Q ∈ ∆ ( κ , l ) there is a cube Q ∈ M ( κ , l ) such that Q ⊆ Q ,(ii) if Q ∈ M ( κ , l ) and there exists some Q (cid:48) ∈ ∆ ( κ , l ) for which Q ⊆ Q (cid:48) , then Q = Q (cid:48) .The proof of the following proposition is inspired by the argument employed in proving Lemma . of [ ] andits counterpart in the first Heisenberg group H , Lemma . of [ ]. Proposition . . Suppose that Q is a cube in ∆ ( E ( ϑ , γ ) , ι ) for some ι ∈ N satisfying the two following conditions:(i) ˜ d c ( Q ) ,4 k diam Q ( φ (cid:120) E ( ϑ , γ ) , M ) ≤ ε ,(ii) there exists a plane W ∈ Gr ( Q − ) such that: diam Q Q− C A Q− ≤ S Q− (cid:120) W (cid:16) P W (cid:104) c ( Q ) − ( Q ∩ E ( ϑ , γ )) (cid:105)(cid:17) . ( ) Let x ∈ E ( ϑ , γ ) ∩ Q and y ∈ B ( x , ( k − ) diam Q /8 ) ∩ E ( ϑ , γ ) be two points for which:R diam Q ≤ d ( x , y ) ≤ N + ζ − R diam Q . ( ) Then, for any α > (cid:16) ζ ε G + N R − k Λ ( )( C + ) (cid:17) − = : α we have y ∈ xC W ( α ) .Remark . . Thanks to the definition of R and k , we have:2 ( N + ) ζ − R = ( N + ) ζ − · − ( N + ) ζ k = k /32 < ( k − ) /8.This implies that B ( x , 2 N + ζ − R diam Q ) (cid:98) B ( x , ( k − ) diam Q /8 ) and thus the request d ( x , y ) ≥ R diam Q iscompatible with the fact that y is chosen in B ( x , ( k − ) diam Q /8 ) . Proof of Proposition . . Suppose by contradiction there are two points x , y ∈ E ( ϑ , γ ) satisfying the hypothesis ofthe proposition such that y (cid:54)∈ xC W ( α ) for some α > α . This implies, since the cone C W ( α ) is closed by definition,that we have π ( x − y ) (cid:54) =
0. Furthermore, Proposition . together with ( ) yield:diam Q ≤ R − Λ ( α ) | π ( x − y ) | ≤ R − Λ ( ) | π ( x − y ) | , ( )where the last inequality comes from the fact that Λ , the function yielded by Proposition . , is decreasing. Let ρ : = diam ( Q ) and note that Proposition . and the fact that B ( x , 4 ( k − ) ρ ) ⊆ B ( c ( Q ) , 4 k ρ ) , imply:˜ d x ,4 ( k − ) ρ ( φ (cid:120) E ( ϑ , γ ) , M ) ≤ ( k / ( k − )) Q− ˜ d c ( Q ) ,4 k ρ ( φ (cid:120) E ( ϑ , γ ) , M ) ≤ Q− ε ,Therefore, thanks to Proposition . and the choice of ε we infer that there exists a plane V ∈ Gr ( Q − ) , that weconsider fixed throughout the proof, such that:sup w ∈ E ( ϑ , γ ) ∩ B ( x , ( k − ) ρ ) dist ( w , xV ) ( k − ) ρ ≤ C ε Q . ( ) he support of codimensional measures with flat tangents is intrinsic rectifiable 30 Since y ∈ B ( x , ( k − ) ρ ) we deduce from ( ), that:dist ( y , xV ) ≤ ( k − ) C ε Q ρ . ( )In this paragraph we prove that if there exists a point v ∈ V such that v (cid:54) = | π ( P W v ) | ≤ ϑ | v | for some0 < ϑ <
1, then: |(cid:104) n ( V ) , n ( W ) (cid:105)| ≤ ϑ / (cid:112) − ϑ . ( )We note that the assumptions on v imply that: | v | − (cid:104) n ( W ) , v (cid:105) = | v − (cid:104) n ( W ) , v (cid:105) n ( W ) | = | π W v | = | π ( P W v ) | ≤ ϑ | v | . ( )By means of few omitted algebraic manipulations of ( ), we conclude that √ − ϑ | v | ≤ |(cid:104) n ( W ) , v (cid:105)| . Finally,since (cid:104) n ( V ) , v (cid:105) =
0, thanks to ( ) and Cauchy-Schwartz inequality, we have: ϑ | v | ≥|(cid:104) π W v , n ( V ) (cid:105)| = |(cid:104) v − (cid:104) n ( W ) , v (cid:105) n ( W ) , n ( V ) (cid:105)| = |(cid:104) n ( W ) , v (cid:105)(cid:104) n ( V ) , n ( W ) (cid:105)| ≥ (cid:112) − ϑ | v ||(cid:104) n ( V ) , n ( W ) (cid:105)| . ( )It is immediate to see that ( ) is equivalent to ( ), proving the claim.Given V , W ∈ Gr ( Q − ) and x , y ∈ E ( ϑ , γ ) as above, let us construct a v with v (cid:54) = | π ( P W v ) | ≤ ϑ | v | for a suitably small ϑ . Since y (cid:54)∈ xC W ( α ) , thanks to Proposition . we have: | π ( P W ( x − y )) | ≤(cid:107) P W ( x − y ) (cid:107) < α − (cid:107) P n ( W ) ( x − y ) (cid:107) = α − |(cid:104) n ( W ) , π ( x − y ) (cid:105)| ≤ α − | π ( x − y ) | .Defined v to be the point of V for which d ( y , xv ) = dist ( y , xV ) , thanks to ( ) and the fact that y ∈ B ( x , ( k − ) ρ /8 ) we have: (cid:107) v (cid:107) ≤ d ( xv , y ) + d ( y , x ) ≤ dist ( y , xV ) + ( k − ) ρ /8 ≤ ( C ε Q + ) k ρ < ( k − ) ρ ,where the last inequality comes from the choice of ε . Furthermore, thanks to ( ) and ( ) we have: R Λ ( ) − ρ /2 ≤ ( R Λ ( ) − − C k ε Q ) ρ ≤| π ( x − y ) | − d ( y , xv ) ≤ | π ( x − y ) | − | π ( y − xv ) |≤| π ( x − y ) − π ( y − xv ) | = | v | , ( )and where the first inequality above, comes from the choice of ε . Let us prove that v satisfies the inequality | π ( P W v ) | ≤ R − Λ ( ) k ( C C ε Q + + N ζ − α − ) | v | . Since v (cid:54)∈ C W ( α ) , thanks to Proposition . : | π ( P W ( v )) | ≤| π ( P W ( v )) − π ( P W ( x − y )) | + | π ( P W ( x − y )) |≤| π ( P W ( y − xv )) | + (cid:107) P W ( x − y ) (cid:107) ≤ | π ( P W ( y − xv )) | + α − (cid:107) P N ( W ) ( x − y ) (cid:107)≤(cid:107) P W ( y − xv ) (cid:107) + α − | π ( x − y ) | ≤ (cid:107) P W ( y − xv ) (cid:107) + + N ζ − R α − ρ , ( )where the last inequality of the last line above comes from ( ). Theorem . together with ( ), ( ) and ( )implies that: | π ( P W ( v )) | ≤(cid:107) P W ( y − xv ) (cid:107) + + N ζ − R α − ≤ C (cid:107) y − xv (cid:107) + + N ζ − R α − ρ = C d ( y , xV ) + + N ζ − R α − ρ ≤ ( C C ( k − ) ε Q + + N ζ − R α − ) ρ ≤ R − Λ ( ) k ( C C ε Q + + N ζ − α − ) | v | = : θ ( α , ε ) | v | . ( )Thanks to the choice of the constants ε , R and k together with some algebraic computations that we omit, it ispossible to prove that (cid:112) − θ ( α , ε ) ≥ | π ( P W ( v )) | ≤ θ ( α , ε ) | π ( v ) | , we deduce thanks ( )that: |(cid:104) n ( V ) , n ( W ) (cid:105)| ≤ θ ( α , ε ) (cid:112) − θ ( α , ε ) ≤ θ ( α , ε ) . ( ) he support of codimensional measures with flat tangents is intrinsic rectifiable 31 Let us prove that ( ) is in contradiction with ( ). Suppose that z ∈ B ( x , ( k − ) ρ /8 ) ∩ E ( ϑ , γ ) and note that: |(cid:104) n ( V ) , π ( P W ( x − z )) (cid:105)| = |(cid:104) n ( V ) , π W ( z − x ) (cid:105)| ≤ |(cid:104) n ( V ) , z − x (cid:105)| + |(cid:104) n ( V ) , π n ( W ) ( z − x ) (cid:105)|≤|(cid:104) n ( V ) , z − x (cid:105)| + |(cid:104) n ( V ) , n ( W ) (cid:105)||(cid:104) z − x , n ( W ) (cid:105)|≤(cid:107) P N ( V ) ( x − z ) (cid:107) + d ( x , z ) |(cid:104) n ( V ) , n ( W ) (cid:105)| = dist ( z , xV ) + d ( x , z ) |(cid:104) n ( V ) , n ( W ) (cid:105)| , ( )where the last identity comes from Proposition . . Inequalities ( ), ( ), ( ) and the choice of z imply: |(cid:104) n ( V ) , π ( P W ( x − z )) (cid:105)| ≤ d ( z , xV ) + d ( x , z ) |(cid:104) n ( V ) , n ( W ) (cid:105)|≤ C k ε Q ρ + θ ( α , ε ) d ( x , z ) ≤ C k ε Q ρ + θ ( α , ε ) k ρ . ( )Furthermore, defined n : = π W ( n ( V )) , it is immediate to see from ( ) that | n − n ( V ) | ≤ θ ( α , ε ) , which yieldsthanks to ( ), ( ), the triangular inequality and Proposition . the following bound: |(cid:104) n , π ( P W ( x − z )) (cid:105)| ≤|(cid:104) n ( V ) , π ( P W ( x − z )) (cid:105)| + | n − n ( V ) || π ( P W ( x − z )) |≤|(cid:104) n ( V ) , π ( P W ( x − z )) (cid:105)| + | n − n ( V ) |(cid:107) P W ( x − z ) (cid:107)≤ ( kC ε Q ρ + θ ( α , ε ) k ρ ) + θ ( α , ε ) C k ρ = ( C ε Q + ( C + ) θ ( α , ε )) k ρ . ( )For the sake of notation, we introduce the following set: S : = { w ∈ W : |(cid:104) n , w (cid:105)| ≤ ( C ε Q + ( C + ) θ ( α , ε )) k ρ } .The bound ( ) implies that the projection of x − E ( ϑ , γ ) ∩ B ( ( k − ) ρ /8 ) on W is contained in S , which is a verynarrow strip around V ∩ W inside W . Furthermore, we recall that thanks to Proposition . we have: P W ( B ( ( k − ) ρ /8 )) ⊆ B ( C ( k − ) ρ /8 ) . ( )Finally, putting together ( ) and ( ), we deduce that: P W (cid:16) x − E ( ϑ , γ ) ∩ B ( ( k − ) ρ /8 ) (cid:17) ⊆ P W (cid:0) x − E ( ϑ , γ ) (cid:1) ∩ P W (cid:0) B ( ( k − ) ρ /8 ) (cid:1) ⊆ S ∩ B ( C ( k − ) ρ /8 ) . ( )Completing { n ( W ) , n } to an orthonormal basis E : = { n ( W ) , n / | n | , e , . . . , e n } of R n satisfying ( ) thanks to Remark . , we have: S ∩ B ( C ( k − ) ρ /8 ) ⊆ S ∩ Box E ( C k ρ /8 ) . ( )The above inclusion together with Tonelli’s theorem yieds: H n − eu (cid:120) W (cid:16) P W (cid:16) x − E ( ϑ , γ ) ∩ B ( ( k − ) ρ /8 ) (cid:17)(cid:17) ≤H n − eu (cid:120) W ( S ∩ B ( C ( k − ) ρ /8 )) ≤ H n − eu (cid:120) W ( S ∩ Box E ( C k ρ /8 )) = (cid:0) C ε Q + ( C + ) θ ( α , ε ) (cid:1) k ρ · n − s ∏ i = (cid:101) − n i i (cid:18) C k ρ (cid:19) Q− = n − Q + C Q− s ∏ i = (cid:101) − n i i (cid:0) C ε Q + ( C + ) θ ( α , ε ) (cid:1) ( k ρ ) Q− . ( )The inclusion ( ), the bound ( ), Proposition . and the definition of A finally imply that: S Q− (cid:120) W (cid:16) P W (cid:16) x − E ( ϑ , γ ) ∩ B ( ( k − ) ρ /8 ) (cid:17)(cid:17) ≤ S Q− (cid:120) W (cid:16) S ∩ B ( C k ρ /8 ) (cid:17) = β − H n − eu (cid:120) W (cid:16) S ∩ B ( C k ρ /8 ) (cid:17) ≤ β − n − Q + C Q− s ∏ i = (cid:101) − n i i (cid:0) C ε Q + ( C + ) θ ( α , ε ) (cid:1) ( k ρ ) Q− = − C − ε − G A − ( Q− ) (cid:0) C ε Q + ( C + ) θ ( α , ε ) (cid:1) ρ Q− , ( ) he support of codimensional measures with flat tangents is intrinsic rectifiable 32 where the last identity comes from the definition of ε G , see Notation . . Furthermore, since S Q− (cid:120) W ( P W ( p ∗ E ) = S Q− (cid:120) W ( P W ( E )) for any measurable set E in G , see for instance the proof of Proposition . . in [ ], we deducethat: S Q− (cid:120) W (cid:16) P W (cid:16) x − E ( ϑ , γ ) ∩ B ( ( k − ) ρ /8 ) (cid:17)(cid:17) = S Q− (cid:120) W (cid:16) P W (cid:16) c ( Q ) − E ( ϑ , γ ) ∩ B ( c ( Q ) − x , ( k − ) ρ /8 ) (cid:17)(cid:17) .Thanks to the choice of k and the fact that x ∈ Q , we infer that B ( ρ ) ⊆ B ( c ( Q ) − x , ( k − ) ρ /8 ) . Together with( ), this allows us to deduce that: S Q− (cid:120) W (cid:16) P W (cid:16) x − E ( ϑ , γ ) ∩ B ( ( k − ) ρ /8 ) (cid:17)(cid:17) ≥S Q− (cid:120) W (cid:16) P W (cid:16) c ( Q ) − E ( ϑ , γ ) ∩ B ( ρ ) (cid:17)(cid:17) ≥S Q− (cid:120) W (cid:16) P W (cid:0) c ( Q ) − ( E ( ϑ , γ ) ∩ Q ) (cid:1)(cid:17) ≥ ρ Q− C A Q− . ( )Putting together ( ) and ( ) we conclude:2 ε G ≤ (cid:0) C ε Q + ( C + ) θ ( α , ε ) (cid:1) = C ε Q + R − k Λ ( )( C + )( C C ε Q + + N ζ − α − ) ,The choice of ε and α imply, with some algebraic computations that we omit, that the above inequality is false,showing that the assumption y (cid:54)∈ xC W ( α ) is false. . Flat tangents imply big projections
This subsection is devoted to the proof of the following result, that asserts that the hypothesis (ii) of Proposition . is satisfied by φ (cid:120) E ( ϑ , γ ) . More precisely we construct a compact subset C of E ( ϑ , γ ) having big measure inside E ( ϑ , γ ) such that: Theorem . . For any cube Q of sufficient small diameter such that ( − ε ) φ ( Q ) ≤ φ ( Q ∩ C ) , we have: S Q− ( P Π ( Q ) ( Q ∩ C )) ≥ diam Q Q− A Q− .In the following it will be useful to reduce to a compact subset C of E ( ϑ , γ ) where the distance of φ from planesis uniformly small under a fixed scale. Proposition . . For any µ ≥ ϑ , there exists a ν ∈ N , a compact subset C of E ( µ , ν ) and a ι ∈ N such that:(i) φ ( K \ C ) ≤ ε φ ( K ) ,(ii) d x ,4 kr ( φ , M ) + d x ,4 kr ( φ (cid:120) E ( ϑ , γ ) , M ) ≤ −Q ε for any x ∈ C and any < r < − ι N + / γ .Proof. Since by assumption Tan Q− ( φ , x ) ⊆ M for φ -almost every x ∈ G , thanks to Proposition . we infer thatthe functions f r ( x ) : = d x ,4 kr ( φ , M ) converge φ -almost everywhere to 0 on K as r goes to 0. Thanks to Proposition . , the same line of reasoning implies also that f ϑ , γ r ( x ) : = d x ,4 kr ( φ (cid:120) E ( ϑ , γ ) , M ) converges φ -almost everywhereto 0 on E ( ϑ , γ ) . Proposition . and Severini-Egoroff’s theorem yield a compact subset C of E ( µ , ν ) such that φ ( E ( ϑ , γ ) \ C ) ≤ ε φ ( E ( ϑ , γ )) and such that the sequence d x ,4 kr ( φ , M ) + d x ,4 kr ( φ (cid:120) E ( ϑ , γ ) , M ) converges uniformlyto 0 on C . This directly implies both (i) and (ii) thanks to the choice of ϑ and γ . Notation . . From now on, we assume µ ≥ ϑ , the set C and the natural numbers ν and ι yielded by Proposition . to be fixed. Furthermore, we define ι : = max { ι , ν } .The construction of a compact set satisfying the thesis of Proposition . can be achieved in the following slightlydifferent conditions on the measure: Proposition . . Suppose ψ is a Radon on G supported on the compact set K and satisfying the following assumptions:(i) there exists a δ ∈ N such that δ − ≤ Θ Q− ∗ ( ψ , x ) ≤ Θ Q− ∗ ( ψ , x ) ≤ δ for ψ -almost every x ∈ G , he support of codimensional measures with flat tangents is intrinsic rectifiable 33 (ii) lim sup r → d x ,4 kr ( ψ , M ) ≤ − ( Q + ) ε ( δ ) for ψ -almost every x ∈ G .Then, there exist ˜ ι ∈ N and γ ∈ N for which for any µ ≥ δ we can find a ν ∈ N and a compact set ˜ C ⊆ E ψ δ , γ ( µ , ν ) suchthat:(i) ψ ( K \ ˜ C ) ≤ ε ψ ( K ) where ε was introduced in Notation . ,(ii) d x ,4 kr ( ψ , M ) + d x ,4 kr ( ψ (cid:120) E ψ ( δ , γ ) , M ) ≤ −Q ε ( δ ) for any x ∈ ˜ C and any < r < − ˜ ι N ( δ )+ / γ ,where ε and ε ( δ ) are the constants introduced in Notation . and . respectively.Proof. First of all, thanks to Propositions . and . we can find a γ ∈ N such that ψ ( K \ E ψϑ , γ ( µ , ν )) ≤ ε ψ ( K ) .Let us now prove that:lim sup r → d x ,4 kr ( ψ (cid:120) E ψ ( δ , γ ) , M ) ≤ − ( Q + ) ε ( δ ) , for ψ -almost every x ∈ E ψ ( δ , γ ) .Recall that for ψ -almost every x ∈ E ψ ( δ , γ ) we have that Tan Q− ( ψ (cid:120) E ψ ( δ , γ ) , x ) = Tan Q− ( ψ , x ) . Thanks to thisand the fact that ε ( δ ) ≤ δ − , Proposition . yields:lim sup r → d x ,4 kr ( ψ , M ) = lim sup r → d x ,4 kr ( ψ (cid:120) E ψ ( δ , γ ) , M ) ≤ − ( Q + ) ε ( δ ) .Therefore, for ψ -almost every x ∈ E ψ ( δ , γ ) , there exists an r ( x ) > d x ,4 kr ( ψ , M ) + d x ,4 kr ( ψ (cid:120) E ψ ( δ , γ ) , M ) ≤ −Q ε ( δ ) , for every 0 < r < r ( x ) . For any j ∈ N , define: E j : = { x ∈ E ψϑ , γ ( µ , ν ) : r ( x ) > j } .Let us show that the sets E j are Borel. Thanks to Proposition . (ii) we know that the map x (cid:55)→ d x , r ( ψ , M ) iscontinuous and thus for any r > Ω r : = { y ∈ G : d y , r ( ψ , M ) < −Q ε ( δ ) } is open. In particular if x ∈ Ω r for any r ∈ (
0, 1/ j ) ∩ Q , then thanks to Proposition . (iv) we have that x ∈ E j , and obviously the viceversa holdsas well. This shows that the sets E j are G δ sets intersected with the compact set E ψϑ , γ ( µ , ν ) and thus are Borel.Furthermore, thanks to the assumption on the measure ψ we infer that: ψ ( E ψ ( δ , γ ) \ (cid:91) j ∈ N E j ) =
0. ( )Finally, ( ) together with the measurability of the sets E j are measurable, implies that we can find a big enough j ∈ N and a compact set C contained in E j satisfying (i) and (ii) of Proposition . .The following lemma rephrases Propositions . and . into the language of dyadic cubes. Lemma . . For any cube Q ∈ ∆ ( C , ι ) we have α ( Q ) ≤ ε . Furthermore, there is a plane Π ( Q ) ∈ Gr ( Q − ) for which:(i) sup w ∈ E ( ϑ , γ ) ∩ B ( c ( Q ) , k diam Q /2 ) dist ( w , c ( Q ) Π ( Q )) k diam Q ≤ C ε Q , (ii) for any w ∈ B ( c ( Q ) , k diam Q /2 ) ∩ c ( Q ) Π ( Q ) we have E ( ϑ , γ ) ∩ B ( w , 3 kC ε ( Q + ) diam Q ) (cid:54) = ∅ .Proof. Let Q ∈ ∆ ( C , ι ) , fix a x ∈ Q ∩ C and define ρ : = diam Q . Thanks to Proposition . we know that: d x ,4 kr ( ψ , M ) + d x ,4 kr ( ψ (cid:120) E ( ϑ , γ ) , M ) ≤ − ( Q + ) ε , ( )for any r < − ι N + / γ . Thanks to Theorem A. (iv) we have that ρ < − ι N + / γ and thus by Proposition . wehave:˜ d c ( Q ) ,2 k ρ ( ψ , M ) ≤ Q ˜ d x ,4 k ρ ( ψ , M ) ≤ −Q ε and ˜ d c ( Q ) ,2 k ρ ( ψ (cid:120) E ( ϑ , γ ) , M ) ≤ Q ˜ d x ,4 k ρ ( ψ (cid:120) E ( ϑ , γ ) , M ) ≤ −Q ε .( ) he support of codimensional measures with flat tangents is intrinsic rectifiable 34 The bounds in ( ) together with Proposition . imply (i) and that α ( Q ) ≤ ε .The proof of the second part of the statement is a little more delicate. Since C is a subset of E ϑ , γ ( µ , nu ) , thanks tothe choice of ι and Proposition A. we have that c ( Q ) ∈ E ( ϑ , γ ) ∩ B ( x , ρ ) . Secondly, Proposition . (iii), Proposition . and ( ) yield for any V ∈ Π ( x , 2 k ρ ) that:dist ( c ( Q ) , xV ) ≤ dist ( c ( Q ) , c ( Q ) V ) + dist ( xV , c ( Q ) V ) = dist ( x , c ( Q ) V ) ≤ − ( Q− ) k ρ C ε Q , ( )For any V ∈ Π ( x , 2 k ρ ) and any w ∈ B ( c ( Q ) , k ρ /2 ) ∩ c ( Q ) V , we define: w ∗ : = P N ( V ) ( c ( Q ) − x ) − P V ( c ( Q ) − x ) − wP N ( V ) ( c ( Q ) − x ) .With few computations that we omit, it is not difficult to see that: d ( c ( Q ) w , xw ∗ ) = (cid:107) P N ( V ) ( c ( Q ) − x ) = dist ( c ( Q ) , xV ) ≤ − ( Q− ) k ρ C ε Q , ( )where the second identity follows from Proposition . and the last one from inequality ( ). Thanks to definitionof w ∗ the triangular inequality, Proposition . and the fact that d ( c ( Q ) , x ) ≤ ρ , the norm of w ∗ can be estimatedas follows: (cid:107) w ∗ (cid:107) ≤ (cid:107) P N ( V ) ( c ( Q ) − x ) (cid:107) + (cid:107) P V ( c ( Q ) − x ) (cid:107) + (cid:107) w (cid:107) ≤ ρ + C ρ + k ρ /2 < k ρ . ( )Thanks to inequalities ( ) and ( ) and Proposition . we infer that B ( xw ∗ , 2 k ρε ( Q + ) ) ∩ E ( ϑ , γ ) (cid:54) = ∅ . Finally,thanks to ( ) and the fact that 2 Q− C <
1, we conclude that: E ( ϑ , γ ) ∩ B ( c ( Q ) w , 3 k ρ C ε ( Q + ) ) ⊇ E ( ϑ , γ ) ∩ B ( xw ∗ , 2 k ρε ( Q + ) ) (cid:54) = ∅ .This concludes the proof of the proposition.As in the case of Proposition . , one can impose slightly different conditions on the measure and obtain afamily of cubes satisfying the same thesis as Proposition . . Proposition . . Suppose ψ is a Radon on G supported on the compact set K satisfying the following assumptions:(i) there exists a δ ∈ N such that δ − ≤ Θ Q− ∗ ( ψ , x ) ≤ Θ Q− ∗ ( ψ , x ) ≤ δ for ψ -almost every x ∈ G ,(ii) lim sup r → d x ,4 kr ( ψ , M ) ≤ − ( Q + ) ε ( δ ) for ψ -almost every x ∈ G .Then, fixed µ ≥ δ if γ , ˜ ι , ν ∈ N and ˜ C (cid:98) E ψϑ , γ ( µ , ν ) are the natural numbers and the compact set yielded by Proposition . , defined ˜ ι : = max { ˜ ι , ν } for any cube Q ∈ ∆ ψ ( ˜ C ; 2 δ , γ , ι ) we have that α ( Q ) ≤ ε ( δ ) and for any such cube Q there isa plane Π ( Q ) ∈ Gr ( Q − ) for which:(i) sup w ∈ E ψ ( δ , γ ) ∩ B ( c ( Q ) , k ( δ ) diam Q /2 ) dist ( w , c ( Q ) Π ( Q )) k diam Q ≤ C ( δ ) ε ( δ ) Q , (ii) for any w ∈ B ( c ( Q ) , k ( δ ) diam Q /2 ) ∩ c ( Q ) Π ( Q ) we have:E ψ ( δ , γ ) ∩ B ( w , 3 kC ( δ ) ε ( δ ) ( Q + ) diam Q ) (cid:54) = ∅ . Proof. If ψ is a Radon measure satisfying the hypothesis of Proposition . , we can find a compact a γ ∈ N and acompact set ˜ C contained in E ψ ( δ , γ ) such that:(i) ψ ( K \ ˜ C ) ≤ ε ψ ( K ) where ε was introduced in Notation . ,(ii) d x ,4 kr ( ψ , M ) + d x ,4 kr ( ψ (cid:120) E ψ ( δ , γ ) , M ) ≤ −Q ε ( δ ) for any x ∈ ˜ C and any 0 < r < − ˜ ι N ( δ )+ / γ .Thus, if ∆ ψ ( δ , γ ) is the family of dyadic cubes relative to the parameters 2 δ , γ and the measure ψ yielded byTheorem A. , one can prove that the cubes of ∆ ψ ( ˜ C ; 2 δ , γ , ι ) satisfy (i) and (ii) by using the verbatim argument weemployed in the proof of Proposition . . he support of codimensional measures with flat tangents is intrinsic rectifiable 35 The arguments we will use in the rest of the subsection to prove Proposition . through Theorem . , followfrom an adaptation of the techniques that can be found in Chapter , § of [ ]. The first of such adaptations isthe following definition, that is a way of saying that two cubes are close both in metric and in size terms: Definition . (Neighbour cubes) . Let A : = A and let Q j ∈ ∆ i j be two cubes with j =
1, 2. We say that, Q and Q are neighbours if:dist ( Q , Q ) : = inf x ∈ Q , y ∈ Q d ( x , y ) ≤ (I) A ( diam Q + diam Q ) and | j − j | ≤ (II) A .Furthermore, in the following for any Q ∈ ∆ ( C , ι ) we let for the sake of notation: n ( Q ) : = n ( Π ( Q )) ,where Π ( Q ) ∈ Gr ( Q − ) is the plane yielded by Lemma . .Finally, two planes V , W ∈ Gr ( Q − ) are said to have compatible orientations if their normals n ( V ) , n ( W ) ∈ V are chosen in such a way that (cid:104) n ( V ) , n ( W ) (cid:105) >
0. By extension, we will say that two cubes Q , Q ∈ ∆ ( C , ι ) have compatible orientation themselves if Π ( Q ) and Π ( Q ) are chosen to have compatible orientation. Proposition . . Suppose that Q j ∈ ∆ i j for j =
1, 2 . Then:(i) if Q is the parent of Q , then Q and Q are neighbors,(ii) if Q and Q are neighbors for any non- negative integer k ≤ min { i , i } their ancestors ˜ Q ∈ ∆ i − k and ˜ Q ∈ ∆ i − k are neighbors,(iii) if Q , Q ∈ ∆ ( E ( ϑ , γ ) , 1 ) are neighbours, then (cid:12)(cid:12)(cid:12) log diam Q diam Q (cid:12)(cid:12)(cid:12) ≤ AN.Proof.
Let us prove (i). Since Q ⊆ Q then (I) of Definition . follows immediately. On the other hand, thanks toProposition A. , we infer: | j − j | ≤ C ≤ A = A ,where the second inequality comes from the choice of A , see Definition . , and this proves (II) of Definition . .In order to prove (ii), we first note that | ( i − k ) − ( i − k ) | = | i − i | ≤ A and secondly that:dist ( ˜ Q , ˜ Q ) ≤ dist ( Q , Q ) ≤ A ( diam Q + diam Q ) ≤ A ( diam ˜ Q + diam ˜ Q ) .In order to prove (iii), note that thanks to Theorem A. (ii) and (v), we infer that: (cid:12)(cid:12)(cid:12) log diam Q diam Q (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) log 2 − Nj + / γζ − Nj − / γ (cid:12)(cid:12)(cid:12) = ( N | j − j | + ) log 2 − ζ ≤ AN ,where the last inequality comes from the choice of A . Remark . . If Q ∈ ∆ ( C , ι ) then c ( Q ) ∈ E ( ϑ , γ ) thanks to Proposition A. and the fact that we chose ι ≥ ϑ . Proposition . . Suppose that Q , Q ∈ ∆ ( C , ι ) are two neighbour cubes. Then: ( − C ε Q + ) = ( − ( n − ) C ε Q + ) ≤ |(cid:104) n ( Q ) , n ( Q ) (cid:105)| . Proof.
Thanks to the definition of k , we have: A ( diam Q + diam Q ) ≤ A max { diam Q , diam Q } ≤ ( k /4 ) max { diam Q , diam Q } .Without loss of generality we can assume that diam Q ≤ diam Q . Since the cubes Q , Q are supposed to beneighbors, we deduce that: dist ( Q , Q ) ≤ ( k /4 ) diam Q . he support of codimensional measures with flat tangents is intrinsic rectifiable 36 This implies that for any z ∈ Q , we have:dist ( z , Q ) ≤ diam Q + inf y ∈ Q dist ( y , Q ) ≤ diam Q + dist ( Q , Q ) ≤ ( k /4 + ) diam Q < ( k /2 ) diam Q . ( )Inequality ( ) implies that for any z ∈ Q we have Q ⊆ B ( z , k diam Q /2 ) . This, together with Lemma . (i),implies that for any w ∈ E ( ϑ , γ ) ∩ Q we have:dist ( w , c ( Q ) Π ( Q )) ≤ C ε Q k diam Q , ( )We now claim that the set B : = { u ∈ G : dist ( u , Q ) ≤ ( k /20 ) diam Q } is contained in the ball B ( c ( Q ) , k diam Q /2 ) .In order to prove such inclusion, let u ∈ B and note that:dist ( u , c ( Q )) ≤ inf w ∈ Q ( d ( u , w ) + d ( w , c ( Q )) ≤ inf w ∈ Q d ( u , w ) + diam Q + dist ( Q , Q ) + diam Q ≤ u ∈ B k
20 diam Q + diam Q + dist ( Q , Q ) + diam Q ≤ k + Q < k Q , ( )where the second last inequality comes from the assumption that Q is the cube with the biggest diameter. Theinequality ( ) concludes the proof of the inclusion B ⊆ B ( c ( Q ) , k diam Q /2 ) . Proposition . (iii) together withthe fact that Q , Q ∈ ∆ ( E ( ϑ , γ ) , ι ) and inequality ( ) imply that for any u ∈ E ( ϑ , γ ) ∩ B we have:dist ( u , c ( Q ) Π ( Q )) ≤ C ε Q k diam Q ≤ C e NA ε Q k diam Q . ( )Furthermore, thanks to Remark . we have c ( Q ) ∈ B ∩ E ( ϑ , γ ) . This also implies, by Proposition . and ( ),that for any u ∈ B ∩ E ( ϑ , γ ) we have:dist ( u , c ( Q ) Π ( Q )) ≤ dist ( u , c ( Q ) Π ( Q )) + dist ( c ( Q ) Π ( Q ) , c ( Q ) Π ( Q ))= dist ( u , c ( Q ) Π ( Q )) + dist ( c ( Q ) , c ( Q ) Π ( Q )) ≤ C e NA ε Q k diam Q . ( )Thanks to Lemma . (ii), we deduce that for any y ∈ B ( c ( Q ) , k diam Q /40 ) ∩ c ( Q ) Π ( Q ) there exists some w ( y ) in E ( ϑ , γ ) ∩ B ( y , 3 kC ε ( Q + ) diam Q ) . Since by definition ε ≤ (( k − ) /20 k ) Q + , we have:dist ( w ( y ) , Q ) ≤ inf p ∈ Q d ( w ( y ) , y ) + d ( y , c ( Q )) + d ( c ( Q ) , p ) ≤ kC ε ( Q + ) diam Q + k
40 diam Q + diam Q ≤ k
20 diam Q , ( )where the last inequality comes from the choice of k . Inequality ( ) implies that w ( y ) ∈ B and thanks to ( ) weinfer that: dist ( w ( y ) , c ( Q ) Π ( Q )) ≤ C e NA ε Q k diam Q .Summing up, for any y ∈ B ( c ( Q ) , k diam Q /40 ) ∩ c ( Q ) Π ( Q ) we have:dist ( y , c ( Q ) Π ( Q )) ≤ C d ( y , w ( y )) + dist ( w ( y ) , c ( Q ) Π ( Q )) ≤ C ε Q + k diam Q + C e NA ε Q k diam Q ≤ ( C + C e NA ε Q ( Q + ) ) ε Q + k diam Q ≤ C ε Q + k diam Q , ( )where the last inequality comes from the choice of ε and few algebraic computations that we omit. Furthermore,inequality ( ) and Proposition . imply that: |(cid:104) π ( y − c ( Q )) , n ( Q ) (cid:105)| = (cid:107) P n ( Q ) ( c ( Q ) − y ) (cid:107) = dist ( y , c ( Q ) Π ( Q )) ≤ C ε Q + k diam Q .Suppose { v i , i =
1, . . . , n − } are the unit vectors of the first layer V spanning the orthogonal of n ( Q ) inside V and let y j : = c ( Q ) δ k diam Q /80 ( v j ) . Then, thanks to inequality ( ), we deduce that:1 = |(cid:104) n ( Q ) , n ( Q ) (cid:105)| + n − ∑ j = |(cid:104) v j , n ( Q ) (cid:105)| = |(cid:104) n ( Q ) , n ( Q ) (cid:105)| + n − ∑ j = |(cid:104) π ( c ( Q ) − y j ) , n ( Q ) (cid:105)| ( k diam Q /80 ) . ≤|(cid:104) n ( Q ) , n ( Q ) (cid:105)| + ( n − ) C ε Q + .This concludes the proof of the proposition. he support of codimensional measures with flat tangents is intrinsic rectifiable 37 Proposition . . Let Q , Q ∈ ∆ ( C , ι ) and suppose that Π ( Q ) and Π ( Q ) , the planes yielded by Lemma . , are chosenwith compatible orientations. Then: | n ( Q ) − n ( Q ) | ≤ (cid:112) C ε ( Q + ) . Furthermore, the planes Π ( Q i ) have compatible orientations if and only if the planes Π ( ˜ Q i ) relative to the parent cubes ˜ Q , ˜ Q of Q and Q respectively, have the compatible orientations.Proof. If Q and Q are neighbors and have the compatible orientations, then (cid:104) n ( Q ) , n ( Q ) (cid:105) ≥ . we infer that: | n ( Q ) − n ( Q ) | = − (cid:104) n ( Q ) , n ( Q ) (cid:105) ≤ − ( − C ε Q + ) ≤ (cid:112) C ε Q + .If Q and Q are neighbors, Proposition . implies that the couples ˜ Q and ˜ Q , Q and ˜ Q , Q and ˜ Q areneighbors. Therefore, Proposition . implies that: (cid:104) n ( ˜ Q ) , n ( ˜ Q ) (cid:105) = (cid:104) n ( Q ) , n ( Q ) (cid:105) + (cid:104) n ( ˜ Q ) − n ( Q ) , n ( Q ) (cid:105) + (cid:104) n ( ˜ Q ) , n ( ˜ Q ) − n ( Q ) (cid:105)≥ ( − C ε Q + ) − (cid:112) C ε Q + ≥ Π ( ˜ Q ) and Π ( ˜ Q ) have the same orientation, the same line of reasoning yields that the planes Π ( Q ) and Π ( Q ) have compatible orientation as well. Proposition . . It is possible to fix an orientation on the planes { Π ( Q ) : Q ∈ ∆ ( C , ι ) } , in such a way that: | n ( Q ) − n ( Q ) | ≤ whenever Q , Q ∈ ∆ ( C , ι ) are neighbors and contained in the same maximal cube Q ∈ M ( C , ι ) .Proof. Let Q i ∈ ∆ j i for i =
1, 2 and suppose without loss of generality that j ≤ j . Fix the normal of the plane Π ( Q ) and determine the normals of all other planes Π ( Q ) as Q varies in ∆ ( C , ι ) by demanding that the orientationof the cube Q is compatible with the one of Π ( ˜ Q ) , where ˜ Q is the parent of Q .If Q = Q , let us consider the finite sequence { ˜ Q i } i = M of ancestors of Q such that ˜ Q = Q , ˜ Q M = Q andsuch that ˜ Q i + is the parent of ˜ Q i . Then, the orientation of Π ( Q ) and Π ( Q ) fixed in the above paragraph mustbe compatible, indeed: (cid:104) n ( Q ) , n ( Q ) (cid:105) ≥ (cid:104) n ( ˜ Q ) , n ( Q ) (cid:105) − M ∑ i = | n ( ˜ Q i ) − n ( ˜ Q i + ) | ≥ ( − C ε Q + ) − (cid:112) C M ε ( Q + ) , ( )where the last inequality comes from Propositions . , . and the fact that the orientation of ˜ Q i and ˜ Q i + werechosen to be compatible. Since Q and Q were assumed to be neighbours, we deduce that M ≤ A and thus,thanks to ( ) and the choice of ε , we have: (cid:104) n ( Q ) , n ( Q ) (cid:105) ≥ ( − C ε Q + ) − (cid:112) C A ε ( Q + ) > Q . The proof of the general case can be obtained with thefollowing argument. Thanks to Proposition . , we know that the orientation of the planes Π ( Q ) and Π ( Q ) iscompatible if and only if the orientation of Π ( ˜ Q ) and Π ( ˜ Q ) , the planes relative to their parent cubes ˜ Q and ˜ Q ,are compatible. Thus, taking the parents of the parents and so on, one can reduce to the case in which one of thecubes is Q . Definition . . For each cube Q ∈ ∆ ( C , ι ) , we let: G ± ( Q ) : = c ( Q ) { u ∈ B ( A diam Q ) : ±(cid:104) π u , n ( Q ) (cid:105) ≥ A − diam Q } = { u ∈ B ( c ( Q ) , A diam Q ) : ±(cid:104) π u − π ( c ( Q )) , n ( Q ) (cid:105) ≥ A − diam Q } .and G ( Q ) = G + ( Q ) ∪ G − ( Q ) . Furthermore, for any Q ∈ M ( C , ι ) we let: G ± ( Q ) : = (cid:91) Q ∈ ∆ ( C , ι ) Q ⊆ Q G ± ( Q ) and G ( Q ) : = (cid:91) Q ∈ ∆ ( C , ι ) Q ⊆ Q G ( Q ) . he support of codimensional measures with flat tangents is intrinsic rectifiable 38 Lemma . . For any cube Q of ∆ ( C , ι ) and any x ∈ G ( Q ) , we have:A − Q ≤ (A) dist ( x , E ( ϑ , γ )) ≤ (B) A diam Q . ( ) Proof.
Since A ≤ k /4, if we let z ∈ E ( ϑ , γ ) be the point realizing the minimum distance of x from E ( ϑ , γ ) , wededuce that: d ( x , z ) = dist ( x , E ( ϑ , γ )) ≤ d ( x , c ( Q )) ≤ A diam Q , ( )where the first inequality above comes from the fact that c ( Q ) ∈ E ( ϑ , γ ) , see Remark . , and the last inequalitycomes from the very definition of G ( Q ) . Note that inequality ( ) proves ( )(B). Furthermore, since 1 + A < k /2the bound ( ) also implies that z ∈ B ( c ( Q ) , k diam Q /2 ) ∩ E ( ϑ , γ ) and thus, thanks to Proposition . we deducethat: dist ( z , c ( Q ) Π ( Q )) ≤ C ε Q k diam Q . ( )Let w be an element of Π ( Q ) satisfying the identity d ( z , c ( Q ) w ) = dist ( z , c ( Q ) Π ( Q )) and note that ( ) impliesthat:dist ( x , E ( ϑ , γ )) = dist ( x , z ) ≥ d ( x , c ( Q ) w ) − d ( c ( Q ) w , z ) ≥ dist ( c ( Q ) − x , Π ( Q )) − dist ( z , c ( Q ) Π ( Q )) ≥|(cid:104) n ( Q ) , π ( c ( Q ) − x ) (cid:105)| − C ε Q k diam Q ≥ A − diam Q − C ε Q k diam Q ≥ A diam Q , ( )where the last inequality comes from the choice of ε and A . Lemma . . For any Q ∈ M ( C , ι ) we have G + ( Q ) ∩ G − ( Q ) = ∅ .Proof. Suppose this is not the case, assume that we can find two cubes Q , Q ∈ ∆ ( C , ι ) such that G + ( Q ) ∩ G − ( Q ) (cid:54) = ∅ and let x be a point of intersection. Thanks to the definition of G ± ( Q ) , we immediately deduce that: B ( c ( Q ) , A diam Q ) ∩ B ( c ( Q ) , A diam Q ) (cid:54) = ∅ . ( )This in particular implies that dist ( Q , Q ) ≤ A ( diam Q + diam Q ) . Therefore, since 2 A ≤ A , Q and Q satisfy the condition (I) of Definition . . Furthermore, Lemma . implies that:diam Q A ≤ dist ( x , E ( ϑ , γ )) ≤ A diam Q , and diam Q A ≤ dist ( x , E ( ϑ , γ )) ≤ A diam Q . ( )Putting together the bounds in ( ), we infer that: ( A ) − ≤ diam Q diam Q ≤ A . ( )Thanks to ( ) and Theorem A. (iv),(vii) we have that: ( A ) − ≤ diam Q diam Q ≤ − j N + / γζ − j N − / γ and ζ − j N − / γ − j + / γ ≤ diam Q diam Q ≤ A ( )Finally, thanks to the bounds in ( ) together with some algebraic computations that we omit, we deduce: | j − j | ≤ log ( ζ − A ) N log 2 ≤ log A ,where the last inequality comes from the choice of A . Since A ≥
2, we infer that | j − j | ≤ A , proving (II)of Definition . . This concludes the proof that Q and Q are neighbors. Since Q and Q are neighbors, ( )together with Proposition . (iii) implies that: d ( c ( Q ) , c ( Q )) ≤ d ( c ( Q ) , x ) + d ( x , c ( Q )) ≤ A ( diam Q + diam Q ) ≤ A ( + e NA ) diam Q < k diam Q /2, he support of codimensional measures with flat tangents is intrinsic rectifiable 39 where the last inequality comes from the choice of k and of A . Since c ( Q ) ∈ E ( ϑ , γ ) ∩ B ( c ( Q ) , k diam Q /2 ) ,thanks to Lemma . (i), we deduce that:dist ( c ( Q ) , c ( Q ) Π ( Q )) ≤ C k ε Q diam Q ≤ C ke NA ε Q diam Q . ( )Furthermore, since Proposition . implies that | n ( Q ) − n ( Q ) | ≤ C ε ( Q + ) , we have that: (cid:104) π ( c ( Q ) − x ) , n ( Q ) (cid:105) = (cid:104) π ( c ( Q ) − x ) , n ( Q ) (cid:105) + (cid:104) π ( c ( Q ) − x ) , n ( Q ) − n ( Q ) (cid:105) + (cid:104) π ( c ( Q ) − c ( Q )) , n ( Q ) (cid:105)≤ − A − diam Q + | π ( c ( Q ) − x ) || n ( Q ) − n ( Q ) | + dist ( c ( Q ) , c ( Q ) Π ( Q )) ≤ − A − diam Q + A diam Q · C ε ( Q + ) + C ke NA ε Q diam Q , ( )where last inequality comes from ( ) and the fact that x ∈ G − ( Q ) . The chain of inequalities in ( ) and thedefinition of A imply: (cid:104) π ( c ( Q ) − x ) , n ( Q ) (cid:105) ≤ ( − A − + A C ε ( Q + ) + C ke NA ε Q ) diam Q ≤
0, ( )where the last inequality comes from the definition of ε and some algebraic computations that we omit. Thiscontradicts the fact that x ∈ G + ( Q ) , proving that the assumption that G ( Q ) + ∩ G − ( Q ) (cid:54) = ∅ was absurd. Proposition . . For any cube Q in M ( C , ι ) we define:I ( Q ) : = (cid:91) Q ∈ ∆ ( C , ι ) Q ⊆ Q B ( c ( Q ) , ( A − ) diam Q ) . Furthermore, for any x ∈ I ( Q ) we let: d ( x ) : = inf Q ∈ ∆ ( C , ι ) Q ⊆ Q dist ( x , Q ) + diam Q . ( ) Then dist ( x , E ( ϑ , γ )) ≤ A − d ( x ) whenever x ∈ I ( Q ) \ G ( Q ) .Proof. Fix some x ∈ I ( Q ) \ G ( Q ) and let Q ⊆ Q be a cube of ∆ ( C , ι ) such that:dist ( x , Q ) + diam Q ≤ d ( x ) /3. ( )Let Q (cid:48) be an ancestor of Q in ∆ ( C , ι ) , possibly Q itself. Since x (cid:54)∈ G ( Q ) , then x (cid:54)∈ G ( Q (cid:48) ) and, thanks to Proposition . , we have: dist ( x , c ( Q (cid:48) ) Π ( Q (cid:48) )) = |(cid:104) π ( c ( Q (cid:48) ) − x ) , n ( Q (cid:48) ) (cid:105)| ≤ A − diam Q (cid:48) , ( )where the last inequality is true provided dist ( x , c ( Q (cid:48) )) < A diam Q (cid:48) . Since x ∈ I ( Q ) , there must exist some˜ Q ∈ ∆ ( C , ι ) such that ˜ Q ⊆ Q and x ∈ B ( c ( ˜ Q ) , ( A − ) diam ˜ Q ) . This implies that:dist ( x , c ( Q )) ≤ d ( x , c ( ˜ Q )) + d ( c ( ˜ Q ) , c ( Q )) ≤ ( A − ) diam ˜ Q + diam Q < A diam Q . ( )Therefore the inequality dist ( x , c ( Q )) < A diam Q is verified and hence ( ) holds for Q (cid:48) = Q . Let Q ⊆ Q ⊆ Q be the smallest cube in ∆ ( C , ι ) for which dist ( x , c ( Q )) < A diam Q holds.Let w ∈ Π ( Q ) be the point for which d ( x , c ( Q ) w ) = dist ( x , c ( Q ) Π ( Q )) , and note that the choice of Q andthe bound ( ) imply: (cid:107) w (cid:107) = dist ( c ( Q ) w , c ( Q )) ≤ d ( c ( Q ) w , x ) + d ( x , c ( Q )) ≤ dist ( x , c ( Q ) Π ( Q )) + A diam Q ≤ A − diam Q + A diam Q ≤ A diam Q < k diam Q /2. ( ) he support of codimensional measures with flat tangents is intrinsic rectifiable 40 Since Q ∈ ∆ ( C , ι ) , thanks to inequality ( ), Lemma . (ii) implies that E ( ϑ , γ ) ∩ B ( c ( Q ) w , 3 kC ε ( Q + ) diam Q ) (cid:54) = ∅ . Therefore, since by definition of Q the bound ( ) holds with Q (cid:48) = Q , we have:dist ( x , E ( ϑ , γ )) ≤ d ( x , c ( Q ) w ) + dist ( c ( Q ) w , E ( ϑ , γ )) = d ( x , c ( Q ) Π ( Q )) + dist ( c ( Q ) w , E ( ϑ , γ )) ≤ A − diam Q + kC ε ( Q + ) diam Q ≤ A − diam Q , ( )where the last inequality comes from the choice of ε .If Q = Q , then ( ) implies that dist ( x , E ( ϑ , γ )) ≤ A − diam Q ≤ A − d ( x ) . Otherwise, let Q be the child of Q , possibly Q itself, that contains Q . Thanks to the choice of Q we have dist ( x , c ( Q )) ≥ A diam Q , and thus:dist ( x , Q ) ≥ d ( x , c ( Q )) − diam Q ≥ ( A − ) diam Q ≥ A − C diam Q ≥ diam Q , ( )where the second last inequality above follows from Proposition A. and the last one from the choice of A .Eventually, thanks to ( ), ( ), ( ) and the fact that Q ⊆ Q , we deduce that:dist ( x , E ( ϑ , γ )) ≤ A − diam Q ≤ A − dist ( x , Q ) ≤ A − dist ( x , Q ) ≤ A − d ( x ) ,concluding the proof of the proposition.We are now ready to prove the main result of this subsection. Theorem . asserts that the compact set C hasbig projections on planes. Theorem . . For any cube Q ∈ ∆ ( C , ι ) such that ( − ε ) φ ( Q ) ≤ φ ( Q ∩ C ) , we have: S Q− ( P Π ( Q ) ( Q ∩ C )) ≥ diam Q Q− A Q− . Proof.
Let Q ∈ ∆ ( C , ι ) be such that ( − ε ) φ ( Q ) ≤ φ ( Q ∩ C ) and define: F ( Q ) : = C ∩ Q ∪ (cid:91) Q ∈ I ( Q ) B ( c ( Q ) , 2 C diam Q ) ,where I ( Q ) is a family of maximal cubes Q ∈ ∆ ( E ( ϑ , γ ) , ι ) such that Q ⊆ Q and Q (cid:54)∈ ∆ ( C , ι ) . As a first step, weestimate the size of the projection of the balls (cid:83) Q ∈ I ( Q ) B ( c ( Q ) , C diam Q ) . Thanks to Proposition . we deducethat: S Q− (cid:18) P Π ( Q ) (cid:18) (cid:91) Q ∈ I ( Q ) B ( c ( Q ) , 2 C diam Q ) (cid:19)(cid:19) ≤ Q− c ( Π ( Q )) C Q− ∑ Q ∈ I ( Q ) diam Q Q− . ( )We now need to estimate the sum in the right-hand side of ( ). Since the cubes in I ( Q ) are disjoint and theyare contained in ∆ ( E ( ϑ , γ ) , ι ) , thanks to Remark A. and the fact that ( − ε ) φ ( Q ) ≤ φ ( Q ∩ C ) we have: C − ∑ Q ∈ I ( Q ) diam Q Q− ≤ ∑ Q ∈ I ( Q ) φ ( Q ) = φ (cid:18) (cid:91) Q ∈ I ( Q ) Q (cid:19) ≤ φ ( Q \ C ) ≤ ε φ ( Q ) ≤ ε C diam Q Q− . ( )Putting together ( ) and ( ), we deduce that: S Q− (cid:18) P Π ( Q ) (cid:18) (cid:91) Q ∈ I ( Q ) B ( c ( Q ) , 2 C diam Q ) (cid:19)(cid:19) ≤ Q− c ( Π ( Q )) C ε C Q− diam Q Q− ≤ c ( Π ( Q )) A Q− diam Q Q− , ( )where the last inequality comes from the choice of ε , see Notation . . he support of codimensional measures with flat tangents is intrinsic rectifiable 41 The next step in the proof of the proposition is to show that: S Q− ( P Π ( Q ) ( F ( Q ))) ≥ c ( Π ( Q )) diam Q Q− A Q− . ( )In order to ease notations in the following we let x = c ( Q ) δ A − diam Q ( n ( Q )) and define: B + : = B ( x , A − diam Q ) and B − : = B ( x , A − diam Q ) δ A − diam Q ( n ( Q ) − ) .Before proceeding further with the proof of ( ), we give a brief outline of what we are going to do, hopingto help the reader in keeping track of what purpose each of our computations serve. As a first step towardsthe proof of ( ), we prove that B ± ⊆ G ± ( Q ) . Note that this implies that each one of the B + and B − are ondifferent sides of the plane Π ( Q ) . Let Q is the element of M ( C , ι ) containing Q and recall that by Lemma . G + ( Q ) and G − ( Q ) are disjoint open sets. Therefore, for any horizontal line parallel to the normal of theplane Π ( Q ) , with starting point in B + and end point in B − , we can find an y in such segment belonging thecomplement of G ( Q ) . Hence, our final step in the proof of ( ) is to show that y ∈ F ( Q ) , thus proving theinclusion P Π ( Q ) ( Q ) ⊆ P Π ( Q ) ( F ( Q )) and in turn our claim. To achieve this we will show that either y belongsto E ( ϑ , γ ) or that there there is a centre c ( ˜ Q ) of a cube ˜ Q in the family I ( Q ) such that d ( y , c ( Q )) ≤ C diam ˜ Q .Let us proceed with the proof of ( ). We will prove that B + ⊆ G + ( Q ) and B − ⊆ G − ( Q ) separately, since thecomputations differ.Let us begin with the proof of the inclusion B + ⊆ G + ( Q ) . For any ∆ ∈ G such that (cid:107) ∆ (cid:107) ≤ A − diam Q , wehave: d ( c ( Q ) , x ∆ ) = (cid:107) δ A − diam Q ( n ( Q )) ∆ (cid:107) ≤ A − diam Q ≤ A diam Q ( )Moreover, from the definition of B + we infer: (cid:104) π ( c ( Q ) − x ∆ ) , n ( Q ) (cid:105) = (cid:104) π ( δ A − diam Q ( n ( Q )) ∆ ) , n ( Q ) (cid:105) = A − diam Q + (cid:104) π ∆ , n ( Q ) (cid:105) ≥ A − diam Q . ( )Inequalities ( ) and ( ) finally imply that B + ⊆ G + ( Q ) .Let us prove that B − ⊆ G − ( Q ) . Similarly to the previous case, for any (cid:107) ∆ (cid:107) ≤ A − diam Q , we have: d ( c ( Q ) , x ∆ δ A − diam Q ( n ( Q ) − )) = (cid:107) δ A − diam Q ( n ( Q )) ∆ δ A − diam Q ( n ( Q ) − ) (cid:107)≤ A − diam Q ≤ A diam Q . ( )Finally, we deduce that: (cid:10) π ( c ( Q ) − x ∆ δ A − diam Q ( n ( Q ) − )) , n ( Q ) (cid:11) = (cid:10) π ( δ A − diam Q ( n ( Q )) ∆ δ A − diam Q ( n ( Q ) − )) , n ( Q ) (cid:11) = − A − diam Q + (cid:104) π ∆ , n ( Q ) (cid:105) ≤ − A − diam Q . ( )Inequalities ( ) and ( ) together finally imply that B − ⊆ G − ( Q ) .Suppose now that Q is the unique cube in M ( C , ι ) containing Q . Thanks to Lemma . we know that thesets G + ( Q ) and G − ( Q ) are open and disconnected. With this in mind, for any a ∈ B + we define the curve γ a : [
0, 1 ] → G as: γ a ( t ) : = a δ A − diam Q t ( n ( Q ) − ) .Note that by definition of B − , we have that γ a ( ) ∈ B − . On the other hand, since γ a ( ) ∈ B + we infer that γ a mustmeet the complement of G ( Q ) at y = γ a ( s ) for some s ∈ (
0, 1 ) . We can estimate the distance of y from c ( Q ) inthe following way: d ( y , c ( Q )) ≤ d ( a δ A − diam Q s ( n ( Q )) , c ( Q )) ≤ d ( a , c ( Q )) + A − diam Q s ≤ d ( x , c ( Q )) + d ( x , a ) + A − diam Q s ≤ A − diam Q + A − diam Q + A − diam Q s ≤ A − diam Q < ( A − ) diam Q , ( ) he support of codimensional measures with flat tangents is intrinsic rectifiable 42 where the first inequality in the last line above comes from the definition of x and the fact that a ∈ B ( x , A − diam Q ) .The above computation, implies that if Q is the cube of M ( C , ι ) containing Q , then y ∈ I ( Q ) . Furthermore, thanksto inequality ( ), the choice of A and Proposition A. we have:dist ( y , E ( ϑ , γ ) \ Q ) ≥ dist ( c ( Q ) , E ( ϑ , γ ) \ Q ) − d ( y , c ( Q )) ≥ − ζ diam Q − A − diam Q ≥ A − diam Q . ( )Thanks to ( ) and ( ) we deduce that:dist ( y , E ( ϑ , γ ) \ Q ) ≥ A − diam Q > d ( y , c ( Q )) ≥ dist ( y , Q ) . ( )Therefore, if z ∈ E ( ϑ , γ ) is the point of minimal distance of y from E ( ϑ , γ ) , ( ) implies that z ∈ Q ∩ E ( ϑ , γ ) .Furthermore, since by assumption y (cid:54)∈ G ( Q ) and by ( ) we have y ∈ I ( Q ) , Proposition . implies: d ( z , y ) = dist ( y , E ( ϑ , γ )) ≤ A − d ( y ) < d ( y ) /10, ( )where the last inequality can be strict only if d ( y ) >
0. In this case, the definition of the function d (see ( ))implies that: d ( z ) ≥ d ( y ) − d ( z , y ) > d ( y )
10 , ( )and thus z cannot be contained in a cube Q ∈ ∆ ( C , ι ) with diam Q ≤ d ( y ) /10, where last inequality is strict onlyif d ( y ) > d ( y ) =
0, the bound ( ) implies that d ( y , z ) = E ( ϑ , γ ) is compact we have y = z ∈ E ( ϑ , γ ) .Therefore: y ∈ E ( ϑ , γ ) ∩ Q ⊆ C ∩ Q ∪ (cid:91) Q ∈ I ( Q ) Q ⊆ C ∩ Q ∪ (cid:91) Q ∈ I ( Q ) B ( c ( Q ) , 2 C diam Q ) = F ( Q ) .If on the other hand d ( y ) >
0, we claim that there is a cube Q ∈ ∆ ( C , ι ) , contained in Q and possibly Q itself,such that:(a) z ∈ Q and for any cube Q ∈ ∆ ( C , ι ) contained in Q we have z (cid:54)∈ Q ,(b) diam Q ≥ d ( y ) /10,(c) there exists a ˜ Q ∈ I ( Q ) , that is a child of Q and for which z ∈ ˜ Q ,Let us verify that such a cube Q exists. Since z ∈ Q , for any cube Q ∈ ∆ ( C , ι ) such that Q ⊆ Q and z ∈ Q wehave: 9 d ( y ) /10 ≤ d ( z ) ≤ dist ( z , Q ) + diam Q = diam Q , ( )where the first inequality above comes from ( ) and the second one from the definition of d . Let Q be thesmallest cube of ∆ ( C , ι ) containing z and note that for any cube Q ⊆ Q belonging to ∆ ( C , ι ) we have that z (cid:54)∈ Q .This proves (a) and (b). In order to prove (c), we note that any ancestor of Q in ∆ ( E ( ϑ , γ ) , ι ) must be containedin ∆ ( C , ι ) . Furthermore, since the condition diam Q ≥ d ( y ) /10 implies that z ∈ E ( ϑ , γ ) \ C , we infer that theremust exist a cube ˜ Q in I ( Q ) for which z ∈ ˜ Q . Such cube must be a child of Q otherwise the maximality of ˜ Q would be contradicted.Let us use (a), (b) and (c) to conclude the proof of the theorem. Items (a), (b) and inequality ( ) imply:dist ( y , Q ) ≤ d ( y , z ) = dist ( y , E ( ϑ , γ )) ≤ d ( y ) /10 ≤ diam Q /9. ( )Therefore, Proposition A. together with ( ) imply: d ( c ( ˜ Q ) , y ) ≤ d ( c ( ˜ Q ) , z ) + d ( z , y ) ≤ diam ˜ Q + diam Q /9 ≤ diam ˜ Q + C diam ˜ Q /9 < C diam ˜ Q . ( ) he support of codimensional measures with flat tangents is intrinsic rectifiable 43 The bound ( ) finally proves that y ∈ F ( Q ) , and more precisely we have shown that for any a ∈ B + the curve γ a meets the set F ( Q ) . In turn, this shows that F ( Q ) has big projections, indeed thanks to Proposition . weinfer: S Q− (cid:16) P Π ( Q ) ( F ( Q )) (cid:17) ≥ S Q− (cid:16) P Π ( Q ) ( B ( x , A − diam Q ) (cid:17) = c ( Π ( Q )) A − ( Q− ) diam Q Q− . ( )This implies thanks to ( ), ( ) and Proposition . that: S Q− ( P Π ( Q ) ( Q ∩ C )) ≥S Q− ( P Π ( Q ) ( F ( Q ))) − S Q− (cid:18) P Π ( Q ) (cid:16) (cid:91) Q ∈ I ( Q ) B ( c ( Q ) , C diam Q ) (cid:17)(cid:19) ≥ c ( Π ( Q )) A Q− diam Q Q− ≥ diam Q Q− A Q− . Remark . . Suppose ψ is a Radon on G supported on the compact set K and satisfying the following assumptions:(i) there exists a δ ∈ N such that δ − ≤ Θ Q− ∗ ( ψ , x ) ≤ Θ Q− ∗ ( ψ , x ) ≤ δ for ψ -almost every x ∈ G ,(ii) lim sup r → d x ,4 kr ( ψ , M ) ≤ − ( Q + ) ε ( δ ) for ψ -almost every x ∈ G .Then Propositions . , . , . , . and Theorem . can proved repeating verbatim their proofs, where inthis case the compact C is substituted with ˜ C , the compact set yielded by Proposition . and instead of using ofProposition . we always use Proposition . . . Construction of the φ -positive intrinsic Lipschitz graph This subsection is devoted to the proof of the main result of Section , Theorem . that we restate here forreader’s convenience: Theorem . . There is an intrinsic Lipschitz graph Γ such that φ ( Γ ) > . The proof of Theorem . follows the following argument. Fixed a cube Q ∈ M ( C , ι ) , we prove that the family B ( Q ) of the maximal sub-cubes of Q having small projection on Π ( Q ) , thanks to Theorem . is small in measure.Therefore, we can find a cube Q (cid:48) ∈ ∆ ( C , ι ) \ B ( Q ) , that is contained in Q , and for which any sub-cube ˜ Q of Q has big projections on Π ( Q ) . This independence on the scales, thanks to Proposition . implies that C ∩ Q is a Π ( Q ) -intrinsic Lipschitz graph. Proposition . . Suppose E is a Borel subset of G and assume there is a plane W ∈ Gr ( Q − ) and an α > such that forany w ∈ E we have: E ⊆ wC W ( α ) . ( ) Then E is contained in an intrinsic Lipschitz graph.Proof.
Thanks to the assumption on E , for any w , w ∈ E we have w − w ∈ C W ( α ) . This implies that for any v ∈ P W ( E ) , there exists a unique w ∈ E such that P W ( w ) = v , otherwise we would have w − w ∈ N ( W ) .Let f : P W ( E ) → N ( V ) be the map associating every w ∈ P W ( E ) to the only element in its preimage P − W ( w ) .With this definition we have that the set gr ( f ) : = { v f ( v ) : v ∈ P − W ( E ) } coincides with E and thus it is an intrinsicLipschitz graph since gr ( f ) ⊆ vC W ( α ) , for any v ∈ E . Proposition . . Defined ε : = min { ε , ( ϑ C C A Q− ) − } . There exists a compact set C ⊆ C and a ι ∈ N such that:(i) φ ( C \ C ) ≤ ε φ ( C ) ,(ii) whenever Q ∈ ∆ ( C , ι ) , we have ( − ε /32 ) φ ( Q ) ≤ φ ( Q ∩ C ) . he support of codimensional measures with flat tangents is intrinsic rectifiable 44 Proof.
First of all, we prove that the set ∆ ( C , ι ) is a φ (cid:120) C Vitali relation. It is immediate to see that the family ∆ ( C , ι ) is a fine covering of C . Furthermore, let E be a Borel set contained in C and suppose A ⊆ ∆ ( C , ι ) is a fine coveringof E . Defined A ∗ : = { Q ∈ A : Q is maximal } , it is immediate too see that: (cid:91) Q ∈A Q = (cid:91) Q ∈A ∗ Q ,and thus the family A ∗ is still a covering of E . The maximality of the elements of A ∗ implies that they are pairwisedisjoint and thus ∆ ( C , ι ) is a φ -Vitali relation in the sense of section . . of [ ]. Therefore, thanks to Theorem . . in [ ], we deduce that: lim Q → x φ ( C ∩ Q ) φ ( Q ) =
1, ( )for φ -almost every x ∈ C . For any j ∈ N , define the functions f j ( x ) : = φ ( C ∩ Q j ( x )) / φ ( Q j ( x )) , where Q j ( x ) isthe unique cube of the generation ∆ j containing x . The identity ( ) implies that lim j → ∞ f j ( x ) = φ -almostevery x ∈ C . Therefore, Severini-Egoroff theorem implies that we can find a compact subset C of C such that φ ( C \ C ) ≤ ε φ ( C ) and f j ( x ) converges uniformly to 1 on C . This proves (i) and (ii) at once. Remark . . If ψ is a Radon measure on G satisfying the hypothesis of Proposition . and ˜ C is the com-pact set yielded by Proposition . with the same argument we employed above we can construct a com-pact subset ˜ C of ˜ C and a ˜ ι ∈ N satisfying (i) and (ii) of Proposition . provided ε is substituted with ε ( δ ) : = min { ε , ( δ C C ( δ ) A Q− ( δ )) − } , ε with ε ( δ ) and ∆ ( C , ι ) with ∆ ψ ( ˜ C ; 2 δ , γ , ˜ ι ) . Theorem . . Let C be the compact set yielded by Proposition . then, there exists a cube Q (cid:48) ∈ ∆ ( C , 2 ι ) such thatQ (cid:48) ∩ C is an intrinsic Lipschitz graph of positive φ -measure.Proof. For any Q ∈ M ( C , 2 ι ) , Theorem . and Proposition . imply that: S Q− ( P Π ( Q ) ( Q ∩ C )) ≥ diam Q Q− A Q− . ( )Therefore, for any Q ∈ M ( C , 2 ι ) we let B ( Q ) be the family of the maximal cubes Q ∈ ∆ ( C , 2 ι ) , contained in Q for which: S Q− (cid:0) P Π ( Q ) ( E ( ϑ , γ ) ∩ Q ) (cid:1) < diam Q Q− C A Q− , ( )and we define B ( Q ) : = (cid:83) Q ∈B ( Q ) Q . The first step of the proof of the theorem is to show that: φ ( C ∩ [ Q \ B ( Q )]) > φ ( Q ) ϑ C C A Q− , for any Q ∈ M ( C , 2 ι ) . ( )Throughout this paragraph we shall assume that Q ∈ M ( C , 2 ι ) is fixed. The maximality of the elements Q of B ( Q ) implies that they are pairwise disjoint and since Q ∩ E ( ϑ , γ ) (cid:54) = ∅ , Remark A. yields: S Q− (cid:0) P Π ( Q ) ( E ( ϑ , γ ) ∩ Q ) (cid:1) < diam Q Q− C A Q− ≤ φ ( Q ) C A Q− . ( )Thanks to Proposition . and Corollary . , we have: φ ( C ∩ [ Q \ B ( Q )]) ≥ S Q− ( C ∩ [ Q \ B ( Q )]) ϑ ≥ S Q− (cid:0) P Π ( Q ) ( C ∩ [ Q \ B ( Q )]) (cid:1) c ( Π ( Q )) ϑ . ( )On the other hand, thanks to ( ) we infer that: S Q− (cid:0) P Π ( Q ) ( C ∩ [ Q \ B ( Q )]) (cid:1) ≥ S Q− ( P Π ( Q ) ( C ∩ Q )) − S Q− (cid:16) P Π ( Q ) (cid:16) E ( ϑ , γ ) ∩ B ( Q ) (cid:17) ≥ diam Q Q− A Q− − ∑ Q ∈B ( Q ) S Q− ( P Π ( Q ) ( E ( ϑ , γ ) ∩ Q ))) . ( ) he support of codimensional measures with flat tangents is intrinsic rectifiable 45 Since Q ∩ E ( ϑ , γ ) (cid:54) = ∅ , Remark A. , ( ), ( ) and the fact that the elements in B ( Q ) are disjoint imply: S Q− (cid:0) P Π ( Q ) ( C ∩ [ Q \ B ( Q )]) (cid:1) ≥ φ ( Q ) C A Q− − C A Q− ∑ Q ∈B ( Q ) φ ( Q )= φ ( Q ) C A Q− − C A Q− φ ( B ( Q )) . ( )Thanks to the bounds ( ) and ( ), we eventually deduce that:2 c ( Π ( Q )) ϑφ ( C ∩ [ Q \ B ( Q )]) ≥ φ ( Q ) C A Q− − C A Q− φ ( B ( Q ))= φ ( Q ) C A Q− + C A Q− φ ( Q \ B ( Q )) , ( )where the last inequality above follows since B ( Q ) ⊆ Q . Inequality ( ) together with the bound from aboveon c ( Π ( Q )) , see Proposition . , immediately imply ( ).Now that ( ) is proved, we want to construct a cube Q (cid:48) ∈ ∆ ( C , 2 ι ) disjoint from (cid:83) Q ∈M ( C ,2 ι ) B ( Q ) andsuch that φ ( C ∩ Q (cid:48) ) >
0. Since the elements of M ( C , 2 ι ) are pairwise disjoint and their union covers C , weinfer that: φ (cid:18) C \ (cid:91) Q ∈M ( C ,2 ι ) B ( Q ) (cid:19) = φ (cid:18) (cid:91) Q ∈M ( C ,2 ι ) C ∩ [ Q \ B ( Q )] (cid:19) = ∑ Q ∈M ( C ,2 ι ) φ ( C ∩ [ Q \ B ( Q )]) ≥ ∑ Q ∈M ( C ,2 ι ) φ ( C ∩ [ Q \ B ( Q )]) − φ (( C \ C ) ∩ Q ) ≥ ∑ Q ∈M ( C ,2 ι ) φ ( Q ) ϑ C C A Q− − φ ( C \ C ) ≥ φ ( C ) ϑ C C A Q− − ε φ ( C ) . ( )where the first inequality of the last line above follows from ( ). Therefore, Proposition . and ( ) imply: φ (cid:18) C \ (cid:91) Q ∈M ( C ,2 ι ) B ( Q ) (cid:19) ≥ − ε ϑ C C A Q− φ ( C ) − ε φ ( C ) ≥ φ ( C ) ϑ C C A Q− . ( )Inequality ( ) implies that there must exist a cube Q (cid:48) ∈ M ( C , 2 ι ) such that φ ( C \ (cid:83) Q ∈B ( Q (cid:48) ) Q ) >
0. Defined G to be the set of maximal cubes in ∆ ( C , 2 ι ) \ B ( Q (cid:48) ) contained in Q (cid:48) , we can find at least a cube Q (cid:48) ∈ G forwhich φ ( C ∩ Q (cid:48) ) >
0. Furthermore, thanks to the maximality of the elements in B ( Q (cid:48) ) we also deduce that anysub-cube of Q (cid:48) is not an element of B ( Q (cid:48) ) .We prove now that C ∩ Q (cid:48) is contained in an intrinsic Lipschitz graph. Indeed, we claim that: x − x ∈ C Π ( Q (cid:48) ) ( α ) for any x , x ∈ C ∩ Q (cid:48) , ( )where α was defined in Proposition . . Fix x , x ∈ C ∩ Q (cid:48) and note that there exists a unique j ∈ N such that: R γ − − jN + ≤ d ( x , x ) ≤ R γ − − ( j − ) N + .For i =
1, 2 we let Q x i be the unique cubes in the j -th layer of cubes ∆ j for which x i ∈ Q x i . Suppose Q (cid:48) ∈ ∆ j andnote that Theorem A. (iv) and the choice of j imply: R γ − − jN + ≤ d ( x , x ) ≤ diam Q (cid:48) ≤ γ − − jN + . ( )The chain of inequalities ( ) implies that j ≤ j and thus by Theorem A. (iii) we infer that Q x i ⊆ Q (cid:48) for i =
1, 2.Furthermore, thanks to Theorem A. (ii), (v), for i =
1, 2 we have: R diam Q x i ≤ R γ − − jN + ≤ d ( x , x ) ≤ R γ − − ( j − ) N + ≤ N + γ − R − jN − ≤ N + ζ − R diam Q x i , ( ) onclusions and discussion of the results 46 Since Q x i ∈ ∆ ( C , 2 ι ) , Lemma . implies that α ( Q x i ) ≤ ε for i =
1, 2. Furthermore, the construction of Q (cid:48) insures that for any cube Q ∈ ∆ ( C , 2 ι ) contained in Q (cid:48) , we have: S Q− (cid:0) P Π ( Q (cid:48) ) ( E ( ϑ , γ ) ∩ Q ) (cid:1) ≥ diam Q Q− C A Q− . ( )This proves that the hypothesis of Proposition . are satisfied and thus x ∈ x C Π ( Q (cid:48) ) ( α ) . Finally C ∩ Q (cid:48) isproved to be contained in an intrinsic Lipschitz graph by means of Proposition . .We observe that is immediate to infer that Theorem . directly implies Theorem . . Finally, [...] Theorem . of Section . Proposition . . Suppose ψ is a Radon measure on G supported on a compact set K satisfying the two following assump-tions:(i) δ − ≤ Θ Q− ∗ ( ψ , x ) ≤ Θ Q− ∗ ( ψ , x ) ≤ δ for ψ -almost every x ∈ G ,(ii) lim sup r → d x ,4 kr ( ψ , M ) ≤ − ( Q + ) ε ( δ ) for ψ -almost every x ∈ G .Then if ˜ C be the compact set yielded by Remark . then, there exists a cube Q (cid:48) ∈ ∆ ψ ( ˜ C ; 2 δ , γ , 2˜ ι ) such that Q (cid:48) ∩ ˜ C is anintrinsic Lipschitz graph of positive ψ -measure.Proof. Thanks to Propositions . , . and Remarks . and . , the verbatim argument we used to prove . can be applied here. In this section we use the main result of Section , i.e. Theorem . , to deduce a number of consequences.First of all we prove Theorem . , that is the main result of this paper, that is a 1-codimensional extension of theMarstrand-Mattila rectifiability criterion to general Carnot groups. Secondly, we provide in Corollary . a rigidityresults for finite perimeter sets in Carnot groups: we are able to show that if locally a Caccioppoli set is not toofar from its natural tangent plane, then its boundary is an intrinsic rectifiable set , see Definition . . Eventually, weuse Theorem . to prove that a 1-codimensional version of Preiss’s rectifiability theorem in the Heisenberg groups H n . . Main results
In this subsection we finally conclude the proof of the main results of this work.
Theorem . . Suppose φ is a Radon measure on G and let ˜ d ( · , · ) be a left invariant, homogeneous distance on G . Assumefurther that for φ -almost all x ∈ G we have:(i) < lim inf r → φ ( ˜ B ( x , r )) r Q− ≤ lim sup r → φ ( ˜ B ( x , r )) r Q− < ∞ , where ˜ B ( x , r ) is the ball relative to the metric ˜ d centred at x of radius r > ,(ii) Tan Q− ( φ , x ) ⊆ M , where M is the family of -codimensional flat measures introduced in Definition . .Then φ is absolutely continuous with respect to S Q− and G can be covered φ -almost all with countably many C G -surfaces.Proof. Since ˜ d is bi-Lipschitz equivalent to d , see for instance Corollary . in [ ], the hypothesis (i) implies:0 < Θ Q− ∗ ( φ , x ) ≤ Θ Q− ∗ ( φ , x ) < ∞ , ( ) onclusions and discussion of the results 47 for φ -almost every x ∈ G . For any ϑ , γ , R ∈ N we define: E ( ϑ , γ , R ) : = { x ∈ B ( R ) : ϑ − r Q− ≤ φ ( B ( x , r )) ≤ ϑ r Q− for any 0 < r < γ } .It is possible to prove, with the same arguments used in the proof of Proposition . , that the E ( ϑ , γ , R ) arecompact sets and: φ ( G \ (cid:91) ϑ , γ , R E ( ϑ , γ , R )) =
0, ( )Thus, if A is an S Q− -null Borel set, Proposition . yields: φ ( A ) ≤ ∑ ϑ , γ , R ∈ N φ ( A ∩ E ( ϑ , γ , R )) ≤ ∑ ϑ , γ , R ∈ N ϑ Q− S Q− ( A ∩ E ( ϑ , γ , R )) = φ is absolutely continuous with respect to S Q− and just to fix notations welet ρ ∈ L ( S Q− ) be such that φ = ρ S Q− .As a second step, we show that G can be covered φ -almost all with countably many intrinsic Lipschitz graphs.Assume by contradiction there are ϑ , γ , R ∈ N for which we can find a subset of E ( ϑ , γ , R ) of positive φ -measurethat we denote, following the notations of Corollary . , with E ( ϑ , γ , R ) u and that has S Q− -null intersection withany intrinsic Lipschitz graph. Thanks to Corollary . . of [ ] it is immediate to see that: ϑ − ≤ Θ Q− ∗ ( φ (cid:120) E ( ϑ , γ , R ) u , x ) ≤ Θ Q− ∗ ( φ (cid:120) E ( ϑ , γ , R ) u , x ) ≤ ϑ ,for φ -almost every x ∈ E ( ϑ , γ , R ) u . Furthermore, thanks to Proposition . , we infer that Tan Q− ( φ (cid:120) E ( ϑ , γ , R ) u , x ) ⊆ M for φ -almost every x ∈ E ( ϑ , γ , R ) u . And since its hypothesis are satisfied, Theorem . implies that there existsan intrinsic Lipschitz graph Γ such that φ ( Γ ∩ E ( ϑ , γ , R ) u ) >
0. However, this is not possible since Proposition . would yield: 0 < φ ( Γ ∩ E ( ϑ , γ , R ) u ) ≤ ϑ Q− S Q− ( E ( ϑ , γ , R ) u ∩ Γ ) ,and this contradicts the fact that E ( ϑ , γ , R ) intersects in a S Q− -null set every intrinsic Lipschitz graph.This concludes the first part of the proof of the proposition. Up to this point we have shown that the sets E ( ϑ , γ , R ) for any choice of ϑ , γ , R are covered S Q− -almost all by countably many intrinsic Lipschitz graphs.Furthermore, since φ (cid:28) S Q− , thanks to ( ) we conclude that φ -almost all G can be covered by countably manyintrinsic Lipschitz graphs.In this paragraph, we assume that ϑ , γ , R ∈ N are fixed. Thanks to Proposition . we infer that S Q− (cid:120) E ( ϑ , γ , R ) is mutually absolutely continuous with respect to φ (cid:120) E ( ϑ , γ ) and in particular: ϑ − ≤ ρ ( x ) ≤ ϑ Q− for S Q− -almost every x ∈ E ( ϑ , γ , R ) .Let { γ i } i ∈ N be the sequence of intrinsic Lipschitz functions γ i : W i → N ( W i ) for which φ ( E ( ϑ , γ , R ) \ (cid:83) i ∈ N gr ( γ i )) = E i : = epi ( γ i ) be the epigraph of the function γ i , that was defined in ( ). Thanks to Proposition . wededuce that for φ -almost every x ∈ E ( ϑ , γ , R ) ∩ gr ( γ i ) we have:Tan Q− ( φ (cid:120) E ( ϑ , γ , R ) ∩ gr ( γ i ) , x ) = ρ ( x ) Tan Q− ( S Q− (cid:120) gr ( γ i ) , x ) = ρ ( x ) d ( x ) Tan Q− ( | ∂ E i | G , x ) ,where d is the density yielded by Remark B. and | ∂ E i | G is the perimeter measure of E i . Finally, Proposition B. implies that: Tan Q− ( φ (cid:120) E ( ϑ , γ , R ) ∩ gr ( γ i ) , x ) ⊆ ρ ( x ) d ( x ) { λ S Q− (cid:120) V i ( x ) : λ ∈ [ L − G , l − G ] } , ( )where V i ( x ) ∈ Gr ( Q − ) is the plane orthogonal to n E i ( x ) , the generalized inward normal introduced in DefinitionB. .We now prove that ( ) implies that for S Q− -almost every x ∈ gr ( γ i ) and every α > r → S Q− ( gr ( γ i ) ∩ B ( x , r ) \ xX V x ( α )) r Q− =
0, ( ) onclusions and discussion of the results 48 where X V x ( α ) : = { w ∈ G : dist ( w , V x ) ≤ α (cid:107) w (cid:107)} . Thanks to ( ), for S Q− -almost every x ∈ gr ( γ i ) and anysequence r i →
0, there exists a λ > T x , r S Q− (cid:120) E ( ϑ , γ , R ) r Q− i (cid:42) λ S Q− (cid:120) V x . ( )The convergence in ( ) implies that:lim i → ∞ S Q− (cid:120) gr ( γ i )( B ( x , r i ) \ xX V x ( α )) r Q− i = lim i → ∞ T x , r i ( S Q− (cid:120) gr ( γ i )) (cid:0) B (
0, 1 ) \ X V x ( α ) (cid:1) r Q− i = λ ( S Q− (cid:120) V x ) (cid:0) B (
0, 1 ) \ X V x ( α ) (cid:1) =
0, ( )where the second last identity above comes from the fact that S Q− (cid:0) V x ∩ ∂ B (
0, 1 ) \ X V x ( α ) (cid:1) = . of [ ].Proposition B. and ( ) together imply that each one of the intrinsic Lipschitz graphs gr ( γ i ) can be covered S Q− -almost all with C G -surfaces. In particular this shows that for any ϑ , γ , R the set E ( ϑ , γ , R ) can be covered S Q− -almost all, and thus φ -almost all, by countably many C G -surfaces. This, thanks to the arbitrariness of ϑ , γ , R ∈ N and ( ) concludes the proof of the theorem.The following theorem trades off the regularity of tangents, that are assumed only to be close enough to flatmeasures, with a strengthened hypothesis on the ( Q − ) -density of φ . Theorem . . Suppose φ is a Radon measure on G and let ˜ d ( · , · ) be a left invariant, homogeneous distance on G . If thereexists a δ ∈ N such that: δ − < lim inf r → φ ( ˜ B ( x , r )) r Q− ≤ lim sup r → φ ( ˜ B ( x , r )) r Q− < δ for φ -almost every x ∈ G , ( ) where ˜ B ( x , r ) is the ball relative to the metric ˜ d centred at x of radius r > , then we can find an ε ( δ , ˜ d ) > such that, if: lim sup r → d x , r ( φ , M ) ≤ ε ( δ , ˜ d ) for φ -almost every x ∈ G , then φ is absolutely continuous with respect to S Q− and G can be covered φ -almost all with countably many intrinsicLipschitz surfaces.Proof. The first step in the proof is to note that since the metric ˜ d and d are bi-Lipschitz equivalent, there exists aconstant c >
1, that we can assume without loss of generality to be a natural number, such that: ( c δ ) − < lim inf r → φ ( B ( x , r )) r Q− ≤ lim sup r → φ ( B ( x , r )) r Q− < c δ for φ -almost every x ∈ G .If we let ε ( δ , ˜ d ) : = − ( Q + ) ε ( c δ ) then the verbatim repetition of the first part of the argument used to proveTheorem . , where instead of Theorem . we make use of Proposition . , proves the claim.An immediate consequence of Theorem . is the following: Corollary . . Let ϑ G : = max { l − G , L G } where l G and L G are the constants yielded by Theorem B. and suppose Ω ⊆ G isa finite perimeter set such that: lim sup r → d x , r ( | ∂ Ω | G , M ) ≤ ε ( ϑ G , d ) for | ∂ Ω | G -almost every x ∈ G , where ε ( ϑ G , d ) is the constant yielded by Theorem . and d is the metric introduced in Definition . . Then G can be covered | ∂ Ω | G -almost all with countably many intrinsic Lipschitz surfaces.Proof. Theorem B. implies that l G < Θ Q− ∗ ( | ∂ Ω | G , x ) ≤ Θ Q− ∗ ( | ∂ Ω | G , x ) < L G for φ -almost every x ∈ G . Theo-rem . directly imply the statement. onclusions and discussion of the results 49 As mentioned at the beginning of this section, the main application of Theorem . is an extension Preiss’srectifiability theorem to 1-codimensional measures in H n . Theorem . . Suppose d is the Koranyi metric in H n and φ is a Radon measure on H n such that: < Θ n + ( φ , x ) : = lim r → φ ( B ( x , r )) r n + < ∞ , for φ -almost every x ∈ H n . ( ) Then φ is absolutely continuous with respect to S Q− and H n can be covered φ -almost all with C H n -surfaces.Proof. Thanks to Theorem . of [ ], the almost sure existence of the limit in ( ) implies that Tan ( φ , x ) ⊆ M , for φ -almost every x ∈ G . Thanks to Theorem . , this proves the claim. . Discussion of the results
Theorem . shows that C G -rectifiability in Carnot groups can be characterized by the same conditions on thedensities and on the tangents as the Lipschitz rectifiability in Euclidean spaces. With this in mind we introducethe following two definitions: Definition . ( P -rectifiable measures) . Suppose that φ is a Radon measure on some Carnot group G endowedwith a left invariant and homogeneous metric d and let m be a positive integer. We say that φ is P m -rectifiable if:(i) 0 < Θ m ∗ ( φ , x ) ≤ Θ m , ∗ ( φ , x ) < ∞ for φ -almost every x ∈ G ,(ii) Tan m ( φ , x ) ⊆ { λµ x : λ > } , for φ -almost every x ∈ G where µ x is some Radon measure on G . Remark . . It was already remarked by P. Mattila in the last paragraph of [ ] that Definition . may be consideredthe correct notion of rectifiability in H . Remark . . Instead of condition (ii) of Definition . , we can assume without loss of generality that µ x = H m (cid:120) V ( x ) for some V ( x ) ∈ Gr ( m ) , where Gr ( m ) is the family of m -dimensional homogeneous subgroups of G introduced inDefinition . . This is due to Theorem . of [ ] and Theorem . of [ ]: the former result tells us that µ x mustbe the Haar measure of a closed, dilation-invariant subgroup of G and the latter that such subgroup is actually aLie subgroup. Definition . ( P ∗ -rectifiable measures) . Suppose that φ is a Radon measure on some Carnot group G endowedwith a left invariant and homogeneous metric d and let m be a positive integer. We say that φ is P ∗ m -rectifiable if:(i) 0 < Θ m ∗ ( φ , x ) ≤ Θ m , ∗ ( φ , x ) < ∞ for φ -almost every x ∈ G ,(ii) Tan m ( φ , x ) ⊆ M ( m ) , for φ -almost every x ∈ G .The difference between Definitions . and . is that in the former the tangent to φ is the same plane at everyscale, while in the latter the tangents are planes that may vary at different scales. Although there is no a priorireason for which these definition should be equivalent in general, we see that our main result Theorem . , maybe rewritten as: Theorem . . Suppose φ is a Radon measure on G . The following are equivalent:(i) φ is P Q− -rectifiable,(ii) φ is P ∗Q− -rectifiable,(iii) φ is absolutely continuous with respect to H Q− and G can be covered φ -almost all with countably many C G -surfaces. The notion of P -rectifiable measures is also relevant since in different contests it appears to imply the rightnotion of rectifiability. This is summarized in the following theorem, that is an immediate consequence of theEucidean Marstrand-Mattila rectifiability criterion and Theorem . : onstruction of dyadic cubes 50 Theorem . . The following two statements hold:(i) A Radon measure φ on R n is P m -rectifiable if and only if it is Euclidean m-rectifiable;(ii) A Radon measure φ on G is P Q− -rectifiable if and only if it is C G -rectifiable. In [ ] P. Mattila, F. Serra Cassano and R. Serapioni proved in Theorems . and . that whenever a goodnotion of regular surface is available in the Heisenberg group, provided the tangents are selected carefully, seeDefinition . of [ ], a P m -rectifiable measure is also rectifiable with respect of the family of regular surfaces ofthe right dimension. However, because of the algebraic structure of the group H n , there is not an a priori (known)good notion of regular surface that includes the vertical line V : = { (
0, 0, t ) : t ∈ R } . For this reason the uniformmeasure S (cid:120) V is considered to be non-rectifiable from the standpoint of [ ]. Up to this point Haar measures of notcomplemented homogeneous subgroups (like the vertical line V in H ) were considered non-rectifiable and thuspreventing a possible extension of Preiss’s Theorem to low dimension even in H . This was already remarked in[ ]. On the other hand, we have: Theorem . . Let φ be a Radon measure on H such that for φ -almost every x ∈ H , we have: < Θ ( φ , x ) : = lim r → φ ( B ( x , r )) r < ∞ , where B ( x , r ) are the metric balls with respect to the Koranyi metric. Then φ is P -rectifiable.Proof. This follows from Proposition . of [ ] and Theorem . of [ ].As remarked in the previous paragraph, to our knowledge in literature there is not a good candidate of rectifia-bility in Carnot groups for which the density problem may have a positive answer. On the other hand, Theorems . , . and . encourage us to state the density problem in Carnot groups in the following way: Dentsity Problem.
Suppose φ is a Radon measure on the Carnot group G . There exists a left invariant distance d on G such that the following are equivalent:(i) there exists an α > such that for φ -almost every x ∈ G we have < Θ α ( φ , x ) : = lim r → φ ( B ( x , r )) / r α < ∞ ,(ii) α ∈ {
0, . . . , Q} and φ is P α - rectifiable. Neither one of the implications of the formulation of the density problem is of easy solution. In [ ] the authorof the present work in collaboration with G. Antonelli prove the implication (ii) ⇒ (i) of the Density Problem whenthe tangents measures to φ are supported on complemented subgroups.Furthermore, as already observed in [ ], if d is a left invariant distance coming from a polynomial norm on G with the same argument used in [ ] and later on in [ ], it is possible to show that if (i) in the Density Problemholds, then α ∈ N . In R n this implies thanks to Theorem . of [ ], that there is an open and dense set Ω in thespace of norms (with the distance induced by the Hausdorff distance of the unit balls) for which for any (cid:107)·(cid:107) ∈ Ω ,Marstrand’s theorem holds. a construction of dyadic cubes Throughout this section we assume φ to be a fixed Radon measure on the Carnot group G , supported on acompact set K , and such that:0 < lim inf r → φ ( B ( x , r )) r m ≤ lim sup r → φ ( B ( x , r )) r m < ∞ , for φ -almost every x ∈ G .In the following, we construct a family of dyadic cubes for the measure φ . There are many constructions in literatureof such objects both in the Euclidean and in (rather general) metric spaces. Unfortunately, in the context of generalmetric spaces, dyadic cubes are only available for AD-regular measures, see for instance [ ]. However, since themeasure φ is not so regular, we need to provide a generalisation. More precisely, for any fixed ξ , τ ∈ N , weconstruct a family of partitions { ∆ φ j ( ξ , τ ) } j ∈ N of K such that: onstruction of dyadic cubes 51 (i) diam Q ≤ − j / τ and φ ( Q ) ≤ − jm / τ for any Q ∈ ∆ φ j ( ξ , τ ) ,(ii) 2 − j (cid:46) diam Q and 2 − jm (cid:46) ξ φ ( Q ) for those cubes Q ∈ ∆ φ j ( ξ , τ ) , for which Q ∩ E φ ( ξ , τ ) (cid:54) = ∅ .The collection ∆ φ ( ξ , τ ) : = { ∆ φ j ( ξ , τ ) : j ∈ N } is said to be a family of dyadic cubes for φ relative to the parameters ξ , τ . The strategy we employ to construct such partitions, is to adapt the construction given by G. David forAD-regular measures that can be found in Appendix I of [ ] to this less regular case.In Subsection A. we briefly recall the definition of the Hausdorff distance of compact sets and prove sometechnical facts used in Section and Subsection A. . In Subsection A. , we state Theorem A. that is the mainresult of this appendix and prove some of its consequences. Finally, in Subsection A. , we prove Theorem A. . a . Hausdorff distance of sets
In this subsection we recall some of the properties of the Hausdorff distance of sets and the Hausdorff conver-gence of compact sets.
Definition A. (Hausdorff distance) . For any couple of sets in A , B ⊆ G , we define their Hausdorff distance as: d H ( A , B ) : = max (cid:110) sup x ∈ A dist ( x , B ) , sup y ∈ B dist ( A , y ) (cid:111) .Furthermore, for any compact set κ in G , we define F ( κ ) : = { A ⊆ κ : A is compact } . Remark A. . Recall that for any couple of sets A , B ⊆ G we have d H ( A , B ) = d H ( A , B ) .The following is a well known property of the Hausdorff metric. Theorem A. . For any compact set κ of G , we have that ( F ( κ ) , d H ) is a compact metric space.Proof. See for instance Theorem VI of § in [ ].The next proposition will be used in the proof of Lemma A. and Proposition A. . It establishes the stabilityof the Hausdorff convergence under finite unions. Proposition A. . Let M ∈ N and assume { D i } i = M and { D (cid:48) i } i = M are finite families of subsets of G . Then:d H (cid:16) M (cid:91) i = D i , M (cid:91) i = D (cid:48) i (cid:17) ≤ max i = M d H ( D i , D (cid:48) i ) . ( ) Let κ be a compact set in G . Suppose that for any j =
1, . . . ,
M, the sequences { A ji } i ∈ N ⊆ F ( κ ) converge in the Hausdorffmetric to some A j ∈ F ( κ ) . Then: lim i → ∞ d H (cid:18) M (cid:91) j = A ji , M (cid:91) j = A j (cid:19) =
0. ( ) Finally, if the set B is contained in (cid:83) Mj = A ji for any i ∈ N , then B ⊆ (cid:83) Mj = A j .Proof. First of all, note that since identity ( ) is an immediate consequence of ( ), we just need to prove theformer. Thanks to the definition of the distance d H , we have: d H (cid:16) M (cid:91) i = D i , M (cid:91) i = D (cid:48) i (cid:17) = max (cid:26) sup x ∈ (cid:83) Mi = D i dist (cid:18) x , M (cid:91) i = D (cid:48) i (cid:19) , sup y ∈ (cid:83) Mi = D (cid:48) i dist (cid:18) y , M (cid:91) i = D i (cid:19)(cid:27) ≤ max (cid:26) max i = M sup x ∈ D i dist (cid:18) x , M (cid:91) i = D (cid:48) i (cid:19) , max i = M sup x ∈ D (cid:48) i dist (cid:18) x , M (cid:91) i = D (cid:48) i (cid:19)(cid:27) ≤ max (cid:26) max i = M sup x ∈ D i dist ( x , D (cid:48) i ) , max i = M sup x ∈ D (cid:48) i dist ( x , D (cid:48) i ) (cid:27) = max i = M max (cid:26) sup x ∈ D i dist ( x , D (cid:48) i ) , sup x ∈ D (cid:48) i dist ( x , D (cid:48) i ) (cid:27) = max i = M dist ( D i , D (cid:48) i ) . onstruction of dyadic cubes 52 This concludes the proof of ( ) and thus of ( ). The proof of the last part of the proposition follows by thepidgeonhole principle. Indeed, for any b ∈ B there exists a j ( b ) ∈ {
1, . . . , M } such that b ∈ A i k j ( b ) for any k ∈ N . Inparticular, we conclude that dist ( b , A j ( b ) ) = A j ( b ) is closed, we infer that b ∈ A j ( b ) . a . Dyadic cubes
In this subsection we state the main theorem of this appendix, Theorem A. and prove, assuming its validity,a couple of consequences that will be used in Section . Throughout the rest of Appendix A, we will alwaysassume that ξ and τ are two fixed natural numbers such that φ ( E φ ( ξ , τ )) >
0, where the set E φ ( ξ , τ ) was definedin Proposition . . Definition A. . For any subset A of G and any δ >
0, we let: ∂ ( A , δ ) : = { u ∈ A : dist ( u , K \ A ) ≤ δ } ∪ { u ∈ K \ A : dist ( u , A ) ≤ δ } ,where we recall that K is the compact set supporting the measure φ .Throughout the rest of this subsection, we simplify the expression of the constant introduced in Notation . to: N : = N ( ξ ) , ζ : = ζ ( ξ ) , C : = C ( ξ ) , C : = C ( ξ ) , C : = C ( ξ ) . Theorem A. . For any j ∈ N there are disjoint partitions ∆ φ j ( ξ , τ ) of K having the following properties:(i) if j ≤ j (cid:48) , Q ∈ ∆ φ j ( ξ , τ ) and Q (cid:48) ∈ ∆ φ j (cid:48) ( ξ , τ ) , then either Q contains Q (cid:48) or Q ∩ Q (cid:48) = ∅ ,(ii) if Q ∈ ∆ φ j ( ξ , τ ) we have diam ( Q ) ≤ − Nj + / τ ,(iii) if Q ∈ ∆ φ j ( ξ , τ ) and Q ∩ E φ ( ξ , τ ) (cid:54) = ∅ , then C − (cid:0) − Nj / τ (cid:1) m ≤ φ ( Q ) ≤ C (cid:0) − Nj / τ (cid:1) m ,(iv) if Q ∈ ∆ φ j ( ξ , τ ) , we have φ (cid:0) ∂ ( Q , ζ − Nj / τ ) (cid:1) ≤ C ζ (cid:0) − Nj / τ ) m ,(v) if Q ∈ ∆ φ j ( ξ , τ ) and Q ∩ E φ ( ξ , τ ) (cid:54) = ∅ , there exists a c ( Q ) ∈ K such that B ( c ( Q ) , ζ − Nj − / τ ) ⊆ Q j ( x ) .We will denote with the symbol ∆ φ ( ξ , τ ) the family of all dyadic cubes, i.e. ∆ φ ( ξ , τ ) = { Q ∈ ∆ φ ( ξ , τ ) : j ∈ N } .Remark A. . Part (iii) of Theorem A. can be rephrased in the following useful way. Recalling that C ( ξ ) = C ( ζ − ) m and putting together Theorem A. (ii), (iii) and (v) we infer that:(iii)’ if Q ∩ E φ ( ξ , τ ) (cid:54) = ∅ then C − diam Q m ≤ φ ( Q ) ≤ C diam Q m .The families of cubes yielded by Theorem A. may have the annoying property that fixed a cube Q ∈ ∆ φ j ( ξ , τ ) ,the only sub-cube of Q in the layer ∆ φ j + ( ξ , τ ) contained in Q , is just Q itself. The following proposition shows thatthis is not much of a problem for the cubes intersecting E φ ( ξ , τ ) . Proposition A. . Suppose that Q ∗ ∈ ∆ φ j ( ξ , τ ) is parent of a cube Q ∈ ∆ φ j + k ( ξ , τ ) such that Q ∩ E φ ( ξ , τ ) (cid:54) = ∅ , i.e. Q ∗ isthe smallest cube in ∆ φ ( ξ , τ ) strictly containing Q. Then k < (cid:4) C / Nm (cid:5) + and: diam Q ∗ diam Q ≤ C . Proof.
Suppose ˜ Q is the ancestor of the cube Q contained in the layer ∆ φ j (cid:48) ( ξ , τ ) for some j (cid:48) for which j (cid:48) − j ≥ (cid:4) C / Nm (cid:5) +
1. Then ˜ Q ∩ E φ ( ξ , τ ) (cid:54) = ∅ and thanks to Theorem A. (i) and (iii), we infer: φ ( ˜ Q \ Q ) = φ ( ˜ Q ) − φ ( Q ) ≥ C − (cid:18) − jN τ (cid:19) m − C (cid:18) − j (cid:48) N τ (cid:19) m = C − (cid:18) − jN τ (cid:19) m ( − C − ( j (cid:48) − j ) Nm ) >
0, ( ) onstruction of dyadic cubes 53 where the last inequality above comes from the choice of j (cid:48) − j . It is immediate to see that inequality ( ) impliesthat Q is strictly contained in ˜ Q . Therefore, the parent cube of Q must be contained in some ∆ φ j + k ( ξ , τ ) with0 ≤ k < (cid:98) C / mN (cid:99) +
1. Hence, thanks to Theorem A. (v) we infer that:diam Q ∗ ≤ − Nj + / τ = Nk + ζ − · ζ − N ( j + k ) − / τ ≤ Nk + ζ − diam Q ≤ C m + N ζ − diam Q = C diam Q .The following result tells us that item (v) of Theorem A. in some cases can be strengthened to assuming thatthe centre of the cube c ( Q ) is contained in E φ ( ξ , τ ) . Proposition A. . Assume that µ ∈ N is such that µ ≥ ξ . Then, for any cube Q ∈ ∆ φ ( E φξ , τ ( µ , ν ) ; ξ , τ , ν ) we can find a c ( Q ) ∈ E φ ( ξ , τ ) ∩ Q such that: B ( c ( Q ) , ζ diam Q /64 ) ∩ K ⊆ Q . Remark A. . Recall that the set E φξ , τ ( µ , ν ) was introduced in Proposition . and ∆ φ ( κ ; ξ , τ , ν ) in Notation . . Proof.
In order to prove the proposition it suffices to show that: E φ ( ξ , τ ) ∩ Q \ ∂ ( Q , ζ diam Q /32 ) (cid:54) = ∅ . ( )In order to fix ideas, we let Q ∈ ∆ φ j ( ξ , τ ) for some j ≥ ν and note that since Q ∩ E φ ( ξ , τ ) (cid:54) = ∅ , thanks to TheoremA. (ii), (iii) and (iv), we have: φ ( E φ ( ξ , τ ) ∩ Q \ ∂ ( Q , ζ diam Q /32 )) ≥ φ ( E φ ( ξ , τ ) ∩ Q ) − φ ( ∂ ( Q , ζ diam Q /32 )) ≥ φ ( E φ ( ξ , τ ) ∩ Q ) − φ ( ∂ ( Q , ζ − jN / τ )) ≥ φ ( E φ ( ξ , τ ) ∩ Q ) − C ζ ( − jN / τ ) Q− = φ ( Q ) − φ ( Q \ E φ ( ξ , τ )) − C ζ ( − jN / τ ) Q− ≥ φ ( Q ) − φ ( Q \ E φ ( ξ , τ )) − C ζφ ( Q ) . ( )Since Q ∈ ∆ φ ( E φξ , τ ( µ , ν ) ; ξ , τ , ν ) we have diam Q ≤ − N ν + / τ and there exists a w ∈ E φξ , τ ( µ , ν ) ∩ Q . Therefore, thedefinition of E φξ , τ ( µ , ν ) and Theorem A. (vi), imply: φ ( Q \ E φ ( ξ , τ )) ≤ φ ( B ( w , 2 − jN + / τ ) \ E φ ( ξ , τ )) ≤ µ − φ ( B ( w , 2 − jN + / τ )) ≤ µ − ξ ( − jN + / τ ) Q− ≤ µ − ξ C φ ( Q ) . ( )Putting together ( ) and ( ), we conclude that: φ ( E φ ( ξ , τ ) ∩ Q \ ∂ ( Q , ζ diam Q )) ≥ ( − µ − ξ − C ζ ) φ ( Q ) ≥ φ ( Q ) /4.where the last inequality follows from the fact that C ζ = Q ξ · − Q ξ − ≤ µ − ξ ≤ ) and in turn the proposition. a . Construction of the dyadic cubes
For the rest of the section, we fix ξ , τ ∈ N in such a way that φ ( E φ ( ξ , τ )) > η : = ξ − − Q . Definition A. . For any j ∈ N , we let:(i) ג ( j ) be a maximal set of points in E φ ( ξ , τ ) such that d ( x , x (cid:48) ) ≥ − j / τ for any couple of distinct x , x (cid:48) ∈ ג ( j ) ,(ii) Ξ ( j ) be a maximal set of points in K , containing ג ( j ) and such that d ( x , x (cid:48) ) ≥ − j / τ for any couple ofdistinct x , x (cid:48) ∈ Ξ ( j ) , onstruction of dyadic cubes 54 Lemma A. . Let x ∈ K be a point such that φ ( B ( x , r )) ≤ ξ r m for any < r < τ . Then for any j ∈ N , there is a ballB j ( x ) centred at x of radius r ∈ ( − j / τ , ( + η ) − j / τ ) such that: φ ( ∂ ( B j ( x ) , η − j / τ )) ≤ m ξη ( − j / τ ) m = : C η ( − j / τ ) m . Proof.
For any p ∈ {
1, . . . , (cid:98) η (cid:99) − } , let: C ( x , p , j ) : = B (cid:16) x , (cid:16) + η p (cid:98) η (cid:99) + η (cid:17) − j / τ (cid:17) \ B (cid:16) x , (cid:16) + η p (cid:98) η (cid:99) − η (cid:17) − j / τ (cid:17) .The coronas C ( x , p , j ) are pairwise disjoint since η was chosen small enough, and their union is contained in theball B ( x , ( + η ) − j / τ ) . This implies, thanks to the choice of x , that there exists a p ∈ {
1, . . . , (cid:98) η (cid:99) − } such that: φ ( C ( x , p , j )) ≤ ξ ( + η ) m (cid:98) η (cid:99) ( − j / τ ) m ,and this concludes the proof of the lemma. Lemma A. . For any x ∈ Ξ ( j ) , we have Card (cid:0) Ξ ( j ) ∩ B ( x , 2 − j + / τ ) (cid:1) ≤ Q .Proof. The balls { B ( y , 2 − j − / τ ) : y ∈ Ξ ( j ) ∩ B ( x , 2 − j + / τ ) } are disjoint and contained in B ( x , 2 − j + / τ ) . Thisimplies that: ( − j − / τ ) Q Card ( Ξ ( j ) ∩ B ( x , 2 − j + / τ )) ≤ S Q ( B ( x , 2 − j + / τ )) = ( − j + / τ ) Q .We let ≺ be a total order on Ξ ( j ) such that x ≺ y whenever x ∈ ג ( j ) and y ∈ Ξ ( j ) \ ג ( j ) . For every x ∈ Ξ ( j ) we define: B (cid:48) j ( x ) : = B j ( x ) ∩ (cid:34) (cid:91) y ≺ x B j ( y ) (cid:35) c .Therefore, thanks to the definitions of the order ≺ , for any x ∈ ג ( j ) and any w ∈ Ξ ( j ) \ ג ( j ) , we have: B j ( x ) ∩ B (cid:48) j ( w ) = ∅ . ( )The sets B (cid:48) j are by construction a disjoint cover of K , however their measure could be very small. To avoid thisproblem, we will glue together some of the B (cid:48) j s. Let ג ( j ) be the subset of those x ∈ ג ( j ) such that: φ ( B (cid:48) j ( x )) ≥ −Q ξ − ( − j / τ ) m . Lemma A. . For any x ∈ ג ( j ) we have ג ( j ) ∩ B ( x , 2 − j + / τ ) (cid:54) = ∅ .Proof. Since the sets B (cid:48) j are a disjoint cover of K , for any x ∈ ג ( j ) we have: K ∩ B j ( x ) ⊆ (cid:91) y ∈ Ξ ( j ) B j ( x ) ∩ B (cid:48) j ( y ) (cid:54) = ∅ K ∩ B (cid:48) j ( y ) = (cid:91) y ∈ ג ( j ) B j ( x ) ∩ B (cid:48) j ( y ) (cid:54) = ∅ K ∩ B (cid:48) j ( y ) , ( )where the last identity above comes from ( ). Furthermore, thanks to the inclusion ( ), we have: ∑ y ∈ ג ( j ) B j ( x ) ∩ B (cid:48) j ( y ) (cid:54) = ∅ φ ( B (cid:48) j ( y )) ≥ φ ( B j ( x )) ≥ ξ (cid:18) − j τ (cid:19) m , ( )where the last inequality comes from the fact that x ∈ E φ ( ξ , τ ) . Finally, Lemma A. together with ( ) and thefact that if B j ( x ) ∩ B (cid:48) j ( y ) (cid:54) = ∅ then y ∈ B ( x , 2 − j + / τ ) , imply that: ξ − ( − j / τ ) ≤ φ ( B j ( x )) ≤ ∑ y ∈ ג ( j ) B j ( x ) ∩ B (cid:48) j ( y ) (cid:54) = ∅ φ ( B (cid:48) j ( y )) ≤ Q max y ∈ ג ( j ) B j ( x ) ∩ B (cid:48) j ( y ) (cid:54) = ∅ φ ( B (cid:48) j ( y )) .This concludes the proof. onstruction of dyadic cubes 55 Let h be a map assigning each y ∈ ג ( j ) \ ג ( j ) to an element of ג ( j ) ∩ B ( y , 2 − j + ) , that exists thanks to LemmaA. and define: D j ( y ) : = K ∩ (cid:20) B (cid:48) j ( y ) ∪ (cid:83) x ∈ h − ( y ) B (cid:48) j ( x ) (cid:21) if y ∈ ג ( j ) , K ∩ B (cid:48) j ( y ) if y ∈ Ξ ( j ) \ ג ( j ) .Finally we let Ξ ( j ) : = ג ( j ) ∪ Ξ ( j ) \ ג ( j ) . Proposition A. . For any j ∈ N the sets { D j ( y ) : y ∈ Ξ ( j ) } are a disjoint cover of K and:(i) diam ( D j ( y )) ≤ − j + / τ and D j ( y ) ⊆ B ( y , 2 − j + / τ ) ,(ii) for any j ∈ N we have E φ ( ξ , τ ) ⊆ (cid:83) y ∈ ג ( j ) D j ( y ) ,(iii) φ ( D j ( y )) ≥ −Q ξ − ( − j / τ ) m provided y ∈ ג ( j ) and j ≥ ,(iv) φ ( ∂ ( D j ( y ) , 2 − j η / τ )) ≤ C η ( − j / τ ) m where C : = · Q C ,(v) for any y ∈ ג ( j ) there exists a c ∈ D j ( y ) such that B ( c , η − j − / τ ) ⊆ D j ( y ) .Proof. The fact that { D j ( y ) : y ∈ Ξ ( j ) } is a partition of K , follows from the fact that the B (cid:48) j s are. In order to prove(ii), it is sufficient to note that ( ) implies that the union of the sets { B (cid:48) j ( y ) : y ∈ ג ( j ) } covers E φ ( ξ , τ ) . Thanksto the definition of the D j s, this also yields that { D j ( y ) : y ∈ ג ( j ) } cover E φ ( ξ , τ ) as well. The claim (iii) followsdirectly from the fact that B (cid:48) j ( y ) ⊆ D j ( y ) for any y ∈ ג ( j ) .In order to prove (i), we distinguish two cases. If y ∈ Ξ ( j ) \ ג ( j ) , we have D j ( y ) = B (cid:48) j ( y ) ∩ K ⊆ B j ( y ) ∩ K and thusthe claim follows directly from the definition of B j ( y ) . On the other hand, if y ∈ ג ( j ) , for any w , w ∈ D j ( y ) thereare a , a ∈ { y } ∪ h − ( y ) such that w i ∈ B (cid:48) j ( a i ) for i =
1, 2. Since by construction we have d ( h ( w ) , w ) ≤ − j + / τ for any w ∈ ג ( j ) and B (cid:48) j ( w ) ⊆ B j ( w ) for any w ∈ Ξ ( j ) , we deduce that: d ( w , w ) ≤ d ( w , a ) + d ( a , y ) + d ( y , a ) + d ( a , w ) ≤ · − j + / τ + · − j + / τ ≤ − j + / τ ,proving that diam ( D j ( y )) ≤ − j + / τ . The proof of the second part of (i) follows similarly.In order to prove (iv), we first estimate φ ( ∂ ( B (cid:48) j ( y ) , 2 − j η / τ )) . Since for any y ∈ Ξ ( j ) , we have: ∂ ( B (cid:48) j ( y ) , 2 − j η / τ ) ⊆ (cid:91) w (cid:22) yB j ( w ) ∩ B j ( y ) (cid:54) = ∅ ∂ ( B j ( y ) , 2 − j η / τ ) ,this implies that: φ ( ∂ ( B (cid:48) j ( y ) , 2 − j η / τ )) ≤ ∑ w (cid:22) yB j ( w ) ∩ B j ( y ) (cid:54) = ∅ φ ( ∂ ( B j ( w ) , 2 − j η / τ )) ≤ Q C η ( − j / τ ) m , ( )where the last inequality comes from Lemmas A. and A. . We immediately see that the bound in ( ) proves(iv) if y ∈ Ξ ( j ) \ ג ( j ) . On the other hand, if y ∈ ג ( j ) thank to the inclusion: ∂ ( D j ( y ) , 2 − j η / τ ) ⊆ ∂ ( B (cid:48) j ( y ) , 2 − j η / τ ) ∪ (cid:91) w ∈ h − ( y ) ∂ ( B (cid:48) j ( y ) , 2 − j η / τ ) ,we infer that: φ ( ∂ ( D j ( y ) , 2 − j η / τ )) ≤ φ ( ∂ ( B (cid:48) j ( y ) , 2 − j η / τ )) + ∑ z ∈ h − ( y ) φ ( ∂ ( B (cid:48) j ( z ) , 2 − j η / τ )) ≤ ∑ w (cid:22) yB j ( w ) ∩ B j ( y ) (cid:54) = ∅ φ ( ∂ ( B j ( w ) , 2 − j η / τ )) + ∑ z ∈ h − ( y ) ∑ w (cid:22) zB j ( w ) ∩ B j ( z ) (cid:54) = ∅ φ ( ∂ ( B j ( w ) , 2 − j η / τ )) ≤ Q C η ( − j / τ ) m + Q Card ( h − ( y )) C η ( − j / τ ) m ≤ · Q C η ( − j / τ ) m , onstruction of dyadic cubes 56 where the first inequality of the last line comes from ( ) and the last one from the estimate on the cardinality of h − ( y ) , which can be bound with the one of ג ( j ) ∩ B ( y , 2 − j + / τ ) , see Lemma A. . The bound in (iv) follows fromthe choice C .In order to verify (v) we first observe that for any y ∈ ג ( j ) we have 2 φ ( ∂ ( D j ( y ) , 2 − j η / τ )) ≤ φ ( D j ( y )) . Thisimplies that: φ ( { u ∈ D j ( y ) : dist ( u , K \ D j ( y )) > − j η / τ } ) > φ ( ∂ ( D j ( y ) , 2 − j η / τ )) > c ∈ K such that K ∩ B ( c , η − j − / τ ) ⊆ D j ( y ) , concluding the proof of the proposition.We need to modify the sets D j in order to make each generation of cubes to interact in the intended way withthe others. Definition A. . Suppose j ∈ { Nl : l ∈ N and l ≥ } and let y ∈ Ξ ( j ) . Since the sets D j are a disjoint cover of K forany j , we can define ϕ ( y ) as the point x ∈ Ξ ( j − N ) such that y ∈ D j − N ( x ) . Furthermore, for any d ∈ N we let: E d ( x ) : = { y ∈ Ξ ( j + Nd ) : ϕ d ( y ) = x } ,where ϕ d = ϕ ◦ ϕ ◦ . . . ◦ ϕ (cid:124) (cid:123)(cid:122) (cid:125) d . Remark A. . Since the sets D j are a partition of K , this implies that for any w ∈ Ξ ( j + Nd ) there exists an x ∈ Ξ ( j ) such that w ∈ D j ( x ) , or more compactly: (cid:91) x ∈ Ξ ( j ) E d ( x ) = Ξ ( j + Nd ) , ( )We define A ( j ) as the set of those x ∈ Ξ ( j ) for which E d ( x ) is not definetely empty, i.e.: A ( j ) : = { x ∈ Ξ ( j ) : Card ( { d ∈ N : E d ( x ) (cid:54) = ∅ } ) = ∞ } .Since Ξ ( j ) is finite, it is immediate to see (thanks to the pidgeonhole principle) that A ( j ) must be non-empty.The following proposition is an elementary consequence of the definition of the sets E d . Proposition A. . For any j ∈ N N , x ∈ Ξ ( j ) and d ∈ N , if E d ( x ) (cid:54) = ∅ , then:E k ( x ) (cid:54) = ∅ for any k ≤ d . ( ) Furthermore, the following identity holds for any l ∈ N :E d + l ( x ) = (cid:91) w ∈ E d ( x ) E l ( w ) = (cid:91) w ∈ E d ( x ) E l ( w ) (cid:54) = ∅ E l ( w ) . ( ) Proof. If p belongs to the right hand side of ( ), there exists a w ∈ E d ( x ) such that p ∈ E l ( w ) . This impliesthat p ∈ E d + l ( x ) , since ϕ d + l ( p ) = ϕ d ( ϕ l ( p )) = ϕ d ( w ) = x . Viceversa, if p ∈ E d + l ( x ) then ϕ d ( p ) ∈ E l ( x ) and p ∈ E ( ϕ d ( p )) (cid:54) = ∅ , proving that p ∈ (cid:83) w ∈ E d ( x ) , E l ( w ) (cid:54) = ∅ E l ( w ) . Eventually, identity ( ) directly implies ( ).Indeed, if there was some 0 < l < k such that E l ( w ) = ∅ for any w ∈ E l ( w ) , we would have by ( ) that: E k ( x ) = (cid:91) w ∈ E l ( x ) E k − l ( w ) (cid:54) = ∅ E k − l ( w ) = ∅ ,since the union over an empty family is the empty set and this is a contradiction with the fact that by assumption E d ( x ) (cid:54) = ∅ . Remark A. . Thanks to Proposition A. , we also have that if x ∈ A ( j ) , then E d ( x ) (cid:54) = ∅ for any d ∈ N . This impliesthat, again thanks to the finiteness of Ξ ( j ) , that for any j ∈ N there exists an M ( j ) ∈ N such that E d ( x ) = ∅ forany x ∈ Ξ ( j ) \ A ( j ) whenever d ≥ M ( j ) . onstruction of dyadic cubes 57 Proposition A. . For any j ∈ N N we have ג ( j ) ⊆ A ( j ) .Proof. Thanks to Proposition A. (iv), for any x ∈ ג ( j ) there exists a c ∈ K such that: K ∩ B ( c , η − j − / τ ) ⊆ { u ∈ D j ( x ) : dist ( u , K \ D j ( x )) > − j η / τ } . ( )Since the sets D j + Nd cover K for any d ∈ N , there exists a z ∈ ג ( j + Nd ) such that c ∈ D j + Nd ( z ) . If d is bigenough, then D j + Nd ( z ) must be contained in D j ( x ) thanks to ( ) and Proposition A. (i). Thus E d ( x ) (cid:54) = ∅ for any d ≥ d . Definition A. . Let j ∈ N N and d ∈ N . For any x ∈ Ξ ( j ) for which E d ( x ) (cid:54) = ∅ , we define: D j , d ( x ) : = (cid:91) y ∈ E d ( x ) D j + Nd ( y ) .For the sake of notation we also let D j ,0 ( x ) : = D j ( x ) . Lemma A. . For any j ∈ N N and any x ∈ Ξ ( j ) for which there is a d ≥ such that E d ( x ) (cid:54) = ∅ , we have:d H ( D j , k − ( x ) , D j , k ( x )) ≤ − j − N ( k − )+ / τ , for any ≤ k ≤ d . ( ) Proof.
We prove the proposition proceeding by induction on k . As a first step, we verify ( ) in the base case k = E d ( x ) (cid:54) = ∅ , thanks to Proposition A. we have that E ( x ) (cid:54) = ∅ . Therefore, if z ∈ D j ,1 ( x ) there exists an y ∈ E ( x ) such that z ∈ D j + N ( y ) . Thanks to Proposition A. (i) we know that d ( z , y ) ≤ − j − N + / τ and in additionto this, thanks to the choice of y we also have that y ∈ D j ( x ) and thus:dist ( z , D j ( x )) ≤ d ( z , y ) ≤ − j − N + / τ , for any z ∈ D j ,1 ( x ) . ( )On the other hand for any w ∈ D j ( x ) , we have:dist ( w , D j ,1 ( x )) ≤ min y ∈ E ( x ) d ( w , y ) ≤ d ( w , x ) + min y ∈ E ( x ) d ( x , y ) ≤ − j + / τ + − j + / τ = − j + / τ , ( )where the last inequality comes from Proposition A. (i). Summing up, ( ) and ( ) imply that for any j ∈ N and any x ∈ Ξ ( j ) for which E ( x ) (cid:54) = ∅ we have: d H ( D j ( x ) , D j ,1 ( x )) ≤ − j + / τ . ( )Suppose now that the bound ( ) holds for k − ≤ k ≤ d . The definition of D j , k ( x ) together withidentity ( ) implies: D j , k ( x ) = (cid:91) y ∈ E k ( x ) D j + Nk ( y ) = (cid:91) w ∈ E k − ( x ) E ( w ) (cid:54) = ∅ (cid:91) y ∈ E ( w ) D j + N ( k − )+ N ( y ) = (cid:91) w ∈ E k − ( x ) E ( w ) (cid:54) = ∅ D j + N ( k − ) ,1 ( w ) . ( )Identity ( ) thanks to inequalities ( ) and ( ), allows us to infer: d H ( D j , k − ( x ) , D j , k ( x )) = d H (cid:32) (cid:91) w ∈ E k − ( x ) E ( w ) (cid:54) = ∅ D j + N ( k − ) ( w ) , (cid:91) w ∈ E k − ( x ) E ( w ) (cid:54) = ∅ D j + N ( k − ) ,1 ( w ) (cid:33) ≤ max w ∈ E k − ( x ) E ( w ) (cid:54) = ∅ d H (cid:0) D j + N ( k − ) ( w ) , D j + N ( k − ) ,1 ( w ) (cid:1) ≤ − j − N ( k − )+ / τ ,concluding the proof of the proposition.Since the sets D j , d ( x ) are contained in the compact set K and, thanks to Lemma A. , they form a Cauchysequence with respect to the Hausdorff metric d H , for any j ∈ N and x ∈ A ( j ) there is a compact set R j ( x ) contained in K such that: lim d → ∞ d H ( D j , d ( x ) , R j ( x )) = onstruction of dyadic cubes 58 Lemma A. . For any j ∈ N and any x ∈ A ( j ) we have R j ( x ) ⊆ B ( x , 2 − j + / τ ) .Proof. Thanks to Lemma A. and the triangular inequality for d H , we have: d H ( R j ( x ) , D j ( x )) ≤ ∞ ∑ d = d H ( D j , d + ( x ) , D j , d ( x )) ≤ − j + / τ ∞ ∑ j = − Nd = − j + / τ − − N . ( )Thanks to Proposition A. (i) we know that D j ( x ) ⊆ B ( x , 2 − j + / τ ) and thus ( ) and the triangular inequalityyield: sup y ∈ R j ( x ) d ( x , y ) ≤ d H ( { x } , R j ( x )) ≤ d H ( { x } , D j ( x )) + d H ( D j ( x ) , R j ( x )) ≤ − j + / τ + − j + / τ − − N < − j + / τ ,where the last inequality comes from the fact that N >
2. The above computation immediately implies that R j ( x ) ⊆ B ( x , 2 − j + / τ ) . Lemma A. . The compact sets { R j ( x ) : x ∈ A ( j ) } are a covering of K. Furthermore, for any j ∈ N N , any x ∈ A ( j ) andany l ∈ N we have: R j ( x ) = (cid:91) z ∈ E l ( x ) ∩ A ( j + Nl ) R j + Nl ( z ) . ( ) Proof.
Thanks to the definition of M ( j ) , see Remark A. , for any d ≥ M ( j ) we have: (cid:91) x ∈ A ( j ) E d ( x ) = Ξ ( j + Nd ) .Therefore, since by Proposition A. the sets { D j + Nd ( y ) : y ∈ Ξ ( j + Nd ) } for any j , d ∈ N are a partition of K , forany d ≥ M ( j ) we deduce that: K = (cid:91) x ∈ Ξ ( j + Nd ) D j + Nd ( x ) = (cid:91) x ∈ A ( j ) (cid:91) w ∈ E d ( x ) D j + Nd ( w ) = (cid:91) x ∈ A ( j ) D j , d ( x ) . ( )The first part of the statement, namely that the R j ’s are a cover of K , follows from ( ), Proposition A. and thefact that lim d → ∞ d H ( D j , d ( x ) , R j ( x )) = x ∈ A ( j ) .Concerning the second part of the statement for any j , l ∈ N , any w ∈ Ξ ( j + lN ) and any d ≥ M ( j + Nl ) thanksto Remark A. , we have E d ( w ) (cid:54) = ∅ if and only if w ∈ A ( j + Nl ) . Thanks to Proposition A. , and the aboveargument we infer that: D j , d + l ( x ) = (cid:91) w ∈ E d + l ( x ) D j + Nl + Nd ( w ) = (cid:91) w ∈ E l ( x ) E d ( w ) (cid:54) = ∅ (cid:91) z ∈ E d ( w ) D j + Nl + Nd ( z )= (cid:91) w ∈ E l ( x ) E d ( w ) (cid:54) = ∅ D j + Nl , d ( w ) = (cid:91) w ∈ E l ( x ) ∩ A ( j + Nl ) D j + Nl , d ( w ) .Since D j , l + d ( x ) converges to R j ( x ) and D j + Nl , d ( w ) to R j + Nl ( w ) in the Hausdorff metric as d goes to infinity, theuniqueness of limit and Proposition A. imply identity ( ). Proposition A. . For any j ∈ N we have E φ ( ξ , τ ) ⊆ (cid:83) x ∈ ג ( j ) R j ( x ) .Proof. Thanks to Proposition A. (ii), for any j , k ∈ N we have E φ ( ξ , τ ) ⊆ (cid:83) w ∈ ג ( j + Nk ) D j + Nk ( w ) . This implies thatfor any y ∈ ג ( j + Nk ) we can find a x ∈ ג ( j ) such that ϕ k ( y ) = x . Thanks to Proposition A. , for any k ∈ N wehave: E φ ( ξ , τ ) ⊆ (cid:91) y ∈ ג ( j + Nk ) D j + Nk ( y ) ⊆ (cid:91) x ∈ ג ( j ) (cid:91) y ∈ E k ( x ) D j + Nk ( y ) = (cid:91) x ∈ ג ( j ) D j , k ( x ) . ( ) onstruction of dyadic cubes 59 If we prove that dist ( w , (cid:83) x ∈ ג ( j ) R j ( x )) = w ∈ E φ ( ξ , τ ) , since (cid:83) x ∈ ג ( j ) R j ( x ) is closed and contained in K ,we have w ∈ (cid:83) x ∈ ג ( j ) R j ( x ) proving the proposition. So, fix some w ∈ E φ ( ξ , τ ) and note that thanks to ( ) thereexists a sequence { y k } ⊆ ג ( j ) such that w ∈ D j , k ( y k ) . Since ג ( j ) is a finite set, thanks to the pidgeonhole principlewe can assume without loss of generality that y k is a fixed element y ∈ ג ( j ) that it is also contained in A ( j ) thanksto Proposition A. . This implies that for any k ∈ N we have:dist ( w , R j ( y )) ≤ dist ( w , D j , k ( y )) + d H ( D j , k ( y ) , R j ( y )) = d H ( D j , k ( y ) , R j ( y )) .The arbitrariness of k and the definition of R j ( y ) concludes dist ( w , R j ( y )) =
0, proving the proposition.Despite the fact that the R j s cover E , they may not be disjoint. In order to correct this, we put a total order (cid:22) j on A ( j ) in such a way that:(i) x ≺ j y for any x ∈ ג ( j ) and any y ∈ A ( j ) ∩ Ξ ( j ) \ ג ( j ) (ii) x ≺ j y if ϕ ( x ) ≺ j − N ϕ ( y ) . Definition A. . For any j ∈ N N and x ∈ A ( j ) , we define: Q j ( x ) : = R j ( x ) ∩ (cid:34) (cid:91) w ≺ j x R j ( w ) (cid:35) c .Furthermore, we let ∆ j : = { Q j ( x ) : x ∈ A ( j ) } . Remark A. . Thanks to the definition of Q j we have R j ( x ) ∩ Q j ( w ) = ∅ for any x ∈ A ( j ) ∩ ג ( j ) and any w ∈ Ξ ( j ) \ ג ( j ) ∩ A ( j ) . Proposition A. . Let j ≤ j (cid:48) , x ∈ A ( j ) and y ∈ A ( j (cid:48) ) . Then either Q j (cid:48) ( y ) ⊆ Q j ( x ) or Q j (cid:48) ( y ) ∩ Q j ( x ) (cid:54) = ∅ .Proof. First of all we note that if j = j (cid:48) , since the sets Q j are pairwise disjoint, either Q j ( x ) = Q j ( y ) , and thus x = y ,or Q j ( x ) ∩ Q j (cid:48) ( y ) = ∅ . From now on, we can assume without loss of generality that j (cid:48) = j + Nl for some l ≥ y ∈ E l ( z ) for some x ≺ j z , then Q j (cid:48) ( y ) ∩ Q j ( x ) = ∅ . Indeed, for any w ∈ E l ( x ) , we have: ϕ l ( w ) = x ≺ j z = ϕ l ( y ) .and thanks to the definition of ≺ j (cid:48) , we deduce that w ≺ j y . Therefore this implies thanks to Lemma A. that: R j ( x ) = (cid:91) w ∈ E l ( x ) ∩ A ( j + Nl ) R j (cid:48) ( w ) ⊆ (cid:91) w ≺ j (cid:48) y R j (cid:48) ( w ) . ( )Thanks to the definition of Q j (cid:48) ( y ) and ( ), we conclude that: Q j (cid:48) ( y ) ∩ Q j ( x ) ⊆ Q j (cid:48) ( y ) ∩ R j ( x ) ⊆ R j (cid:48) ( y ) ∩ (cid:91) w ≺ j (cid:48) y R j (cid:48) ( w ) = ∅ .On the other hand, if y ∈ E l ( z ) for some z ≺ j x , since by assumption y ∈ A ( j (cid:48) ) , thanks to Lemma A. , we infer: Q j (cid:48) ( y ) ∩ Q j ( x ) ⊆ R j (cid:48) ( y ) ∩ Q j ( x ) = R j (cid:48) ( y ) ∩ R j ( x ) ∩ (cid:34) (cid:91) w ≺ j x R j ( w ) (cid:35) c = R j (cid:48) ( y ) ∩ R j ( x ) ∩ (cid:34) (cid:91) w ≺ j x (cid:18) (cid:91) z ∈ E l ( w ) ∩ A ( j (cid:48) ) R j (cid:48) ( z ) (cid:19)(cid:35) c ⊆ R j (cid:48) ( y ) ∩ R j ( x ) ∩ R j (cid:48) ( y ) c = ∅ .Eventually, if y ∈ E l ( x ) we deduce thanks to Lemma A. , that R j (cid:48) ( y ) ⊆ R j ( x ) . Thanks to the definition of ≺ j weknow that if w ∈ E l ( z ) for some z ≺ j x , then ϕ l ( w ) = z ≺ j x = ϕ l ( y ) , and thus w ≺ j (cid:48) y thanks to the definition ofthe order relation ≺ j (cid:48) . This and Lemma A. imply that: (cid:91) z ≺ j x R j ( z ) = (cid:91) z ≺ j x (cid:91) w ∈ E l ( z ) ∩ A ( j + Nl ) R j (cid:48) ( p ) ⊆ (cid:91) w ≺ j (cid:48) y R j (cid:48) ( w ) . onstruction of dyadic cubes 60 The above inclusion, shows that R j (cid:48) ( y ) ∩ (cid:83) z ≺ j x R j ( z ) = ∅ and thus, thanks to the definition of the sets Q j , wededuce that Q j (cid:48) ( y ) ⊆ Q j ( x ) . Proposition A. . For any j ∈ N N and any x ∈ A ( j ) we have: sup v ∈ R j ( x ) dist ( v , D j ( x )) ≤ − j − N + / τ . ( ) Furthermore, the following inclusions hold:D j ( x ) (cid:52) R j ( x ) ⊆ ∂ ( D j ( x ) , η − j − / τ ) and D j ( x ) (cid:52) Q j ( x ) ⊆ ∂ ( D j ( x ) , η − j − / τ ) . ( ) Proof. If u ∈ D j ,1 ( x ) , there is a y ∈ Ξ ( j + N ) ∩ D j ( x ) such that u ∈ D j + N ( y ) and thanks to Proposition A. (i) wehave: dist ( u , D j ( x )) ≤ dist ( u , y ) ≤ diam ( D j + N ( y )) ≤ − j − N + / τ .Therefore, for any v ∈ R j ( x ) , we have:dist ( v , D j ( x )) ≤ inf u ∈ D j ,1 ( x ) d ( u , v ) + dist ( u , D j ( x )) ≤ inf u ∈ D j ,1 ( x ) d ( u , v ) + − j − N + / τ = dist ( v , D j ,1 ( x )) + − j − N + / τ . ( )Since dist ( v , D j ,1 ( x )) ≤ d H ( R j ( x ) , D j ,1 ( x )) , Lemma A. together with ( ) implies that:dist ( v , D j ( x )) ≤ d H ( D j ,1 ( x ) , R j ( x )) + − j − N + / τ ≤ ∞ ∑ d = d H ( D j , d ( x ) , D j , d + ( x )) + − j − N + / τ ≤ ∞ ∑ d = − j − Nd + / τ + − j − N + / τ ≤ − j − N + / τ ,for any v ∈ R j ( x ) , where the last inequality follows since N > ). For any u ∈ D j ( x ) ∩ Q j ( x ) c , we can assume without loss of generality thatthere exists an x ( u ) ∈ A ( j ) \ { x } such that u ∈ R j ( x ( u )) . This is due to the following argument. The fact thatthe sets R j are a cover of K shows that we can always find a y ∈ A ( j ) such that y ∈ R j ( y ) . Furthermore, if u ∈ R ( x ) ∩ Q j ( x ) c , then by definition of Q j ( x ) there must exists another x ( u ) ≺ j x such that u ∈ R j ( x ( u )) , andthis shows that we can always choose x ( u ) different from x .Therefore, since D j ( x ( u )) ⊆ K \ D j ( x ) , thanks to ( ) we deduce that for any u ∈ D j ( x ) ∩ Q j ( x ) c , we have:dist ( u , K \ D j ( x )) ≤ dist ( u , D j ( x ( u ))) ≤ sup v ∈ R j ( x ( u )) dist ( v , D j ( x ( u ))) ≤ − j − N + / τ . ( )Thanks to inequality ( ), we infer that: D j ( x ) ∩ R j ( x ) c ⊆ D j ( x ) ∩ Q j ( x ) c ⊆ { w ∈ D j ( x ) : dist ( w , K \ D j ( x )) ≤ − j − N + / τ } . ( )On the other hand, for any u ∈ D j ( x ) c ∩ R j ( x ) , inequality ( ) implies that: D j ( x ) c ∩ Q j ( x ) ⊆ D j ( x ) c ∩ R j ( x ) ⊆ { w ∈ K \ D j ( x ) : dist ( w , D j ( x )) ≤ − j − N + / τ } . ( )The inclusions ( ) and ( ) yield both of the inclusion of ( ), and this concludes the proof, since thanks to thechoice of η we have 2 − N + ≤ η /2.We are left to prove that the families of sets ∆ j satisfy the properties of Theorem A. : Theorem A. . Defined C : = Q ξ and C : = Q ξ , the families { ∆ j } j ∈ N have the following properties:(i) K = (cid:83) Q ∈ ∆ j Q for any j ∈ N and the elements of ∆ j are pairwise disjoint, onstruction of dyadic cubes 61 (ii) for any j ∈ N we have E φ ( ξ , τ ) ⊆ (cid:83) x ∈ ג ( j ) Q j ( x ) ,(iii) if j ≤ j (cid:48) , Q ∈ ∆ j and Q (cid:48) ∈ ∆ j (cid:48) , then either Q contains Q (cid:48) or Q ∩ Q (cid:48) = ∅ ,(iv) for any Q ∈ ∆ j we have diam ( Q ) ≤ − j + / τ ,(v) if x ∈ ג ( j ) , then C − (cid:0) − j / τ (cid:1) m ≤ φ ( Q j ( x )) ≤ C (cid:0) − j / τ (cid:1) m ,(vi) if Q ∈ ∆ j , we have φ (cid:0) ∂ ( Q , η − j / τ ) (cid:1) ≤ C η (cid:0) − j / τ ) m ,(vii) for any x ∈ ג ( j ) there is a c ∈ K such that B ( c , η − j − / τ ) ∩ K ⊆ Q j ( x ) .Proof. Point (i) is satisfied by the definition of the sets Q j and Lemma A. and (ii) follows from Proposition A. and Remark A. . Furthermore, (iii) is implied by Proposition A. and (iv) by Lemma A. . proof of ( v ) If x ∈ ג ( j ) the upper bound follows immediately thanks to the fact that x ∈ E φ ( ξ , τ ) andProposition A. : φ ( Q j ( x )) ≤ φ ( B ( x , 2 − j + / τ )) ≤ ξ ( − j + / τ ) m .For the lower bound note that: φ ( Q j ( x )) = φ ( D j ( x )) − φ ( D j ( x ) \ Q j ( x )) + φ ( Q j ( x ) \ D j ( x )) ≥ φ ( D j ( y )) − φ ( Q j ( x ) (cid:52) D j ( x )) .Thanks to Proposition A. we know that Q j ( x ) (cid:52) D j ( x ) ⊆ ∂ ( D j ( x ) , η − j / τ ) . Therefore, Proposition A. (iii) and(iv) imply that: φ ( Q j ( x )) ≥ φ ( D j ( y )) − φ ( ∂ ( D j ( x ) , η − j / τ )) ≥ ( −Q ξ − − C η )( − j / τ ) m ≥ −Q ξ − ( − j / τ ) m ,where the last inequality follows from the fact that we assumed η ≤ − Q ξ − , see the beginning of SubsectionA. . This concludes the proof of (v) thanks to the choice of C . proof of ( vi ) For any j ∈ N N and any x ∈ A j ( x ) let us define the sets: (cid:91) Q j ( x ) : = { u ∈ Q j ( x ) : dist ( u , K \ Q j ( x )) ≤ η − j − / τ } , (cid:91) Q j ( x ) : = { u ∈ K \ Q j ( x ) : dist ( u , Q j ( x )) ≤ η − j − / τ } .First of all for any x ∈ A ( j ) , we estimate the measure of the set (cid:91) Q j ( x ) . If u ∈ (cid:91) Q j ( x ) , since the Q j s are a partitionof K , we can find x (cid:54) = x (cid:48) ∈ A ( j ) such that u ∈ (cid:91) Q j ( x (cid:48) ) . Thanks Proposition A. and the triangular inequality, onthe one hand we deduce that:dist ( u , D j ( x )) ≤ min w ∈ Q j ( x ) d ( u , w ) + dist ( w , D j ( x )) ≤ − j − N + ≤ − j − η / τ .where the last inequality follows from the choice of η . On the other:dist ( u , D j ( x (cid:48) )) ≤ inf w ∈ Q j ( x (cid:48) ) d ( u , w ) + dist ( w , D j ( x (cid:48) )) ≤ − j − η / τ + − j − N + ≤ η − j / τ ,where once again the last inequality follows from the choice of η . Therefore the two above bounds imply inparticular that u ∈ ∂ ( D j ( x (cid:48) ) , η − j / τ ) . Therefore, thanks to the arbitrariness of u , we conclude that: (cid:91) Q j ( x ) ⊆ (cid:91) x (cid:48) ∈ A ( j ) ∂ ( D j ( x (cid:48) ) , η − j / τ ) ∩ (cid:91) Q j ( x ) (cid:54) = ∅ ∂ ( D j ( x (cid:48) ) , η − j / τ ) ( )Thanks to Proposition A. (i), Lemma A. and using the same argument we employed to prove Lemma A. , onecan show that:Card ( { y ∈ A ( j ) : Q j ( y ) ∩ ∂ ( D j ( x ) , η − j / τ ) (cid:54) = ∅ } ) ≤ Card ( { y ∈ A ( j ) : R j ( y ) ∩ ∂ ( D j ( x ) , η − j / τ ) (cid:54) = ∅ } ) ≤ Card ( { y ∈ A ( j ) : B ( x , 2 − j + / τ ) ∩ B ( x , 2 − j + / τ ) (cid:54) = ∅ } ) ≤ Q . ( ) inite perimeter sets in carnot groups 62 Therefore, the inclusion ( ), the bound ( ) and Proposition A. (iv) imply that: φ ( (cid:91) Q j ( x )) ≤ Q max y : Q j ( y ) ∩ D j ( x ) (cid:54) = ∅ φ ( ∂ ( D j ( x ) , η − j / τ )) ≤ Q C η ( − j / τ ) m . ( )We are left to estimate the measure of (cid:91) Q j ( x ) . Since the Q j s are a partition of K , we infer that: φ ( (cid:91) Q j ( x )) = ∑ y ∈ A ( j ) \{ x } φ ( { u ∈ Q j ( y ) : dist ( u , Q j ( x )) ≤ η − j − } ) . ( )The cardinality of those y ∈ A ( j ) for which there exists u ∈ Q j ( y ) such that dist ( u , K \ Q j ( x )) ≤ η − j − can bebounded by 2 Q and this can be shown with the same argument used for ( ). Thanks to the bounds ( ) and( ) we infer: φ ( (cid:91) Q j ( x )) ≤ Q · max y ∈ A ( j ) \{ x } Q j ( y ) ∩ B ( Q j ( x ) , η − j − ) (cid:54) = ∅ φ ( { u ∈ Q j ( y ) : dist ( u , Q j ( x )) ≤ η − j − } ) ≤ Q C η ( − j / τ ) m . ( )Hence putting together ( ) and ( ), we deduce that: φ ( ∂ ( Q j ( x ) , η − j − / τ )) = φ ( (cid:91) Q j ( x )) + φ ( (cid:91) Q j ( x )) ≤ Q C η ( − j / τ ) m + Q C η ( − j / τ ) m ≤ Q C η ( − j / τ ) m .Thus, (v) follows with the choice C : = Q C . proof of ( vii ) Thanks for (iv) and (v), for any x ∈ ג ( j ) we have: C − ( − j / τ ) m ≤ φ ( Q j ( x )) ≤ φ ( ∂ ( Q j ( x ) , η − j − / τ )) + φ ( { u ∈ Q j ( x ) : dist ( u , K \ Q j ( x )) > η − j − / τ } ) ≤ C η ( − j / τ ) m + φ ( { u ∈ Q j ( x ) : dist ( u , K \ Q j ( x )) > η − j − / τ } ) Since η ≤ ξ − − Q , we deduce that C − − C η ≥ ξ − − Q and thus: ξ − − Q ( − j / τ ) m ≤ φ ( { u ∈ Q j ( x ) : dist ( u , K \ Q j ( x )) > η − j − / τ } ) .This implies that there exists c ∈ K such that B ( c , η − j − / τ ) ⊆ Q j ( x ) . b finite perimeter sets in carnot groups Throughout this second part of the appendix if not otherwise stated, we will always endow G with the smooth-box metric introduced in Definition . . b . Finite perimeter sets, intrinsic Lipschitz graphs and regular surfaces
In this subsection we recall the definitions of functions of bounded variations and finite perimeter sets and wecollect from various papers some results that will be useful in the following.
Definition B. . We say that a function f : G → R is of local bounded variation if f ∈ L loc ( G ) and: (cid:107)∇ G f (cid:107) ( Ω ) : = sup (cid:110) ˆ Ω f ( x ) div G ϕ ( x ) dx : ϕ ∈ C ( Ω , H G ) , | ϕ ( x ) | ≤ (cid:111) < ∞ ,for any bounded open set Ω ⊆ G , where div G ϕ : = ∑ n i = X i ϕ i and where X , . . . , X n are the vector fields introducedin Definition . . We denote by BV G , loc ( G ) the set of all functions of locally bounded variation. As usual a Borelset E ⊆ G is said to be of finite perimeter if χ E is of bounded variation. inite perimeter sets in carnot groups 63 Theorem B. (Theorem . . , [ ]) . If f is a function of bounded variation, then (cid:107)∇ G f (cid:107) is a Radon measure on G .Moreover there exists a (cid:107)∇ G f (cid:107) -measurable horizontal section σ f : G → H G such that | σ f ( x ) | = for (cid:107)∇ G f (cid:107) -a.e. x ∈ G ,and for any open set Ω we have: ˆ Ω f ( x ) div G ϕ ( x ) dx = ˆ Ω (cid:104) ϕ , σ f (cid:105) d (cid:107)∇ G f (cid:107) , for every ϕ ∈ C ( Ω , H G ) .As in the Euclidean spaces functions of bounded variation are compactly embedded in L : Theorem B. (Theorem . , [ ]) . The set BV G , loc ( G ) is compactly embedded in L loc ( G ) . Definition B. . If E ⊆ G is a Borel set of locally finite perimeter, we let | ∂ E | G : = (cid:107)∇ G χ E (cid:107) . Furthermore we call generalized horizontal inward G - normal to ∂ E the horizontal vector n E ( x ) : = σ χ E ( x ) . Finally, we define the reducedboundary ∂ ∗ G E to be the set of those x ∈ G for which:(i) | ∂ E | G ( B ( x , r )) > r > r → ffl B ( x , r ) n E d | ∂ E | G exists,(iii) lim r → (cid:13)(cid:13)(cid:13) ffl B ( x , r ) n E d | ∂ E | G (cid:13)(cid:13)(cid:13) = Lemma B. . Assume E is a set of finite perimeter in G and let x ∈ G and r > . Then: | ∂ ( δ r ( x − E )) | G = r − ( Q− ) T x , r | ∂ E | G . Proof.
For any ϕ ∈ C ( G , H G ) , any x ∈ G and any r >
0, defined ˜ ϕ ( z ) : = ϕ ( δ r ( x − z )) , the following identityholds: div G ˜ ϕ ( z ) = r − div G ϕ ( δ r ( x − z )) . ( )This, indeed is due to the fact that: X j ˜ ϕ j ( z ) : = lim h → ˜ ϕ j ( z δ h ( e j )) − ˜ ϕ j ( z ) h = lim h → ϕ j ( δ r ( x − z δ h ( e j ))) − ϕ j ( δ r ( x − z )) h = r − X j ϕ j ( δ r ( x − z )) .Thanks to identity ( ) and the fact that the Lebesgue measure is a Haar measure for G , we infer that: ˆ χ δ r ( x − E ) ( y ) div G ϕ ( y ) dy = r −Q ˆ χ E div G ϕ ( δ r ( x − y )) dy = r − ( Q− ) ˆ χ E ( y ) div G ˜ ϕ ( y ) dy .It is not hard to see that ϕ ∈ C ( Ω , H G ) if and only if ˜ ϕ ∈ C ( x δ r Ω , H G ) and thus for any open set Ω we have: | ( ∂δ r ( x − E )) | G ( Ω ) = r − ( Q− ) | ∂ E | G ( x δ r Ω ) = r − ( Q− ) T x , r | ∂ E | G ( Ω ) .This concludes the proof. Theorem B. (Theorem . , [ ]) . Let E ⊆ G be a set of locally finite perimeter. Then | ∂ E | G is asymptotically doubling,and more precisely the following hold. For | ∂ E | G -a.e. x ∈ G there exists an r ( x ) > such that:l G r Q− ≤ | ∂ E | G ( B ( x , r )) ≤ L G r Q− , for any r ∈ ( r ( x )) , ( ) where the constants l G and L G depend only on G and the metric d. As a consequence | ∂ E | G is concentrated on ∂ ∗ G E, i.e. | ∂ E | G ( G \ ∂ ∗ G E ) = .Remark B. . Proposition . and Theorem B. imply that l G S Q− (cid:120) ∂ ∗ G E ≤ | ∂ E | G ≤ L G S Q− (cid:120) ∂ ∗ G E . Therefore, themeasures S Q− (cid:120) ∂ ∗ G E and | ∂ E | G are mutually absolutely continuous. In particular there exists a d ∈ L ( | ∂ E | G ) suchthat: S Q− (cid:120) ∂ ∗ G E = d | ∂ E | G ,and for | ∂ E | G -almost every x ∈ G we have L − G ≤ d ( x ) ≤ l − G . inite perimeter sets in carnot groups 64 Theorem B. (Theorem . . , [ ]) . If f : V → N ( V ) is an intrinsic Lipschitz map, the sub-graph:epi ( f ) : = (cid:110) v ∗ δ t ( n ( V )) : t < (cid:104) π f ( v ) , n ( V ) (cid:105) (cid:111) , ( ) is a set with locally finite G -perimeter. Since the topological boundary of epi ( f ) coincides with gr ( f ) , thanks to Theorem . . of [ ], we infer that | ∂ epi ( f ) | G ( G \ ∂ ∗ G epi ( f )) = | ∂ epi ( f ) | G ( gr ( f ) \ ∂ ∗ G epi ( f )) =
0. In particular, thanks to Remark B. , we deduce thefollowing: Proposition B. . S Q− ( gr ( f ) \ ∂ ∗ G epi ( f )) = . It is convenient to associate to every intrinsic Lipschitz function f : V → N ( V ) a normal vector field to its graph: Definition B. . For any f : V → N ( V ) intrinsic Lipschitz function, we denote by n f : ∂ ∗ G epi ( f ) → H G the inwardinner G -normal of epi ( f ) . b . Tangents measures versus tangent sets to finite perimeter sets
In this subsection we connect the notion of tangent sets to Caccioppoli sets, that is extensively used in the theoryof finite perimeter sets, to the notion of tangent measures. This will help us to prove that if the perimeter measureof a Caccioppoli set has flat tangents, then it has a unique tangent, that coincides with the plane in Gr ( Q − ) orthogonal to the normal. Definition B. . (Tangent sets) Let E ⊆ G be a set of locally finite perimeter and assume x ∈ ∂ ∗ G E . We denote byTan ( E , x ) the limit points in the topology of the local convergence in measure of the sets { δ r ( x − E ) } r > as r → ] and in particular to Proposition . . Proposition B. . If E is a set of finite perimeter, for S Q− -almost every x ∈ ∂ ∗ G E we have:(i)
Tan ( E , x ) (cid:54) = ∅ ,(ii) the elements of Tan ( E , x ) are finite perimeter sets,(iii) for any F ∈ Tan ( E , x ) we have n F ( y ) = n E ( x ) for | ∂ F | G -almost every y ∈ G . The following proposition is a characterisation of the tangent measures of perimeter measures:
Proposition B. (observation ( . ), [ ]) . If E is a set of locally finite perimeter, for | ∂ E | G -almost every x ∈ ∂ ∗ G E we havethat:(i) if { r i } i ∈ N is an infinitesimal sequence such that δ r i ( x − E ) converges locally in measure to some Borel set L,then L is a finite perimeter set and r − ( Q− ) i T x , r i | ∂ E | G (cid:42) | ∂ L | G . In particular, if L ∈ Tan ( E , x ) then | ∂ L | G ∈ Tan Q− ( | ∂ E | G , x ) ,(ii) if ν ∈ Tan Q− ( | ∂ E | G , x ) , then there is an L ∈ Tan ( E , x ) such that ν = | ∂ L | G .Proof. Let us first prove (i). Proposition B. (ii) implies that for | ∂ E | G -almost all x ∈ ∂ ∗ E , every limit (locally inmeasure) of a sequence of the form δ r ( x − E ) is of finite perimeter. Therefore, from now on we assume withoutloss of generality that x is a fixed point where Proposition B. (ii) holds.Fix now any open and bounded set Ω ⊆ G and note that for any ϕ ∈ C ( Ω , H G ) a simple computation yields: (cid:12)(cid:12)(cid:12)(cid:12) ˆ δ ri ( x − E ) div G ϕ ( z ) dz − ˆ L div G ϕ ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) div G ϕ (cid:107) ∞ L n ( δ r i ( x − E ) (cid:52) L ∩ Ω ) . ( ) inite perimeter sets in carnot groups 65 Suppose ˜ ϕ ∈ C ( Ω , H G ) is a vector field such that ´ E div G ˜ ϕ ( w ) dw ≥ | ∂ E | G ( x δ r i Ω ) − (cid:101) and define ϕ ( z ) : = ˜ ϕ ( x δ r i ( z )) . The bound ( ) implies that: r − ( Q− ) i | ∂ E | G ( x δ r i Ω ) − | ∂ L | G ( Ω ) − (cid:101) ≤ r Q− i ˆ E div G ˜ ϕ ( z ) dz − ˆ L div G ϕ ( z ) dz = ˆ δ r ( x − E ) div G ϕ ( z ) dz − ˆ L div G ϕ ( z ) dz = (cid:107) div G ϕ (cid:107) ∞ L n ( δ r i ( x − E ) (cid:52) L ∩ Ω ) , ( )where the first inequality of the second line follows from ( ). On the other hand, if ˜ ϕ is such that ´ L div G ϕ ( w ) dw ≥| ∂ L | G ( Ω ) − (cid:101) we let ϕ ( z ) : = ˜ ϕ ( δ r i ( x − z )) . Eventually, the same argument we used in ( ), imply: | ∂ L | G ( Ω ) − r − ( Q− ) i | ∂ E | G ( x δ r i Ω ) − (cid:101) ≤ ˆ L div G ϕ ( z ) dz − r Q− i ˆ E div G ˜ ϕ ( z ) dz ≤(cid:107) div G ϕ (cid:107) ∞ L n ( δ r i ( x − E ) (cid:52) L ∩ Ω ) . ( )Putting together ( ) and ( ) we infer that:lim i → ∞ (cid:12)(cid:12)(cid:12) | ∂ L | G ( Ω ) − r − ( Q− ) i | ∂ E | G ( x δ r i Ω ) (cid:12)(cid:12)(cid:12) − (cid:101) ≤ ( (cid:107) div G ϕ (cid:107) ∞ + (cid:107) div G ϕ (cid:107) ∞ ) lim i → ∞ L n ( δ r i ( x − E ) (cid:52) L ∩ Ω ) = (cid:101) >
0, implies that for any bounded open set Ω we have:lim r i → (cid:12)(cid:12)(cid:12)(cid:12) | ∂ L | G ( Ω ) − T x , r i | ∂ E | G r Q− i ( Ω ) (cid:12)(cid:12)(cid:12)(cid:12) =
0. ( )Let ν i : = r − ( Q− ) i T x , r i | ∂ E | G and for any ρ > µ i : = ν i ( B ( ρ )) − ν i (cid:120) B ( ρ ) and µ : = | ∂ L | G ( B ( ρ )) − | ∂ L | G (cid:120) B ( ρ ) .Thanks to identity ( ), we infer that for any bounded open set Ω we have:lim i → ∞ µ i ( Ω ) = lim i → ∞ ν i ( B ( ρ ) ∩ Ω ) ν i ( B ( ρ )) = | ∂ L | G ( B ( ρ ) ∩ Ω ) | ∂ L | G ( B ( ρ )) = µ ( Ω ) . ( )Thanks to Theorem . of [ ], we deduce that µ i (cid:42) µ . Therefore, for any f ∈ C c ( G ) we have:lim i → ∞ ˆ f ( x ) d ν i ( x ) = lim i → ∞ ν i ( B ( ρ )) ˆ f ( x ) d µ i ( x ) = ˆ f ( x ) d | ∂ L | G ( x ) ,proving that r − ( Q− ) i T x , r i | ∂ E | G (cid:42) | ∂ L | G . The second part of the statement of (i) follows immediately.We now prove (ii). We can assume without loss of generality that x = andthat { r i } is an infinitesimal sequence such that: r − ( Q− ) i T r i | ∂ E | G (cid:42) ν ∈ Tan Q− ( | ∂ E | G , x ) ,Now let E i : = δ r i ( E ) , so that | ∂ E i | G = r Q− i T r i | ∂ E | G . For any open and bounded open set Ω we can find an R > Ω ⊆ B ( R ) . Therefore, thanks to Lemma B. we have: | ∂ ( δ r i ( x − E )) | G ( Ω ) ≤ | ∂ ( δ r i ( x − E )) | G ( B ( R )) = r − ( Q− ) i T x , r i | ∂ E | G ( B ( R )) = | ∂ E | G ( B ( x , Rr i )) r Q− i .This implies, thanks to Theorem B. that:lim sup i → ∞ | ∂ ( δ r ( x − E )) | G ( Ω ) ≤ lim sup i → ∞ | ∂ E | G ( B ( x , Rr i )) r Q− i ≤ L G R Q− .Thus, thanks to Theorem B. the sequence { δ r i ( x − E ) } i ∈ N is precompact in L loc ( G ) and since we assumed δ r i ( x − E ) converges locally in measure to L then δ r i ( x − E ) converges in L loc ( G ) to L . In particular, thanks toTheorem . of [ ], we infer that L is of local finite perimeter. Thus, by definition of the tangent sets we have L ∈ Tan ( E , 0 ) and thanks to item (i), we conclude that r − ( Q− ) i T r i | ∂ E | G (cid:42) | ∂ L | G . Thanks to the uniqueness of thelimit we conclude that | ∂ L | G = ν . inite perimeter sets in carnot groups 66 Proposition B. . If E is an open set of finite perimeter in G , for S Q− -almost any x ∈ ∂ E and any L ∈ Tan ( E , x ) we have L n ( L \ int ( L )) = . In particular the measures | ∂ L | G and | ∂ ( int ( L )) | G coincide on Borel sets.Proof. This follows for instance from Proposition B. and Theorem . of [ ]. Remark B. . Let V ± : = { w ∈ G : ±(cid:104) n ( V ) , w (cid:105) > } . Thanks to identity ( . ) in [ ], it is immediate to see that V ± areopen sets of finite perimeter in G and that ∂ V ± = ∓ n ( V ) H n − eu (cid:120) V . This implies that the horizontal normal of eachof the half spaces determined by V coincides, up to a sign, | ∂ V ± | G -almost everywhere with n ( V ) . Proposition B. . Let V ∈ Gr ( Q − ) and let f : V → N ( V ) be an intrinsic Lipschitz function. Suppose that: Tan Q− ( | ∂ epi(f) | G , x ) ⊆ M , for | ∂ epi(f) | G -almost every x ∈ G . Then for | ∂ epi(f) | G -almost every x ∈ G , we have: Tan Q− ( | ∂ epi(f) | G , x ) ⊆ { λ S Q− (cid:120) V ( x ) : λ ∈ [ L − G , l − G ] } , where V ( x ) ∈ Gr ( Q − ) is the plane orthogonal to n f ( x ) , that is the normal to gr ( f ) introduced in Definition B. .Proof. Proposition B. implies that for S Q− -almost every x ∈ ∂ ∗ G epi ( f ) and for every L ∈ Tan ( epi ( f ) , x ) , we have: | ∂ L | G = λ S Q− (cid:120) V L , x for some V L , x ∈ Gr ( Q − ) and λ >
0. ( )Furthermore Remark B. , Proposition . and a simple computation that we omit, imply that λ ∈ [ l G , L G ] .Fix now an x ∈ ∂ ∗ epi ( f ) at which ( ) hold and that satisfies the thesis of Proposition B. and let L ∈ Tan ( epi ( f ) , x ) . Thanks to these choices, L is a finite perimeter set with constant horizontal normal and Propo-sition B. and ( ) tell us that its topological boundary must coincide up to S Q− -null sets with the plane V L , x .Therefore, since by Proposition B. we can assume without loss of generality that L is an open set, we concludethat L must coincide with one of the two halfspaces determined by V L , x . This implies however, thanks to RemarkB. , that: n ( V L , x ) = n L ( y ) for S Q− -almost every y ∈ ∂ L . ( )Furthermore, Proposition B. (iii) and ( ) imply that n ( V L , x ) = n L ( y ) = n f ( x ) for S Q− -almost all y ∈ ∂ L . Thisshows however that for S Q− -almost all x ∈ gr ( f ) , every element of Tan ( epi ( f ) , x ) is an halfspace whose boundaryis the plane orthogonal to n f ( x ) and Proposition B. concludes the proof. Proposition B. . Suppose γ : V → N ( V ) is an intrinsic Lipschitz function such that for S Q− -almost every x ∈ V thereexists a plane V ( x ) ∈ Gr ( Q − ) for which: lim r → S Q− (cid:0) gr ( γ ) ∩ B ( x γ ( x ) , r ) \ x γ ( x ) X V γ ( x γ ( x )) ( α ) (cid:1) r Q− =
0, ( ) whenever α > and where X V γ ( x γ ( x )) ( α ) : = { w ∈ G : dist ( w , V ( x γ ( x ))) ≤ α (cid:107) w (cid:107)} . Then, gr ( γ ) can be covered withcountably many C G -surfaces.Proof. For any i ∈ N we define: A i : = { x ∈ gr ( γ ) : ( ) holds at x and S h ( B ( x , r ) ∩ gr ( γ )) ≥ L − G l G r Q− for any 0 < r < i } .Let us prove that the sets A i are S h (cid:120) gr ( γ ) -measurable. It is immediate to see that if we show that the set:˜ A i : = { x ∈ gr ( γ ) : S h ( B ( x , r ) ∩ gr ( γ )) ≥ L − G l G r Q− for any 0 < r < i } ,is closed the measurability of A i immediately follows since ( ) holds on a set of full S h (cid:120) gr ( γ ) -measure. Sincegr ( γ ) is closed, to prove the closedness of ˜ A i it is sufficient to show that if a sequence { x j } j ∈ N ⊆ ˜ A i converges tosome x ∈ gr ( γ ) , then x ∈ ˜ A i . So, let 0 < r < i and note that if d ( x , x j ) < r we have: L − G l G ( r − d ( x , x j )) Q− ≤ S h (cid:120) gr ( γ )( B ( x j , r − d ( x , x j ))) ≤ S h (cid:120) gr ( γ )( B ( x , r )) . inite perimeter sets in carnot groups 67 The arbitrariness of j implies that for any 0 < r < i we have S h (cid:120) gr ( γ )( B ( x , r )) ≥ L − G l G r Q− , proving that x ∈ ˜ A i .We now prove that the sets A i cover S h -almost all gr ( γ ) . Recall that gr ( γ ) is the boundary of the set of locallyfinite perimeter epi ( γ ) . Thanks to Theorem B. , this implies that for | ∂ epi ( γ ) | G -almost every x ∈ G there exists a r ( x ) > < r < r ( x ) we have: L G S h (cid:120) gr ( γ )( B ( x , r )) ≥ | ∂ epi ( γ ) | G ( B ( x , r )) ≥ l G r Q− ,where the first inequality above comes from Remark B. . Therefore, if r ( x ) > i and ( ) holds at x , then x ∈ A i and this concludes the proof of the fact that S h ( gr ( γ ) \ (cid:83) i ∈ N A i ) = i , j ∈ N and any x ∈ A i we let: ρ i , j ( x ) : = sup (cid:26) |(cid:104) n γ ( x ) , π ( x − y ) (cid:105)| d ( x , y ) : y ∈ A i and 0 < d ( x , y ) < j (cid:27) .We want to prove that for any i ∈ N and any x ∈ A i we have:lim j → ∞ ρ i , j ( x ) =
0. ( )Assume by contradiction this is not the case and that there exists a i ∈ N and a z ∈ A i for which ( ) fails. Then,there is a 0 < c ≤ { j k } k ∈ N such that for any k ∈ N there isa y k ∈ A i for which y ∈ B ( z , 1/ j k ) and |(cid:104) n γ ( z ) , π ( z − y k ) (cid:105)| > c d ( z , y k ) . Thanks to Proposition . we infer that y i (cid:54)∈ zX V γ ( z ) ( c /2 ) , indeed: dist ( V γ ( z ) , z − y k ) = |(cid:104) n γ ( z ) , π ( z − y k ) (cid:105)| > c d ( z , y k ) . ( )We now claim that for any k ∈ N we have: B ( y k , c d ( z , y k ) /4 ) ⊆ B ( z , 2 d ( z , y k )) \ zX V γ ( z ) ( c /4 ) . ( )In order to prove the inclusion ( ) we fix a k ∈ N and let w : = y k v ∈ B ( y k , c d ( z , y k ) /8 ) . With these choicesProposition . and the triangular inequality imply:dist ( V γ ( z ) , z − w ) = |(cid:104) n γ ( z ) , π ( z − w ) (cid:105)| ≥ |(cid:104) n γ ( z ) , π ( z − y k ) (cid:105)| − |(cid:104) n γ ( z ) , π ( y − k w ) (cid:105)|≥ c d ( z , y k ) − d ( y k , w ) ≥ c d ( z , w ) − ( + c ) d ( y k , w ) . ( )Furthermore, thanks to the choice of w we have: d ( y k , w ) ≤ c d ( z , y k ) /4 ≤ c d ( z , w ) /4 + c d ( y k , w ) /4, ( ) d ( z , w ) ≤ d ( z , y k ) + d ( y k , w ) ≤ ( + c /8 ) d ( z , y k ) ≤ d ( z , y k ) . ( )Putting together ( ), ( ) and some algebraic computations that we omit, we infer that:dist ( V γ ( z ) , z − w ) ≥ c d ( z , w ) /4.The inclusion ( ) follows immediately from the above bound and ( ). Therefore, ( ) and ( ) imply:lim sup r → S Q− (cid:0) gr ( γ ) ∩ B ( z , r ) \ zX V γ ( z ) ( c /8 ) (cid:1) r Q− ≥ lim k → ∞ S Q− (cid:0) gr ( γ ) ∩ B ( z , 2 d ( z , y k )) \ zX V γ ( z ) ( c /8 ) (cid:1) ( d ( z , y k )) Q− ≥ lim k → ∞ S Q− (cid:0) gr ( γ ) ∩ B ( y k , c d ( z , y k ) /8 ) (cid:1) Q− d ( z , y k ) Q− ≥ lim k → ∞ L − G l G ( c d ( z , y k ) /8 ) Q− Q d ( z , y k ) = l G L − G (cid:18) c (cid:19) Q− , ( )where the second last inequality comes from Theorem B. and the fact that y k ∈ A i . However, since by construction( ) holds at any point of A i , ( ) is in contradiction with ( ) and thus ( ) must hold at any x ∈ A i . Define f i to be the function identically 0 on A i and for any ι ∈ N we let K i ( ι ) be a compact subset of A i for which: eferences (i) S Q− ( A i \ K i ( ι )) ≤ ι ,(ii) n γ is continuous on K i ( ι ) ,(iii) ρ i , j converges uniformly to 0 on K i ( ι ) .The existence of K i ( ι ) is implied by Lusin’s theorem and Severini-Egoroff’s theorem. Thanks to Whitney extensiontheorem, see for instance Theorem . in [ ], we infer that we can find a C G -function such that f i , ι | K = ∇ H f i , ι ( x ) = n γ ( x ) for any x ∈ K i ( ι ) . This imples that A i and thus gr ( γ ) , can be covered S Q− -almost all with C G -surfaces. references [ ] Amir Ali Ahmadi, Etienne de Klerk, and Georgina Hall. “Polynomial norms”. In: SIAM J. Optim. . ( ),pp. – . issn : - .[ ] Luigi Ambrosio, Bruce Kleiner, and Enrico Le Donne. “Rectifiability of sets of finite perimeter in Carnotgroups: existence of a tangent hyperplane”. In: J. Geom. Anal. . ( ), pp. – . issn : - .[ ] Gioacchino Antonelli and Andrea Merlo. On rectifiable measures in Carnot groups: structure and representations .In preparation.[ ] Costante Bellettini and Enrico Le Donne. “Sets with constant normal in Carnot groups: properties andexamples”. In: arXiv: . ( ), arXiv: . .[ ] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni. Stratified Lie groups and potential theory for their sub-Laplacians .Springer Monographs in Mathematics. Springer, Berlin, , pp. xxvi+ . isbn : - - - - ; - - - .[ ] Vasileios Chousionis, Katrin Fässler, and Tuomas Orponen. “Intrinsic Lipschitz graphs and vertical β -numbersin the Heisenberg group”. In: Amer. J. Math. . ( ), pp. – . issn : - .[ ] Vasilis Chousionis, Valentino Magnani, and Jeremy T. Tyson. “On uniform measures in the Heisenberggroup”. In: Adv. Math. ( ), pp. , . issn : - .[ ] Vasilis Chousionis and Jeremy T. Tyson. “Marstrand’s density theorem in the Heisenberg group”. In: Bull.Lond. Math. Soc. . ( ), pp. – . issn : - .[ ] Michael Christ. “A T ( b ) theorem with remarks on analytic capacity and the Cauchy integral”. In: Colloq.Math. / . ( ), pp. – . issn : - .[ ] Guy David. Wavelets and singular integrals on curves and surfaces . Vol. . Lecture Notes in Mathematics.Springer-Verlag, Berlin, , pp. x+ . isbn : - - - .[ ] Guy David and Stephen Semmes. Analysis of and on uniformly rectifiable sets . Vol. . Mathematical Surveysand Monographs. American Mathematical Society, Providence, RI, , pp. xii+ . isbn : - - - .[ ] Guy David and Stephen Semmes. “Quantitative rectifiability and Lipschitz mappings”. In: Transactions of theAmerican Mathematical Society . ( ), pp. – . issn : - .[ ] Camillo De Lellis. Rectifiable sets, densities and tangent measures . Zurich Lectures in Advanced Mathematics.European Mathematical Society (EMS), Zürich, , pp. vi+ . isbn : - - - - .[ ] Herbert Federer. Geometric measure theory . Die Grundlehren der mathematischen Wissenschaften, Band .Springer-Verlag New York Inc., New York, , pp. xiv+ .[ ] Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano. “On the structure of finite perimeter sets instep Carnot groups”. In:
J. Geom. Anal. . ( ), pp. – . issn : - .[ ] Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano. “Rectifiability and perimeter in the Heisenberggroup”. In: Math. Ann. . ( ), pp. – . issn : - .[ ] Bruno Franchi and Raul Paolo Serapioni. “Intrinsic Lipschitz graphs within Carnot groups”. In: J. Geom.Anal. . ( ), pp. – . issn : - . eferences [ ] Bruno Franchi, Raul Paolo Serapioni, and Francesco Serra Cassano. “Area formula for centered Hausdorffmeasures in metric spaces”. In: Nonlinear Anal. ( ), pp. – . issn : - X.[ ] Felix Hausdorff. Set theory . Second edition. Translated from the German by John R. Aumann et al. ChelseaPublishing Co., New York, , p. .[ ] Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, and Jeremy T. Tyson. Sobolev spaces on metricmeasure spaces . Vol. . New Mathematical Monographs. An approach based on upper gradients. CambridgeUniversity Press, Cambridge, , pp. xii+ . isbn : - - - - .[ ] Antoine Julia, Sebastiano Nicolussi Golo, and Davide Vittone. “Area of intrinsic graphs and coarea formulain Carnot groups”. In: arXiv: . ( ), arXiv: . .[ ] Bernd Kirchheim and David Preiss. “Uniformly distributed measures in Euclidean spaces”. In: Math. Scand. . ( ), pp. – . issn : - .[ ] Bernd Kirchheim and Francesco Serra Cassano. “Rectifiability and parameterization of intrinsic regular sur-faces in the Heisenberg group”. In: Ann. Sc. Norm. Super. Pisa Cl. Sci. ( ) . ( ), pp. – . issn : - X.[ ] Achim Klenke. Probability theory . Second. Universitext. A comprehensive course. Springer, London, ,pp. xii+ . isbn : - - - - ; - - - - .[ ] Lie groups and Lie algebras. I . Vol. . Encyclopaedia of Mathematical Sciences. Foundations of Lie theory.Lie transformation groups, A translation of ıt Current problems in mathematics. Fundamental directions.Vol. (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, [ MR ( m: )], Translation by A. Kozlowski, Translation edited by A. L. Onishchik. Springer-Verlag, Berlin, , pp. vi+ . isbn : - - - .[ ] Valentino Magnani. “A new differentiation, shape of the unit ball, and perimeter measure”. In: Indiana Univ.Math. J. . ( ), pp. – . issn : - .[ ] Valentino Magnani. “Unrectifiability and rigidity in stratified groups”. In: Arch. Math. (Basel) . ( ),pp. – . issn : - X.[ ] J. M. Marstrand. “Hausdorff two-dimensional measure in 3-space”. In: Proc. London Math. Soc. ( ) ( ),pp. – . issn : - .[ ] Pertti Mattila. Geometry of sets and measures in Euclidean spaces . Vol. . Cambridge Studies in AdvancedMathematics. Fractals and rectifiability. Cambridge University Press, Cambridge, , pp. xii+ . isbn : - - - ; - - - .[ ] Pertti Mattila. “Hausdorff m regular and rectifiable sets in n -space”. In: Trans. Amer. Math. Soc. ( ),pp. – . issn : - .[ ] Pertti Mattila. “Measures with unique tangent measures in metric groups”. In: Math. Scand. . ( ),pp. – . issn : - .[ ] Pertti Mattila, Raul Serapioni, and Francesco Serra Cassano. “Characterizations of intrinsic rectifiability inHeisenberg groups”. In: Ann. Sc. Norm. Super. Pisa Cl. Sci. ( ) . ( ), pp. – . issn : - X.[ ] Andrea Merlo. “Geometry of 1-codimensional measures in Heisenberg groups”. In: arXiv e-prints , arXiv: . (Aug. ), arXiv: . .[ ] David Preiss. “Geometry of measures in R n : distribution, rectifiability, and densities”. In: Ann. of Math. ( ) . ( ), pp. – . issn : - X.[ ] Alexander Schechter. “On the centred Hausdorff measure”. In: J. London Math. Soc. ( ) . ( ), pp. – . issn : - .[ ] Francesco Serra Cassano. “Some topics of geometric measure theory in Carnot groups”. In: Geometry, analysisand dynamics on sub-Riemannian manifolds. Vol. . EMS Ser. Lect. Math. Eur. Math. Soc., Zürich, , pp. – .[ ] Davide Vittone. “Lipschitz surfaces, perimeter and trace theorems for BV functions in Carnot-Carathéodoryspaces”. In: Ann. Sc. Norm. Super. Pisa Cl. Sci. ( ) . ( ), pp. – . issn : - X. eferences acknowledgements I am deeply grateful to Roberto Monti, without whose guidance and constant encouragment I would have neverbeen able to tackle the density problem in the Carnot setting. I would like to also thank Joan Verdera, who invitedme in Barcelona during the February of2019