On rectifiable measures in Carnot groups: structure theory
OO N R E C T I F I A B L E M E A S U R E S I NC A R N O T G R O U P S : S T R U C T U R E T H E O R Y g i oac c h i n o a n t o n e l l i * a n d a n d r e a m e r l o ** abstract In this paper we prove the one-dimensional Preiss’ theorem in the first Heisenberg group H . Moreprecisely we show that a Radon measure φ on H with positive and finite one-density with respect to the Koranyidistance is supported on a one-rectifiable set in the sense of Federer, i.e., it is supported on the countable union ofthe images of Lipschitz maps A ⊆ R → H .The previous theorem is a consequence of a Marstrand-Mattila type rectifiability criterion, which we prove inarbitrary Carnot groups for measures with tangent planes that admit a normal complementary subgroup. Namely,in this co-normal case, even if we a priori ask that the tangent planes at a point might rotate at different scales, aposteriori the measure has a unique tangent almost everywhere. Since every horizontal subgroup has a normalcomplement, our criterion applies in the particular case of one-dimensional horizontal subgroups.These results are the outcome of a detailed study of a new notion of rectifiability: we say that a Radon measureon a Carnot group is P h -rectifiable, for h ∈ N , if it has positive h -lower density and finite h -upper density almosteverywhere, and, at almost every point, it admits as tangent measures only (multiple of) the Haar measure of ahomogeneous subgroup of Hausdorff dimension h .We also prove several structure properties of P h -rectifiable measures. First, we compare P h -rectifiability withother notions of rectifiability previously known in the literature in the setting of Carnot groups and we realizethat it is strictly weaker than them. Furthermore, we show that a P h -rectifiable measure has almost everywherepositive and finite h -density whenever the tangents admit at least one complementary subgroup. Finally, we provethat a P h -rectifiabile measure with complemented tangents is supported on the union of intrinsically differentiablegraphs. keywords Carnot groups, Heisenberg groups, Rectifiability, Preiss’ Theorem, Rectifiable set, Rectifiable mea-sure, Marstrand-Mattila rectifiability criterion. msc ( ) C , E , A , Q , A . In Euclidean spaces a Radon measure φ is said to be k - rectifiable if it is absolutely continuous with respect tothe k -dimensional Hausdorff measure H k and it is supported on a countable union of k -dimensional Lipschitzsurfaces, for a reference see [ , § . . ]. This notion of regularity for a measure is an established, thoroughlystudied and well understood concept and its versatility is twofold. On the one hand it can be effortlessly extendedto general metric spaces. On the other, it can be shown, at least in Euclidean spaces, that the global regularityproperties arise as a consequence of the local structure of the measure, as it is clear from the following classicalproposition, see e.g., [ , Theorem . ]. Proposition . . Assume φ is a Radon measure on R n and k is a natural number such that ≤ k ≤ n. Then, φ is ak-rectifiable measure if and only if for φ -almost every x ∈ R n we have * Scuola Normale Superiore, Piazza dei Cavalieri, , Pisa, Italy, ** Univerité Paris-Saclay,
Rue Michel Magat Bâtiment,
Orsay, France. a r X i v : . [ m a t h . M G ] S e p ntroduction 2 (i) < Θ k ∗ ( φ , x ) ≤ Θ k , ∗ ( φ , x ) < + ∞ ,(ii) Tan k ( φ , x ) ⊆ { λ H k (cid:120) V : λ > and V is a k-dimensional vector subspace } ,where Θ k ∗ ( φ , x ) and Θ k , ∗ ( φ , x ) are, respectively, the lower and the upper k-density of φ at x, see Definition . , and Tan k ( φ , x ) is the set of k-tangent measures to φ at x, see Definition . , while H k is the Hausdorff measure. As mentioned above one can define rectifiable measures in arbitrary metric spaces: however one quickly under-stands that there are some limitations to what the classical rectifiability can achieve.The first example of this is the curve in L ([
0, 1 ]) that at each t ∈ [
0, 1 ] assigns the indicator function of theinterval [ t ] . This curve is Lipschitz continuous, however it fails to be Fréchet differentiable at every point of [
0, 1 ] ,thus not admitting a tangent. This shows that we cannot expect anything like Proposition . to hold in general.For the second example we need to briefly introduce Carnot groups, see Section for details. A Carnot group G is a simply connected nilpotent Lie group, whose Lie algebra is stratified and generated by its first layer. Carnotgroups are a generalization of Euclidean spaces, we remark that (quotients of) Carnot groups arise as the infinitesi-mal models of sub-Riemannian manifolds and their geometry, even at an infinitesimal scale, might be very differentfrom the Euclidean one. We endow G with an arbitrary left-invariant homogeneous distance d , we recall that anytwo of them are bi-Lipschitz equivalent. These groups have finite Hausdorff dimension, that is commonly denotedby Q , and any Lipschitz map f : R Q − → ( G , d ) has H Q − -null image, see for instance [ ] and [ , Theorem . ].This from an Euclidean perspective means that there are no Lipschitz regular parametrized one-codimensionalsurfaces inside ( G , d ) . However, as shown in the fundational paper [ ], in Carnot groups there is a well definednotion of finite perimeter sets and their boundary can be covered up to H Q − -negligible sets by countably many intrinsic C -regular hypersurfaces, C hypersurfaces from now on, see [ , Definition . ]. The success of theapproach attempted in [ ] has started an effort to study Geometric Measure Theory in sub-Riemannian Carnotgroups, and in particular to study various notion(s) of rectifiability, see [ , , , , , , , , , , , , , ]. The big effort represented by the aformentioned papers in trying to understand rectifiability in Carnot groupshas given rise to a multiplication of definitions, each one suiting some particular cases.As we shall see in the subsequent paragraphs, not only one could consider our approach reversed with respectto the ones known in the literature but it also has a twofold advantage. On the one hand the definition of P -rectifiable measure is natural and intrinsic with respect to the (homogeneous) structure of Carnot groups and itis equivalent to the usual one in the Euclidean setting; on the other hand we do not have to handle with theproblem of distingushing, in the definition, between the low-dimensional and the low-codimensional rectifiability.Nevertheless, we are able to prove, for arbitrary Carnot groups, strong structure results for rectifiable measures,see Section . . These structure results directly lead to the proof of the one-dimensional Preiss’ theorem in the firstHeisenberg group H , see Theorem . . . P -rectifiable measures In this paper we study structure results in the class of P -rectifiable measures, which have been introduced in[ , Definition . & Definition . ]. Definition . ( P -rectifiable measures) . Fix a natural number 1 ≤ h ≤ Q . A Radon measure φ on G is said to be P h -rectifiable if for φ -almost every x ∈ G we have(i) 0 < Θ h ∗ ( φ , x ) ≤ Θ h , ∗ ( φ , x ) < + ∞ ,(ii) Tan h ( φ , x ) ⊆ { λ S h (cid:120) V ( x ) : λ ≥ } , where V ( x ) is a homogeneous subgroup of G of Hausdorff dimension h ,where Θ h ∗ ( φ , x ) and Θ h , ∗ ( φ , x ) are, respectively, the lower and the upper h -density of φ at x , see Definition . ,Tan h ( φ , x ) is the set of h -tangent measures to φ at x , see Definition . , and S h is the spherical Hausdorff measureof dimension h , see Definition . . Furthermore, we say that φ is P ∗ h -rectifiable if (ii) is replaced with the weaker ( ii ) ∗ Tan h ( φ , x ) ⊆ { λ S h (cid:120) V : λ ≥
0, and V is a homogeneous subgroup of G Hausdorff dimension h } .If we impose more regularity on the tangents we can define different subclasses of P -rectifiable or P ∗ -rectifiablemeasures, see Definition . for details. We notice that, a posteriori, in the aformentioned definitions we can andwill restrict to λ >
0, see Remark . . ntroduction 3 The definition of P -rectifiable measure seems a natural one in the setting of Carnot groups. Indeed, we haveon G a natural family of dilations { δ λ } λ > , see Section , that we can use to give a good definition of blow-up ofa measure. Hence we ask, for a measure to be rectifiable, that the tangents are flat . The natural class of flat spaces,i.e., the analogous of vector subspaces of the Euclidean space, seems to be the class of homogeneous subgroupsof G . This latter assertion is suggested also from the result in [ , Theorem . ] according to which on everylocally compact group G endowed with dilations and isometric left translations, if a Radon measure µ has unique(up to multiplicative constants) tangent µ -almost everywhere then this tangent is µ -almost everywhere (up tomultiplicative constants) the left Haar measure on a closed dilation-invariant subgroup of G . As a consequence, inthe definition of P h -rectifiable measure we can equivalently substitute item (ii) of Definition . with the weakerrequirement ( ii ) (cid:48) Tan h ( φ , x ) ⊆ { λν x : λ > } , where ν x is a Radon measure on G .Moreover, it is interesting to stress that if a metric group is locally compact, isometrically homogeneous and admitsone dilation, as it is for the class of metric group studied in [ ], and moreover the distance is geodesic, then it isa sub-Finsler Carnot group, see [ , Theorem . ].As already mentioned, according to one of the approaches to rectifiability in Carnot groups, the good parametriz-ing objects for the notion of rectifiability are C -regular surfaces with complemented tangents in G , i.e., sets thatare locally the zero-level sets of C functions f - see Definition . - with surjective Pansu differential d f andsuch that Ker ( D f ) admits a complementary subgroup in G . This approach has been taken to its utmost level ofgenerality through the works [ , , ]. In particular in [ , Definition . ] the authors give the most general,and available up to now, definition of ( G , G (cid:48) ) -rectifiable sets, see Definition . and Definition . , and they provearea and coarea formulae within this class of rectifiable sets. We remark that our definition is strictly weaker thanthe one in [ ], see Proposition . and Remark . . Moreover for discussions on the converse of the followingProposition . we refer the reader to Remark . . Proposition . . Let us fix G and G (cid:48) two arbitrary Carnot groups of homogeneous dimensions Q and Q (cid:48) respectively. Let ustake Σ ⊆ G a ( G , G (cid:48) ) -rectifiable set. Then S Q − Q (cid:48) (cid:120) Σ is a P Q − Q (cid:48) -rectifiable measure with complemented tangents. Moreover,there exists G a Carnot group, Σ ⊆ G , and ≤ h ≤ Q such that S h (cid:120) Σ is a P h -rectifiable measure and, for every Carnotgroup G (cid:48) , Σ is not ( G , G (cid:48) ) -rectifiable. Let us stress that the second part of Proposition . is not surprising. Indeed, the approach to rectifiabilitydescribed above and used in [ ] is selecting rectifiable sets whose tangents are complemented normal subgroupsof G , see [ , Section . ] for a more detailed discussion. This can be easily understood if one thinks that, ac-cording to this approach to rectifiability, the parametrizing class of objects is given by C -regular surfaces Σ withcomplemented tangents, whose tangent at p ∈ Σ , being Ker ( d f p ) , is a complemented normal subgroup of G .In some sense we could say that the approach of [ ] is covering, in the utmost generality known up to now,the case of low-codimensional rectifiable sets in a Carnot group G . It has been clear since the works [ , ] that,already in the Heisenberg groups H n , one should approach the low-dimensional rectifiability in a different waywith respect to the low-codimensional one. Indeed, in the low-dimensional case in H n , the authors in [ , ]choose as a parametrizing class of objects the images of C -regular (or Lipschitz-regular) functions from subsetsof R d to H n , with 1 ≤ d ≤ n , see [ , Definition . & Definition . ], and [ , Definition . and Definition . ].The bridge between the definition of P -rectifiability and the ones disscused above is done in [ ] in the settingof Heisenberg groups and in [ ] in arbitrary groups but in the case of horizontal tangents . Let us stress that theresult in [ , (i) ⇔ (iv) of Theorem . ] shows precisely that on the Heisenberg groups the P -rectifiability withtangents that are vertical subgroups is equivalent to the rectifiability given in terms of C -regular surfaces. More-over [ , (i) ⇔ (iv) of Theorem . ] shows that on the Heisenberg groups the P -rectifiability with tangents that arehorizontal subgroups is equivalent to the rectifiability given in terms of Lipschitz-regular images. Moreover, veryrecently, in [ , Theorem . ], the authors prove a generalization of [ , Theorem . ] in arbitrary homogeneousgroups. Namely they prove that in a homogeneous group the k -rectifiability of a set in the sense of Federer canbe characterized with the fact that the tangent measures to the set are horizontal subgroups, or equivalently withthe fact that there exists an approximate tangent that is a horizontal subgroup almost everywhere. In our settingthis implies that the P -rectifiability with tangents that are horizontal subgroups is equivalent to the rectifiabilitygiven in terms of Lipschitz-regular images, which is Federer’s one. We stress that the latter results leave open the ntroduction 4 challenging question of understanding what is the precise structure of a measure φ on H such that the tangentsare φ -almost everywhere the vertical line. . Main results
One of the main contributions of this paper is a co-normal Marstrand-Mattila rectifiability criterion in the settingof Carnot groups. Already in the Euclidean case, it is not trivial to prove that a P ∗ -rectifiable measure is rectifiable,see [ , Theorem . ], and [ , Theorem . ]. Proving that a P ∗ Q − -rectifiable measure in a Carnot group ofHausdorff dimension Q is supported on the countably union of C -regular hypersurfaces is a challenging problemthat has been solved by the second-named author in [ , Theorem ]. In this paper we adapt Preiss’ techniquesin [ ] to prove that P ∗ -rectifiable measures with co-normal tangents, i.e., with tangents that admit a normalcomplementary subgroup , are P -rectifiable, see Theorem . . This means that in this co-normal case, even ifthe tangents at a point might a priori rotate at different scales, then a posteriori the tangent is unique almosteverywhere. We recall that when we say that a homogeneous subgroup V admits a complementary subgroup , wemean that there exists a homogeneous subgroup L such that G = V · L and V ∩ L = { } . Theorem . (Co-normal Marstrand-Mattila rectifiability criterion) . Let φ be a P ∗ h -rectifiable measure on G withtangents that admit at least one normal complementary subgroup . Then φ is a P h -rectifiable measure. Furthermore,at φ -almost every point x ∈ G the tangent measure is unique and it is a Haar measure of some homogeneous subgroup V ( x ) of Hausdorff dimension h. Moreover, there are countably many homogeneous subgroups V i and Lipschitz maps Φ i : A i ⊆ V i → G , where A i ’s are compact, such that φ (cid:16) G \ (cid:91) i ∈ N Φ i ( A i ) (cid:17) = . because of the following key observations: whenever V admits a normal complementary subgroup L , then the projection P V : G → V related to this splitting is aLipschitz homogeneous homomorphism, see Proposition . , and moreover V is a Carnot subgroup, see [ ,Remark . ]. This allows us to adapt Preiss’ strategy in [ ] not without some difficulties that are essentially dueto the fact that, on the contrary with respect to the Euclidean setting, we do not have a canonical choice of anormal complementary subgroup to V when it eventually admits one. For further discussions about Marstrand-Mattila rectifiability criterion in the very different codimension-one case we refer the reader to Remark . . We alsostress that, for the Marstrand-Mattila rectifiability criterion, the assumption on the strictly positive lower densityis necessary already in the Euclidean case, see [ , Example . ].The hypotheses of Theorem . are satisfied whenever we have a P ∗ h -rectifiable measure with horizontal tan-gents. Thus, the previous co-normal Marstrand-Mattila rectifiability criterion, jointly with the result of [ , Theorem . ], can be used to give the proof of Preiss’ theorem for measures with one-density in the Heisenberg group H endowed with the Koranyi norm. For the sake of clarity, let us recall that if we identify H ≡ R = { ( x , t ) : x ∈ R , t ∈ R } through exponential coordinates, then the Koranyi norm is (cid:107) ( x , t ) (cid:107) : = ( (cid:107) x (cid:107) + t ) , where (cid:107) · (cid:107) eu isthe standard Euclidean norm. Theorem . (One-dimensional Preiss’ theorem in H ) . Let H be the first Heisenberg group endowed with the Koranyinorm. Let φ be a Radon measure on H such that the one-density Θ ( φ , x ) exists positive and finite for φ -almost every x ∈ H . Then H can be covered φ -almost all with countably many images Φ i ( A i ) of Lipschitz functions Φ i : A i ⊆ R → H .Proof. From the fact that the one-density exists at φ -almost every x ∈ H we deduce that at φ -almost every x ∈ H the tangent measures are uniform measures, see [ , Proposition . ]. Then from [ , Theorem . ] we get that thetangent measures, at φ -almost every x ∈ H , are S (cid:120) L , where L is a horizontal line. Finally from Theorem . ,since every horizontal line admits a normal complementary subgroup, we get the sought conclusion.Let us remark that the latter theorem is the last step needed to completely solve in H the implication (i) ⇒ (ii)of the density problem formulated in [ , page ]. Let us explain this and give a scheme here. If in H endowedwith the Koranyi norm we have a Radon measure φ such that there exists α ≥ α -density Θ α ( φ , x ) exists positive and finite for φ -almost every x ∈ H we first get that α is an integer, see [ , Theorem . ]. Thus theonly non-trivial cases are ntroduction 5 • α =
1. In this case φ is P -rectifiable, see Theorem . . Moreover we can cover φ -almost all of H withcountably many images of Lipschitz maps from subsets of R to H . Note that we can improve the latterconclusion. Indeed, we can cover φ -almost all of H with countably many images of C -functions definedon open subsets of R to H . This last improvement comes from Pansu-Rademacher theorem for Lipschitzmaps between Carnot groups, see [ ], and the Whitney exstension theorem proved in [ , Theorem . ]. • α =
2. In this case φ is P -rectifiable, see [ , Theorem . ]. This means that the tangent measure is φ -almosteverywhere unique and it is a Haar measure of the vertical line in H . • α =
3. In this case φ is P -rectifiable, see [ , Theorem ]. Moreover we can cover φ -almost G with countablymany C -hypersurfaces, see [ , Theorem ].As it is clear from the list above, an interesting problem could be a finer study of the structure of P -rectifiablemeasures in H .Another main contribution of this paper is the proof of the fact that a P h -rectifiable measure with complementedtangents has density, see Corollary . , and Remark . . Theorem . (Existence of the density) . Let φ be a P h -rectifiable measure with complemented tangents on G andassume d is a homogeneous left-invariant metric on G . Then, if B d ( x , r ) is the closed metric ball relative to d of centre x andradius r, for φ -almost every x ∈ G we have < lim inf r → φ ( B d ( x , r )) r h = lim sup r → φ ( B d ( x , r )) r h < + ∞ . Moreover, for φ -almost every x ∈ G we haver − h T x , r φ (cid:42) Θ h ( φ , x ) C h (cid:120) V ( x ) , as r goes to where T x , r is defined in Definition . , the convergence is understood in the duality with the continuous functions withcompact support on G , Θ h ( φ , x ) is the h-density with respect to the smooth-box norm (cid:107)·(cid:107) , see Definition . , and C h (cid:120) V ( x ) is the h-dimensional centered Hausdorff measure, with respect to the smooth-box norm, restricted to V ( x ) , see Definition . . Let us remark that the previous theorem solves the implication (ii) ⇒ (i) of the density problem formulated in[ , page ] in the setting of P h -rectifiable measures with complemented tangents. In Euclidean spaces the proofof Theorem . is an almost immediate consequence of the fact that projections on linear spaces are 1-Lipschitzin conjunction with the area formula. In our context we neither have at our disposal the Lipschitz property ofprojections nor an area formula for P h -rectifiable measures with complemented tangents, so the proof requirenew ideas. In order to obtain Theorem . first of all one reduces to the case of the surface measure on anintrinsically Lipschitz graph with very small Lipschitz constant thanks to the structure result Theorem . below.Secondly, one needs to show that the surface measures of the tangents and their push-forward on the graphare mutually absolutely continuous. For this last point to hold it will be crucial on the one hand that a P h -rectifiable measure with complemented tangents can be covered almost everyhwere with intrinsic graphs , see theforthcoming Theorem . , on the other hand that intrinsic Lipschitz graphs have big projections on their tangents,see Proposition . . Third, one exploits the fact that the density exists for the tangents to infer its existence for theoriginal measure.The last contributions of this paper are structure results for P -rectifiable measures. Since they will be givenin terms of sets that satisfy a cone property, let us clarify which cones we are choosing. For any α > V of G , the cone C V ( α ) is the set of points w ∈ G such that dist ( w , V ) ≤ α (cid:107) w (cid:107) , where (cid:107) · (cid:107) is the homogeneous norm relative to the fixed distance d on G . Moreover a set E ⊆ G is a C V ( α ) -set if E ⊆ pC V ( α ) for every p ∈ E . We refer the reader to Section . for such definitions and some properties of them. We stress thatthe cones C V ( α ) are used to give the definition of intrinsically Lispchitz graphs and functions, see [ , Definition and Proposition . ]. The first result reads as follows, see Theorem . . ntroduction 6 Theorem . (Structure : covering with sets with the cone property) . Let φ be a P h -rectifiable measure on G . Then G can be covered φ -almost everywhere with countably many compact sets with the cone property with arbitrarily small opening.In other words for every α > we have φ (cid:18) G \ + ∞ (cid:91) i = Γ i (cid:19) = where Γ i are compact C V i ( α ) -sets, where V i are homogeneous subgroups of G of Hausdorff dimension h. If we ask that the tangents are complemented subgroups, we can improve the previous result. In particular wecan take the Γ i ’s to be intrinsic graphs , see Theorem . and Proposition . . For the definition of intrinsicallyLipschitz function, we refer the reader to Definition . . Let us remark that the fact that the Γ i ’s can be taken tobe graphs will be crucial for the proof of the existence of the density in Theorem . . Theorem . (Structure : covering with intrinsically Lipschitz graphs) . Let φ be a P h -rectifiable measure withcomplemented tangents on G . Then G can be covered φ -almost everywhere with countably many compact graphs ofintrinsically Lipschitz functions with arbitrarily small Lipschitz constant. In other words for every α > we have φ (cid:18) G \ + ∞ (cid:91) i = Γ i (cid:19) = where Γ i = graph ( ϕ i ) are compact sets, with ϕ i : A i ⊆ V i → L i being an intrinsically α -Lipschitz function between acompact subset A i of V i , which is a homogeneous subgroup of G of Hausdorff dimension h, and L i , which is a subgroupcomplementary to V i . By pushing a little bit further the information about the fact that the tangent measures at φ -almost every x areconstant multiples of S h (cid:120) V ( x ) , we can give a structure result within the class of intrinsically differentiable graphs.Roughly speaking we say that the graph of a function between complementary subgroups ϕ : U ⊆ V → L is intrinsically differentiable at a · ϕ ( a ) if graph ( ϕ ) admits a homogeneous subgroup as Hausdorff tangent at a · ϕ ( a ) , see Definition . for details. For the forthcoming theorem, see Corollary . . Theorem . (Structure : covering with intrinsically differentiable graphs) . Let φ be a P h -rectifiable measure withcomplemented tangents on G . Then G can be covered φ -almost everywhere with countably many compact graphs that areintrinsically differentiable almost everywhere . In other words φ (cid:18) G \ + ∞ (cid:91) i = Γ i (cid:19) = where Γ i = graph ( ϕ i ) are compact sets, with ϕ i : A i ⊆ V i → L i being a function between a compact subset A i of V i , whichis a homogeneous subgroup of G of Hausdorff dimension h, and L i , which is a subgroup complementary to V i ; in addition graph ( ϕ i ) is an intrinsically differentiable graph at a · ϕ i ( a ) for S h (cid:120) A i -almost every a ∈ V i . Let us briefly remark that if a Rademacher-type theorem holds, i.e., if an intrinsically Lipschitz function isintrinsically differentiable almost everywhere, the latter theorem would simply come from Theorem . . We remarkthat a Rademacher-type theorem at such level of generality, i.e., between arbitrary complementary subgroups of aCarnot group, is not available up to now. On the other hand, some results in particular cases have been providedin [ , , ] for intrinsically Lipschitz functions with one-dimensional target in groups in which De Giorgi C -rectifiability for finite perimeter sets holds, and for functions with normal targets in arbitrary Carnot groups. Westress that very recently in [ ] the author proves the Rademacher theorem at any codimension in the Heisenberggroups H n .Let us briefly comment on the results listed above. Theorem . generalizes the implication in [ , (iv) ⇒ (ii)of Theorem . ] to the setting of P h -rectifiable measures whose tangents are complemented in arbitrary Carnotgroups. Indeed, in [ , (iv) ⇒ (ii) of Theorem . ] the authors prove that if n + ≤ h ≤ n , and S h (cid:120) Γ is a P h -rectifiable measure with tangents that are vertical subgroups in the Heisenberg group H n , then the h -density of S h (cid:120) Γ exists almost everywhere and the tangent is unique almost everywhere. The analogous property in H n , but ntroduction 7 with P h -rectifiable measures with tangents that are horizontal subgroups is obtained in [ , (iv) ⇒ (ii) of Theorem . ], and in arbitrary homogeneous groups in the recent [ , (iii) ⇒ (ii) of Theorem . ]. However, in this specialhorizontal case treated in [ , Theorem . ] and [ , Theorem . ] the authors do not assume Θ h ∗ ( S h (cid:120) Γ , x ) > , Theorem . ], while the authors in [ ] areable to overcome this issue by adapting [ , Lemma . . ] in [ , Theorem . ]. We do not address in this paperthe problem of obtaining the same general results as in Theorem . , Theorem . , Theorem . , and Theorem . removing the hypothesis on the strictly positive lower density in item (i) of Definition . when the tangent isunique (up to a mutiplicative constant).We also mention that, in some particular cases, one could prove the converse implications of Theorem . ,Theorem . , and Theorem . . These converse implications are of the same type that are proved, in the specificcase of H n , in [ , (i) ⇒ (iv) of Theorem . and Theorem . ] and in arbitrary homogeneous groups but withhorizontal tangents in [ , (i) ⇒ (iii) of Theorem . ]. For example if a set Γ is, up to null sets, the countable union of C ( G , G (cid:48) ) -surfaces, then S h (cid:120) Γ is P h -rectifiable, see Proposition . . Moreover, if Γ is, up to null sets, the countableunion of Lipschitz images of functions between subsets of V to G , where V is a Carnot subgroup of G , then S h (cid:120) Γ is P h -rectifiable. We do not discuss explicitly this latter statement, but it comes from the final argument of theproof of Theorem . . Acknowledgments : The first author is partially supported by the European Research Council (ERC StartingGrant
GeoMeG ‘
Geometry of Metric Groups ’). notation
We add below a list of frequently used notations, together with the page of their first appearance: (cid:107)·(cid:107) smooth-box norm, V i layers of the stratification of the Lie algebra of G , κ number of layers 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 , Q i coefficients of the coordinate representation of the group operation δ λ intrinsic dilations, T x , r φ dilated of a factor r > φ at the point x ∈ G , Tan h ( φ , x ) set of h -dimensional tangent measures to the measure φ at x , (cid:42) weak convergence of measures, Gr ( h ) the h -dimensional Grassmanian, i.e., the family of homogeneous subgroups of G of homogeneous dimension h , Gr c ( h ) the h -dimensional complemented Grassmanian, i.e., the family of homogeneouscomplemented subgroups of G of homogeneous dimension h , Gr (cid:69) ( h ) the h -dimensional co-normal Grassmanian, i.e., the family of homogeneous sub-groups of homogeneous dimension h of G complemented by one normal sub-group of G , s ( V ) the stratification vector relative to the homogeneous subgroup V , Gr s (cid:69) ( h ) the family of subgroups of Gr (cid:69) ( h ) having stratification vector s , P h class of Radon measures with dimension h and unique flat tangent almost ev-erywhere, P ∗ h class of Radon measure with dimension h and flat tangents almost everywhere, P ch P h -rectifiable measures with complemented tangents, P (cid:69) , ∗ h P ∗ h -rectifiable measures with co-normal tangents, P (cid:69) , ∗ , s h P ∗ h -rectifiable measures with co-normal tangents with stratification s , S ( h ) the set of all stratification vectors relative to the h -dimensional homogeneoussubgroups of G , reliminaries 8 P V splitting projection onto the homogeneous subgroup V , T ( u , r ) cylinder of centre u and radius r with co-normal axis, e ( V ) function that measures the minimum possible deformation of splitting projec-tions on the homogeneous complemented subgroup V , M ( h ) the set of the Haar measures of the elements of Gr ( h ) , M ( h , G ) the set of the Haar measures of the elements of G ⊆ Gr ( h ) , C W ( α ) cone of amplitude α with axis W , Lip + ( K ) non-negative 1-Lipschitz functions with support contained in the compact set K . S h h -dimensional spherical Hausdorff measure, C h h -dimensional centred spherical Hausdorff measure, H h h -dimensional Hausdorff measure, H h eu Euclidean h -dimensional Hausdorff measure, Θ h ∗ ( φ , x ) h -dimensional lower density of the measure φ at x , Θ h , ∗ ( φ , x ) h -dimensional upper density of the measure φ at x , d x , r ( · , M ) distance of the Radon measure φ inside the ball B ( x , r ) from flat measures, F K ( · , · ) metric on measures on the compact set K ⊆ G , d G , H ( · , · ) Hausdorff distance between closed sets in G , d G ( · , · ) distance between homogeneous subgroups,
132 preliminaries2 . 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 we refer to [ ].A Carnot group G of step κ is a connected and simply connected Lie group whose Lie algebra g admits astratification g = V ⊕ V ⊕ · · · ⊕ V κ . We say that V ⊕ V ⊕ · · · ⊕ V κ is a stratification of g if g = V ⊕ V ⊕ · · · ⊕ V κ , [ V , V i ] = V i + , for any i =
1, . . . , κ −
1, and [ V , V κ ] = { } ,where [ A , B ] : = span { [ a , b ] : a ∈ A , b ∈ B } . We call V the horizontal layer of G . We denote by n the topologicaldimension of g , by n j the dimension of V j for every j =
1, . . . , κ . Furthermore, we define π i : G → V i to be theprojection maps on the i -th strata. We will often 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 · thegroup operation determined by the Baker-Campbell-Hausdorff formula. From now on, we will always assumethat 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 κ ) : = ( λ v , λ v , . . . , λ κ v κ ) , for any λ > v i ∈ V i . The stratification of the Lie algebra g naturally induces a gradation on each of its homogeneous Liesub-algebras h , i.e., a sub-algebra that is δ λ -invariant for any λ >
0, that is h = V ∩ h ⊕ . . . ⊕ V κ ∩ h . ( )We say that h = W ⊕ · · · ⊕ W κ is a gradation of h if [ W i , W j ] ⊆ W i + j for every 1 ≤ i , j ≤ κ , where we mean that W (cid:96) : = { } for every (cid:96) > κ . Since the exponential map acts as the identity, the Lie algebra automorphisms δ λ arealso group 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 . reliminaries 9 We recall the following basic terminology: a horizontal subgroup of a Carnot group G is a homogeneous subgroupof it that is contained in exp ( V ) ; a Carnot subgroup W = exp ( h ) of a Carnot group G is a homogeneous subgroupof it such that the first layer V ∩ h of the grading of h inherited from the stratification of g is the first layer of astratification of h .Homogeneous Lie subgroups of G are in bijective correspondence through exp with the Lie sub-algebras of g that are invariant under the dilations δ λ . For any Lie algebra h with gradation h = W ⊕ . . . ⊕ W κ , we define its homogeneous dimension as dim hom ( h ) : = κ ∑ i = i · dim ( W i ) .Thanks to ( ) we infer that, if h is a homogeneous Lie sub-algebra of g , we have dim hom ( h ) : = ∑ κ i = i · dim ( h ∩ V i ) .It is well-known that the Hausdorff dimension (for a definition of Hausdorff dimension see for instance [ ,Definition . ]) of a graded Lie group G with respect to a left-invariant homogeneous distance coincides with thehomogeneous dimension of its Lie algebra. For a reference for the latter statement, see [ , Theorem . ]. Fromnow on, if not otherwise stated, G will be a fixed Carnot group .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, the group operation · is determined by the Campbell-Hausdorff formula, and it hasthe form (see [ , Proposition . ]) p · q = p + q + Q ( p , q ) , for all p , q ∈ R n ,where Q = ( Q , . . . , Q κ ) : R n × R n → V ⊕ . . . ⊕ V κ , and the Q i ’s have the following properties. For any i =
1, . . . κ 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 · as p · q = ( p + q , p + q + Q ( p , q ) , . . . , p κ + q κ + Q κ ( p , . . . , p κ − , q , . . . , q κ − )) . ( ) Definition . (Homogeneous left-invariant distance) . A metric d : G × G → R is said to be homogeneous and leftinvariant if for any x , y ∈ G we have(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. We remark that two homogeneous left-invariant distances on a Carnot group arealways bi-Lipschitz equivalent. Definition . (Smooth-box metric) . For any g ∈ G , we let (cid:107) g (cid:107) : = max { ε | g | , ε | g | , . . . , ε κ | g κ | κ } ,where ε = ε , . . . ε κ are suitably small parameters depending only on the group G . For the proof of thefact that we can choose the ε i ’s in such a way that (cid:107)·(cid:107) is a left invariant, homogeneous norm on G that induces adistance we refer to [ , Section ]. Furthermore, we define d ( x , y ) : = (cid:107) x − · y (cid:107) ,and let U ( x , r ) : = { z ∈ G : d ( x , z ) < r } be the open metric ball relative to the distance d centred at x at radius r >
0. The closed ball will be denoted with B ( x , r ) : = { z ∈ G : d ( x , z ) ≤ r } . Moreover, for a subset E ⊆ G and r >
0, we denote with B ( E , r ) : = { z ∈ G : dist ( z , E ) ≤ r } the closed r-tubular neighborhood of E and with U ( E , r ) : = { z ∈ G : dist ( z , E ) < r } the open r-tubular neighborhood of E . reliminaries 10 The following estimate on the norm of the conjugate will be useful later on.
Lemma . . For any k > there exists a constant C : = C ( k , G ) > such that if x , y ∈ B ( k ) , then: (cid:107) y − · x · y (cid:107) ≤ C (cid:107) x (cid:107) κ . Proof.
This follows immediately from [ , Lemma . ]. Definition . (Hausdorff Measures) . Throughout the paper we define the h -dimensional spherical Hausdorff mea-sure relative to the left invariant homogeneous metric d as S h ( A ) : = sup δ > inf (cid:26) ∞ ∑ j = r hj : A ⊆ ∞ (cid:91) j = B ( x j , r j ) , r j ≤ δ (cid:27) .We define the h -dimensional Hausdorff measure relative to d as H h ( A ) : = sup δ > inf ∞ ∑ j = − h ( diam E j ) h : A ⊆ ∞ (cid:91) j = E j , diam E ≤ δ .We define the h -dimensional centered Hausdorff measure relative to d as C h ( A ) : = sup E ⊆ A C ( E ) ,where C h ( E ) : = sup δ > inf (cid:26) ∞ ∑ j = r mj : E ⊆ ∞ (cid:91) j = B ( x j , r j ) , x j ∈ E , r j ≤ δ (cid:27) .We stress that C h is an outer measure, and thus it defines a Borel regular measure, see [ , Proposition . ], andthat the measures S h , H h , C h are all equivalent measures, see [ , Section . . ] and [ , Proposition . ]. Definition . (Hausdorff distance) . For any couple of sets in A , B ⊆ G , we define the Hausdorff distance of A from B as: d H , G ( A , B ) : = max (cid:110) sup x ∈ A dist ( x , B ) , sup y ∈ B dist ( A , y ) (cid:111) .Furthermore, for any compact set K in G , we define F ( K ) : = { A ⊆ K : A is compact } . . Densities and tangents of Radon measures
Throughout the rest of the paper we will always assume that G is a fixed Carnot group endowed with thesmooth box norm (cid:107) · (cid:107) , defined in Definition . , which induces a left-invariant homogeneous distance d . The ho-mogeneous, and thus Hausdorff, dimensione with respect to d will be denoted with Q . Furthermore as discussedin the previous subsection, we will assume without loss of generality that G coincides with R n endowed with theproduct induced by the Baker-Campbell-Hausdorff formula relative to Lie ( G ) . Definition . (Weak convergence of measures) . Given a family { φ i } i ∈ N of Radon measures on G we say that φ i weakly converges to a Radon measure φ , and we write φ i (cid:42) φ , if ˆ f d φ i → ˆ f d φ for any f ∈ C c ( G ) . reliminaries 11 Definition . (Tangent measures) . Let φ be a Radon measure on G . For any x ∈ G and any r > T x , r φ ( E ) : = φ ( x · δ r ( E )) , for any Borel set E .Furthermore, we define Tan h ( φ , x ) , the h -dimensional tangents to φ at x , to be the collection of the Radon measures ν for which there is an infinitesimal sequence { r i } i ∈ N such that r − hi T x , r i φ (cid:42) ν . Remark . . (Zero as a tangent measure) We remark that our definition potentially admits the zero measure as atangent measure, as in [ ], while the definitions in [ ] and [ ] do not. Definition . (Lower and upper densities) . If φ is a Radon measure on G , and α >
0, we define Θ h ∗ ( φ , x ) : = lim inf r → φ ( B ( x , r )) r h , and Θ h , ∗ ( φ , x ) : = lim sup r → φ ( B ( x , r )) r h ,and we say that Θ h ∗ ( φ , x ) and Θ h , ∗ ( φ , x ) are the lower and upper h -density of φ at the point x ∈ G , respectively.Furthermore, we say that measure φ has h -density if0 < Θ h ∗ ( φ , x ) = Θ h , ∗ ( φ , x ) < ∞ , for φ -almost any x ∈ G .A very useful property of measures with positive lower density and finite upper density is that Lebesguetheorem holds and thus local properties are stable under restriction to Borel subsets. Proposition . . Suppose φ is a Radon measure on G such that < Θ h ∗ ( φ , x ) ≤ Θ h , ∗ ( φ , x ) < ∞ for φ -almost every x ∈ G .Then, for any Borel set B ⊆ G and for φ -almost every x ∈ B we have Θ h ∗ ( φ (cid:120) B , x ) = Θ h ∗ ( φ , x ) and Θ h , ∗ ( φ (cid:120) B , x ) = Θ h , ∗ ( φ , x ) . Proof.
This is a direct consequence of Lebesgue differentiation Theorem of [ , page ], that can be applied since ( G , d , φ ) is a Vitali metric measure space due to [ , Theorem . . ].We stress that whenever the h -lower density of φ is stricly positve and the h -upper density of φ is finite φ -almosteverywhere, the set Tan h ( φ , x ) is nonempty for φ -almost every x ∈ G , see [ , Proposition . ]. The followingproposition has been proved in [ , Proposition . ]. Proposition . (Locality of tangents) . Let α > , and let φ be a Radon measure such that for φ -almost every x ∈ G wehave < Θ h ∗ ( φ , x ) ≤ Θ h , ∗ ( φ , x ) < ∞ . Then for every ρ ∈ L ( φ ) that is nonnegative φ -almost everywhere we have Tan h ( ρφ , x ) = ρ ( x ) Tan h ( φ , x ) for φ -almostevery x ∈ G . More precisely the following holds: for φ -almost every x ∈ G thenif r i → is such that r − hi T x , r i φ (cid:42) ν then r − hi T x , r i ( ρφ ) (cid:42) ρ ( x ) ν . ( )Let us introduce a useful split of the support of a Radon measure φ on G . Definition . . Let φ be a Radon measure on G that is supported on the compact set K . For any ϑ , γ ∈ N wedefine E ( ϑ , γ ) : = (cid:8) x ∈ K : ϑ − r h ≤ φ ( B ( x , r )) ≤ ϑ r h for any 0 < r < γ (cid:9) . ( ) Proposition . . For any ϑ , γ ∈ N , the set E ( ϑ , γ ) defined in Definition . is compact.Proof. This is [ , Proposition . ]. Proposition . . Assume φ is a Radon measure supported on the compact set K such that < Θ h ∗ ( φ , x ) ≤ Θ h , ∗ ( φ , x ) < ∞ for φ -almost every x ∈ G . Then φ ( G \ (cid:83) ϑ , γ ∈ N E ( ϑ , γ )) = . reliminaries 12 Proof.
Let w ∈ K \ (cid:83) ϑ , γ E ( ϑ , γ ) and note that this implies that either Θ h ∗ ( φ , x ) = Θ h , ∗ ( φ , x ) = ∞ . Since0 < Θ h ∗ ( φ , x ) ≤ Θ h , ∗ ( φ , x ) < ∞ for φ -almost every x ∈ G , this concludes the proof.We recall here a useful proposition about the structure of Radon measures Proposition . ([ , Proposition . and Corollary . ]) . Let φ be a Radon measure supported on a compact set on G such that < Θ h ∗ ( φ , x ) ≤ Θ h , ∗ ( φ , x ) < ∞ for φ -almost every x ∈ G . For every ϑ , γ ∈ N we have that φ (cid:120) E ( ϑ , γ ) ismutually absolutely continuous with respect to S h (cid:120) E ( ϑ , γ ) . . Intrinsic Grassmannian in Carnot groups
Let us recall the definition of the Euclidean Grassmannian, along with some of its properties.
Definition . (Euclidean Grassmannian) . Given k ≤ n we let Gr ( n , k ) to be the set of the k -vector subspaces of R n . We can endow Gr ( n , k ) with the following distance d eu ( V , V ) : = d H ,eu ( V ∩ B eu (
0, 1 ) , V ∩ B eu (
0, 1 )) ,where B eu (
0, 1 ) is the (closed) Euclidean unit ball, and d H ,eu is the Hausdorff distance between sets induced by theEuclidean distance on R n . Remark . (Euclidean Grassmannian and convergence) . It is well-known that ( Gr ( n , k ) , d H ,eu ) is a compact metricspace. Moreover, the following hold(i) if V n → V , then for every v ∈ V there exist v n ∈ V n such that v n → v ;(ii) if V n → V and there is a sequence v n ∈ V n such that v n → v , then v ∈ V .The proof of the two items below is left to the reader as an exercise. It is a simple outcome of the definition ofHausdorff distance.We now give the definition of the intrinsic Grassmannian on Carnot groups and introduce the classes of com-plemented and co-normal homogeneous subgroups. Definition . (Intrinsic Grassmanian on Carnot groups) . For any 1 ≤ h ≤ Q , we define Gr ( h ) to be the familyof homogeneous subgroups V of G that have Hausdorff dimension h .Let us recall that if V is a homogeneous subgroup of G , any other homogeneous subgroup such that V · L = G and V ∩ L = { } .is said to be a complement of G . Finally, we let(i) Gr c ( h ) to be the subfamily of those V ∈ Gr ( h ) that have a complement and we will refer to Gr c ( h ) as the h -dimensional complemented Grassmanian,(ii) Gr (cid:69) ( h ) the subfamily of those V ∈ Gr c ( h ) having a normal complement and we will refer to Gr (cid:69) ( h ) as the h -dimensional co-normal Grassmanian.Let us introduce the stratification vector of a homogeneous subgroup.
Definition . (Stratification vector) . Let h ∈ {
1, . . . , Q } and for any V ∈ Gr ( h ) we denote with s ( V ) the vector s ( V ) : = ( dim ( V ∩ V ) , . . . , dim ( V κ ∩ V )) ,that with abuse of language we call the stratification , or the stratification vector , of V . Furthermore, we define S ( h ) : = { s ( V ) ∈ N κ : V ∈ Gr ( h ) } .We remark that the cardinality of S ( h ) is bounded by ∏ κ i = ( dim V i + ) for any h ∈ {
1, . . . , Q } . reliminaries 13 Definition . ( s - co - normal Grassmannian) . For any s ∈ S ( h ) we let Gr s (cid:69) ( h ) : = { V ∈ Gr (cid:69) ( h ) : s ( V ) = s } ,and we will refer to Gr s (cid:69) ( h ) as the s - co - normal Grassmannian.We now collect in the following some topological properties of the Grassmanians introduced above.
Proposition . (Compactness of the Grassmannian) . For any ≤ h ≤ Q the functiond G ( W , W ) : = d H , G ( W ∩ B (
0, 1 ) , W ∩ B (
0, 1 )) , with W , W ∈ Gr ( h ) , is a distance on Gr ( h ) . Moreover ( Gr ( h ) , d G ) is a compact metric space.Proof. The fact that d G is a distance comes from well-known properties of the Hausdorff distance. Let us considera sequence { W j } j ∈ N ⊆ Gr ( h ) , with W j = W j ,1 ⊕ · · · ⊕ W j , κ , where W j , i : = V i ∩ W j for any j ∈ N and 1 ≤ i ≤ κ .By extracting a (non re-labelled) subsequence we can suppose that there exist { k i } i = κ natural numbers suchthat the topological dimension is dim W j , i = k i for all j ∈ N , and for all 1 ≤ i ≤ κ . In particular the topologicaldimension of W j is constant. Exploiting the compactness of the Euclidean Grassmannian, see Remark . , we getthat up to a (non re-labelled) subsequence, W j , i → W i , i.e. d eu ( W j , i , W i ) → ≤ i ≤ κ , ( )where the convergence is meant in the Euclidean Grassmannian Gr ( k i , V i ) . As a consequence W j = W j ,1 ⊕ · · · ⊕ W j , κ → W = W ⊕ · · · ⊕ W κ , i.e., d H ,eu ( W j , W ) →
0, ( )where the convergence is meant in the Euclidean Grassmannian Gr ( ∑ κ i = k i , n ) . The previous equality is a conse-quence of ( ) and the following observation: if V and W are two orthogonal linear subspaces such that R n = V ⊕ W ,and A , B are vector subspaces of V , and C , D are vector subspaces of W , then d eu ( A ⊕ C , B ⊕ D ) ≤ d eu ( A , B ) + d eu ( C , D ) ,where the direct sums above are orthogonal too. Let us notice that, from ( ) it follows that d H ,eu ( W j ∩ B (
0, 1 ) , W ∩ B (
0, 1 )) →
0, ( )where we stress that B (
0, 1 ) is the closed unit ball in the homogeneous left-invariant metric d . The proof of ( ) canbe reached by contradiction exploiting ( ) and the fact that B (
0, 1 ) is compact. We leave the routine details to thereader.In order to conclude the proof, we need to show that d G ( W j ∩ B (
0, 1 ) , W ∩ B (
0, 1 )) →
0. ( )Indeed, on the compact set B (
0, 1 ) , one has d ≤ Cd κ eu for some constant C >
0, see for instance [ , Proposition . ]. This means that for subsets contained in B (
0, 1 ) one has d H ≤ Cd sH ,eu . This last inequality with ( ) gives ( ).Finally from ( ) we get, by the very definition of d G , d G ( W j , W ) → W is a homogeneous subgroup of homogeneous dimension h we are done. The homogeneitycomes from the fact that W admits a stratification ( ), while the homogeneous dimension is fixed because itdepends on the dimensions of W i that are all equal to k i . Let us prove W is a subgroup. First of all W is inverse-closed, because W = exp W , and W is a vector space. Now take a , b ∈ W . By the first point of Remark . we find a n , b n ∈ W n such that a n → a , and b n → b . Then, by continuity of the operation, a n · b n → a · b , and a n · b n ∈ W n .Then from the second point of Remark . we get that a · b ∈ W . reliminaries 14 Proposition . . There exists a constant ¯ h G > , depending only on G , such that if W , V ∈ Gr ( h ) and d G ( V , W ) ≤ ¯ h G ,then s ( V ) = s ( W ) .Proof. Let us fix 1 ≤ h ≤ Q . Let us suppose by contradiction that there exist V i and W i in Gr ( h ) such that, forevery i ∈ N , the stratification of V i is different from W i and such that d G ( V i , W i ) →
0. Up to extract two (nonre-labelled) subsequences we can assume that the V i ’s have the same stratification for every i ∈ N , as well as the W i ’s. Then, by compactness, see the proof of Proposition . , we can assume up to passing to a (non re-labelled)subsequence that W i → W where W has the same stratification of the W i ’s, and V i → V where V has thesame stratification of the V i ’s. Since d G ( V i , W i ) → d G ( V , W ) = V = W but this is acontradiction since they have different stratifications. This proves the existence of a constant ¯ h that depends bothon G and h . However, taking the minimum over h of such ¯ h ’s, the dependence on h is eliminated. Proposition . . Suppose V ∈ Gr ( h ) is a homogeneous subgroup of topological dimension d. Then S h (cid:120) V , H h (cid:120) V , C h (cid:120) V and H d eu (cid:120) V are Haar measures of V . Furthermore, any Haar measure λ of V is h-homogeneous in the sense that λ ( δ r ( E )) = r h λ ( E ) , for any Borel set E ⊆ V . Proof.
This follows from the fact that the Hausdorff, the spherical Hasudorff, and the centered Hausdorff measuresintroduced in Definition . are invariant under left-translations and thus on the one hand they are Haar measuresof V . Furthermore, one can show by an explicit computation that the Lebesgue measure L d of the vector space V is a Haar measure. Indeed, this last assertion comes from the fact that for every v ∈ V the map p → vp hasunitary Jacobian determinant when seen as a map from V to V , see [ , Lemma . ]. Thus since when seen V as immersed in R n we have that the Lebesgue measure of V coincides with H d eu (cid:120) V , we conclude that H d eu (cid:120) V isa Haar measure of V as well. The last part of the proposition comes from the fact that the property is obviousby definition for the spherical Hausdorff measure, and the fact that all the Haar measures are the same up to aconstant. Definition . (Projections related to a splitting) . For any V ∈ Gr c ( h ) , if we choose a complement L , we can findtwo unique elements g V : = P V g ∈ V and g L : = P L g ∈ L such that g = P V ( g ) · P L ( g ) = g V · g L .We will refer to P V ( g ) and P L ( g ) as the splitting projections , or simply projections , of g onto V and L , respectively.We recall here below a very useful fact on splitting projections. Proposition . . Let us fix V ∈ Gr c ( h ) and L two complementary homogeneous subgroups of a Carnot group G . Then,for any x ∈ G the map Ψ : V → V defined as Ψ ( z ) : = P V ( xz ) is invertible and it has unitary Jacobian. As a consequence S h ( P V ( E )) = S h ( P V ( xP V ( E ))) = S h ( P V ( x E )) for every x ∈ G and E ⊆ G Borel.Proof.
The first part is a direct consequence of [ , Proof of Lemma . ]. For the second part it is sufficient to usethe first part and the fact that for every x , y ∈ G we have P V ( xy ) = P V ( xP V y ) . Proposition . . Let W ∈ Gr ( h ) be an h-homogeneous subgroup of topological dimension d. Then(i) there exists a constant C : = C ( s ( W )) such that for any p ∈ W and any r > we have H d eu ( B ( p , r ) ∩ W ) = C r h , ( ) (ii) there exists a constant β ( W ) such that C h (cid:120) W = β ( W ) H d eu (cid:120) W ,(iii) β ( W ) = H d eu (cid:120) W ( B (
0, 1 )) − and in particular β ( W ) = β ( s ( W )) .Proof. Thanks to Proposition . , we have H d eu ( B ( p , r ) ∩ W ) = H d eu ( B ( r ) ∩ W ) = H d eu ( δ r ( B (
0, 1 ) ∩ W )) = r h H d eu ( B (
0, 1 ) ∩ W ) . reliminaries 15 Furthermore, if V is another homogeneous subgroup such that s ( W ) = s ( V ) , we can find a linear map T that actsas an orthogonal transformation on each of the V i ’s and that maps W to V . Since we are endowing G with thebox metric Definition . , we get that T ( B (
0, 1 ) ∩ W ) = B (
0, 1 ) ∩ V . Since T is an orthogonal transformation itself,it is an isometry of R n and this implies that H d eu ( B (
0, 1 ) ∩ W ) = H d eu ( T ( B (
0, 1 ) ∩ W )) = H d eu ( B (
0, 1 ) ∩ V ) .Concerning (ii) thanks to Proposition . we have that both S h (cid:120) W and H d eu (cid:120) W are Haar measures of W . Thisimplies that there must exist a constant β ( W ) such that H d eu (cid:120) W = β ( W ) S h (cid:120) W .Finally in order to prove (iii) let us fix an ε >
0, let us take A ⊆ W ∩ B (
0, 1 ) such that C h ( A ) ≥ C h ( W ∩ B (
0, 1 )) − ε , δ > A with closed balls B i : = { B ( x i , r i ) } i ∈ N centred on A ⊆ W and with radii r i ≤ δ suchthat ∑ i ∈ N r hi ≤ C h ( A ) + ε .This implies that C h ( B (
0, 1 ) ∩ W ) (cid:0) C h ( B (
0, 1 ) ∩ W ) + ε (cid:1) ≥ C h ( B (
0, 1 ) ∩ W ) (cid:0) C h ( A ) + ε (cid:1) ≥ ∑ i ∈ N C h ( B (
0, 1 ) ∩ W ) r hi = ∑ i ∈ N C h ( B ( x i , r i ) ∩ W ) ≥ C h ( A ) ≥ C h ( A ) ≥ C h ( W ∩ B (
0, 1 )) − ε ,where the first inequality is true since C h ( B (
0, 1 ) ∩ W ) ≥ C h ( A ) ≥ C h ( A ) , and the third equality is true since x i ∈ W and C h (cid:120) W is a Haar measure on W . Thanks to the arbitrariness of ε we finally infer that C h ( W ∩ B (
0, 1 )) ≥ , item (ii) of Theorem . and Remark . ], we have that, calling B t : = { x ∈ W ∩ B (
0, 1 ) : Θ ∗ , h ( C h (cid:120) W , x ) > t } for every t >
0, we infer that C h ( B t ) ≥ t C h ( B t ) for every t >
0. Thus, for every t > C h ( B t ) = C h (cid:120) W -almost every x ∈ W ∩ B (
0, 1 ) we have that Θ ∗ , h ( C h (cid:120) W , x ) ≤ x ∈ W ∩ B (
0, 1 ) we can write C h ( B (
0, 1 ) ∩ W ) = lim sup r → C h ( B ( x , r ) ∩ W ) r h = Θ ∗ , h ( C h (cid:120) W , x ) ≤ . . Thus C h ( W ∩ B (
0, 1 )) = β ( W ) depends only on s ( W ) follows with the same argument weused to prove that C depends only on the stratification. Remark . . The above proposition can be proved whenever the distance is a multiradial distance , see [ , Definition . ].The following proposition is a consequence of the choice of the norm in Definition . , since it is based onProposition . . 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 ( h ) and any radially symmetric, positive function ϕ we have ˆ ϕ d C h (cid:120) V = h ˆ s h − 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 suppose V has topological dimension d and let ϕ ( z ) : = ∑ Ni = a i χ B ( r i ) ( z ) andnote that thanks to Proposition . (iii) for any V ∈ Gr ( h ) we have that C h (cid:120) V ( B ( r i )) = r hi and then ˆ ϕ ( z ) d C h (cid:120) V = N ∑ i = a i C h (cid:120) V ( B ( r i )) = N ∑ i = a i r hi = h N ∑ i = a i ˆ r i s h − ds = h ˆ N ∑ i = a i s h − χ [ r i ] ( s ) ds = h ˆ s h − g ( s ) ds . reliminaries 16 Proposition . (Corollary . of [ ]) . If V and L are two complementary subgroups, there exists a constant C ( V , L ) such that for any g ∈ G we haveC ( V , L ) (cid:107) P L ( g ) (cid:107) ≤ dist ( g , V ) ≤ (cid:107) P L ( g ) (cid:107) , for any g ∈ G . ( ) In the following, whenever we write C ( V , L ) , we are choosing the supremum of all the constants such that inequality ( ) is satisfied. Proposition . . For any V ∈ Gr c ( h ) with complement L there is a constant C ( V , L ) > such that for any p ∈ G andany r > we have S h (cid:120) V (cid:0) P V ( B ( p , r )) (cid:1) = C ( V , L ) r h . Furthermore, for any Borel set A ⊆ G for which S h ( A ) < ∞ , we have S h (cid:120) V ( P V ( A )) ≤ C ( V , L ) S h ( A ) . ( ) Proof.
The existence of such C ( V , L ) is yielded by [ , Lemma . ]. Suppose { B ( x i , r i ) } i ∈ N is a countablecovering of A with closed balls for which ∑ i ∈ N r hi ≤ S h ( A ) . Then S h ( P V ( A )) ≤ S h (cid:16) P V (cid:16) (cid:91) i ∈ N B ( x i , r i ) (cid:17)(cid:17) ≤ C ( V , L ) ∑ i ∈ N r hi ≤ C ( V , L ) S h ( A ) . . Cones over homogeneous subgroups and cylinder with co-normal axis
In this subsection, we introduce the intrinsic cone C W ( α ) and the notion of C W ( α ) -set, and prove some of theirproperties. Definition . (Intrinsic cone) . For any α > W ∈ Gr ( h ) , we define the cone C W ( α ) as C W ( α ) : = { w ∈ G : dist ( w , W ) ≤ α (cid:107) w (cid:107)} . Definition . ( C W ( α ) -set) . Given W ∈ Gr ( h ) , and α >
0, we say that a set E ⊆ G is a C W ( α ) -set if E ⊆ p · C W ( α ) , for any p ∈ E . Lemma . . For any W , W ∈ Gr ( h ) , ε > and α > if d G ( W , W ) < ε /4 , thenC W ( α ) ⊆ C W ( α + ε ) . Proof.
We prove that any z ∈ C W ( α ) is contained in the cone C W ( α + ε ) . Thanks to the triangle inequality weinfer dist ( z , W ) ≤ d ( z , b ) + inf w ∈ W d ( b , w ) , for any b ∈ W .Thus, choosing b ∗ ∈ W in such a way that d ( z , b ∗ ) = dist ( z , W ) , and evaluating the previous inequality at b ∗ weget dist ( z , W ) ≤ dist ( z , W ) + dist ( b ∗ , W ) ≤ α (cid:107) z (cid:107) + dist ( b ∗ , W ) , ( )where in the second inequality we used z ∈ C W ( α ) .Let us notice that, given W an arbitrary homogeneous subgroup of G , p ∈ G an arbitrary point such that p ∗ ∈ W is one of the points at minimum distance from W to p , then the following inequality holds (cid:107) p ∗ (cid:107) ≤ (cid:107) p (cid:107) . ( )Indeed, (cid:107) p ∗ (cid:107) − (cid:107) p (cid:107) ≤ (cid:107) ( p ∗ ) − · p (cid:107) = d ( p , W ) ≤ (cid:107) p (cid:107) ⇒ (cid:107) p ∗ (cid:107) ≤ (cid:107) p (cid:107) . reliminaries 17 Now, by homogeneity, since b ∗ ∈ W is the point at minimum distance from W of z , we get that D (cid:107) z (cid:107) ( b ∗ ) is the point at minimum distance from W of D (cid:107) z (cid:107) ( z ) . Thus, since (cid:107) D (cid:107) z (cid:107) ( z ) (cid:107) =
1, from ( ) we get that (cid:107) D (cid:107) z (cid:107) ( b ∗ ) (cid:107) ≤
2. Finally we obtaindist ( b ∗ , W ) = (cid:107) z (cid:107) dist (cid:0) D (cid:107) z (cid:107) ( b ∗ ) , W (cid:1) = (cid:107) z (cid:107) dist (cid:0) D (cid:107) z (cid:107) ( b ∗ ) , W ∩ B (
0, 4 ) (cid:1) ≤≤ (cid:107) z (cid:107) d H ( W ∩ B (
0, 4 ) , W ∩ B (
0, 4 )) = (cid:107) z (cid:107) d H ( W ∩ B (
0, 1 ) , W ∩ B (
0, 1 )) < ε (cid:107) z (cid:107) , ( )where the first equality follows from the homogeneity of the distance, and the second is a consequence of thefact that (cid:107) D (cid:107) z (cid:107) ( b ∗ ) (cid:107) ≤
2, and thus, from ( ), the point at minimum distance of D (cid:107) z (cid:107) ( b ∗ ) from W has normbounded above by ; the third inequality comes from the definition of Hausdorff distance, the fourth equality istrue by homogeneity and the last inequality comes from the hypothesis d G ( W , W ) < ε /4. Joining ( ), and ( )we get z ∈ C W ( α + ε ) , that was what we wanted. Lemma . . Let V ∈ Gr c ( h ) , and let L be a complementary subgroup of V . There exists ε : = ε ( V , L ) > such that L ∩ C V ( ε ) = { } . Moreover we can, and will, choose ε ( V , L ) : = C ( V , L ) /2 .Proof. We prove that it suffices to take ε ( V , L ) : = C ( V , L ) /2. Let us suppose the statement is false. Thus thereexists 0 (cid:54) = v ∈ L ∩ C V ( ε ) . From Proposition . and from the very definition of the cone C V ( ε ) we have C ( V , L ) (cid:107) v (cid:107) ≤ dist ( v , V ) ≤ ε (cid:107) v (cid:107) = C ( V , L ) (cid:107) v (cid:107) /2,which is a contradiction with the fact that v (cid:54) = Remark . . Let V ∈ Gr c ( h ) and let L be a complement of V . Let us notice that if there exists α > L ∩ C V ( α ) = { } , then C ( V , L ) ≥ α . Ineed it is enough to prove that α (cid:107) P L ( g ) (cid:107) ≤ dist ( g , V ) for every g ∈ G . If g ∈ V the latter in equality is trivially verified. Hence suppose by contradiction that there exists g / ∈ V such that α (cid:107) P L ( g ) (cid:107) > dist ( g , V ) . Since dist ( g , V ) = dist ( P L ( g ) , V ) we conclude that P L ( g ) ∈ L ∩ C V ( α ) = { } , that is acontradiction since g / ∈ V .We will not use the following proposition in the paper, but it is worth mentioning it. Proposition . . The family of the complemented subgroups Gr c ( h ) is an open subset of Gr ( h ) .Proof. Fix a W ∈ Gr c ( h ) and let L be one complementary subgroup of W and set ε < min { ε ( V , L ) , ¯ h G } . Then, if W (cid:48) ∈ Gr ( h ) is such that d G ( W , W (cid:48) ) < ε /4, Lemma . implies that W (cid:48) ⊆ C W ( ε ) and in particular L ∩ W (cid:48) ⊆ L ∩ C W ( ε ) = { } .Moreover, since ε < ¯ h G from Proposition . , we get that W (cid:48) has the same stratification of W and thus thesame topological dimension. This, jointly with the previous equality and the Grassmann formula, means that ( W (cid:48) ∩ V i ) + ( L ∩ V i ) = V i for every i =
1, . . . , κ . This, jointly with the fact that L ∩ W (cid:48) = { } , implies that L and W (cid:48) are complementary subgroups in G due to the triangular structure of the product · on G , see ( ). For analternative proof of the fact that L and W (cid:48) are complementary subgroups, see also [ , Lemma . ]. Proposition . . Let W ∈ Gr c ( h ) and assume L is one of the complementary subgroups of W . Any other subgroup V ∈ Gr ( h ) on which P W is injective and satisfying the identity s ( V ) = s ( W ) is contained in Gr c ( h ) and admits L as acomplement.Proof. The hypothesis s ( V ) = s ( W ) implies that V and W have the same topological dimension. If by contradic-tion there exists a 0 (cid:54) = y ∈ L ∩ V , then P W ( y ) = = P W ( ) .This however is not possible since we assumed that P W is injective on V . The fact that L ∩ V = { } concludes theproof by the same argument we used in the proof of the previous Proposition . . reliminaries 18 The following definition of intrinsically Lipschitz functions is equivalent to the classical one in [ , Definition ]because the cones in [ , Definition ] and the cones C V ( α ) are equivalent whenever V admits a complementarysubgroup, see [ , Proposition . ]. Definition . (Intrinsically Lipschitz functions) . Let W ∈ Gr c ( h ) and assume L is a complement of W and let E ⊆ W be a subset of V . A function f : E → L is said to be an intrinsically Lipschitz function if is there exists an α > ( f ) : = { v · f ( v ) : v ∈ E } is a C W ( α ) -set. Proposition . . Let us fix W ∈ Gr c ( h ) with complement L . If Γ ⊂ G is a C W ( α ) -set for some α ≤ ε ( W , L ) , then themap P W : Γ → W is injective. As a consequence Γ is the intrinsic graph of an intrinsically Lipschitz map defined on P W ( Γ ) .Proof. Suppose by contradiction that P W : Γ → W is not injective. Then, there exist p (cid:54) = q with p , q ∈ Γ such that P W ( p ) = P W ( q ) . Thus p − · q ∈ L . Moreover, since Γ is a C W ( α ) -set, we have that p − · q ∈ C W ( α ) . Eventually weget p − · q ∈ L ∩ C W ( α ) ⊆ L ∩ C W ( ε ( W , L )) ,where the last inclusion follows since α ≤ ε ( W , L ) . The above inclusion, jointly with Lemma . , gives that p − · q = P L ◦ (cid:0) ( P W ) | Γ (cid:1) − is well-defined from P W ( Γ ) to L and its intrinsic graph is Γ by definition. Moreover, since Γ is a C W ( α ) -set, the latter map is intrinsically Lipschitz by Definition . .The following two lemmata will play a fundamental role in the proof that P ch -rectifiable measures have h -density. Lemma . . Let V ∈ Gr c ( h ) and L be one of its complementary subgroups. For any < α < C ( V , L ) /2 , let c ( α ) : = α / ( C ( V , L ) − α ) . ( ) Then we have B (
0, 1 ) ∩ V ⊆ P V ( B (
0, 1 ) ∩ C V ( α )) ⊆ B (
0, 1/ ( − c ( α ))) ∩ V . ( ) Proof.
The first inclusion comes directly from the definition of projections and cones. Concerning the second, if v ∈ B (
0, 1 ) ∩ C V ( α ) , thanks to Proposition . we have C ( V , L ) (cid:107) P L ( v ) (cid:107) ≤ dist ( v , V ) ≤ α (cid:107) v (cid:107) ≤ α ( (cid:107) P L ( v ) (cid:107) + (cid:107) P V ( v ) (cid:107) ) . ( )This implies in particular that (cid:107) P L ( v ) (cid:107) ≤ c ( α ) (cid:107) P V ( v ) (cid:107) and thus1 ≥ (cid:107) P V ( v ) P L ( v ) (cid:107) ≥ (cid:107) P V ( v ) (cid:107) − (cid:107) P L ( v ) (cid:107) ≥ ( − c ( α )) (cid:107) P V ( v ) (cid:107) .This concludes the proof of the lemma. Lemma . . Let V ∈ Gr c ( h ) and L be one of its complementary subgroups. Suppose Γ is a C V ( α ) -set with α < C ( V , L ) /2 , and let C ( α ) : = − c ( α ) + c ( α ) , ( ) where c ( α ) is defined in ( ) . Then S h ( P V ( B ( x , r ) ∩ Γ )) ≥ S h (cid:16) P V (cid:0) B ( x , C ( α ) r ) ∩ xC V ( α ) (cid:1) ∩ P V ( Γ ) (cid:17) , for any x ∈ Γ . The same inequality above holds if we substitute S h with any other Haar measure on V , see Proposition . , because all ofthem are equal up to a constant.Proof. First of all, let us note that we have S h (cid:0) P V ( B ( x , r ) ∩ Γ ) (cid:1) = S h (cid:16) P V (cid:0) B ( r ) ∩ x − Γ (cid:1)(cid:17) , ( ) reliminaries 19 where the last equality is true since S h ( P V ( E )) = S h ( P V ( x − E )) for any Borel E ⊆ G , see Proposition . . Wewish now to prove the following inclusion P V (cid:0) B ( C ( α ) r ) ∩ C V ( α ) (cid:1) ∩ P V ( x − Γ ) ⊆ P V ( B ( r ) ∩ x − Γ ) . ( )Indeed, fix an element y of P V ( B ( C ( α ) r ) ∩ C V ( α )) ∩ P V ( x − Γ ) . Thanks to our choice of y there are a w ∈ x − Γ and a w ∈ B ( C ( α ) r ) ∩ C V ( α ) such that P V ( w ) = y = P V ( w ) .Furthermore, since Γ is a C V ( α ) -set, we infer that w ∈ C V ( α ) and thus with the same computations as in ( ) weobtain that (cid:107) P L ( w ) (cid:107) ≤ c ( α ) (cid:107) P V ( w ) (cid:107) and thus (cid:107) w (cid:107) ≤ ( + c ( α )) (cid:107) P V w (cid:107) ≤ ( + c ( α )) (cid:107) y (cid:107) . ( )Furthermore, since by assumption w ∈ B ( C ( α ) r ) ∩ C V ( α ) , Lemma . yields (cid:107) y (cid:107) = (cid:107) P V ( w ) (cid:107) ≤ C ( α ) r / ( − c ( α )) = r / ( + c ( α )) . ( )The bounds ( ) and ( ) together imply that (cid:107) w (cid:107) ≤ r , and thus w ∈ B ( r ) ∩ x − Γ and this concludes the proofof the inclusion ( ). Finally ( ), ( ) imply S h ( P V ( B ( x , r ) ∩ Γ )) ≥ S h (cid:16) P V ( B ( C ( α ) r ) ∩ C V ( α )) ∩ P V ( x − Γ ) (cid:17) . ( )Furthermore, for any Borel subset E of G we have P V ( x E ) = P V ( xP V ( E )) , since for every g ∈ E we have thefollowing simple equality P V ( xg ) = P V ( xP V g ) . Therefore, by using the latter observation and Proposition . ,we get, denoting with Ψ the map Ψ ( v ) = P V ( x − v ) for every v ∈ V , that S h (cid:16) P V (cid:0) B ( C ( α ) r ) ∩ C V ( α ) (cid:1) ∩ P V (cid:0) x − Γ (cid:1)(cid:17) = S h (cid:16) P V (cid:0) x − P V ( B ( x , C ( α ) r ) ∩ xC V ( α )) (cid:1) ∩ P V (cid:0) x − P V ( Γ ) (cid:1)(cid:17) = S h (cid:16) Ψ (cid:0) P V ( B ( x , C ( α ) r ) ∩ xC V ( α )) (cid:1) ∩ Ψ (cid:0) P V ( Γ ) (cid:1)(cid:17) = S h (cid:16) P V ( B ( x , C ( α ) r ) ∩ xC V ( α )) ∩ P V ( Γ ) (cid:17) . ( )Joining together ( ) and ( ) gives the sought conclusion.We conclude this subsection with a more careful study of the co-normal Grassmanian. These results will turnout to be fundamental when approaching the Marstrand-Mattila rectifiability criterion in Section . Proposition . . For any s ∈ S ( h ) the function e : Gr s (cid:69) ( h ) → R defined as e ( V ) : = sup { ε ( V , L ) : L is a normal complement of V } , ( ) is lower semicontinuous. Moreover the following conclusion holdsif G ⊆ Gr s (cid:69) ( h ) is compact with respect to d G , then there exists a e G > such that e ( V ) ≥ e G for any V ∈ G ⊆ Gr s (cid:69) ( h ) .Proof. Let us prove that the function e is lower semincontinuous. Since ε ( V , L ) = C ( V , L ) /2, see the proofof Lemma . , it is enough to prove the proposition with 2 e ( V ) instead of e ( V ) , and with C ( V , L ) instead of ε ( V , L ) . Let us fix V ∈ Gr s (cid:69) ( h ) and 0 < ε < e ( V ) , and denote with L one of the normal complement subgroupsof V for which C ( V , L ) > e ( V ) − ε . For any W ∈ Gr s (cid:69) ( h ) thanks to Lemma . we have C W ( C ( V , L ) − d G ( V , W ) − ε ) ⊆ C V ( C ( V , L ) − ε ) , whenever d G ( V , W ) is small enough. ( ) reliminaries 20 Therefore if d G ( V , W ) is sufficiently small, the latter inclusion and the same proof as in Lemma . imply that L ∩ W ⊆ L ∩ C V ( C ( V , L ) − ε ) = { } . Since L ∩ W = { } , L and V are complementary subgroups and V and W have the same stratification vector, and thus the same topological dimension, we have that L is acomplement of W for the same argument used in the proof of Proposition . . Thus, taking ( ) into accountwe get that L ∩ C W ( C ( V , L ) − d G ( V , W ) − ε ) = { } and thus, from Remark . , we get that C ( W , L ) ≥ C ( V , L ) − d G ( V , W ) − ε whenever d G ( V , W ) is sufficiently small. This implies that2 e ( W ) ≥ C ( W , L ) ≥ C ( V , L ) − d G ( V , W ) − ε ≥ e ( V ) − d ( V . W ) − ε , whenever d G ( V , W ) is small enough.and thus lim inf d G ( W , V ) → e ( W ) ≥ e ( V ) − ε ,from which the lower semicontinuity follows due to the arbitrariness of ε . The conclusion in item (i) follows since G ⊆ Gr s (cid:69) ( h ) is compact and e is lower semincontinuous. Remark . . We observe that in the previous proposition we did not use the fact L is normal, but we stated theproposition in this specific case since we are going to use this formulation in the paper. The same proof works inthe more general case when V ∈ Gr s c ( h ) and e ( V ) = sup { ε ( V , L ) : L is a complement of V } . Proposition . . Let C > and V ∈ Gr s (cid:69) ( h ) be such that e ( V ) ≥ C. Then there exists a normal complement L of V such that (cid:107) P V ( g ) (cid:107) ≤ ( + C ) (cid:107) g (cid:107) , and (cid:107) P L ( g ) (cid:107) ≤ ( C ) (cid:107) g (cid:107) , for all g ∈ G , ( ) provided P V and P L are the projections relative to the splitting G = VL .Proof. Thanks to the definition of e ( V ) , see ( ), there exists a normal complementary subgroup L of V such that ε ( V , L ) ≥ C /2. Thus, from Lemma . , we get L ∩ C V ( C /2 ) = { } . This implies, arguing as in Remark . , thatfor any g ∈ G we have C (cid:107) P L ( g ) (cid:107) /2 ≤ dist ( V , P L ( g )) = dist ( V , g ) ≤ (cid:107) g (cid:107) . ( )Furthermore, thanks to the triangle inequality we have (cid:107) g (cid:107) ≥ (cid:107) P V ( g ) (cid:107) − (cid:107) P L ( g ) (cid:107) ≥ (cid:107) P V ( g ) (cid:107) − ( C ) (cid:107) g (cid:107) ,thus concluding the proof of the proposition. Proposition . . Let C > and V ∈ Gr s (cid:69) ( h ) be such that e ( V ) ≥ C. Let L be a normal complementary subgroup to V as in Proposition . . Then the projection P V : G → V related to the splitting G = V · L is a ( + C ) -Lipschitzhomogeneous homomorphism.Proof. Thanks to the fact that L is normal, we have that for any x , y ∈ G the following equality holds P V ( xy ) = P V ( x V x L y V y L ) = P V ( x V y V · y − V x L y V · y L ) = P V ( x ) P V ( y ) .Since P V is always an homogeneous map, we infer that P V is a homogeneous homomorphism. Moreover, fromProposition . we have that (cid:107) P V ( g ) (cid:107) ≤ ( + C ) (cid:107) g (cid:107) ,for every g ∈ G . Hence from the fact that P V is a homomorphism we have (cid:107) P V ( x ) − P V ( y ) (cid:107) = (cid:107) P V ( x − y ) (cid:107) ≤ ( + C ) (cid:107) x − y (cid:107) ,for every x , y ∈ G and thus P V is ( + C ) -Lipschitz. Remark . . Notice that in the proof of the above proposition we proved that whenever L is normal, then P V is ahomomorphism. Definition . (Cylinder) . Let V , L be two complementary subgroups of G . For any u ∈ G , and r > T ( u , r ) : = P − V ( P V ( B ( u , r ))) . reliminaries 21 In the following proposition we study the structure of cylinders T ( · , · ) when L is normal. Proposition . . Let C > and V ∈ Gr s (cid:69) ( h ) be such that e ( V ) ≥ C. Let L be a normal complementary subgroup to V as in Proposition . . Thus, for any u ∈ G we have T ( u , r ) = P V ( u ) δ r T (
0, 1 ) . Furthermore, we haveT ( u , r ) ⊆ P V ( u ) δ r P − V ( B ( ( + C )) ∩ V ) = P − V ( B ( P V ( u ) , ( + C ) r ) ∩ V ) . Finally, for any h ∈ L we have B ( uh , r ) ⊆ T ( u , r ) .Proof. First of all, we note that thanks to Proposition . we have that w ∈ P V ( B ( u , r )) if and only if there existsa v ∈ B (
0, 1 ) such that w = P V ( u ) δ r P V ( v ) . Therefore, given u ∈ G and r >
0, we have that y ∈ T ( u , r ) if and onlyif y = P V ( u ) δ r P V ( v ) h for some v ∈ B (
0, 1 ) and h ∈ L . Thus we conclude that T ( u , r ) = P V ( u ) δ r T (
0, 1 ) for every u ∈ G and r > . we infer that P V ( B (
0, 1 )) ⊆ V ∩ B ( ( + C )) and thus combining suchinclusion with the first part of the proposition we deduce that T ( u , r ) ⊆ P V ( u ) δ r P − V ( B ( ( + C )) ∩ V ) = P − V ( B ( P V ( u ) , ( + C ) r ) ∩ V ) ,where the last equality is true since P V is a homogeneous homomorphism. Finally, thanks to the first part of theproposition, for any u ∈ V and any h ∈ L we have B ( uh , r ) ⊆ T ( uh , r ) = T ( u , r ) ,and this concludes the proof of the proposition. . Rectifiable measures in Carnot groups
In what follows we are going to define the class of h -flat measures on a Carnot group and then we will giveproper definitions of rectifiable measures on Carnot groups. Definition . (Flat measures) . For any h ∈ {
1, . . . , Q } we let M ( h ) to be the family of flat h-dimensional measures in G , i.e. M ( h ) : = { λ S h (cid:120) W : for some λ > W ∈ Gr ( h ) } .Furthermore, if G is a subset of the h -dimensional Grassmanian Gr ( h ) , we let M ( h , G ) to be the set M ( h , G ) : = { λ S h (cid:120) W : for some λ > W ∈ G } . ( )We stress that in the previous definitions we can use any of the Haar measures on W , see Proposition . , sincethey are the same up to a constant. Definition . ( P h and P ∗ h -rectifiable measures) . Let h ∈ {
1, . . . , Q } . A Radon measure φ on G is said to be a P h -rectifiable measure if for φ -almost every x ∈ G we have(i) 0 < Θ h ∗ ( φ , x ) ≤ Θ h , ∗ ( φ , x ) < + ∞ ,(ii) there exists a V ( x ) ∈ Gr ( h ) such that Tan h ( φ , x ) ⊆ { λ S h (cid:120) V ( x ) : λ ≥ } .Furthermore, we say that φ is P ∗ h -rectifiable if (ii) is replaced with the weaker(ii)* Tan h ( φ , x ) ⊆ { λ S h (cid:120) V : λ ≥ V ∈ Gr ( h ) } . Remark . . (About λ = . ) It is readily noticed that, since in Definition . we are asking Θ h ∗ ( φ , x ) > φ -almost every x , we can not have the zero measure as a tangent measure. As a consequence,a posteriori, we have that in item (ii) and item (ii)* above we can restrict to λ >
0. We will tacitly work in thisrestriction from now on.On the contrary, if we only know that for φ -almost every x ∈ G we have Θ h , ∗ ( φ , x ) < + ∞ , and Tan h ( φ , x ) ⊆ { λ S h (cid:120) V ( x ) : λ > } , ( ) reliminaries 22 for some V ( x ) ∈ Gr ( h ) , hence Θ h ∗ ( φ , x ) > φ -almost every x ∈ G , and the same property holds with theitem (ii)* above. Indeed, if at some x for which ( ) holds we have Θ h ∗ ( φ , x ) =
0, then there exists r i → r − hi φ ( B ( x , r i )) =
0. Since Θ h , ∗ ( φ , x ) < + ∞ , up to subsequences (see [ , Theorem . ]), we have r − hi T x , r i φ → λ S h (cid:120) V ( x ) , for some λ >
0. Hence, by applying [ , Proposition . (b)] we conclude that r − hi T x , r i φ ( B (
0, 1 )) → λ S h (cid:120) V ( x )( B (
0, 1 )) >
0, that is a contradiction.Throughout the paper it will be often convenient to restrict our attention to some subclasses of P h - and P ∗ h -rectifiable measures, imposing different restrictions on the algebraic nature of the tangents. More precisely wegive the following definition. Definition . (Subclasses of P h and P ∗ h -rectifiable measures) . Let h ∈ {
1, . . . , Q } . In the following we denoteby P ch the family of those P h -rectifiable measures such that for φ -almost every x ∈ G we haveTan h ( φ , x ) ⊆ M ( h , Gr c ( h )) .Furthermore, the family of those P ∗ h -rectifiable measures φ such that for φ -almost any x ∈ G we have(i) Tan h ( φ , x ) ⊆ M ( h , Gr c ( h )) is said P ∗ , ch ,(ii) Tan h ( φ , x ) ⊆ M ( h , Gr (cid:69) ( h )) is said P ∗ , (cid:69) h ,(iii) Tan h ( φ , x ) ⊆ M ( h , Gr s (cid:69) ( h )) is said P ∗ , (cid:69) , s h .The following proposition is a consequence of the choice of the norm in Definition . , since it is based onProposition . . Proposition . . Let h ∈ {
1, . . . , Q } and assume φ is a Radon measure on G . If { r i } i ∈ N is an infinitesimal sequence suchthat r − hi T x , r i φ (cid:42) λ C h (cid:120) V for some λ > and V ∈ Gr ( h ) then lim i → ∞ φ ( B ( x , r i )) / r hi = λ . Proof.
Since C h (cid:120) V ( x )( ∂ B (
0, 1 )) =
0, see e.g., [ , Lemma . ], thanks to Proposition . (iii) and to [ , Proposition . (b)] we have λ = λ C h (cid:120) V ( x )( B (
0, 1 )) = lim i → ∞ T x , r i φ ( B (
0, 1 )) r hi = lim i → ∞ φ ( B ( x , r i )) r hi ,and this concludes the proof.The above proposition has the following immediate consequence. Corollary . . Let h ∈ {
1, . . . , Q } and assume φ is a P ∗ h -rectifiable. Then for φ -almost every x ∈ G we have Tan h ( φ , x ) ⊆ { λ C h (cid:120) W : λ ∈ [ Θ h ∗ ( φ , x ) , Θ h , ∗ ( φ , x )] and W ∈ Gr ( h ) } .We introduce now a way to estimate how far two measures are. Definition . . Given φ and ψ two Radon measures on G , and given K ⊆ G a compact set, we define F K ( φ , ψ ) : = sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) ˆ f d φ − ˆ f d ψ (cid:12)(cid:12)(cid:12)(cid:12) : f ∈ Lip + ( K ) (cid:27) . ( )We also write F x , r for F B ( x , r ) . Remark . . With few computations that we omit, it is easy to see that F x , r ( φ , ψ ) = rF ( T x , r φ , T x , r ψ ) . Furthermore, F K enjoys the triangular inequality, indeed if φ , φ , φ are Radon measures and f ∈ Lip + ( K ) , then (cid:12)(cid:12)(cid:12) ˆ f d φ − ˆ f d φ (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ˆ f d φ − ˆ f d φ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˆ f d φ − ˆ f d φ (cid:12)(cid:12)(cid:12) ≤ F K ( φ , φ ) + F K ( φ , φ ) .The arbitrariness of f concludes that F K ( φ , φ ) ≤ F K ( φ , φ ) + F K ( φ , φ ) . reliminaries 23 The proof of the following criterion is contained in [ , Proposition . ] and we omit the proof. Proposition . . Let { µ i } be a sequence of Radon measures on G . Let µ be a Radon measure on G . The following areequivalent . µ i (cid:42) µ ; . F K ( µ i , µ ) → , for every K ⊆ G compact. The following proposition is a consequence of the choice of the norm in Definition . , since it is based onProposition . . Proposition . . Let h ∈ {
1, . . . , Q } and suppose that { V i } i ∈ N is a sequence of planes in Gr ( h ) converging in the metricd G to some V ∈ Gr ( h ) . Then, C h (cid:120) V i (cid:42) C h (cid:120) V .Proof. First of all note that Proposition . implies that there exists a i ∈ N such that for any i ≥ i we have that V i and V have the same stratification and thus the same topological dimension d . Since the V i ’s have the samestratification if i ≥ i , Proposition . (iii) implies that β ( V i ) = β ( V ) for any i ≥ i . Thus, for any continuousfunction f with compact support thanks to Proposition . we havelim i → ∞ ˆ f d C h (cid:120) V i − ˆ f d C h (cid:120) V = lim i → ∞ β ( V ) (cid:18) ˆ f d H d eu (cid:120) V i − ˆ f d H d eu (cid:120) V (cid:19) = H d eu (cid:120) V i (cid:42) H d eu (cid:120) V .Now we are going to define a functional that in some sense tells how far is a measure from being flat around apoint x ∈ G and at a certain scale r > Definition . . For any x ∈ G , any h ∈ {
1, . . . , Q } and any r > d x , r ( φ , M ( h )) : = inf Θ > V ∈ Gr ( h ) F x , r ( φ , Θ S h (cid:120) x V ) r h + . ( )Furthermore, if G is a subset of the h -dimensional Grassmanian Gr ( h ) , we also define d x , r ( φ , M ( h , G )) : = inf Θ > V ∈ G F x , r ( φ , Θ S h (cid:120) x V ) r h + . Remark . . It is a routine computation to prove that, whenever h ∈ N and r > x (cid:55)→ d x , r ( φ , M ( h , G )) is a continuous function. The proof can be reached as in [ , Item (ii) of Proposition . ]. Moreover,from the invariance property in Remark . and Proposition . , if in ( ) we use the measure C h (cid:120) x V we obtainthe same functional. Proposition . . Let φ be a Radon measure on G satisfying item (i) in Definition . . Further, let G be a subfamily ofGr ( h ) and let M ( h , G ) be the set defined in ( ) . If for φ -almost every x ∈ G we have Tan h ( φ , x ) ⊆ M ( h , G ) , then for φ -almost every x ∈ G and every every k > we have lim r → d x , kr ( φ , M ( h , G )) = Proof.
Let us fix x ∈ G a point for which Tan h ( φ , x ) ⊆ M ( h , G ) and let us assume by contradiction that there exist k > r i → ε > d x , kr i ( φ , M ( h , G )) > ε . ( )Since φ satisfies item (i) in Definition . , we can use [ , Proposition . (b)] and then, up to subsequences, thereare Θ ∗ > V ∗ ∈ G such that r − hi T x , r i φ (cid:42) Θ ∗ S h (cid:120) V ∗ . ( ) reliminaries 24 Thus, d x , kr i ( φ , M ( h , G )) = d k ( r − hi T x , r i φ , M ( h , G )) ≤ k − h − F k ( r − hi T x , r i φ , Θ ∗ S h (cid:120) V ∗ ) → . , and the last convergence follows from ( ), andProposition . . This is in contradiction with ( ).The following proposition is an adaptation of [ , . ( )] and it will be crucial in the proof of Marstrand-Mattila’srectifiability criterion in Section . We stress that it is a consequence of the choice of the norm in Definition . ,since it is based on Proposition . . Proposition . . Suppose that h ∈ {
1, . . . , Q } , φ is a Radon measure supported on a compact set, and let G ⊆ Gr ( h ) . Ifthere exists an x ∈ E ( ϑ , γ ) , a σ ∈ (
0, 2 − ( h + ) ϑ − ) and a < t < ( γ ) such thatd x , t ( φ , M ( h , G )) ≤ σ h + , then there is a V ∈ G such that(i) whenever y , z ∈ B ( x , t /2 ) ∩ x V and σ t ≤ r , s ≤ t /2 we have φ ( B ( y , r ) ∩ B ( x V , σ t )) ≥ ( − ( h + ) ϑσ )( r / s ) h φ ( B ( z , s )) ; (ii) furthermore, if the plane V yielded by item (i) above admits a complementary normal subgroup L , denote by P V the splitting projection on V according to this splitting. Then for any k > with σ k < − h ϑ − , if we defineT V ( t /4 k ) : = P − V ( P V ( B ( t /4 k ))) we have φ ( B ( x , t /4 ) ∩ xT V ( t /4 k )) ≤ ( + σ ( kh + )) C h ( P ( B (
0, 1 ))) k − h φ ( B ( x , t /4 )) . Proof.
First of all, we notice that by the definition of d x , t ( φ , M ( h , G )) there exist V ∈ G and λ > F x , t ( φ , λ C h (cid:120) x V ) ≤ σ h + t h + . proof of ( i ) The key of the proof of item (i) is to show that for any w ∈ B ( x , t /2 ) ∩ x V , any τ ∈ ( t /2 ] andany ρ ∈ ( τ ] we have φ ( B ( w , τ )) ≤ λ C h (cid:120) ( x V )( B ( w , τ + ρ )) + σ h + t h + / ρ , ( ) λ C h (cid:120) ( x V )( B ( w , τ − ρ )) ≤ φ ( B ( w , τ ) ∩ B ( x V , ρ )) + σ h + t h + / ρ . ( )Before proving that ( ) and ( ) together imply the claim, we need to give a lower bound for λ . Since x ∈ E ( ϑ , γ ) ,with the choice w = x , τ = t /4, and ρ : = σ t we have, from ( ), that the following inequality holds ϑ − ( t /4 ) h ≤ φ ( B ( x , t /4 )) ≤ λ C h (cid:120) ( x V )( B ( x , ( + σ ) t )) + σ h + t h = λ ( + σ ) h t h + σ h + t h , ( )where the last equality comes from item (iii) of Proposition . . Since σ ≤ ( ( h + ) ϑ ) , we obtain that σ h + ≤ ( h ϑ ) , and then from ( ) we infer ϑ − − h ≤ λ ( + σ ) h + σ h + and in particular λ ≥ ϑ − − h , ( )where we exploited the fact that 1/4 + σ <
1, the fact that σ h + ≤ ( h ϑ ) and the fact that 4 − h − − h ≥ − h .Let us now prove that ( ) and ( ) imply the claim. Since by hypothesis r , s ≥ σ t with the choice ρ = σ t wehave ρ < r , s . Furthermore since σ t ≤ r , s ≤ t /2 and y , z ∈ B ( x , t /2 ) ∩ x V , the bounds ( ) and ( ) imply φ ( B ( y , r ) ∩ B ( x V , ρ )) φ ( B ( z , s )) ≥ λ C h (cid:120) ( x V )( B ( y , r − ρ )) − σ h + t h + / ρλ C h (cid:120) ( x V )( B ( z , s + ρ )) + σ h + t h + / ρ = r h s h λ ( − σ t / r ) h − σ h + ( t / r ) h λ ( + σ t / s ) h + σ h + ( t / s ) h ≥ r h s h λ ( − σ ) h − σ h + ( t / r ) h λ ( + σ ) h + σ h + ( t / s ) h ≥ r h s h λ ( − σ ) h − σλ ( + σ ) h + σ , reliminaries 25 where the equality in the first line comes from item (iii) of Proposition . and we are using σ t / r ≤
1, and σ t / s ≤
1. Since 2 h σ ≤
1, we have that ( + σ ) h ≤ + h σ , that can be easily proved by induction on h . Thistogether with ( ) and Bernoulli’s inequality ( − σ ) h ≥ − σ h allows us to finally infer that φ ( B ( y , r ) ∩ B ( x V , ρ )) φ ( B ( z , s )) ≥ r h s h − ( λ h + ) σ / λ + ( h λ + ) σ / λ ≥ ( − ( h + ) ϑσ ) r h s h ,where the last inequality comes from the fact that σ ≤ ( h + ) ϑ , from ( ) and some easy algebraic computationsthat we omit. An easy way to verify the last inequality is to show that ( − (cid:101) ασ ) / ( + (cid:101) βσ ) ≥ − (cid:101) γσ , where (cid:101) α : = ( λ h + ) / λ , (cid:101) β : = ( h λ + ) / λ and (cid:101) γ : = ( h + ) ϑ , and observe that the latter inequality is implied by the factthat (cid:101) α + (cid:101) β − (cid:101) γ ≤ ) and ( ). In order to prove ( ), we let g ( z ) : = min {
1, dist ( z , G \ U ( w , τ + ρ )) / ρ } and note that φ ( B ( w , τ )) ≤ ˆ g ( z ) d φ ( z ) ≤ ˆ g ( z ) d λ C h (cid:120) ( x V )( z ) + Lip ( g ) F x , t ( φ , λ C h (cid:120) ( x V )) ≤ λ C h (cid:120) ( x V )( B ( w , τ + ρ )) + σ h + t h + / ρ .On the the other hand, to prove ( ) we let h ( z ) : = min {
1, dist ( z , G \ ( U ( w , τ ) ∩ U ( x V , ρ ))) / ρ } and λ C h (cid:120) ( x V )( B ( w , τ − ρ )) ≤ ˆ h ( z ) d λ C h (cid:120) ( x V )( z ) ≤ ˆ h ( z ) d φ ( z ) + Lip ( h ) F x , t ( φ , λ C h (cid:120) ( x V )) ≤ φ ( B ( w , τ ) ∩ B ( x V , ρ )) + σ h + t h + / ρ . proof of ( ii ): In this proof let us fix τ : = t /4 and define the function (cid:96) ( z ) : = min {
1, dist ( z , G \ U ( U ( x , τ ) ∩ xT ( τ / k ) , ρ )) / ρ } , where 0 < ρ < τ . With this definition we have the following chain of inequalities φ ( B ( x , τ ) ∩ xT ( τ / k )) ≤ ˆ (cid:96) ( z ) d φ ( z ) ≤ ˆ (cid:96) ( z ) d λ C h (cid:120) ( x V )( z ) + Lip ( (cid:96) ) F x , t ( φ , λ C h (cid:120) ( x V )) ≤ λ C h (cid:120) ( x V )( B ( x , τ + ρ ) ∩ xT ( τ / k + ρ )) + h + σ h + τ h + / ρ ≤ λ C h (cid:120) V ( P ( B (
0, 1 )))( τ / k + ρ ) h + h + σ h + τ h + / ρ , ( )where the third inequality above comes from the fact that, according to the proof of Proposition . , the projection P is a homomorphism, and then the following chain of equalities holds P ( B ( T ( τ / k ) , ρ )) = P ( T ( τ / k ) B ( ρ )) = P ( B ( t / k )) P ( B ( ρ )) = P ( B ( τ / k + ρ )) .Putting together ( ) when specialized to the case w = x and τ = t /4, with ( ) and item (iii) of Proposition . ,we infer that φ ( B ( x , τ ) ∩ xT ( τ / k )) φ ( B ( x , τ )) ≤ λ C h (cid:120) V ( P ( B (
0, 1 )))( τ / k + ρ ) h + h + σ h + τ h + / ρλ ( τ − ρ ) h − h + σ h + τ h + / ρ . ( )Since σ < ρ : = σ τ and note that since σ k < − h ϑ − , the following proposition yields φ ( B ( x , τ ) ∩ xT ( τ / k )) φ ( B ( x , τ )) ≤ λ C h ( P ( B (
0, 1 )))( k + σ ) h + h + σ h + λ ( − σ ) h − h + σ h + ≤ ( + σ ( kh + )) C h ( P ( B (
0, 1 ))) k − h ,where we omit the computations that lead to the last inequality but we stress that we need C h ( P ( B (
0, 1 ))) ≥
1, thatin turns comes from the fact that P ( B (
0, 1 )) ⊇ B (
0, 1 )) and C h ( B (
0, 1 )) =
1, due to item (iii) of Proposition . ;and also the bound on λ in ( ). The last inequality concludes the proposition.We prove the following compactness result that will be of crucial importance in the proof of the co-normalMarstrand-Mattila rectifiability criterion later on. tructure of P h - rectifiable measures 26 Proposition . . Let h ∈ {
1, . . . , Q } and assume φ is a P ∗ h -rectifiable measure. Then, for φ -almost all x ∈ G the set Tan h ( φ , x ) is weak- ∗ compact.Proof. Let x ∈ G be such that 0 < Θ h ∗ ( φ , x ) ≤ Θ h , ∗ ( φ , x ) < ∞ and Tan h ( φ , x ) ⊆ M ( h ) . We now prove that forany sequence { λ j C h (cid:120) V j } j ∈ N ⊆ Tan h ( φ , x ) , there are a λ > V ∈ Gr ( h ) such that, up to non-relabelledsubsequences we have λ j C h (cid:120) V j (cid:42) λ C h (cid:120) V .Indeed, thanks to Corollary . we have that λ j ∈ [ Θ h ∗ ( φ , x ) , Θ h , ∗ ( φ , x )] for any j ∈ N and thus we can assumewithout loss of generality that λ j → λ ∈ [ Θ h ∗ ( φ , x ) , Θ h , ∗ ( φ , x )] up to a non-relabelled subsequence. Furthermore,thanks to Proposition . there exists a V ∈ Gr ( h ) such that V j → V with respect to the metric d G . Thus, thanksto Proposition . and a simple computation that we omit, we conclude that λ j C h (cid:120) V j (cid:42) λ C h (cid:120) V .Since we assumed { λ j C h (cid:120) V j } ⊆ Tan h ( φ , x ) then, for any j ∈ N there is a sequence { r (cid:96) ( j ) } (cid:96) ∈ N such that r (cid:96) ( j ) − h T x , r (cid:96) ( j ) φ (cid:42) λ j C h (cid:120) V j .Thus, Proposition . implies that lim (cid:96) → ∞ F ( r (cid:96) ( j ) − h T x , r (cid:96) ( j ) φ , λ j C h (cid:120) V j ) =
0, and in particular for any j ∈ N thereexists an (cid:96) j ∈ N such that defined r j : = r (cid:96) j ( j ) we have F ( r − hj T x , r j φ , λ j C h (cid:120) V j ) ≤ j .Since lim sup j → ∞ r − hj T x , r j φ ( B ( r )) ≤ Θ h , ∗ ( φ , x ) r h for any r >
0, thanks to [ , Corollary . ], we can assumewithout loss of generality that there exists a Radon measure ν such that r − hj T x , r j φ (cid:42) ν . On the other hand, bydefinition we have that ν ∈ Tan h ( φ , x ) and thus by hypothesis on φ there is a η > W ∈ Gr ( h ) such that ν = η C h (cid:120) W . This implies that for any j ∈ N we have F ( η C h (cid:120) W , λ C h (cid:120) V ) ≤ F ( η C h (cid:120) W , r − hj T x , r j φ ) + F ( r − hj T x , r j φ , λ j C h (cid:120) V j ) + F ( λ j C h (cid:120) V j , λ C h (cid:120) V ) ≤ F ( η C h (cid:120) W , r − hj T x , r j φ ) + j + F ( λ j C h (cid:120) V j , λ C h (cid:120) V ) .The arbitrariness of j and Proposition . implies that F ( η C h (cid:120) W , λ C h (cid:120) V ) = . , we conclude that η C h (cid:120) W = λ C h (cid:120) V . This shows that λ C h (cid:120) V ∈ Tan h ( φ , x ) and then the proof isconcluded. P h - rectifiable measures In what follows we set G a Carnot group of homogeneous dimension Q and we fix ≤ h ≤ Q . We alsoassume that φ is a fixed Radon measure on G and we suppose that it is supported on a compact set K . Moreoverwe fix ϑ , γ ∈ N and we freely use the notation E ( ϑ , γ ) introduced in Definition . . In this section we prove Theorem . and Theorem . whose precise statements can be found in Theorem . and Theorem . , respectively.The first step in order to prove Theorem . is to observe the following general property, that can be madequantitative at arbitrary points x ∈ E ( ϑ , γ ) : if the measure S h (cid:120) x V , with V ∈ Gr ( h ) , is sufficiently near to φ in aprecise Measure Theoretic sense at the scale r around x , then in some ball of center x and with radius comparablewith r , the points in the set E ( ϑ , γ ) are not too distant from x V . Roughly speaking, if we denote with F x , r thefunctional that measures the distance between measures on the ball B ( x , r ) , see Definition . , we prove that thefollowing implication holdsif there exist a Θ , δ > F x , r ( φ , Θ S h (cid:120) x V ) ≤ δ r h + ,then E ( ϑ , γ ) ∩ B ( x , r ) ⊆ B ( x V , ω ( δ ) r ) where ω ∈ C and ω ( ) =
0. ( ) tructure of P h - rectifiable measures 27 For the precise statement of ( ), see Proposition . . Let us remark that when φ is a P h -rectifiable measure, thenfor φ -almost every x ∈ G the bound on F x , r in the premise of ( ) is satisfied with V ( x ) ∈ Gr ( h ) , for arbitrarilysmall δ > r < r ( x , δ ) . Thus for P h -rectifiable measures we deduce that the estimate in the conclusionof ( ) holds for arbitrarily small δ , and with r < r ( x , δ ) . This latter estimate easily implies, by a very generalgeometric argument, that E ( ϑ , γ ) ∩ B ( x , r ) ⊆ xC V ( x ) ( α ) for arbitrarily small α and for all r < r ( x , α ) . For the latterassertion we refer the reader to Proposition . . The proof of Theorem . is thus concluded by joining togetherthe previous observations and by the general cone-rectifiability criterion in Proposition . .There is a difference between the Euclidean case and the Carnot case that we discuss now. In the Euclidean caseit is easy to see that whenever we are given a vector subspace V , an arbitrary C V ( α ) -set, with α sufficiently small,is actually the graph of a (Lipschitz) map f : A ⊆ V → V ⊥ . The main reason behind this latter statement is thefollowing: we have a canonical choice of a complementary subgroup V ⊥ of V , and moreover V ⊥ ∩ C V ( α ) = { } for α small enough. Already in the first Heisenberg group H if we take the vertical line V H , we notice thatthere is no choice of a complementary subgroup of V H in H . One could try to bypass this problem by definingproperly some coset projections that would play the role of the projection over a splitting, see Definition . . Thiswill be the topic of further investigations.Nevertheless, if we work in an arbitrary Carnot group G and one of its homogeneous subgroups V admitsa complementary subgroup L we already proved that there exists a constant ε : = ε ( V , L ) such that every C V ( ε ) -set is the intrinsic graph of a function f : A ⊆ V → L . This last statement is precisely the analogous of theEuclidean property that we discussed above, see Proposition . . As a consequence, in order to prove Theorem . we follow the path of the proof of Theorem . , which we discussed above, but we have to pay attention to onetechnical detail. We have to split the subset of the Grassmannian Gr ( h ) made by the homogeneous subgroups V that admit at least one complementary subgroup L into countable subsets according to the value of ε ( V , L ) . Thenwe have to write the proof of Theorem . by paying attention to the fact that we want to control the opening of thefinal C V i ( α i ) -sets with α i < ε ( V i , L i ) . This is what we do in Theorem . : we prove a refinement of Theorem . in which we further ask that the opening of the cones is controlled above also by some a priori defined function F ( V , L ) . Definition . . Let us fix x ∈ G , r > φ a Radon measure on G . We define Π δ ( x , r ) to be the subset of planes V ∈ Gr ( h ) for which there exists a Θ > F x , r ( φ , Θ S h (cid:120) x V ) ≤ δ r h + . ( ) Definition . . For any ϑ ∈ N we define δ G = δ G ( h , ϑ ) : = ϑ − − ( h + ) .In the following proposition we prove that if φ is sufficiently d x , r -near to M ( h ) , see Definition . for thedefinition of d x , r , then E ( ϑ , γ ) is at a controlled distance from a plane V . Proposition . . Let x ∈ E ( ϑ , γ ) , fix δ < δ G , where δ G is defined in Definition . , and set < r < γ . Then for every V ∈ Π δ ( x , r ) , see Definition . , we have sup w ∈ E ( ϑ , γ ) ∩ B ( x , r /4 ) dist (cid:0) w , x V (cid:1) r ≤ + ( h + ) ϑ ( h + ) δ ( h + ) = : C ( ϑ , h ) δ ( h + ) . ( ) Proof.
Let V be any element of Π δ ( x , r ) and suppose Θ > (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d φ − Θ ˆ f d S h (cid:120) x V (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ r h + , for any f ∈ Lip + ( B ( x , r )) .Since the function g ( w ) : = min { dist ( w , U ( x , r ) c ) , dist ( w , x V ) } belongs to Lip + ( B ( x , r )) , we deduce that2 δ r h + ≥ ˆ g ( w ) d φ ( w ) − Θ ˆ g ( w ) d S h (cid:120) x V = ˆ g ( w ) d φ ( w ) ≥ ˆ B ( x , r /2 ) min { r /2, dist ( w , x V ) } d φ ( w ) . tructure of P h - rectifiable measures 28 Suppose that y is a point in B ( x , r /4 ) ∩ E ( ϑ , γ ) furthest from x V and let D : = dist ( y , x V ) . If D ≥ r /8, this wouldimply that 2 δ r h + ≥ ˆ B ( x , r /2 ) min { r /2, dist ( w , x V ) } d φ ( w ) ≥ ˆ B ( y , r /16 ) min { r /2, dist ( w , x V ) } d φ ( w ) ≥ r φ ( B ( y , r /16 )) ≥ r h + ϑ h + ,where the last inequality follows from the definition of E ( ϑ , γ ) . The previous inequality would imply δ ≥ ϑ − − ( h + ) , which is not possible since δ < δ G = ϑ − − ( h + ) , see Definition . . This implies that D ≤ r /8and as a consequence, we have2 δ r h + ≥ ˆ B ( x , r /2 ) min { r /2, dist ( w , x V ) } d φ ( w ) ≥ ˆ B ( y , D /2 ) min { r /2, dist ( w , x V ) } d φ ( w ) ≥ D φ ( B ( y , D /2 )) ≥ ϑ − (cid:18) D (cid:19) h + , ( )where the second inequality comes from the fact that B ( y , D /2 ) ⊆ B ( x , r /2 ) . This implies thanks to ( ), thatsup w ∈ E ( ϑ , γ ) ∩ B ( x , r /4 ) dist ( w , x V ) r ≤ Dr ≤ + ( h + ) ϑ ( h + ) δ ( h + ) = C ( ϑ , h ) δ ( h + ) . Remark . . Notice that a priori Π δ ( x , r ) in the statement of Proposition . may be empty. Nevertheless it is easyto notice, by using the definitions, that if d x , r ( φ , M ) ≤ δ then Π δ ( x , r ) is nonempty.In the following proposition we show that if we are at a point x ∈ E ( ϑ , γ ) for which the h -tangents are flat, thenlocally around x the set E ( ϑ , γ ) enjoys an appropriate cone property with arbitrarily small opening. Proposition . . For any α > and any x ∈ E ( ϑ , γ ) for which Tan h ( φ , x ) ⊆ { λ S h (cid:120) V ( x ) : λ > } for some V ( x ) ∈ Gr ( h ) , there exists a ρ ( α , x ) > such that whenever < r < ρ we haveE ( ϑ , γ ) ∩ B ( x , r ) ⊆ xC V ( x ) ( α ) . Proof.
Let us fix α >
0. Let us fix x ∈ E ( ϑ , γ ) and V ( x ) ∈ Gr ( h ) such that Tan h ( φ , x ) ⊆ { λ S h (cid:120) V ( x ) : λ > } . Thus,by using Proposition . , we conclude thatlim r → inf Θ > F x , r ( φ , Θ S h (cid:120) x V ( x )) r h + = ε > γ > r ( ε ) > Θ > F x , r ( φ , Θ S h (cid:120) x V ( x )) ≤ ε r h + , whenever 0 < r ≤ r ( ε ) . ( )Now we aim at proving that, for ε > E ( ϑ , γ ) ∩ B ( x , r ( ε ) /4 ) ⊆ xC V ( x ) ( α ) . In order to prove thiswe notice that ( ) and Proposition . imply that, for ε > p ∈ E ( ϑ , γ ) ∩ B ( x , r /4 ) dist ( p , x V ( x )) ≤ C ( h , ϑ ) ε ( h + ) r , whenever 0 < r ≤ r ( ε ) . ( )Indeed, from ( ) it follows that V ( x ) ∈ Π ε ( x , r ) for every 0 < r ≤ r , see Definition . ; so that it suffices tochoose ε < δ G = ϑ − − ( h + ) , see Definition . , in order to apply Proposition . and conclude ( ).Now let us take ε < δ G so small that the following inequality holds 8 C ( h , ϑ ) ε ( h + ) < α . We finally prove E ( ϑ , γ ) ∩ B ( x , r ( ε ) /4 ) ⊆ xC V ( x ) ( α ) . Indeed, let p ∈ E ( ϑ , γ ) ∩ B ( x , r ( ε ) /4 ) , and k ≥ r − k < (cid:107) x − · p (cid:107) ≤ r − k + . Since p ∈ E ( ϑ , γ ) ∩ B ( x , ( r − k + ) /4 ) , from ( ) we get d ( p , x V ( x )) ≤ C ( h , ϑ ) ε ( h + ) r − k + ≤ C ( h , ϑ ) ε ( h + ) (cid:107) x − · p (cid:107) ≤ α (cid:107) x − · p (cid:107) ,thus showing the claim. tructure of P h - rectifiable measures 29 We now prove a cone-type rectifiability criterion that will be useful in combination with the previous results inorder to split the support of a P h or a P ch -rectifiable measures with sets that have the cone property. Proposition . (Cone-rectifiability criterion) . Suppose that E is a closed subset of G for which there exists a countablefamily F ⊆ Gr ( h ) and a function α : F → (
0, 1 ) such that for every x ∈ E there exist ρ ( x ) > , and V ( x ) ∈ F for whichB ( x , r ) ∩ E ⊆ xC V ( x ) ( α ( V ( x ))) , ( ) whenever < r < ρ ( x ) . Then, there are countably many compact C V i ( β i ) -sets Γ i such that V i ∈ F , and α ( V i ) < β i < α ( V i ) for which E = (cid:91) i ∈ N Γ i . ( ) Proof.
Let us split E in the following way. Let G ( i , j , k ) be the subset of those x ∈ E ∩ B ( k ) for which B ( x , r ) ∩ E ⊆ xC V i ( α ( V i )) ,for any 0 < r < j . Then, from the hypothesis, it follows E = ∪ i , j , k ∈ N G ( i , j , k ) . Since E is closed, it is not difficultto see that G ( i , j , k ) is closed too. Let us fix i , j , k ∈ N , some β i < α ( V i ) < β i < α ( V i ) , and let us provethat G ( i , j , k ) can be covered with countably many compact C V i ( β i ) -sets. Since i , j , k ∈ N are fixed from now onwe assume without loss of generality that G ( i , j , k ) = E so that we can drop the indeces.Let us take { q (cid:96) } a dense subset of E , and let us define the closed tubular neighbourhood of q (cid:96) V S ( (cid:96) ) : = B ( q (cid:96) V , 2 − κ j − κ C ( k , G ) − κ β κ ) , ( )where we recall that κ is the step of the group, and where C is defined in ( . ). We will now prove that S ( (cid:96) ) ∩ E is a C V ( β ) -set, or equivalently that for any p ∈ S ( (cid:96) ) ∩ E we have S ( (cid:96) ) ∩ E ⊆ p · C V ( β ) . ( )If q ∈ S ( (cid:96) ) ∩ E ∩ B ( p , 1/ ( j )) , the inclusion ( ) holds thanks to our assumptions on E . If on the other hand q ∈ S ( (cid:96) ) ∩ E \ B ( p , 1/ ( j )) , let p ∗ , q ∗ ∈ V be such that d ( p , q (cid:96) V ) = (cid:107) ( p ∗ ) − q − (cid:96) p (cid:107) , and d ( q , q (cid:96) V ) = (cid:107) ( q ∗ ) − q − (cid:96) q (cid:107) .Let us prove that (cid:107) q ∗ (cid:107) ≤ k and (cid:107) p ∗ (cid:107) ≤ k . This is due to the fact that (cid:107) q ∗ (cid:107) − (cid:107) q (cid:96) (cid:107) − (cid:107) q (cid:107) ≤ (cid:107) ( q ∗ ) − q − (cid:96) q (cid:107) = d ( q , q (cid:96) V ) ≤ S ( (cid:96) ) , see ( ). From the previous inequality it follows that (cid:107) q ∗ (cid:107) ≤ k +
1, since q , q (cid:96) ∈ B ( k ) . A similar computation proves the bound for (cid:107) p ∗ (cid:107) and this implies that (cid:107) p − · q (cid:96) · p ∗ (cid:107) + (cid:107) ( p ∗ ) − · q ∗ (cid:107) ≤ (cid:107) p − (cid:107) + (cid:107) q (cid:96) (cid:107) + (cid:107) p ∗ (cid:107) + (cid:107) q ∗ (cid:107) ≤ k .The application of Lemma . and the fact that ( q ∗ ) − p ∗ and p − q (cid:96) p ∗ are in B (
0, 14 k ) , due to the previous inequal-ity, imply that d ( p − q , V ) ≤ (cid:107) ( q ∗ ) − p ∗ · p − q (cid:107) = (cid:107) ( q ∗ ) − p ∗ · p − · q (cid:96) p ∗ ( p ∗ ) − q ∗ ( q ∗ ) − q − (cid:96) · q (cid:107)≤ (cid:107) ( q ∗ ) − p ∗ · p − q (cid:96) p ∗ · ( p ∗ ) − q ∗ (cid:107) + (cid:107) ( q ∗ ) − q − (cid:96) q (cid:107)≤ C ( k , G ) (cid:107) p − q (cid:96) p ∗ (cid:107) κ + d ( q , q (cid:96) V ) = C ( k , G ) d ( p , q (cid:96) V ) κ + d ( q , q (cid:96) V ) . ( )Finally, thanks to ( ) and ( ) we infer d ( p − q , V ) ≤ C ( k , G ) + jC ( k , G ) β ≤ β j − ≤ β (cid:107) p − q (cid:107) ,thus showing ( ) in the remaining case. In conclusion we have proved that for any i , j , k , (cid:96) ∈ N , the sets G ( i , j , k ) ∩ S ( (cid:96) ) are C V i ( β i ) -sets. This concludes the proof since E ⊆ (cid:91) i , j , k , (cid:96) ∈ N G ( i , j , k ) ∩ S ( (cid:96) ) ,and on the other hand every G ( i , j , k ) ∩ S ( (cid:96) ) is a bounded and closed, thus compact, C V i ( β i ) -set. The fact that thesets G ( i , j , k ) ∩ S ( (cid:96) ) are contained in E follows by definition, thus concluding the proof of the equality. tructure of P h - rectifiable measures 30 In the following, with the symbol Sub ( h ) , we denote the subset of Gr c ( h ) × Gr c ( h ) defined by { ( V , L ) : V ∈ Gr c ( h ) and L is a homogeneous subgroup that is a complement of V } , ( )we fix a function F : Sub ( h ) → (
0, 1 ) , and for every (cid:96) ∈ N with (cid:96) ≥ Gr F c ( h , (cid:96) ) : = { V ∈ Gr c ( h ) : ∃ L complement of V s.t. 1/ (cid:96) < F ( V , L ) ≤ ( (cid:96) − ) } .Observe that Proposition . implies that Gr F c ( h , (cid:96) ) is separable for any (cid:96) ∈ N , since Gr F c ( h , (cid:96) ) ⊆ Gr ( h ) and ( Gr ( h ) , d G ) is a compact metric space, see Proposition . . Let D (cid:96) : = { V i , (cid:96) } i ∈ N , ( )be a countable dense subset of Gr F c ( h , (cid:96) ) andfor every i ∈ N , choose a complement L i , (cid:96) of V i , (cid:96) such that 1/ (cid:96) < F ( V i , (cid:96) , L i , (cid:96) ) ≤ ( (cid:96) − ) . ( )The following theorem is a more detailed version of Theorem . . Theorem . . Let F : Sub ( h ) → (
0, 1 ) be a function, where Sub ( h ) is defined in ( ) , and for every (cid:96) ∈ N define D (cid:96) asin ( ) , set F : = { V i , (cid:96) } i , (cid:96) ∈ N , and choose L i , (cid:96) as in ( ) . Furthermore, let β : N → (
0, 1 ) and define β ( V i , (cid:96) ) : = β ( (cid:96) ) forevery i , (cid:96) ∈ N . For the ease of notation we rename F : = { V k } k ∈ N . Then the following holds.Let φ be a P ch -rectifiable measure. There are countably many compact sets Γ k that are C V k ( min { F ( V k , L k ) , β ( V k ) } ) -setsfor some V k ∈ F , and such that φ (cid:16) G \ + ∞ (cid:91) k = Γ k (cid:17) = Proof.
Let us notice that without loss of generality, by restricting the measure on balls with integer radius, we cansuppose that φ has a compact support. Fix ϑ , γ ∈ N and let E ( ϑ , γ ) be the set introduced in Definition . withrespect to φ . Furthermore, for any (cid:96) , i , j ∈ N , we let F (cid:96) ( i , j ) : = { x ∈ E ( ϑ , γ ) : B ( x , r ) ∩ E ( ϑ , γ ) ⊆ xC V i , (cid:96) ( − min { F ( V i , (cid:96) , L i , (cid:96) ) , β ( V i , (cid:96) ) } ) for any 0 < r < j } . ( )It is not hard to prove, since E ( ϑ , γ ) is compact, see Proposition . , that for every (cid:96) , i , j the sets F (cid:96) ( i , j ) are compact.We claim that φ (cid:16) E ( ϑ , γ ) \ (cid:91) (cid:96) , i , j ∈ N F (cid:96) ( i , j ) (cid:17) =
0. ( )Indeed, let w ∈ E ( ϑ , γ ) be such that Tan h ( φ , w ) ⊆ { λ S h (cid:120) V ( w ) : λ > } for some V ( w ) ∈ Gr c ( h ) ; this can be donefor φ -almost every point w in E ( ϑ , γ ) since φ is P ch -rectifiable. Let (cid:96) ( w ) ∈ N be the smallest natural number forwhich there exists L complementary to V ( w ) with 1/ (cid:96) ( w ) < F ( V ( w ) , L ) ≤ ( (cid:96) ( w ) − ) . Then by definition wehave V ( w ) ∈ Gr F c ( h , (cid:96) ( w )) . By density of the family D (cid:96) ( w ) in Gr F c ( h , (cid:96) ( w )) there exists a plane V i , (cid:96) ( w ) ∈ D (cid:96) ( w ) suchthat d G ( V i , (cid:96) ( w ) , V ( w )) < − min { (cid:96) ( w ) , β ( V i , (cid:96) ( w ) ) } ;for this last observation to hold it is important that β only depends on (cid:96) ( w ) , as it is by construction. The previousinequality, jointly with Lemma . , imply that C V ( w ) ( − min { (cid:96) ( w ) , β ( V i , (cid:96) ( w ) ) } ) ⊆ C V i , (cid:96) ( w ) ( − min { (cid:96) ( w ) , β ( V i , (cid:96) ( w ) ) } ) ⊆ C V i , (cid:96) ( w ) ( − min { F ( V i , (cid:96) ( w ) , L i , (cid:96) ( w ) ) , β ( V i , (cid:96) ( w ) ) } ) , ( )where the last inclusion follows from the fact that by definition of the family D (cid:96) ( w ) it holds F ( V i , (cid:96) ( w ) , L i , (cid:96) ( w ) ) > (cid:96) ( w ) . Thanks to Proposition . we can find a ρ ( w ) > < r < ρ ( w ) we have B ( w , r ) ∩ E ( ϑ , γ ) ⊆ wC V ( w ) ( − min { (cid:96) ( w ) , β ( V i , (cid:96) ( w ) ) } ) . ( ) Actually this is an abuse of notation. We mean that F is the disjoint union of the families D (cid:96) in ( ). Thus it may happen that the samesubgroup is in the family F more than once, but this is clearly not a problem. ounds for the densities of S h on C V ( α ) - sets 31 In particular, putting together ( ) and ( ) we infer that for φ -almost every w ∈ E ( ϑ , γ ) there are an i = i ( w ) > (cid:96) ( w ) ∈ N and a ρ ( w ) > < r < ρ ( w ) we have B ( w , r ) ∩ E ( ϑ , γ ) ⊆ wC V i , (cid:96) ( w ) ( − min { F ( V i , (cid:96) ( w ) , L i , (cid:96) ( w ) ) , β ( V i , (cid:96) ( w ) ) } ) .This concludes the proof of ( ).Now, if we fix (cid:96) , i , j ∈ N , we can apply Proposition . to the set F (cid:96) ( i , j ) . It suffices to take the family F inthe statement of Proposition . to be the singleton { V i , (cid:96) } and the function α in the statement of Proposition . to be α ( V i , (cid:96) ) : = − min { F ( V i , (cid:96) , L i , (cid:96) ) , β ( V i , (cid:96) ) } . As a consequence we can write each F (cid:96) ( i , j ) as the union ofcountably many compact C V i , (cid:96) ( min { F ( V i , (cid:96) , L i , (cid:96) ) , β ( V i , (cid:96) ) } ) -sets. Thus the same holds φ -almost everywhere for E ( ϑ , γ ) , allowing i , (cid:96) to vary in N , since ( ) holds. Finally, we have φ ( G \ ∪ ϑ , γ ∈ N E ( ϑ , γ )) = . . Thus we can cover φ -almost all of G with compact C V i , (cid:96) ( min { F ( V i , (cid:96) , L i , (cid:96) ) , β ( V i , (cid:96) ) } ) -setsfor i , (cid:96) that vary in N , concluding the proof of the proposition.The following theorem is a more detailed version of Theorem . . Theorem . . There exists a countable subfamily F : = { V k } k ∈ N of Gr ( h ) such that the following holds. Let φ be a P h -rectifiable measure. For any < β < there are countably many compact sets Γ k that are C V k ( β ) -sets for some V k ∈ F ,and such that φ (cid:16) G \ + ∞ (cid:91) k = Γ k (cid:17) = Proof.
The proof is similar to the one of Theorem . . It suffices to choose, as a family F , an arbitrary countabledense subset of Gr ( h ) and then one can argue as in the proof of Theorem . without the technical effort ofintroducing the parameter (cid:96) . We skip the deatils. S h on C V ( α ) - sets Throughout this subsection we assume that V ∈ Gr c ( h ) and that V · L = G . In this chapter whenever wedeal with C V ( α ) -sets we are always assuming that α < ε ( V , L ) , where ε is defined in Lemma . .This section is devoted to the proof of Theorem . , that is obtained through three different steps. Let Γ be acompact C V ( ε ( V , L )) set, and recall that by Proposition . we can write Γ = graph ( ϕ ) with ϕ : P V ( Γ ) → L . Letus denote Φ ( v ) : = v · ϕ ( v ) for every v ∈ P V ( Γ ) .We first show that if we assume that Θ h ∗ ( S h (cid:120) Γ , x ) > at S h (cid:120) Γ -almost every point x , then the push-forwardmeasure ( Φ ) ∗ ( S h (cid:120) V ) is mutually absolutely continuous with respect to S h (cid:120) Γ , see Proposition . . In other wordswe are proving that whenever an intrinsically Lipschitz graph over a subset of an h -dimensional subgroup hasstrictly positive lower density almost everywhere, then the push-forward of the measure S h on the subgroup bymeans of the graph map is mutually absolutely continuous with respect to the measure S h on the graph. Westress that we do not address the issue of removing the hypothesis on the strict positivity of the lower density inProposition . as it is out of the aims of this paper. We remark that in the Euclidean case the analogous statementholds true without this assumption: this is true because in the Euclidean case every Lipschitz graph over a subsetof a vector subspace of dimension h has stricitly positive lower h -density almost everywhere. We also stress thatevery intrinsically Lipschitz graph over a open subset of a h -dimensional homogeneous subgroupshas strictlypositive lower h -density almost everywhere, see [ , Theorem . ].As a second step in order to obtain the proof of Theorem . we prove the following statement that can bemade quantitative: if V ∈ Gr c ( h ) , Γ is a compact C V ( α ) -set with α sufficiently small, and S h (cid:120) Γ is a P h -rectifiablemeasure with complemented tangents, which we called P ch -rectifiable, then we can give an explicit lower boundof the ratio of the lower and upper h -densities of S h (cid:120) Γ . We refer the reader to Proposition . for a more precisestatement and the proof of the following proposition. ounds for the densities of S h on C V ( α ) - sets 32 Proposition . (Bounds on the ratio of the densities) . Let V be in Gr c ( h ) . There exists C : = C ( V ) such that thefollowing holds. Suppose Γ is a compact C V ( α ) -set with α < C ( V ) and such that S h (cid:120) Γ is a P ch -rectifiable measure. Thenthere exists a continuous function ω : = ω ( V ) of α , with ω ( ) = , such that for S h -almost every x ∈ Γ we have − ω ( α ) ≤ Θ h ∗ ( S h (cid:120) Γ , x ) Θ h , ∗ ( S h (cid:120) Γ , x ) ≤
1. ( )The previous result is obtained through a blow-up analysis and a careful use of the mutually absolute continuityproperty that we discussed above, and which is contained in Proposition . . We stress that in order to differentiatein the proof of Proposition . , we need to use proper S h (cid:120) P V ( Γ ) and S h (cid:120) V -Vitali relations, see Proposition . ,and Proposition . , respectively.As a last step of the proof of Theorem . we first use the result in Proposition . in order to prove thatTheorem . holds true for measures of the type S h (cid:120) Γ , see Theorem . . Then we conclude the proof for arbitrarymeasures by reducing ouserlves to the sets E ( ϑ , γ ) , see Corollary . . The last part about the convergence inTheorem . readily comes from the first part and Proposition . .We start this chapter with some lemmata. Lemma . . There exists an A : = A ( V , L ) > such that for any w ∈ B (
0, 1/5 A ) , any y ∈ ∂ B (
0, 1 ) ∩ C V ( ε ( V , L )) andany z ∈ B ( y , 1/5 A ) , we have w − z (cid:54)∈ L .Proof. By contradiction let us assume that we can find sequences { w n } , { y n } ⊆ ∂ B (
0, 1 ) ∩ C V ( ε ) and z n ∈ B ( y n , 1/ n ) such that w n converges to 0 and w − n z n ∈ L . By compactness without loss of generality we can as-sume that the sequence y n converges to some y ∈ ∂ B (
0, 1 ) ∩ C V ( ε ) . Furthermore, by construction we also havethat z n must converge to y . This implies that w − n z n converges to y and since by hypothesis w − n z n ∈ L , thanks tothe fact that L is closed we infer that y ∈ L . This however is a contradiction since y has unit norm and at the sametime we should have y ∈ C V ( ε ) ∩ L = { } by Lemma . . Proposition . . Let α < ε ( V , L ) and suppose Γ is a compact C V ( α ) -set. For any x ∈ Γ let ρ ( x ) to be the biggest numbersatisfying the following condition. For any y ∈ B ( x , ρ ( x )) ∩ Γ we haveP V ( B ( x , r )) ∩ P V ( B ( y , s )) = ∅ for any r , s < d ( x , y ) /5 A , where A = A ( V , L ) is the constant yielded by Lemma . . Then, the function x (cid:55)→ ρ ( x ) is positive everywhere on Γ andupper semicontinuous.Proof. Let x ∈ Γ and suppose by contradiction that there is a sequence of points { y i } i ∈ N ⊆ Γ converging to x and P V ( B ( x , r i )) ∩ P V ( B ( y i , s i )) (cid:54) = ∅ , ( )for some r i , s i < d ( x , y i ) /5 A . We note that ( ) is equivalent to assuming that there are z i ∈ B ( x , r i ) and w i ∈ B ( y i , s i ) such that P V ( w i ) = P V ( z i ) . ( )Identity ( ) implies in particular that for any i ∈ N we have w − i z i ∈ L and let us denote ρ i : = d ( x , y i ) . Thanks tothe assumptions on y i , z i and w i we have that( ) d ( δ ρ i ( x − y i )) = y ∈ ∂ B (
0, 1 ) suchthat lim i → ∞ δ ρ i ( x − y i ) = y ,( ) d ( δ ρ i ( x − z i )) ≤ A and thus up to passing to a non-relabelled subsequence we can assume that thereexists a z ∈ B (
0, 1/5 A ) such that lim i → ∞ δ ρ i ( x − z i ) = z ,( ) d ( δ ρ i ( x − y i ) , δ ρ i ( x − w i )) ≤ A and thus, up to passing to a non re-labelled subsequence, we cansuppose that there exists a w ∈ B ( y , 1/5 A ) such thatlim i → ∞ δ ρ i ( x − w i ) = w . ounds for the densities of S h on C V ( α ) - sets 33 Since Γ is supposed to be a C V ( α ) -set, we have that for any i ∈ N the point x − y i is contained in the cone C V ( α ) and, since C V ( α ) is closed, we infer that y ∈ C V ( α ) . Since we assumed α < ε ( V , L ) , we have y ∈ ∂ B (
0, 1 ) ∩ C V ( ε ( V , L )) . Since δ ρ i ( x − z i ) and δ ρ i ( x − w i ) converge to z and w , respectively, we havelim i → ∞ δ ρ i ( w − i z i ) = lim i → ∞ δ ρ i ( w − i x ) δ ρ i ( x − z i ) = w − z .Furthermore since w − i z i ∈ L for any i ∈ N , we infer that w − z ∈ L since L is closed. Applying Lemma . to y , z , w we see that the fact that w − z ∈ L , z ∈ B (
0, 1/5 A ) and w ∈ B ( y , 1/5 A ) results in a contradiction. Thisconcludes the proof of the first part of the proposition.In order to show that ρ is upper semicontinuous we fix an x ∈ Γ and we assume by contradiction that thereexists a sequence { x i } i ∈ N ⊆ Γ converging to x such thatlim sup i → ∞ ρ ( x i ) > ( + τ ) ρ ( x ) , ( )for some τ >
0. Fix an y ∈ B ( x , ( + τ /2 ) ρ ( x )) ∩ Γ and assume s , r < d ( x , y ) /5 A . Thus, thanks to ( ) and the factthat the x i converge to x , we infer that there exists a i ∈ N such that, up to non re-labelled subsequences, for any i ≥ i we have ρ ( x i ) > ( + τ ) ρ ( x ) , d ( x i , x ) < τρ ( x ) /4 and s , r + d ( x i , x ) < d ( x i , y ) /5 A . Therefore, for any i ≥ i we have y ∈ B ( x i , ( + τ /4 ) ρ ( x )) ⊆ B ( x i , ρ ( x i )) , and s , r + d ( x i , x ) < d ( x i , y ) /5 A .This however, thanks to the definition of ρ ( x i ) , implies that P V ( B ( x , r )) ∩ P V ( B ( y , s )) ⊆ P V ( B ( x i , r + d ( x i , x ))) ∩ P V ( B ( y , s )) = ∅ .Summing up, we have proved that for any y ∈ B ( x , ( + τ /2 ) ρ ( x )) ∩ Γ whenever r , s < d ( x , y ) /5 A we have P V ( B ( x , r )) ∩ P V ( B ( y , s )) = ∅ ,and this contradicts the maximality of ρ ( x ) . This concludes the proof. Corollary . . Let us fix α < ε ( V , L ) and suppose that Γ is a compact C V ( α ) -set. Let us fix x ∈ Γ and choose ρ ( x ) > as in the statement of Proposition . . Then there is a < r ( x ) < such that the following holdsif < r < r ( x ) and y ∈ Γ are such that P V ( B ( x , 2 r )) ∩ P V ( B ( y , 10 r )) (cid:54) = ∅ , then y ∈ B ( x , ρ ( x )) and d ( x , y ) ≤ Ar ,( ) where A = A ( V , L ) is the constant yielded by Lemma . .Proof. Let us first prove that there exists (cid:101) α : = (cid:101) α ( α , x ) such that whenever y ∈ Γ is such that d ( x , y ) ≥ ρ ( x ) then d ( P V ( x ) , P V ( y )) ≥ (cid:101) α . Indeed if it is not the case, we have a sequence { y i } i ∈ N ⊆ Γ such that d ( x , y i ) ≥ ρ ( x ) forevery i ∈ N and d ( P V ( x ) , P V ( y i )) → i → + ∞ . Since Γ is compact we can suppose, up to passing to a non re-labelled subsequence, that y i → y ∈ Γ . Moreover since d ( x , y i ) ≥ ρ ( x ) and d ( P V ( x ) , P V ( y i )) → d ( x , y ) ≥ ρ ( x ) , and hence x (cid:54) = y , and moreover P V ( x ) = P V ( y ) . Then y − · x ∈ L ∩ C V ( α ) that is a contradictionwith Lemma . because y (cid:54) = x and α < ε .Since P V is uniformly continuous on the closed tubular neighborhood B ( Γ , 1 ) , there exists a r ( x ) > (cid:101) α = (cid:101) α ( α , x ) such that for any y ∈ Γ and any r < r ( x ) , we have P V ( B ( y , 10 r )) ⊆ B ( P V ( y ) , (cid:101) α /10 ) . ( )Let us show the first part of the statement. It is sufficient to prove that if r < r ( x ) and y ∈ Γ is such that d ( x , y ) ≥ ρ ( x ) , then P V ( B ( x , 2 r )) ∩ P V ( B ( y , 10 r )) = ∅ . Indeed if d ( x , y ) ≥ ρ ( x ) then d ( P V ( x ) , P V ( y )) ≥ (cid:101) α . More-over, from ( ) we deduce that P V ( B ( x , 10 r )) ⊆ B ( P V ( x ) , (cid:101) α /10 ) and P V ( B ( y , 10 r )) ⊆ B ( P V ( y ) , (cid:101) α /10 ) . Since d ( P V ( x ) , P V ( y )) ≥ (cid:101) α we conclude that B ( P V ( x ) , (cid:101) α /10 ) ∩ B ( P V ( y ) , (cid:101) α /10 ) = ∅ and then also P V ( B ( x , 10 r )) ∩ P V ( B ( y , 10 r )) = ∅ , from which the sought conclusion follows. In order to prove d ( x , y ) ≤ Ar , once we have y ∈ B ( x , ρ ( x )) , the conclusion follows thanks to Proposition . . ounds for the densities of S h on C V ( α ) - sets 34 Lemma . . Fix some N ∈ N and assume that F is a family of closed balls of G with uniformly bounded radii. Then wecan find a countable disjoint subfamily G of F such that(i) if B , B (cid:48) ∈ G then N B and N B (cid:48) are disjoint,(ii) (cid:83) B ∈ F B ⊆ (cid:83) B ∈ G N + B.Proof. If N =
0, there is nothing to prove, since it is the classical 5-Vitali’s covering Lemma.Let us assume by inductive hypothesis that the claim holds for N = k and let us prove that it holds for k +
1. Let G k be the family of balls satisfying (i) and (ii) for N = k , and apply the 5-Vitali’s covering Lemma to the family ofballs (cid:102) F : = { k + B : B ∈ G k } . We obtain a countable subfamily (cid:101) G of (cid:102) F such that if 5 k + B , 5 k + B (cid:48) ∈ (cid:101) G then 5 k + B and 5 k + B (cid:48) are disjoint and that satisfies (cid:83) B ∈ (cid:102) F B ⊆ (cid:83) B ∈ (cid:101) G B . Therefore, if we define G k + : = { B ∈ G k : 5 k + B ∈ (cid:101) G } ,point (i) directly follows and thanks to the inductive hypothesis we have (cid:91) B ∈ F B ⊆ (cid:91) B ∈ G k k + B ⊆ (cid:91) B ∈ G k + k + B ,proving the second point of the statement. Proposition . . Let α < ε ( V , L ) and suppose Γ is a compact C V ( α ) -set of finite S h -measure such that Θ h ∗ ( S h (cid:120) Γ , x ) > for S h -almost every x ∈ Γ . Then, there exists a constant C > depending only on V , L and the smooth-box distance d on G , such that for S h -almost every x ∈ Γ there exists an R : = R ( x ) > such that for any < (cid:96) ≤ R we have S h ( P V ( Γ ∩ B ( x , (cid:96) ))) ≥ C Θ h ∗ ( S h (cid:120) Γ , x ) (cid:96) h . ( ) Proof.
First of all, let us recall that two homogeneous left-invariant distances are always bi-Lipschitz equivalent on G . Therefore if d c is a Carnot-Carathéodory distance on G , which is in particular geodetic, see [ , Section . ]there exists a constant L ( d , d c ) ≥ L ( d , d c ) − d c ( x , y ) ≤ d ( x , y ) ≤ L ( d , d c ) d c ( x , y ) for any x , y ∈ G .We claim that if for any ϑ , γ ∈ N for which S h ( E ( ϑ , γ )) > S h -almost any w ∈ E ( ϑ , γ ) thereexists a R ( w ) > S h ( P V ( Γ ∩ B ( w , (cid:96) ))) ≥ C ( V , L ) (cid:96) h · h A h L ( d , d c ) h ϑ , ( )whenever 0 < (cid:96) < R ( w ) , then the proposition is proved. This is due to the following reasoning. First of all,thanks to [ , Proposition . . ( )], we know that Θ h , ∗ ( S h (cid:120) Γ , x ) ≤
1. Secondly, if we set, for any k ∈ N , Γ k : = { w ∈ Γ : 1/ ( k + ) < Θ h ∗ ( S h (cid:120) Γ , x ) ≤ k } , we have that S h ( Γ \ (cid:91) k ∈ N Γ k ) =
0. ( )We observe now that if S h ( Γ k ) >
0, then S h -almost every w ∈ Γ k belongs to some E ( k + γ ) provided γ is bigenough, or in other words S h (cid:0) Γ k \ (cid:91) γ ∈ N E ( k + γ ) (cid:1) =
0. ( )If our claim ( ) holds true, whenever S h ( E ( k + γ )) >
0, we have that for S h (cid:120) E ( k + γ ) -almost every w thereexists R ( w ) such that whenever 0 < (cid:96) < R ( w ) the following chain of inequalities holds S h ( P V ( Γ ∩ B ( w , (cid:96) ))) ≥ C ( V , L ) (cid:96) h · h A h L ( d , d c ) h ( k + ) ≥ C ( V , L ) (cid:96) h · h A h L ( d , d c ) h k ≥ C ( V , L ) Θ h ∗ ( S h (cid:120) Γ , x ) (cid:96) h · h A h L ( d , d c ) h = C Θ h ∗ ( S h (cid:120) Γ , x ) (cid:96) h . ( ) ounds for the densities of S h on C V ( α ) - sets 35 Identities ( ) and ( ) together with ( ) imply that our claim suffices to prove the proposition. Therefore, in thefollowing we will assume that ϑ , γ ∈ N are fixed and such that S h ( E ( ϑ , γ )) >
0, and we want to prove ( ).Let N ∈ N be the unique natural number for which 5 N − ≤ A L ( d , d c ) < N − and for any k ∈ N and0 < δ < ρ ( x ) is defined in Proposition . , A ϑ , γ ( k ) : = { x ∈ E ( ϑ , γ ) : ρ ( x ) > k } , D ϑ , γ ( k ) : = (cid:26) x ∈ A ϑ , γ ( k ) : lim r → S h ( B ( x , r ) ∩ A ϑ , γ ( k )) S h ( B ( x , r ) ∩ E ( ϑ , γ )) = (cid:27) , F δ ( k ) : = (cid:26) B ( x , r ) : x ∈ D ϑ , γ ( k ) and r ≤ min { ϑ − , γ − , k − , δ } A L ( d , d c ) (cid:27) .For any ϑ , γ ∈ N the sets A ϑ , γ ( k ) are Borel since thanks to Proposition . , the function ρ is upper semicontin-uous. Before going on we observe that S h (cid:120) E ( ϑ , γ )( A ϑ , γ ( k ) \ D ϑ , γ ( k )) =
0. This comes from the fact that thepoints of D ϑ , γ ( k ) are exactly the points of density one of A ϑ , γ ( k ) with respect to the measure S h (cid:120) E ( ϑ , γ ) , thatis asymptotically doubling at S h (cid:120) E ( ϑ , γ ) -almost every point because it has positive lower density and finite up-per density at S h (cid:120) E ( ϑ , γ ) -almost every point, see Proposition . . Moreover observe that from Proposition . S h ( E ( ϑ , γ ) \ ∪ + ∞ k = A ϑ , γ ( k )) =
0. Let us apply Lemma . to N and F δ ( k ) , and thus we infer that there exists asubfamily G δ ( k ) such that( α ) for any B , B (cid:48) ∈ G δ ( k ) we have that 5 N B ∩ N B (cid:48) = ∅ ,( β ) (cid:83) B ∈ F δ ( k ) B ⊆ (cid:83) B ∈ G δ ( k ) N + B .The point ( α ) above implies in particular that whenever B ( x , r ) , B ( y , s ) ∈ G δ ( k ) we have d ( x , y ) > L ( d , d c ) − N ( r + s ) , since d is L ( d , d c ) -Lipschitz equivalent to the geodetic ditance d c , and thanks to the choice of N we deduce that r + s < d ( x , y ) A .Throughout the rest of the proof we fix a w ∈ D ϑ , γ ( k ) and a 0 < R ( w ) < min { ϑ − , γ − , k − } /8 such that S h (cid:120) Γ ( B ( w , (cid:96) )) (cid:96) h ≥ ϑ , and S h (cid:120) D ϑ , γ ( k )( B ( w , (cid:96) )) S h (cid:120) Γ ( B ( w , (cid:96) )) ≥
12 , for any 0 < (cid:96) ≤ R ( w ) . ( )For the ease of notation we continue the proof fixing the radius (cid:96) = R ( w ) = R . We stress that the forthcomingestimates are verified, mutatis mutandis, also for any 0 < (cid:96) < R . The first inequality above comes from thedefinition of E ( ϑ , γ ) , see Definition . , while the second is true, up to choose a sufficiently small R ( w ) , because S h (cid:120) Γ -almost every point of D ϑ , γ ( k ) has density one with respect to the asymptotically doubling measure S h (cid:120) Γ .Let us stress that if we prove our initial claim for such w and R ( w ) we are done since S h (cid:120) Γ -every point of D ϑ , γ ( k ) satisfies ( ), S h (cid:120) E ( ϑ , γ )( A ϑ , γ ( k ) \ D ϑ , γ ( k )) =
0, and S h ( E ( ϑ , γ ) \ ∪ + ∞ k = A ϑ , γ ( k )) = F δ ( k ) implies that there must exist a ball B ∈ G δ ( k ) such that w ∈ N + B . Wenow prove that for any couple of closed balls B ( x , r ) , B ( y , s ) ∈ G δ ( k ) such that B ( w , R ) intersects both B ( x , 5 N + r ) and B ( y , 5 N + s ) , we have P V ( B ( x , r )) ∩ P V ( B ( y , s )) = ∅ . ( )Indeed, Suppose that p ∈ B ( x , 5 N + r ) ∩ B ( w , R ) and note that d ( x , w ) ≤ d ( x , p ) + d ( p , w ) ≤ R + N + r ≤ (cid:16) + N + A L ( d , d c ) (cid:17) min { ϑ − , γ − , k − } ≤ min { ϑ − , γ − , k − } )where the last inequality comes from the choice of N . The bound ( ) shows in particular that d ( x , y ) ≤ d ( x , w ) + d ( w , y ) ≤ min { ϑ − , γ − , k − } < ρ ( x ) , ounds for the densities of S h on C V ( α ) - sets 36 where the last inequality comes from the fact that by construction x is supposed to be in D ϑ , γ ( k ) . Thanks to thefact that r + s < d ( x , y ) /5 A and y ∈ B ( x , ρ ( x )) ∩ E ( ϑ , γ ) we have that Proposition . implies that ( ) holds.In order to proceed with the conclusion of the proof, let us define F δ ( w , R ) : = { B ∈ F δ ( k ) : 5 N + B ∩ B ( w , R ) ∩ D ϑ , γ ( k ) (cid:54) = ∅ } , G δ ( w , R ) : = { B ∈ G δ ( k ) : 5 N + B ∩ B ( w , R ) ∩ D ϑ , γ ( k ) (cid:54) = ∅ } ,Thanks to our choice of R , see ( ), and the definition of G δ ( w , R ) we have R h ϑ ≤ S h (cid:120) Γ ( B ( w , R )) ≤ S h (cid:120) D ϑ , γ ( k )( B ( w , R )) ≤ S h (cid:120) D ϑ , γ ( k ) (cid:18) (cid:91) B ∈ G δ ( w , R ) N + B (cid:19) .Let G δ ( w , R ) = { B ( x i , r i ) } i ∈ N and recall that x i ∈ D ϑ , γ ( k ) and that 5 N + r i ≤ γ . This implies, thanks to Proposi-tion . , that S h (cid:120) D ϑ , γ ( k ) (cid:18) (cid:91) B ∈ G δ ( w , R ) N + B (cid:19) ≤ ϑ h ( N + ) ∑ i ∈ N r hi = ϑ h ( N + ) C ( V , L ) − ∑ i ∈ N S h ( P V ( B ( x i , r i )))= ϑ h ( N + ) C ( V , L ) − S h (cid:18) P V (cid:18) (cid:91) i ∈ N B ( x i , r i ) (cid:19)(cid:19) ≤ ϑ h ( N + ) C ( V , L ) − S h (cid:18) P V (cid:18) (cid:91) B ∈ F δ ( w , R ) B (cid:19)(cid:19) ,where the first inequality comes from the subadditivity and the upper estimate that we have in the definition of E ( ϑ , γ ) , see Definition . ; while the first identity of the second line above comes from ( ). Summing up, for any δ > C ( V , L ) R h · h ( N + ) ϑ ≤ S h (cid:18) P V (cid:18) (cid:91) B ∈ F δ ( w , R ) B (cid:19)(cid:19) .We now prove that the projection under P V of the closure of (cid:83) B ∈ F δ ( w , R ) B converges in the Hausdorff sense to P V ( D ϑ , γ ( k ) ∩ B ( w , R )) as δ goes to 0. Since the set (cid:83) B ∈ F δ ( w , R ) B is a covering of D ϑ , γ ( k ) ∩ B ( w , R ) we have that D ϑ , γ ( k ) ∩ B ( w , R ) (cid:98) (cid:91) B ∈ F δ ( w , R ) B . ( )On the other hand, since by definition the balls of F δ ( w , R ) have radii smaller than δ /4 and centre in D ϑ , γ ( k ) , wealso have (cid:91) B ∈ F δ ( w , R ) B (cid:98) B ( D ϑ , γ ( k ) ∩ B ( w , R ) , 5 N + δ ) . ( )Putting together ( ) and ( ), we infer that the closure of (cid:83) B ∈ F δ ( w , R ) B converges in the Hausdorff metric to theclosure of B ( w , R ) ∩ D ϑ , γ ( k ) . Furthermore, since P V restricted to the ball B ( w , R + ) is uniformly continuous, weinfer that P V (cid:18) (cid:91) B ∈ F δ ( w , R ) B (cid:19) −→ H P V (cid:18) D ϑ , γ ( k ) ∩ B ( w , R ) (cid:19) .Thanks to the upper semicontinuity of the Lebesgue measure with respect to the Hausdorff convergence weeventually infer that C ( V , L ) R h · h ( N + ) ϑ ≤ lim sup δ → S h (cid:18) P V (cid:18) (cid:91) B ∈ F δ ( w , R ) B (cid:19)(cid:19) ≤ S h ( P V ( D ϑ , γ ( k ) ∩ B ( w , R ))) ≤ S h ( P V ( E ( ϑ , γ ) ∩ B ( w , R ))) ,where the last inequality above comes from the fact that by construction D ϑ , γ ( k ) ⊆ E ( ϑ , γ ) and the compactnessof E ( ϑ , γ ) . Finally, since C = − − h A − h L ( d , d c ) − h C ( V , L ) , we infer S h ( P V ( E ( ϑ , γ ) ∩ B ( w , R ))) ≥ C ( V , L ) R h · h ( N + ) ϑ ≥ C R h ϑ ,thus showing the claim Eq. ( ) and then the proof. ounds for the densities of S h on C V ( α ) - sets 37 Proposition . . Let us fix α < ε ( V , L ) and suppose Γ is a compact C V ( α ) -set of finite S h -measure such that Θ h ∗ ( S h (cid:120) Γ , x ) > for S h -almost every x ∈ Γ . Let us set ϕ : P V ( Γ ) → L the map whose graph is Γ , see Proposition . , and set Φ : P V ( Γ ) → G to be the graph map of ϕ . Let us define Φ ∗ S h (cid:120) V to be the measure on Γ such that for every measurable A ⊆ Γ we have Φ ∗ S h (cid:120) V ( A ) : = S h (cid:120) V ( Φ − ( A )) = S h (cid:120) V ( P V ( A )) . Then Φ ∗ S h (cid:120) V is mutually absolutely continuous with respect to S h (cid:120) Γ .Proof. The fact that Φ ∗ S h (cid:120) V is absolutely continuous with respect to S h (cid:120) Γ is an immediate consequence of Propo-sition . . Viceversa, suppose by contradiction that there exists a compact subset C of Γ of positive S h -measuresuch that 0 = Φ ∗ S h (cid:120) V ( C ) = S h ( P V ( C )) . ( )Since by assumption Θ h ∗ ( S h (cid:120) C , x ) > S h -almost every x ∈ C , see Proposition . , Proposition . shows that( ) is false.In the following propositions we are going to introduce two fine coverings of P V ( Γ ) and V , respectively, thatwill be used in the proof of Proposition . to differentiate with respect to the measure S h (cid:120) P V ( Γ ) . Definition . ( φ -Vitali relation) . Let ( X , d ) be a metric space with a Borel measure φ on it and let B ( X ) be thefamily of Borel sets of X . We say that S ⊆ X × B ( X ) is a covering relation if S = { ( x , B ) : x ∈ B ⊆ X } .Furthermore for any Z ⊆ X we let S ( Z ) : = { B : ( x , B ) ∈ S for some x ∈ Z } . ( )Finally a covering S is said to be fine at x ∈ X if and only ifinf { diam ( B ) : ( x , B ) ∈ S } = φ -Vitali relation we mean a covering relation that is fine at every point of X and the following condition holdsIf C is a subset of S and Z is a subset of X such that C is fine at each point of Z , then C ( Z ) has a countabledisjoint subfamily covering φ -almost all of Z .If δ is a nonnegative function on S ( X ) , for any B ∈ S ( X ) we define its δ -enlargement asˆ B : = (cid:91) { B (cid:48) : B (cid:48) ∈ S ( X ) , B (cid:48) ∩ B (cid:54) = ∅ and δ ( B (cid:48) ) ≤ δ ( B ) } . ( )In the remaining part of this section we use the following general result due to Federer: it contains a cryterionto show that a fine covering relation is a φ -Vitali relation, and a Lebesgue theorem for φ -Vitali relations. Proposition . ([ , Theorem . . , Corollary . . and Theorem . . ]) . Let X be a metric space, and let φ be aBorel regular measure on X that is finite on bounded sets. Let S be a covering relation such that S ( X ) is a family of boundedclosed sets, S is fine at each point of X, and let δ be a nonnegative function on S ( X ) such that lim ε → + sup (cid:26) δ ( B ) + φ ( ˆ B ) φ ( B ) : ( x , B ) ∈ S , diam B < ε (cid:27) = for φ -almost every x ∈ X. Then S is a φ -Vitali relation.Moreover, if S is a φ -Vitali relation on X, and f is a φ -measurable real-valued function with ´ K | f | d φ < + ∞ on everybounded φ -measurable K, we have lim ε → + sup (cid:26) ´ B | f ( z ) − f ( x ) | d φ ( z ) φ ( B ) : ( x , B ) ∈ S , diam B < ε (cid:27) = for φ -almost every x ∈ X. In addition, given A ⊆ X, if we defineP : = (cid:26) x ∈ X : lim ε → + inf (cid:26) φ ( B ∩ A ) φ ( B ) : ( x , B ) ∈ S , diam B < ε (cid:27) = (cid:27) , then P is φ -measurable and φ ( A \ P ) = . ounds for the densities of S h on C V ( α ) - sets 38 Proposition . . Let α < ε ( V , L ) and suppose that Γ is a compact C V ( α ) -set of finite S h -measure such that Θ h ∗ ( S h (cid:120) Γ , x ) > for S h -almost every x ∈ Γ . As in the statement of Proposition . , let us denote with Φ : P V ( Γ ) → G the graph map of ϕ : P V ( Γ ) → L whose intrinsic graph is Γ . Then the covering relationS : = (cid:110)(cid:0) z , P V ( B ( Φ ( z ) , r ) ∩ Γ ) (cid:1) : z ∈ P V ( Γ ) and < r < min { R ( Φ ( z )) } (cid:111) , is a S h (cid:120) P V ( Γ ) -Vitali relation, where R ( Φ ( z )) is defined as in Proposition . for S h (cid:120) P V ( Γ ) -almost every z ∈ V and it is + ∞ on the remaining null set where Proposition . eventually does not hold.Proof. First of all, it is readily noticed that S is a fine covering of P V ( Γ ) sine P V is continuous. Let us prove that S is a S h (cid:120) P V ( Γ ) -Vitali relation in ( P V ( Γ ) , d ) with the distance d induced form G . For x ∈ P V ( Γ ) and r >
0, define G ( x , r ) : = P V ( B ( Φ ( x ) , r ) ∩ Γ ) . Notice that an arbitrary element of S ( P V ( Γ )) , see ( ), is of the form G ( x , r ) forsome x ∈ P V ( Γ ) and some 0 < r < min { R ( Φ ( x )) } . Let δ (cid:0) G ( x , r ) (cid:1) : = r and note that the δ -enlargement, see ( ),of G ( x , r ) isˆ G ( x , r ) : = (cid:91) { G ( y , s ) : y ∈ P V ( Γ ) , 0 < s < min { R ( Φ ( y )) } , G ( y , s ) ∩ G ( x , r ) (cid:54) = ∅ and δ ( G ( y , s )) ≤ δ ( G ( x , r )) } = (cid:91) { G ( y , s ) : y ∈ P V ( Γ ) , 0 < s < min { R ( Φ ( y )) } , G ( y , s ) ∩ G ( x , r ) (cid:54) = ∅ and s ≤ r } . ( )Whenever G ( x , r ) ∩ G ( y , s ) (cid:54) = ∅ we have that d ( Φ ( x ) , Φ ( y )) ≤ r + s : indeed, since P V is injective on Γ , seeProposition . , we have P V ( B ( Φ ( x ) , r ) ∩ Γ ) ∩ P V ( B ( Φ ( y ) , s ) ∩ Γ ) (cid:54) = ∅ if and only if B ( Φ ( x ) , r ) ∩ B ( Φ ( y ) , s ) ∩ Γ (cid:54) = ∅ . In particular, since s ≤ r we have B ( Φ ( y ) , s ) (cid:98) B ( Φ ( x ) , 12 r ) , and thus ˆ G ( x , r ) ⊆ G ( x , 12 r ) for every x ∈ P V ( Γ ) and 0 < r < min { R ( Φ ( x )) } .Finally, thanks to Proposition . and Proposition . , for S h -almost every x ∈ P V ( Γ ) we havelim ξ → sup (cid:40) δ ( G ( x , r )) + S h ( ˆ G ( x , r )) S h ( G ( x , r )) : 0 < r < min { R ( Φ ( x )) } , diam ( G ( x , r )) ≤ ξ (cid:41) ≤ + lim ξ → sup S h ( G ( x , 12 r )) S h ( G ( x , r )) ≤ + lim ξ → sup S h ( P V ( B ( Φ ( x ) , 12 r ))) S h ( P V ( B ( Φ ( x ) , r ) ∩ Γ )) ≤ + ( r ) h S h ( P V ( B (
0, 1 ))) C Θ h ∗ ( S h (cid:120) Γ , Φ ( x )) r h = + h S h ( P V ( B (
0, 1 ))) C Θ h ∗ ( S h (cid:120) Γ , Φ ( x )) , ( )where we explicitly mentioned the set over which we take the supremum only in the first line for the ease ofnotation, and where the first inequality in the third line follows from the fact that S h ( P V ( E )) = S h ( P V ( x E )) for any x ∈ G and any Borel set E ⊆ G , see Proposition . . Thanks to ( ) we can apply the first part ofProposition . and thus we infer that S is a S h (cid:120) P V ( Γ ) -Vitali relation. Proposition . . Let α < ε ( V , L ) and let Γ be a compact C V ( α ) -set of finite S h -measure. As in the statement ofProposition . , let us denote with Φ : P V ( Γ ) → G the graph map of ϕ : P V ( Γ ) → L whose intrinsic graph is Γ . Then for S h -almost every w ∈ P V ( Γ ) we have lim r → S h (cid:0) P V (cid:0) B ( Φ ( w ) , r ) ∩ Φ ( w ) C V ( α ) (cid:1) ∩ P V ( Γ ) (cid:1) S h (cid:0) P V (cid:0) B ( Φ ( w ) , r ) ∩ Φ ( w ) C V ( α ) (cid:1)(cid:1) =
1. ( ) Proof.
For any w ∈ V \ P V ( Γ ) we let ρ ( w ) : = inf { r ≥ B ( w , r ) ∩ P V ( B ( Γ , r κ )) (cid:54) = ∅ } .It is immediate to see that ρ ( w ) ≤ dist ( w , P V ( Γ )) and that ρ ( w ) = w ∈ P V ( Γ ) . Throughout the restof the proof we let S be the fine covering of V given by the couples ( w , G ( w , r )) for which Actually R ( x ) is defined for S h -almost every x ∈ Γ , and thus, taking into account Proposition . , R ( Φ ( z )) is defined for S h -almost every z ∈ P V ( Γ ) . ounds for the densities of S h on C V ( α ) - sets 39 ( α ) if w ∈ V \ P V ( Γ ) then r ∈ (
0, min { ρ ( w ) /2, 1 } ) and G ( w , r ) : = B ( w , r ) ∩ V ,( β ) if w ∈ P V ( Γ ) then r ∈ (
0, 1 ) and G ( w , r ) : = P V ( B ( Φ ( w ) , r ) ∩ Φ ( w ) C V ( α )) .Furthermore, for any w ∈ V we define the function δ on S ( V ) , see ( ), as δ (cid:0) G ( w , r ) (cid:1) : = r . ( )If we prove that S is a S h (cid:120) V -Vitali relation, the second part of Proposition . directly implies that ( ) holds. Iffor S h -almost every w ∈ V we prove thatlim ξ → sup ( w , G ( w , r )) ∈ S , diam ( G ( w , r )) ≤ ξ (cid:40) δ (cid:0) G ( w , r ) (cid:1) + S h ( ˆ G ( w , r )) S h ( G ( w , r )) (cid:41) ≤ + lim ξ → sup S h ( ˆ G ( w , r )) S h ( G ( w , r )) < ∞ , ( )where we explicitly mentioned the set over which we take the supremum only the first time for the ease of notation,and where ˆ G ( w , r ) is the δ -enlargement of G ( w , r ) , see ( ); thus, thanks to the first part of Proposition . wewould immediately infer that S is a S h (cid:120) V -Vitali relation. In order to prove that ( ) holds, we need to get a betterunderstanding of the geometric structure of the δ -enlargement of G ( w , r ) .If w ∈ V \ P V ( Γ ) , we note that there must exist an 0 < r ( w ) < min { ρ ( w ) /2, 1 } such that for any 0 < r < r ( w ) we have B ( w , r ) ∩ P V ( B ( Γ , 5 r )) = ∅ .Indeed, if this is not the case there would exist a sequence r i ↓ { z i } i ∈ N such that z i ∈ B ( w , r i ) ∩ P V ( B ( Γ , 5 r i )) .Since P V ( Γ ) is compact and P V is continuous on the closed tubular neighborhood B ( Γ , 1 ) , up to passing to a nonre-labelled subsequence we have that the z i ’s converge to some z ∈ P V ( Γ ) and on the other hand by constructionthe z i ’s converge to w which is not contained in P V ( Γ ) , and this is a contradiction. This implies that if 0 < r < r ( w ) ,we have ˆ G ( w , r ) = (cid:91) { G ( y , s ) : y ∈ V , s > ( y , G ( y , s )) ∈ S , G ( y , s ) ∩ G ( w , r ) (cid:54) = ∅ , and s ≤ r }⊆ (cid:91) { B ( y , s ) ∩ V : B ( y , s ) ∩ B ( w , r ) ∩ V (cid:54) = ∅ and s ≤ r } ⊆ B ( w , 11 r ) ∩ V , ( )where in the inclusion we are using the fact that if y were in V \ P V ( Γ ) , and s ≤ r , then G ( y , s ) ⊆ P V ( B ( Γ , s )) ⊆ P V ( B ( Γ , 5 r )) which would be in contradiction with G ( y , s ) ∩ G ( w , r ) (cid:54) = ∅ , since we chose 0 < r < r ( w ) . Summingup, if w ∈ V \ P V ( Γ ) the bound ( ) immediately follows thanks to ( ) and the homogeneity of S h .If on the other hand w ∈ P V ( Γ ) the situation is more complicated. If y ∈ V \ P V ( Γ ) and s ≤ r are such that G ( y , s ) ∩ P V ( B ( Φ ( w ) , r )) = B ( y , s ) ∩ P V ( B ( Φ ( w ) , r )) (cid:54) = ∅ , ( )since by construction of the covering S we also assumed that 0 < s < ρ ( y ) /2, we infer that we must have r ≥ s κ for ( ) to be satisfied. This allows us to infer that, for every w ∈ P V ( Γ ) and 0 < r <
1, we haveˆ G ( w , r ) = (cid:91) { G ( y , s ) : y ∈ V , s > ( y , G ( y , s )) ∈ S , G ( y , s ) ∩ G ( w , r ) (cid:54) = ∅ , and s ≤ r }⊆ (cid:91) { P V ( B ( Φ ( y ) , s )) : y ∈ P V ( Γ ) , P V ( B ( Φ ( y ) , s )) ∩ P V ( B ( Φ ( w ) , r )) (cid:54) = ∅ , and s ≤ r }∪ (cid:91) { B ( y , s ) ∩ V : y ∈ V \ P V ( Γ ) , B ( y , s ) ∩ P V ( B ( Φ ( w ) , r )) (cid:54) = ∅ , and s ≤ min { r , ρ ( y ) /2 }}⊆ (cid:91) { P V ( B ( Φ ( y ) , s )) : y ∈ P V ( Γ ) , P V ( B ( Φ ( y ) , s )) ∩ P V ( B ( Φ ( w ) , r )) (cid:54) = ∅ , and s ≤ r }∪∪ ( B ( P V ( B ( Φ ( w ) , r )) , 3 r κ ) ∩ V ) , ( )where in the last inclusion we are using the observation right after ( ) according to which s ≤ r κ . We now studyindependently each of the two terms of the union of the last line above. Let us first note that if w , y ∈ P V ( Γ ) , s ≤ r and P V ( B ( Φ ( y ) , s )) ∩ P V ( B ( Φ ( w ) , r )) (cid:54) = ∅ , ounds for the densities of S h on C V ( α ) - sets 40 then P V ( B ( Φ ( y ) , 10 r )) ∩ P V ( B ( Φ ( w ) , 2 r )) (cid:54) = ∅ . This observation and Corollary . imply that if 0 < r < r ( w ) issufficiently small we have d ( Φ ( w ) , Φ ( y )) ≤ Ar , where the constant A = A ( V , L ) is yielded by Lemma . . Inparticular we deduce that for every 0 < r < r ( w ) sufficiently small (cid:91) { P V ( B ( Φ ( y ) , s )) : y ∈ P V ( Γ ) , P V ( B ( Φ ( y ) , s )) ∩ P V ( B ( Φ ( w ) , r )) (cid:54) = ∅ , and s ≤ r } ⊆ P V ( B ( Φ ( w ) , 50 ( A + ) r )) .In order to study the second term in the last line of ( ), we prove the following claim: for every 0 < r <
1, every z ∈ P V ( B ( Φ ( w ) , r )) , and every ∆ ∈ B (
0, 3 r κ ) ∩ V we have z ∆ ∈ P V ( B ( Φ ( w ) , C ( Γ ) r )) , where C ( Γ ) is a constantdepending only on Γ . Indeed, since Γ is compact and P L is continuous, there exists a constant K (cid:48) : = K (cid:48) ( Γ ) such thatwhenever 0 < r <
1, and z ∈ P V ( B ( Φ ( w ) , r )) , there exsits an (cid:96) ∈ L such that z (cid:96) ∈ B ( Φ ( w ) , r ) and (cid:107) (cid:96) (cid:107) ≤ K (cid:48) . Thusthere exists a constant K : = K ( Γ ) > < r < z ∈ P V ( B ( Φ ( w ) , r )) , and ∆ ∈ B (
0, 3 r κ ) ∩ V ,there exists (cid:96) ∈ L with z (cid:96) ∈ B ( Φ ( w ) , r ) and (cid:107) ∆ (cid:107) + (cid:107) (cid:96) (cid:107) ≤ K . Thus we can estimate d ( Φ ( w ) , z ∆ (cid:96) ) ≤ d ( Φ ( w ) , z (cid:96) ) + d ( z (cid:96) , z ∆ (cid:96) ) ≤ r + C ( K , G ) (cid:107) ∆ (cid:107) κ ≤ C ( Γ ) r ,where the second inequality in the last equation comes from Lemma . . Thus z ∆ ∈ P V ( B ( Φ ( w ) , C ( Γ ) r )) , and theclaim is proved. Summing up, we have proved that whenever w ∈ P V ( Γ ) and 0 < r < r ( w ) is sufficiently small wehave ˆ G ( w , r ) ⊆ P V ( B ( Φ ( w ) , 50 ( A + ) r )) ∪ P V ( B ( Φ ( w ) , C ( Γ ) r )) ,and thus ( ) immediately follows by the homogeneity of S h (cid:120) V and the fact that S h ( P V ( x E )) = S h ( P V ( E )) forevery x ∈ G and E a Borel subset of G , see Proposition . . This concludes the proof of the proposition.We prove below a more precise version of Proposition . . Proposition . . Let us fix α < ε ( V , L ) . Suppose Γ is a compact C V ( α ) -set such that S h (cid:120) Γ is P ch -rectifiable. For S h -almost every x ∈ Γ we have ( − c ( α )) h ( + c ( α )) − h ≤ Θ m ∗ ( S h (cid:120) Γ , x ) Θ m , ∗ ( S h (cid:120) Γ , x ) ≤
1, ( ) where c ( α ) is defined in Lemma . .Proof. Let us preliminarly observe that since S h (cid:120) V and C h (cid:120) V are both Haar measures on V , they coincide up to aconstant. Since for S h -almost every x ∈ Γ we have Θ m , ∗ ( S h (cid:120) Γ , x ) >
0, the upper bound is trivial. Let us proceedwith the lower bound. Thanks to Proposition . and the Radon-Nikodym Theorem, see [ , page ], there exists ρ ∈ L ( Φ ∗ C h (cid:120) V ) such that(i) ρ ( x ) > Φ ∗ C h (cid:120) V -almost every x ∈ Γ ,(ii) S h (cid:120) Γ = ρ Φ ∗ C h (cid:120) V .We stress that the following reasoning holds for S h (cid:120) Γ -almost every x ∈ Γ . Let { r i } i ∈ N be an infinitesimal sequencesuch that r − hi T x , r i S h (cid:120) Γ (cid:42) λ C h (cid:120) V ( x ) for some λ >
0. First of all, we immediately see that Corollary . impliesthat λ ∈ [ Θ h ∗ ( S h (cid:120) Γ , x ) , Θ h , ∗ ( S h (cid:120) Γ , x )] and that1 = lim i → ∞ S h (cid:120) Γ ( B ( x , r i )) S h (cid:120) Γ ( B ( x , r i )) = lim i → ∞ ´ P V ( B ( x , r i ) ∩ Γ ) ρ ( Φ ( y )) d C h (cid:120) V ( y ) S h (cid:120) Γ ( B ( x , r i )) = ρ ( x ) λ lim i → ∞ C h (cid:120) V ( P V ( B ( x , r i ) ∩ Γ )) r hi ,where the last identity comes from Proposition . , that allows us to differentiate by using the second part ofProposition . , and Proposition . . Thanks to Lemma . , item (iii) of Proposition . and the fact that Γ is a C V ( α ) -set, we have λρ ( x ) ≤ lim i → ∞ C h (cid:120) V ( P V ( B ( x , r i ) ∩ xC V ( α ))) r hi = C h ( P V ( B (
0, 1 ) ∩ C V ( α ))) ≤ C h (cid:120) V ( B (
0, 1 ))( − c ( α )) h = ( − c ( α )) − h , ( ) ounds for the densities of S h on C V ( α ) - sets 41 where in the second equality we are using the homogeneity of C h and the fact that C h ( P V ( x E )) = C h ( P V ( E )) forevery x ∈ G and E a Borel subset of G , see Proposition . . On the other hand, thanks to Lemma . we have λρ ( x ) = lim i → ∞ C h (cid:120) V ( P V ( B ( x , r i ) ∩ Γ )) r hi ≥ lim i → ∞ C h (cid:0) P V (cid:0) B ( x , C ( α ) r i ) ∩ xC V ( α ) (cid:1) ∩ P V ( Γ ) (cid:1) C h (cid:0) P V (cid:0) B ( x , C ( α ) r i ) ∩ xC V ( α ) (cid:1)(cid:1) C h (cid:0) P V (cid:0) B ( x , C ( α ) r i ) ∩ xC V ( α ) (cid:1)(cid:1) r hi = C ( α ) h C h (cid:0) P V ( B (
0, 1 ) ∩ C V ( α )) (cid:1) ≥ C ( α ) h , ( )where the first identity in the last line comes from Proposition . and the last inequality from Lemma . , item(iii) of Proposition . , and C ( α ) is defined in ( ). Putting together ( ) and ( ), we have ( − c ( α )) h ( + c ( α )) h ≤ λρ ( x ) ≤ ( − c ( α )) h . ( )Thanks to the definition of Θ h ∗ ( S h (cid:120) Γ , x ) and Θ h , ∗ ( S h (cid:120) Γ , x ) we can find two sequences { r i } i ∈ N and { s i } i ∈ N suchthat Θ h ∗ ( S h (cid:120) Γ , x ) = lim i → ∞ S h (cid:120) Γ ( B ( x , r i )) r hi , and Θ h , ∗ ( S h (cid:120) Γ , x ) = lim i → ∞ S h (cid:120) Γ ( B ( x , s i )) s hi ,and without loss of generality, taking Proposition . into account, we can assume that r − hi T x , r i S h (cid:120) Γ (cid:42) Θ h ∗ ( S h (cid:120) Γ , x ) C h (cid:120) V ( x ) , and s − hi T x , s i S h (cid:120) Γ (cid:42) Θ h , ∗ ( S h (cid:120) Γ , x ) C h (cid:120) V ( x ) .The bounds ( ) imply therefore that ( − c ( α )) h ( + c ( α )) h ≤ Θ h ∗ ( S h (cid:120) Γ , x ) ρ ( x ) ≤ ( − c ( α )) h , and ( − c ( α )) h ( + c ( α )) h ≤ Θ h , ∗ ( S h (cid:120) Γ , x ) ρ ( x ) ≤ ( − c ( α )) h . ( )Finally the bounds in ( ) yield ( − c ( α )) h ( + c ( α )) − h ≤ Θ h ∗ ( S h (cid:120) Γ , x ) Θ h , ∗ ( S h (cid:120) Γ , x ) ≤ P ch -rectifiable measures, see Theorem . . We first prove an algebraiclemma, then we prove the existence of the density for measures of the type S h (cid:120) Γ , and then we conclude with theproof of the existence of the density for arbitrary P ch -rectifiable measures. Lemma . . Let us fix < ε < a real number, (cid:96) , h ∈ N , and let f be the function defined as followsf : { ( α , C ) ∈ ( + ∞ ) : α < C } → ( + ∞ ) , f ( α , C ) : = α C − α . Then, there exists (cid:101) α : = (cid:101) α ( ε , (cid:96) , h ) > such that the following implication holdsif < α ≤ (cid:101) α and C > (cid:96) , then α < C and ( − f ( α , C )) h ( + f ( α , C )) − h ≥ − ε . Proof.
Let us choose 0 < (cid:101) ε : = (cid:101) ε ( ε , h ) < ( − x ) h ( + x ) − h ≥ − ε , for all 0 ≤ x ≤ (cid:101) ε .Let us show that the sought constant (cid:101) α ( ε , (cid:96) , h ) is (cid:101) α : = (cid:101) ε / ( (cid:96) ( + (cid:101) ε )) . Indeed, if α ≤ (cid:101) α and C > (cid:96) we infer that α < C and α ≤ (cid:101) ε (cid:96) ( + (cid:101) ε ) ≤ C (cid:101) ε + (cid:101) ε , and then f ( α , C ) = α C − α ≤ (cid:101) ε .This implies that if α ≤ (cid:101) α and C > (cid:96) , then ( − f ( α , C )) h ( + f ( α , C )) − h ≥ − ε ,where the last inequality above comes from the choice of (cid:101) ε . This concludes the proof. ounds for the densities of S h on C V ( α ) - sets 42 Theorem . . Let Γ be a compact subset of G such that S h (cid:120) Γ is a P ch -rectifiable measure. Then < Θ h ∗ ( S h (cid:120) Γ , x ) = Θ ∗ , h ( S h (cid:120) Γ , x ) < + ∞ , for S h (cid:120) Γ -almost every x ∈ G .Proof. In the following, for any ε >
0, we will construct a measurable set A ε ⊆ Γ such that S h ( Γ \ A ε ) = − ε ≤ Θ ∗ , h ( S h (cid:120) Γ , x ) Θ h ∗ ( S h (cid:120) Γ , x ) ≤
1, for every x ∈ A ε . ( )If ( ) holds then we are free to choose ε = n for every n ∈ N and then the density of S h (cid:120) Γ exists on the set ∩ + ∞ n = A n , that has full S h (cid:120) Γ -measure. So we are left to construct A ε as in ( ). Let us define the function F ( V , L ) : = ε ( V , L ) , for all V ∈ Gr c ( h ) with complement L .Let us take the family F : = { V i } + ∞ i = ⊆ Gr c ( h ) and let us choose L i complementary subgroups to V i as in thestatement of Theorem . . We remark that the choices of the family F and of the complementary subgroupsdepend on the function F previously defined, see the discussion before Theorem . . Let us define β : N → (
0, 1 ) , β ( (cid:96) ) : = (cid:101) α ( ε , (cid:96) , h ) ,where (cid:101) α ( ε , (cid:96) , h ) is the constant in Lemma . , and with an abuse of notation let us lift β to a function on F as wedid in the statement of Theorem . . From Theorem . we conclude that there exist countably many Γ i ’s that arecompact C V i ( min { ε ( V i , L i ) , β ( V i ) } ) -sets contained in Γ such that S h (cid:0) Γ \ ∪ + ∞ i = Γ i (cid:1) =
0. ( )Let us write, for the ease of notation, α i : = min { ε ( V i , L i ) , β ( V i ) } for every i ∈ N . Since Γ i ⊆ Γ and S h (cid:120) Γ is P ch -rectifiable, we conclude, by exploiting the locality of tangents, see Proposition . , and the Lebesgue differentiationtheorem in Proposition . , that the measures S h (cid:120) Γ i are P ch -rectifiable as well for every i ∈ N . Thus, since α i ≤ ε ( V i , L i ) , we can apply Proposition . and conclude that, for every i ∈ N , we have ( − c ( α i )) h ( + c ( α i )) − h ≤ Θ ∗ , h ( S h (cid:120) Γ i , x ) Θ h ∗ ( S h (cid:120) Γ i , x ) ≤
1, for S h (cid:120) Γ i -almost every x ∈ G ,where c ( α i ) : = α i / ( C ( V i , L i ) − α i ) . Since Θ ∗ , h ( S h (cid:120) Γ i , x ) = Θ ∗ , h ( S h (cid:120) Γ , x ) and Θ h ∗ ( S h (cid:120) Γ i , x ) = Θ h ∗ ( S h (cid:120) Γ , x ) for S h (cid:120) Γ i -almost every x ∈ G , see Proposition . , for every i ∈ N we conclude that ( − c ( α i )) h ( + c ( α i )) − h ≤ Θ ∗ , h ( S h (cid:120) Γ , x ) Θ h ∗ ( S h (cid:120) Γ , x ) ≤
1, for S h (cid:120) Γ i -almost every x ∈ G . ( )Let us now fix i ∈ N and note there exists a unique (cid:96) ( i ) ∈ N such that1/ (cid:96) ( i ) < ε ( V i , L i ) ≤ ( (cid:96) ( i ) − ) .Moreover, from the definition of β and F we see that β ( V i ) = β ( ε , (cid:96) ( i ) , h ) . This allows us to infer that . α i ≤ β ( V i ) = β ( ε , (cid:96) ( i ) , h ) , since α i : = min { ε ( V i , L i ) , β ( V i ) } , . C ( V i , L i ) > (cid:96) ( i ) , since 1/ (cid:96) ( i ) < ε ( V i , L i ) = C ( V i , L i ) /2, where the last equality comes from Lemma . .Thus we can apply Lemma . and conclude that ( − c ( α i )) h ( + c ( α i )) − h ≥ − ε .This shows, thanks to ( ), that for any i ∈ N , we have1 − ε ≤ Θ ∗ , h ( S h (cid:120) Γ , x ) Θ h ∗ ( S h (cid:120) Γ , x ) ≤
1, for S h (cid:120) Γ i -almost every x ∈ G .Thus by taking into account ( ) and the previous equation we conclude ( ), that is the sought claim. arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 43 Remark . . It is a classical result that if E ⊆ R n is a h -rectifiable set, with 1 ≤ h ≤ n , then Θ h ( S h (cid:120) E , x ) = S h -almost every point x ∈ E , see [ , Theorem . . ]. This is true also in the setting of Heisenberg groups forarbitrary P ch -rectifiable measures, and it is a direct consequence of [ , (iv) ⇒ (ii) of Theorem . & Theorem . ].We remark that we do not address explicitly in this paper the problem of proving that if Γ is a compact set suchthat S h (cid:120) Γ is a P ch -rectifiable measure, then Θ h ( S h (cid:120) Γ , x ) = S h (cid:120) Γ -almost every x ∈ G . Corollary . . Let φ be a P ch -rectifiable measure on a Carnot group G . Then < Θ h ∗ ( φ , x ) = Θ ∗ , h ( φ , x ) < + ∞ , for φ -almost every x ∈ G .Proof. We stress that by restricting ouserlves on balls of integer radii, by using Proposition . and Proposition . ,we can assume that φ has compact support. Let us first recall that, by Proposition . , we have φ G \ (cid:91) ϑ , γ ∈ N E ( ϑ , γ ) =
0. ( )Let us fix ϑ , γ ∈ N . From Lebesgue’s differentiation theorem, see Proposition . , and the locality of tangents,see Proposition . , we deduce that φ being P ch -rectifiable implies that φ (cid:120) E ( ϑ , γ ) is P ch -rectifiable. From Propo-sition . we deduce that φ (cid:120) E ( ϑ , γ ) is mutually absolutely continuous with respect to S h (cid:120) E ( ϑ , γ ) , and thus, byRadon-Nikodym theorem, see [ , page ], there exists a positive ρ ∈ L ( S h (cid:120) E ( ϑ , γ )) such that φ (cid:120) E ( ϑ , γ ) = ρ S h (cid:120) E ( ϑ , γ ) . We stress that we can apply Lebesgue-Radon-Nikodym theorem since φ (cid:120) E ( ϑ , γ ) is asymptoticallydoubling because it has positive h -lower density and finite h -upper density almost everywhere. By Lebesgue-Radon-Nikodym theorem, see [ , page ], and the locality of tangents again, we deduce that S h (cid:120) E ( ϑ , γ ) is a P ch -rectifiable measure, since φ (cid:120) E ( ϑ , γ ) is a P ch -rectifiable measure. Thus we can apply Theorem . to S h (cid:120) E ( ϑ , γ ) and obtain that for every ϑ , γ ∈ N we have that0 < Θ h ∗ ( S h (cid:120) E ( ϑ , γ ) , x ) = Θ ∗ , h ( S h (cid:120) E ( ϑ , γ ) , x ) < + ∞ , for S h (cid:120) E ( ϑ , γ ) -almost every x ∈ G .Since φ (cid:120) E ( ϑ , γ ) = ρ S h (cid:120) E ( ϑ , γ ) we thus conclude from the previous equality and by Lebesgue-Radon-Nikdoymtheorem that for every ϑ , γ ∈ N we have that0 < Θ h ∗ ( φ (cid:120) E ( ϑ , γ ) , x ) = Θ ∗ , h ( φ (cid:120) E ( ϑ , γ ) , x ) < + ∞ , for φ (cid:120) E ( ϑ , γ ) -almost every x ∈ G .The previous equality, jointly with Proposition . and together with ( ) allows us to conclude the proof. Remark . . The existence of the density for P ch -rectifiable measures is independent on the metric. Indeed Corol-lary . and Corollary . imply that for φ -almost every x ∈ G we have r − h T x , r φ (cid:42) Θ h ( φ , x ) C h (cid:120) V ( x ) ,for some V ( x ) ∈ Gr c ( h ) and where Θ h ( φ , x ) is the density of φ at x with respect to the smooth-box metric. Thus,let d be a left-invariant homogeneous metric on G and let B d ( x , r ) be its metric ball of centre x and radius r . Notethat since C h (cid:120) V ( x ) is a Haar measure of V ( x ) we have that C h (cid:120) V ( x )( ∂ B d ( y , s )) = y ∈ V ( x ) and s > , Lemma . ]. Thus, using [ , Proposition . ], we infer that Θ h ( φ , x ) C h (cid:120) V ( x )( B d (
0, 1 )) = lim r → T x , r φ ( B d (
0, 1 )) r h = lim r → φ ( B d ( x , r )) r h = : Θ hd ( φ , x ) . - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures This chapter is devoted to the proof of the following result, which is a more precise version of Theorem . . Theorem . (Co-normal Marstrand-Mattila rectifiability criterion) . Assume φ is a P ∗ , (cid:69) h -rectifiable measure on aCarnot group G . Then there are countably many W i ∈ Gr (cid:69) ( h ) , compact sets K i (cid:98) W i and Lipschitz functions f i : K i → G such that φ ( G \ (cid:91) i ∈ N f i ( K i )) = In particular φ is P ch -rectifiable. arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 44 We briefly discuss the strategy of the proof of Theorem . , which is ultimately an adaptation of Preiss’ tech-nique in [ , Section . ( ), Lemma . , Theorem . , and Corollary . ] to our setting, see Proposition . ,Proposition . , and Proposition . , respectively. In particular we show that whenever a Radon measure satisfiesprecise structure conditions, see Proposition . , that are always verified whenever φ is P ∗ h -rectifiable with tan-gents that admit at least one normal complementary subgroup , see Proposition . , then it is possible to find aLipschitz function f : K (cid:98) V → G , with V ∈ Gr (cid:69) ( h ) , such that φ ( f ( K )) >
0. This implies that G can be covered φ -almost all with ∪ i ∈ N f i ( K i ) , where f i : K i (cid:98) V i → G are Lipschitz functions, see the first part of the proof ofTheorem . .The last part of Theorem . is reached from the first part and the following key observation: if a homogeneoussubgroup of a Carnot group admits a normal complementary subgroup, then it is a Carnot subgroup, see [ ,Remark . ]. Thus the maps f i are Lipschitz maps between Carnot groups and we can apply Pansu-Rademachertheorem, see [ ], Magnani’s area formula, see [ ], and a classical argument to conclude that S h (cid:120) f i ( K i ) is a P ch -rectifiable measure, see the last part of the proof of Theorem . . From this latter observation, the proof ofTheorem . is concluded. . Rigidity of the stratification of P ∗ h -rectifiable measures Throughout this subsection we let G be a Carnot group of homogeneous dimension Q . Moreover, we let ϕ : G → [
0, 1 ] be a positive, smooth radially symmetric function supported in B (
0, 2 ) and such that ϕ ≡ B (
0, 1 ) . Furthermore, we will denote by g its profile function, that is defined in the statement of Proposition . . Proposition . . For any h ∈ {
1, . . . , Q } there exists a constant ג ( G , h ) = ג > such that for any V ∈ Gr ( h ) and any s ∈ S ( h ) \ { s ( V ) } , we have inf W ∈ Gr ( h ) s ( W )= s ˆ ϕ ( z ) dist ( z , W ) d C h (cid:120) V > ג , where the stratification vector s ( · ) was introduced in Definition . .Proof. Suppose by contradiction this is not the case. Thus there are two sequences { W i } ⊆ Gr ( h ) and { V i } ⊆ Gr ( h ) such that for any i ∈ N we have s ( W i ) (cid:54) = s ( V i ) and ˆ ϕ ( z ) dist ( z , W i ) d C h (cid:120) V i ≤ i . ( )Thanks to the pidgeonhole principle and the fact that S ( h ) , see Definition . , is a finite set we can assume up topassing to a non re-labelled subsequence that s ( W i ) = s (cid:54) = s = s ( V i ) , for any i ∈ N .Furthermore, thanks to Proposition . , we can also assume, up to passing to a non re-labelled subsequence, that W i → d G W ∈ Gr ( h ) , and V i → d G V ∈ Gr ( h ) .Furthermore, thanks to Proposition . , we also deduce that s ( W ) = s (cid:54) = s = s ( V ) .In order to conclude the proof of the proposition we first note for any U ∈ Gr ( h ) and any R >
0, if z ∈ B ( R ) ,then every element u ∈ U for which dist ( z , U ) = d ( u , z ) is contained in B (
0, 2 R ) . The same argument as in ( )and ( ) allows us to conclude that for every z ∈ B (
0, 2 ) the following inequality holdsdist ( z , W i ) ≥ dist ( z , W ) − d G ( W , W i ) , for all i ∈ N . ( )Putting together ( ) and ( ) thanks to Proposition . we infer1/ i ≥ ˆ ϕ ( z ) dist ( z , W i ) d C h (cid:120) V i ≥ ˆ ϕ ( z ) dist ( z , W ) d C h (cid:120) V i − d G ( W , W i ) ˆ ϕ ( z ) d C h (cid:120) V i = ˆ ϕ ( z ) dist ( z , W ) d C h (cid:120) V i − d G ( W , W i ) h ˆ s h − g ( s ) ds . ( ) arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 45 Therefore, since ϕ ( z ) dist ( z , W ) is a continuous function with compact support, thanks to Proposition . andsending i to + ∞ in the previous inequality we conclude ˆ ϕ ( z ) dist ( z , W ) d C h (cid:120) V = ( z , W ) = S h (cid:120) V -almost every z ∈ V , and since both Lie ( V ) and Lie ( W ) are vector subspacesof Lie ( G ) we have V ⊆ W . On the one hand this allows us to infer thatdim ( V i ∩ V ) ≤ dim ( V i ∩ W ) , for any i ∈ {
1, . . . , κ } ,and on the other hand, since s ( V ) (cid:54) = s ( W ) , there must exist an (cid:96) ∈ {
1, . . . , κ } such that dim ( V (cid:96) ∩ V ) < dim ( V (cid:96) ∩ W ) . This however contradicts the fact that W ∈ Gr ( h ) , indeed h = dim hom V = κ ∑ i = i · dim ( V i ∩ V ) < κ ∑ i = i · dim ( V i ∩ W ) = dim hom ( W ) . Proposition . . Let s ∈ S ( h ) . For any Radon measure ψ we define F s ( ψ ) : = inf W ∈ Gr ( h ) s ( W )= s ˆ ϕ ( z ) dist ( z , W ) d ψ . Then, the functional F s : M → R on Radon measures is continuous with respect to the weak-* topology in the duality withthe functions with compact support on G .Proof. Let ψ i (cid:42) ψ and note that for any V ∈ Gr ( h ) for which s ( V ) = s , we havelim i → + ∞ ˆ ϕ ( z ) dist ( z , V ) d ψ i = ˆ ϕ ( z ) dist ( z , V ) d ψ , ( )since ϕ ( z ) dist ( z , V ) is a continuous function with compact support. Let us first prove that F s ( ψ ) ≤ lim inf i → ∞ F s ( ψ i ) .Indeed, if by contradiction F s ( ψ ) > lim inf i → ∞ F s ( ψ i ) , up to passing to a non re-labelled subsequence in i thatrealizes the lim inf and up to choosing a quasi-minimizer for F s ( ψ i ) , we can find δ >
0, and W i ∈ Gr ( h ) with s ( W i ) = s such that F s ( ψ ) > ˆ ϕ ( z ) dist ( z , W i ) d ψ i + δ , for all i ∈ N . ( )We can assume that W i → W ∈ Gr ( h ) , with s ( W ) = s , up to a non re-labelled subsequence, see Proposition . and Proposition . . Thus since ψ i (cid:42) ψ passing to the limit the right hand side of ( ) we obtain F s ( ψ ) > ´ ϕ ( z ) dist ( z , W ) d ψ , that is a contradiction with the definition of F s . The proof of the proposition is concluded ifwe prove that F s ( ψ ) ≥ lim sup i → ∞ F s ( ψ i ) .In order to prove the previous inequality let us fix ε > V ε ∈ Gr ( h ) with s ( V ε ) = s such that ˆ ϕ ( z ) dist ( z , V ε ) d ψ − ε ≤ F s ( ψ ) . ( )Putting together ( ) and ( ), we inferlim sup i → ∞ F s ( ψ i ) − ε ≤ lim sup i → ∞ ˆ ϕ ( z ) dist ( z , V ε ) d ψ i − ε = ˆ ϕ ( z ) dist ( z , V ε ) d ψ − ε ≤ F s ( ψ ) . ( )The arbitrariness of ε concludes the limsup inequality and thus the proof of the proposition. Setting f i ( z ) : = ϕ ( z ) dist ( z , W i ) and f ( z ) : = ϕ ( z ) dist ( z , W ) we notice that f i → f uniformly on B (
0, 2 ) since W i → W . Thus | ´ f d ψ − ´ f i d ψ i | ≤ | ´ f d ψ − ´ f d ψ i | + | ´ f d ψ i − ´ f i d ψ i | and the limit is zero because ψ i (cid:42) ψ , sup i ψ i ( B (
0, 2 )) < + ∞ and f i → f uniformly on B (
0, 2 ) . arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 46 Definition . . For any
T ⊆ M ( h ) we define s ( T ) to be the set s ( T ) : = { s ( V ) : there exists a non-null Haar measure of V in T } .Namely we are considering all the possible stratification vectors of the homogeneous subgroups that are thesupport of some element of T . Theorem . . Assume φ is a P ∗ h -rectifiable measure. Then, for φ -almost every x ∈ G the set s ( Tan h ( φ , x )) ⊆ S ( h ) is asingleton.Remark . . In the notation of the above proposition, since for φ -almost every x ∈ G we have Tan h ( φ , x ) ⊆ M ( h ) ,the symbol s ( Tan h ( φ , x )) is well defined φ -almost everywhere. Proof.
Suppose by contradiction there exists a point x ∈ G where(i) 0 < Θ h ∗ ( φ , x ) ≤ Θ h , ∗ ( φ , x ) < ∞ ,(ii) Tan h ( φ , x ) ⊆ M ( h ) ,(iii) there are V , V ∈ Gr ( h ) with s ( V ) (cid:54) = s ( V ) and λ , λ ≥ λ C h (cid:120) V , λ C h (cid:120) V ∈ Tan h ( φ , x ) .Assume that { r i } i ∈ N and { s i } i ∈ N are two infinitesimal sequences such that r i ≤ s i and for which T x , r i φ r hi (cid:42) λ C h (cid:120) V , and T x , s i φ s hi (cid:42) λ C h (cid:120) V .Note that thanks to Proposition . , we have in particular that Θ h ∗ ( φ , x ) ≤ λ , λ ≤ Θ h , ∗ ( φ , x ) . Throughout therest of the proof we let s : = s ( V ) and we define f ( r ) : = inf W ∈ Gr ( h ) s ( W )= s ˆ ϕ ( z ) dist ( z , W ) d T x , r φ r h .Thanks to Proposition . and Proposition . we infer that the function f is continuous on ( ∞ ) and thatlim i → ∞ f ( r i ) = i → ∞ f ( s i ) > ג λ ≥ ג Θ h ∗ ( φ , x ) .Let us choose, for i sufficiently large, σ i ∈ [ r i , s i ] in such a way that f ( σ i ) = ג Θ h ∗ ( φ , x ) /2 and f ( r ) ≤ ג Θ h ∗ ( φ , x ) /2for any r ∈ [ r i , σ i ] . Up to passing to a non re-labelled subsequence, since φ is P ∗ h -rectifiable, we can assumethat σ − hi T x , σ i (cid:42) λ C h (cid:120) V for some λ > V ∈ Gr ( h ) . Thanks to Proposition . , we infer that Θ h ∗ ( φ , x ) ≤ λ ≤ Θ h , ∗ ( φ , x ) and thanks to the continuity of the functional F s in Proposition . , we conclude that ג Θ h ∗ ( φ , x ) /2 = lim i → ∞ f ( σ i ) = lim i → ∞ F s ( σ − hi T x , σ i φ ) = λ F s ( C h (cid:120) V ) . ( )The chain of identities ( ) together with the bounds on λ imply0 < ג Θ h ∗ ( φ , x ) /2 Θ h , ∗ ( φ , x ) ≤ F s ( C h (cid:120) V ) ≤ ג /2. ( )Since V ∈ Gr ( h ) , ( ) on the one hand implies by means of Proposition . that s ( V ) = s . On the other hand,since F s ( C h (cid:120) V ) >
0, we have that s ( V ) (cid:54) = s , resulting in a contradiction. Definition . . Assume φ is a P ∗ h -rectifiable measure. For every x ∈ G we define the map s ( φ , x ) ∈ N κ in thefollowing way s ( φ , x ) : = (cid:40) s if Tan h ( φ , x ) ⊆ M ( h ) and s ( Tan h ( φ , x )) is the singleton { s } ,0 otherwise. Remark . . The map s ( φ , · ) is well defined and non-zero φ -almost everywhere thanks to Theorem . . arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 47 Proposition . . Assume φ is a P ∗ h -rectifiable measure. Then, the map x (cid:55)→ s ( φ , x ) is φ -measurable.Proof. Let ¯ h G be the constant introduced in Proposition . . Let us first prove that there exists (cid:101) α : = (cid:101) α ( G ) such thatthe following assertion holdsfor every 1 ≤ h ≤ Q and for every V , W ∈ Gr ( h ) , if V ⊆ C W ( (cid:101) α ) , then d G ( V , W ) ≤ ¯ h G . ( )Indeed, if this was not the case, we can find an 1 ≤ h ≤ Q and sequences { V i } , { W i } in Gr ( h ) such that V i ⊆ C W i ( i − ) and for which d G ( V i , W i ) > ¯ h G , for all i ∈ N . Thus, up to non re-labelled subsequences, wecan assume that V i → V and W i → W , for some V , W ∈ Gr ( h ) , thanks to Proposition . . Thanks to theaformentioned convergences and the fact that V i ⊆ C W i ( i − ) for every i ∈ N we deduce that V ⊆ W andthus V = W since they both have homogeneous dimension h . But this latter equality is readily seen to be incontradiction with the fact that d G ( V i , W i ) > ¯ h G , for all i ∈ N , since W i → W and V i → V .Let { V (cid:96) } (cid:96) = N be a finite (cid:101) α /3-dense set in Gr ( h ) , where (cid:101) α is defined above. For any r ∈ (
0, 1 ) ∩ Q and (cid:96) =
1, . . . , N we define the functions on G f r , (cid:96) ( x ) : = r − h φ ( { w ∈ B ( x , r ) : dist ( x − w , V (cid:96) ) ≥ (cid:101) α (cid:107) x − w (cid:107)} ) = : r − h φ ( I ( x , r , (cid:96) )) .We claim that the functions f r , (cid:96) are upper semicontinuous. Let { x i } i ∈ N be a sequence of points converging to some x ∈ G and without loss of generality we assume that lim i → ∞ r − h φ ( I ( x i , r , (cid:96) )) exists. Since the sets I ( x i , r , (cid:96) ) arecontained in B ( x , 1 ) provided i is sufficiently big, we infer thanks to Fatou’s Lemma thatlim sup i → ∞ f r , (cid:96) ( x i ) = r h lim sup i → ∞ ˆ χ I ( x i , r , (cid:96) ) ( z ) d φ ( z ) ≤ r h ˆ lim sup i → ∞ χ I ( x i , r , (cid:96) ) ( z ) d φ ( z ) . ( )Furthermore, since x i → x and the sets I ( x i , r , (cid:96) ) and I ( x , r , (cid:96) ) are closed, we havelim sup i → ∞ χ I ( x i , r , (cid:96) ) = χ lim sup i → + ∞ I ( x i , r , l ) ≤ χ I ( x , r , l ) ,where the first equality is true in general. Then, from ( ), we infer thatlim sup i → ∞ f r , (cid:96) ( x i ) ≤ r h ˆ lim sup i → ∞ χ I ( x i , r , (cid:96) ) ( z ) d φ ( z ) ≤ r h ˆ χ I ( x , r , (cid:96) ) ( z ) d φ ( z ) = f r , (cid:96) ( x ) ,and this concludes the proof that f r , (cid:96) is upper semicontinuous. This implies that the function f (cid:96) : = lim inf r ∈ Q , r → f r , (cid:96) ,is φ -measurable and as a consequence, since Tan h ( φ , x ) ⊆ M ( h ) for φ -almost every x ∈ G , we infer that the set B (cid:96) : = { x ∈ G : f (cid:96) ( x ) = } ∩ { x ∈ G : Tan h ( φ , x ) ⊆ M ( h ) } ,is φ -measurable as well. If we prove that for φ -almost any x ∈ B (cid:96) there exists a non-zero Haar measure ν inTan h ( φ , x ) relative to a homogeneous subgroup V of G such that d G ( V , V (cid:96) ) ≤ ¯ h G , we infer that s ( Tan h ( φ , x )) = { s ( V (cid:96) ) } , for φ -almost any x ∈ B (cid:96) , ( )and thus s ( φ , x ) = s ( V (cid:96) ) for φ -almost every x ∈ B (cid:96) . Indeed, if we are able to find such a measure ν relative to V ,( ) is an immediate consequence of the fact that if d G ( V , V (cid:96) ) ≤ ¯ h G , Proposition . implies that V and V (cid:96) havethe same stratification; and the fact that, from Theorem . , φ -almost everywhere the tangent subgroups have thesame stratification.In order to construct such a non-zero Haar measure ν , we fix a point x ∈ B (cid:96) in the φ -full-measure subset of B (cid:96) such that the following conditions hold(i) 0 < Θ h ∗ ( φ , x ) ≤ Θ h , ∗ ( φ , x ) < ∞ ,(ii) Tan h ( φ , x ) ⊆ M ( h ) , arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 48 and we let { r i } i ∈ N be an infinitesimal sequence of rational numbers such that lim i → ∞ f r i , (cid:96) ( x ) =
0. Thanks to item(i) above and the compactness of measures, see [ , Proposition . ], we can find a non re-labelled subsequence of r i such that r − hi T x , r i φ (cid:42) ν .Such a ν belongs by definition to Tan h ( φ , x ) and thus there is a λ > V ∈ Gr ( h ) such that ν = λ C h (cid:120) V .Thanks to [ , Proposition . ], we infer that ν ( { w ∈ U (
0, 1 ) : dist ( w , V (cid:96) ) > (cid:101) α (cid:107) w (cid:107)} ) ≤ lim inf i → ∞ r − hi T x , r i φ ( { w ∈ U (
0, 1 ) : dist ( w , V (cid:96) ) > (cid:101) α (cid:107) w (cid:107)} )= lim inf i → ∞ r − hi φ ( { w ∈ U ( x , r i ) : dist ( x − w , V (cid:96) ) > (cid:101) α (cid:107) x − w (cid:107)} ) = r i . This shows in particular that V ⊆ { w ∈ G : dist ( w , V (cid:96) ) ≤ (cid:101) α (cid:107) w (cid:107)} = C V (cid:96) ( (cid:101) α ) ,and then, from ( ) we conclude that d G ( V , V (cid:96) ) ≤ ¯ h G , that was what we wanted to prove.An immediate consequence of ( ) is thatif (cid:96) , m ∈ {
1, . . . , N } and s ( V (cid:96) ) (cid:54) = s ( V m ) then φ ( B (cid:96) ∩ B m ) =
0. ( )On the other hand, the B (cid:96) ’s cover φ -almost all G . To prove this latter assertion, we note that since φ is P ∗ h -rectifiable, for φ -almost all x ∈ G there is an infinitesimal sequence r i →
0, a λ > V ∈ Gr ( h ) such that r − hi T x , r i φ (cid:42) λ C h (cid:120) V . Since the set { V (cid:96) : (cid:96) =
1, . . . , N } is (cid:101) α /3-dense in Gr ( h ) , there must exist an (cid:96) ∈ {
1, . . . , N } such that V ⊆ { w ∈ G : dist ( w , V (cid:96) ) < (cid:101) α (cid:107) w (cid:107)} . ( )This last inclusion follows since there exists (cid:96) such that d G ( V , V (cid:96) ) ≤ (cid:101) α /3 and the observation that every point in ∂ B (
0, 1 ) ∩ V is such that every point at minimum distance of it from V (cid:96) is in B (
0, 2 ) ∩ V (cid:96) . The previous inclusion,jointly with [ , Proposition . ], implies that f (cid:96) ( x ) = lim inf r ∈ Q , r → f r , (cid:96) ( x ) ≤ lim inf i → ∞ f r i , (cid:96) ( x ) = lim inf i → ∞ r − hi φ ( { w ∈ B ( x , r i ) : dist ( x − w , V (cid:96) ) ≥ (cid:101) α (cid:107) x − w (cid:107)} ) ≤ lim sup i → ∞ r − hi T x , r i φ ( { w ∈ B (
0, 1 ) : dist ( w , V (cid:96) ) ≥ (cid:101) α (cid:107) w (cid:107)} ) ≤ λ C h (cid:120) V ( { w ∈ B (
0, 1 ) : dist ( w , V (cid:96) ) ≥ (cid:101) α (cid:107) w (cid:107)} ) =
0, ( )where the last inequality is true since ( ) holds. This proves that x ∈ B (cid:96) and as a consequence that the B (cid:96) ’s cover φ -almost all G .We are ready to prove the measurability of the map x (cid:55)→ s ( φ , x ) . Fix an s ∈ S ( h ) and let D ( s ) : = { x ∈ G : s ( φ , x ) = s } ∩ (cid:83) N (cid:96) = B (cid:96) . Since by the previous step the B (cid:96) ’s cover φ -almost all G we know that { x ∈ G : s ( φ , x ) = s } \ (cid:83) Nl = B (cid:96) is φ -null and thus it is φ -measurable. Furthermore, thanks to ( ) and ( ) we know that up to φ -null sets we have D ( s ) = (cid:91) s ∈ S ( h ) { B (cid:96) : s ( V (cid:96) ) = s } .Since the sets B (cid:96) are φ -measurable, this concludes the proof that { x ∈ G : s ( φ , x ) = s } is φ -measurable for every s ∈ S ( h ) , taking also into account that s ( φ , · ) − ( ) is φ -null. . Proof of Theorem . This long and technical section is devoted to the proof of Theorem . . Definition . . Let C > C ( C ) : = + C ,and C ( C ) : = ( ( + C )) ( Q + ) . arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 49 Remark . . Let s ∈ S ( h ) be fixed and let V ∈ G s (cid:69) ( h ) with e ( V ) ≥ C , where e is defined in ( ). Let L be acomplement of V and P : = P V the projection on V related to this splitting. Note that with the previous choices of C and C , for any h ∈ {
1, . . . , Q } , thanks to Proposition . and item (iii) of Proposition . , we have2 ( + C ) h C h ( P ( B (
0, 1 ))) < C /2 h + ,since C h ( P ( B (
0, 1 ))) ≤ C h (cid:120) V ( B ( C )) = C h . Proposition . . Let h ∈ {
1, . . . , Q } , s ∈ S ( h ) , and let G be a subset of Gr s (cid:69) ( h ) such that there exists a constant C > for which e ( V ) ≥ C for all V ∈ G , where we recall that e was defined in ( ) . Further let r > , ε ∈ (
0, 5 − h − C − h ] , r : = ( − ε / h ) r, and µ : = − h − C − h ε ,where C and C are defined in terms of C in Definition . .Let φ be a Radon measure and let z ∈ supp ( φ ) . We define Z ( z , r ) to be the set of the triplets ( x , s , V ) ∈ B ( z , C r ) × ( C r ] × Gr s (cid:69) ( h ) such that φ ( B ( y , t )) ≥ ( − ε )( t / C r ) h φ ( B ( z , C r )) , ( ) whenever y ∈ B ( x , C s ) ∩ x V and t ∈ [ µ s , C s ] . The geometric assumption we make on φ is that we can find a compactsubset E of B ( z , C r ) such that z ∈ E, φ ( B ( z , C r ) \ E ) ≤ µ h + C − h φ ( B ( z , C r )) , ( ) and such that for any x ∈ E and every s ∈ ( C r − d ( x , z )] there is a V ∈ Gr s (cid:69) ( h ) such that ( x , s , V ) ∈ Z ( z , r ) .Furthermore we assume that there exists W ∈ G such that ( z , r , W ) ∈ Z ( z , r ) , and let us fix L a normal complementarysubgroup of W such that Proposition . holds. Let us denote P : = P W the projection on W related to the splitting G = WL .Let us recall that with the notation T ( u , r ) we mean the cylinder with center u ∈ G and radius r > related to theprojection P = P W , see Definition . . For any u ∈ P ( B ( z , r )) let s ( u ) ∈ [ r ] be the smallest number with the followingproperty: for any s ( u ) < s ≤ r we have . E ∩ T ( u , s /4 h ) (cid:54) = ∅ , and . φ (cid:0) B ( z , C r ) ∩ T ( u , C s )) ≤ µ − h ( s / C r ) h φ ( B ( z , C r )) .Finally, we define( α ) A : = { u ∈ P ( B ( z , r )) : s ( u ) = } ,( β ) A : = (cid:110) u ∈ P ( B ( z , r )) : s ( u ) > and φ (cid:0) B ( z , C r ) ∩ T ( u , C s ( u )) (cid:1) ≥ ε − ( s ( u ) / C r ) h φ ( B ( z , C r )) (cid:111) ,( γ ) A : = (cid:110) u ∈ P ( B ( z , r )) : s ( u ) > and φ (cid:0) ( B ( z , C r ) \ E ) ∩ T ( u , s ( u ) /4 h ) (cid:1) ≥ − ( s ( u ) /4 hC r ) h φ ( B ( z , C r )) (cid:111) .Then we have(i) s ( u ) ≤ C h µ r for every u ∈ P ( B ( z , r )) ,(ii) The function u (cid:55)→ s ( u ) is lower semicontinuous on P ( B ( z , r )) and as a consequence A is compact,(iii) P ( B ( z , r )) ⊆ A ∪ A ∪ A ,(iv) C h ( P ( B ( z , r )) \ A ) ≤ h + C h C h ( P ( B (
0, 1 ))) ε r h ,(v) P ( E ∩ P − ( A )) = A, S h ( E ∩ P − ( A )) > and there is a constant C > such that C − S h ( E ∩ P − ( A )) ≤ φ ( E ∩ P − ( A )) ≤ C S h ( E ∩ P − ( A )) . arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 50 Proof.
We prove each point of the proposition in a separate paragraph. For the sake of notation we write Z : = Z ( z , r ) , and without loss of generality we will always assume that z =
0, since P W is a homogeneoushomomorphism, see Proposition . , and thus the statement is left-invariant. Since it will be used here and therein the proof, we estimate φ ( B ( C r ) \ B ( C r )) . Since ( r , W ) ∈ Z , we infer that φ ( B ( C r )) ≥ ( − ε )( r / r ) h φ ( B ( C r )) .This implies that φ ( B ( C r ) \ B ( C r )) = φ ( B ( C r )) − φ ( B ( C r )) ≤ φ ( B ( C r ))( − ( − ε )( r / r ) h )= φ ( B ( C r ))( − ( − ε )( − ε / h ) h ) ≤ εφ ( B ( C r )) , ( )where in the last inequality we used that h (cid:55)→ ( − ε / h ) h is increasing. proof of ( i ): Let u ∈ P ( B ( r )) and let C µ hr < s ≤ r . Then φ ( B ( C r ) ∩ T ( u , C s )) ≤ φ ( B ( C r )) ≤ µ − h ( s / C r ) h φ ( B ( C r )) ,where the last inequality comes from the fact that C µ hr < s . Defined v : = u δ µ ( u − ) , we immediately note that v ∈ W and that, from Proposition . , d ( v , u ) = µ d ( u , 0 ) ≤ C µ r . Furthermore, for every ∆ ∈ B ( µ r ) we have d ( u δ µ ( u − ) ∆ ) ≤ µ (cid:107) u (cid:107) + (cid:107) u (cid:107) + (cid:107) ∆ (cid:107) ≤ µ C r + C r + µ r ≤ ( C ( + µ ) + µ ) r ≤ C r , ( )where in the inequality above we used the fact that r > r /2, and C > ( C + ) > C ( + µ ) + µ . Thus, on theone hand we have B ( v , µ r ) ⊆ B ( u , ( + C ) µ r ) and on the other, thanks to ( ), we deduce that B ( v , µ r ) ⊆ B ( C r ) . ( )Since ( r , W ) ∈ Z , this implies thanks to the definition of Z and E that φ ( B ( v , µ r )) ≥ ( − ε ) µ h C − h φ ( B ( C r )) > φ ( B ( C r ) \ E ) . ( )Furthermore, thanks to ( ), ( ) and the definition of T ( · , · ) , we also infer that ∅ (cid:54) = E ∩ B ( v , µ r ) ⊆ E ∩ B ( u , ( + C ) µ r ) ⊆ E ∩ T ( u , s /4 h ) ,where the last inclusion is true since ( + C ) µ r ≤ C µ r /4 < s / ( h ) . proof of ( ii ): Let u ∈ P ( B ( r )) and let 0 < s ≤ s ( u ) . By definition of s ( u ) , up to eventually increasing s such that it still holds 0 < s ≤ s ( u ) , there are two cases. Either φ ( B ( C r ) ∩ T ( u , C s )) > ( + τ ) h µ − h ( s / C r ) h φ ( B ( C r )) , ( )for some τ > E ∩ T ( u , s /4 h ) = ∅ . ( )If v ∈ P ( B ( r )) is sufficiently close to u then s + C − d ( u , v ) ≤ ( + τ ) s and s + C − d ( u , v ) ≤ r , since s ( u ) ≤ r thanks to point (i). If ( ) holds, this implies that φ ( B ( C r ) ∩ T ( v , C ( s + C − d ( u , v )))) > φ ( B ( C r ) ∩ T ( u , C s )) ≥ ( + τ ) h µ − h ( s / C r ) h φ ( B ( C r )) ≥ µ − h (( s + C − d ( u , v )) / C r ) h φ ( B ( C r )) , ( )where the last inequality is true provided d ( u , v ) is suitably small. On the other hand, if ( ) holds, then E ∩ T ( v , ( s − hd ( u , v )) /4 h ) ⊆ E ∩ T ( u , s /4 h ) = ∅ . ( ) arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 51 Taking into account ( ) and ( ), this shows that s ( v ) ≥ min { s − hd ( u , v ) , s + C − d ( u , v ) } = s − hd ( u , v ) provided v is sufficiently close to u . This implies that lim inf v → u s ( v ) ≥ s for any s ≤ s ( u ) for which at least onebetween ( ) and ( ) holds. In particular, from the definition of s ( u ) , we deduce that there exists a sequence s i → s ( u ) − such that at each s i at least one between ( ) and ( ) holds. In conclusion we inferlim inf v → u s ( v ) ≥ s ( u ) . proof of ( iii ): Suppose that u ∈ P ( B ( r )) \ ( A ∪ A ) . Since u (cid:54)∈ A ∪ A , then s ( u ) > φ (cid:0) B ( C r ) ∩ T ( u , C s ( u )) (cid:1) < ε − ( s ( u ) / C r ) h φ ( B ( C r )) . ( )Thanks to the definition of s ( u ) , for any 0 < s < s ( u ) , up to eventually increasing s in such a way that it still holds0 < s < s ( u ) , we have either φ ( B ( C r ) ∩ T ( u , C s )) > µ − h ( s / C r ) h φ ( B ( C r )) , ( )or E ∩ T ( u , s /4 h ) = ∅ . ( )Let us assume that ( ) does not hold for some s < s ( u ) . Then ( ) does not hold for any t such that s ≤ t < s ( u ) .Thus, in this case, we deduce the existence of t i < s ( u ) such that t i → s ( u ) for which ( ) holds. Thus we have µ − h ( s ( u ) / C r ) h φ ( B ( C r )) = lim i → + ∞ µ − h ( t i / C r ) h φ ( B ( C r )) ≤ lim sup i → + ∞ φ ( B ( C r ) ∩ T ( u , C t i )) ≤ φ ( B ( C r ) ∩ T ( u , C s ( u ))) ≤ ε − ( s ( u ) / C r ) h φ ( B ( C r )) , ( )that is a contradiction thanks to the choice of µ and ε . This proves that for any 0 < ρ < s ( u ) we have E ∩ T ( u , v /4 h ) = ∅ and thus E ∩ int ( T ( u , s ( u ) /4 h )) = ∅ .Let us now define the constants s : = hs ( u ) / ε , and σ : = ( h − ) ε /32 h .Thanks to item (i), from which s ( u ) ≤ C h µ r , and from the very definition of µ , we deduce that0 < s ( u ) ≤ s = hs ( u ) / ε ≤ r − r , and µ ≤ σ ≤
1. ( )Thanks to the compactness of E and the definition of s ( u ) we have that E ∩ T ( u , s ( u ) /4 h ) (cid:54) = ∅ . Let us fix x ∈ E ∩ T ( u , s ( u ) /4 h ) and assume V ∈ Gr s (cid:69) ( h ) to be such that ( x , s , V ) ∈ Z . We claim that (cid:107) P ( x − y ) (cid:107) ≥ σ (cid:107) x − y (cid:107) , for every y ∈ x V . ( )Assume by contradiction that there is a y ∈ x V such that (cid:107) x − y (cid:107) = (cid:107) P ( x − y ) (cid:107) < σ . Let us fix w ∈ B ( σ s ) and let t ∈ R be such that | t | ≤ C s ( u ) / ( h σ ) . Then, we have d ( x δ t ( x − y ) w ) ≤ d ( x ) + | t |(cid:107) x − y (cid:107) + σ s ≤ d ( x ) + C s ( u ) h σ + σ s . ( )Thanks to the choice of the constants and item (i), according to which s ( u ) ≤ C h µ r , we infer that C s ( u ) h σ + σ s ≤ C s ( u )( − h + h / (( h − ) ε )) ≤ C − h − ε r ( − h + h / (( h − ) ε )) ≤ C ε r / h , ( ) arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 52 where in the first inequality above we are using the fact that C ≥
1, and in the second we are using the explicitexpression µ = − h − C − h ε and the fact that C − h + <
1. Hence, since x ∈ B ( C r ) putting together ( ) and( ) we infer that d ( x δ t ( x − y ) w ) ≤ C r + C ε r / h < C r , ( )where the second inequality comes from the definition of r and the fact that C ≥ C . As a consequence of theprevious computations we finally deduce that B ( x δ t ( x − y ) , σ s ) ⊆ B ( C r ) , for any | t | ≤ C s ( u ) / ( h σ ) .We now prove that for any | t | ≤ C s ( u ) / ( h σ ) and any w ∈ B ( σ s ) , we have x δ t ( x − y ) w ∈ T ( u , C s ( u )) . ( )Indeed, thanks to Proposition . , we have that P ( x δ t ( x − y ) w ) = P ( x ) δ t ( P ( x − y )) P ( w ) and thus since x ∈ T ( u , s ( u ) /4 h ) by means of Proposition . we infer that d ( u , P ( x )) ≤ C s ( u ) /4 h . Thanks to this, and togetherwith the fact that (cid:107) P ( w ) (cid:107) ≤ C σ s due to Proposition . , we can estimate d ( u , P ( x ) δ t ( P ( x − y )) P ( w )) ≤ d ( u , P ( x )) + | t |(cid:107) P ( x − y ) (cid:107) + C σ s ≤ C s ( u ) h + C s ( u ) h + C σ s ≤ C s ( u ) h + C (cid:16) − h (cid:17) s ( u ) ≤ C s ( u ) ,where in the second inequality of the last line we are using σ s = s ( u )( − ( h )) . Summing up, the abovecomputations yield that B ( x δ t ( x − y ) , σ s ) ⊆ B ( C r ) ∩ T ( u , C s ( u )) , for any | t | ≤ C s ( u ) / ( h σ ) . ( )Now we are in a position to write the following chain of inequalities φ ( B ( C r ) ∩ T ( u , C s ( u ))) ≥ ( σ s ) − ˆ s ( u ) /4 h σ − s ( u ) /4 h σ φ ( B ( x δ t ( x − y ) , σ s )) dt ≥ ( σ s ) − ( s ( u ) /2 h σ )( − ε )( σ s / rC ) h φ ( B ( C r ))= ( − ε )( − h ) h h ( h − ) − ε − ( s ( u ) / C r ) h φ ( B ( C r )) ≥ ε − ( s ( u ) / C r ) h φ ( B ( C r )) ( )where the first inequality is true by applying Fubini theorem to the function F ( t , z ) : = χ B ( σ s ) ( δ t ( y − x ) x − z ) onthe domain [ − s ( u ) / ( h σ ) , s ( u ) / ( h σ )] × G , and by noticing that when | t | ≤ s ( u ) / ( h σ ) we have ( ); the secondinequality is true since x ∈ E and then ( x , s , V ) ∈ Z for some V ∈ Gr s (cid:69) ( h ) ; and the last inequality is true since ( − ε )( − ( h )) h h ( h − ) − ≥
1. Since ( ) is a contradiction with the assumption u / ∈ A we get that ( )holds and thus P | V is injective, since it is also a homomorphism. Furthermore, since V has the same stratificationas W , Proposition . implies that VL = G , where L is the chosen normal complement of W . Thanks to [ ,Proposition . . ], there exists an intrinsically linear function (cid:96) : W → L such that V = graph ( (cid:96) ) and thus P | V is also surjective. In particular we can find a w ∈ x V in such a way that P ( w ) = u and, by using ( ) and d ( u , P ( x )) ≤ C s ( u ) /4 h , that follows from Proposition . , and the fact that P is a homogeneous homomorphism,we conclude that the following inequality holds (cid:107) x − w (cid:107) ≤ σ − (cid:107) P ( x ) − P ( w ) (cid:107) = σ − (cid:107) P ( x ) − u (cid:107) ≤ C s ( u ) h σ . ( )We now claim that the inclusion U ( w , s ( u ) /4 h ) ⊆ ( B ( C r ) \ E ) ∩ int ( T ( u , s ( u ) /4 h )) , ( )concludes the proof of item (iii). Indeed, we have ( x , s , V ) ∈ Z , and since w ∈ B ( x , C s ) ∩ x V , see ( ), and wehave µ s ≤ s ( u ) /4 h ≤ C s , we infer, by approximation and using the hypothesis, that φ ( U ( w , s ( u ) /4 h )) ≥ ( − ε )( s ( u ) /4 hC r ) h φ ( B ( C r )) . ( ) arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 53 Putting together ( ) and ( ) we deduce that φ (cid:0) ( B ( C r ) \ E ) ∩ int ( T ( u , s ( u ) /4 h )) (cid:1) ≥ ( − ε )( s ( u ) /4 hC r ) h φ ( B ( C r )) .and thus u ∈ A , which proves item (iii). In order to prove the inclusion ( ) we note that since (cid:107) x − w (cid:107) ≤ C s ( u ) / ( h σ ) , see ( ), we have thanks to the same computation we performed in ( ), ( ), and ( ), that B ( w , s ( u ) / ( h )) ⊆ B ( C r ) . Furthermore, since P ( w ) = u the inclusion ( ) follows thanks to the fact that B ( w , s ( u ) /4 h ) ⊆ T ( u , s ( u ) /4 h ) , see Proposition . , and the fact that int ( T ( u , s ( u ) /4 h )) ∩ E = ∅ . proof of ( iv ): Let τ >
1. Thanks to [ , Theorem . . ], we deduce that there exists a countable set D ⊆ A such that the following two hold( α ) { B ( w , C s ( w )) ∩ W : w ∈ D } is a disjointed subfamily of { B ( w , C s ( w )) ∩ W : w ∈ A } ,( β ) for any w ∈ A there exists a u ∈ D such that B ( w , C s ( w )) ∩ B ( u , C s ( u )) ∩ W (cid:54) = ∅ and s ( w ) ≤ τ s ( u ) .Furthermore, if we define for every u ∈ A the setˆ B ( u , C s ( u )) : = (cid:91) { B ( w , C s ( w )) ∩ W : w ∈ A , B ( u , C s ( u )) ∩ B ( w , C s ( w )) ∩ W (cid:54) = ∅ , s ( w ) ≤ τ s ( u ) } , ( )we have, thanks to [ , Corollary . . ], that A ⊆ (cid:83) u ∈ A B ( u , C s ( u )) ∩ W ⊆ (cid:83) w ∈ D ˆ B ( w , C s ( w )) . An easycomputation based on the triangle inequality, which we omit, leads to the following inclusionˆ B ( u , C s ( u )) ⊆ W ∩ B ( u , ( + τ ) C s ( u )) , for every u ∈ A . ( )Since D ⊆ A , and since T ( u , C s ( u )) ⊆ P − ( B ( u , C s ( u )) ∩ W ) for every u ∈ A , see Proposition . , weconclude, by exploiting the fact that { B ( w , C s ( w )) ∩ W : w ∈ D } is a disjointed family, the following inequality φ ( B ( C r )) ≥ ∑ u ∈ D φ ( B ( C r ) ∩ T ( u , C s ( u ))) ≥ ε − ∑ u ∈ D ( s ( u ) / C r ) h φ ( B ( C r )) ,where the last inequality above comes from the fact that D ⊆ A . The above inequality can be rewritten as ∑ u ∈ D s ( u ) h ≤ C h ε r h . In particular, thanks to item (iii) of Proposition . , and ( ) we infer that C h ( A ) ≤ ∑ u ∈ D C h ( B ( u , ( + τ ) C s ( u )) ∩ W ) = C h ( + τ ) h ∑ u ∈ D s ( u ) h ≤ C h C h ( + τ ) h ε r h . ( )With a similar argument we used to prove the existence of D , we can construct a countable set D (cid:48) ⊆ A such thatthe family { B ( u , C s ( u ) /4 h ) ∩ W : u ∈ D (cid:48) } is disjointed and the family { ˆ B ( u , C s ( u ) /4 h ) : u ∈ D (cid:48) } , constructedas in ( ), covers A . In a similar way as in ( ) we have ˆ B ( u , C s ( u ) / ( h )) ⊆ W ∩ B ( u , ( + τ ) C s ( u ) /4 h ) forevery u ∈ A . Moreover, since T ( u , s ( u ) /4 h ) ⊆ P − ( B ( u , C s ( u ) /4 h ) ∩ W ) for every u ∈ A , see Proposition . ,we conclude by exploiting the fact that { B ( u , C s ( u ) / ( h )) ∩ W : w ∈ D (cid:48) } is a disjointed family, the followinginequality φ ( B ( C r ) \ E ) ≥ ∑ u ∈ D (cid:48) φ (( B ( C r ) \ E ) ∩ T ( u , s ( u ) /4 h )) ≥ − φ ( B ( C r )) ∑ u ∈ D (cid:48) ( s ( u ) /4 hC r ) h ,where the last inequality holds since D (cid:48) ⊆ A . From the previous inequality, ( ), and the fact that 0 ∈ E , weinfer that ∑ u ∈ D (cid:48) ( s ( u ) /4 hC r ) h ≤ φ ( B ( C r ) \ E ) φ ( B ( C r )) ≤ · φ ( B ( C r ) \ B ( C r )) + φ ( B ( C r ) \ E ) φ ( B ( C r )) ≤ · εφ ( B ( C r )) + µ h + C − h φ ( B ( C r )) φ ( B ( C r )) ≤ ε . ( )Consequently, we deduce that C h ( A ) ≤ ∑ u ∈ D (cid:48) C h ( W ∩ B ( u , ( + τ ) C s ( u ) /4 h )) = ( + τ ) h C h ∑ u ∈ D (cid:48) ( s ( u ) /4 h ) h ≤ ( + τ ) h C h C h ε r h . ( ) arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 54 Finally, putting together ( ), ( ), item (iii) of this proposition, and item (iii) of Proposition . we concludethe following inequality C h ( P ( B ( r )) \ A ) ≤ C h ( P ( B ( r )) \ P ( B ( r ))) + C h ( A ) + C h ( A ) ≤ C h ( P ( B (
0, 1 ))) r h ( − ( − ε / h ) h ) + C h C h ( + τ ) h ε r h + ( + τ ) h C h C h ε r h ≤ ( + τ ) h C h C h ( P ( B (
0, 1 ))) ε r h ,where in the last inequality we used that 1 ≤ C ≤ C , and that C h ( P ( B (
0, 1 )) ≥ P ( B (
0, 1 )) ⊇ B (
0, 1 ) ∩ W and C h ( B (
0, 1 ) ∩ W ) =
1, thanks to item (iii) of Proposition . . With the choice τ =
2, item (iv) follows. proof of ( v ): Let u ∈ A and note that since s ( u ) =
0, for any s > E ∩ T ( u , s /4 h ) (cid:54) = ∅ .Since the sets E ∩ T ( u , s /4 h ) are compact we infer the following equality thanks to the finite intersection property ∅ (cid:54) = E ∩ (cid:92) s > T ( u , s /4 h ) = E ∩ P − ( u ) .This implies that u ∈ P ( E ∩ P − ( u )) for every u ∈ A , and as a consequence A ⊆ P ( E ∩ P − ( A )) . Since the inclusion P ( E ∩ P − ( A )) ⊆ A is obvious we finally infer that A = P ( E ∩ P − ( A )) . Moreover, thanks to item (iv) and to thechoice of ε < − h − C − h , we conclude that S h ( A ) > C h (cid:120) W and S h (cid:120) W are equivalent, seeProposition . , and thanks to the following chain of inequalities C h ( A ) ≥ C h ( P ( B ( r ))) − C h ( P ( B ( r )) \ A ) ≥ C h ( P ( B (
0, 1 ))) r h − h + C h C h ( P ( B (
0, 1 ))) ε r h ≥ r h .Thanks to the fact that P is C -Lipschitz, see Proposition . , we further infer that0 < S h ( A ) = S h ( P ( E ∩ P − ( A ))) ≤ C h S h ( E ∩ P − ( A )) .For any s sufficiently small and u ∈ A , by definition of s ( u ) and A , we have the following chain of inequalities φ ( B ( x , C s )) ≤ φ (cid:0) B ( C r ) ∩ T ( u , C s )) ≤ µ − h ( s / C r ) h φ ( B ( C r )) ,whenever x ∈ E ∩ P − ( u ) , where the first inequality comes from the fact that x ∈ E ⊆ B ( C r ) , and Proposi-tion . . Finally by [ , . . ( )] and the previous inequality we infer φ (cid:120) ( E ∩ P − ( A )) ≤ C − h C − h µ − h φ ( B ( C r )) r h S h (cid:120) ( E ∩ P − ( A )) . ( )On the other hand, if we assume x ∈ E and s sufficiently small, we have ( x , s , V ) ∈ Z for some V ∈ Gr s (cid:69) ( h ) . Thisimplies that, by using the very definition of Z , that φ ( B ( x , s )) ≥ ( − ε )( s / C r ) h φ ( B ( C r )) ,and thus by [ , . . ( )], we have φ (cid:120) E ≥ ( − ε ) φ ( B ( C r ))( C r ) h S h (cid:120) E . ( )Putting together ( ) and ( ), we conclude the proof of item (v). Proposition . . Let φ be a P ∗ , (cid:69) h -rectifiable measure such that there exists an s ∈ N κ for which for φ -almost every x ∈ G we have Tan h ( φ , x ) ⊆ { λ S h (cid:120) V : λ > and V ∈ Gr s (cid:69) ( h ) } . ( ) arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 55 Then, the set G ( x ) : = { V ∈ Gr s (cid:69) ( h ) : there exists Θ > such that Θ S h (cid:120) V ∈ Tan h ( φ , x ) } , ( ) is a compact subset of Gr s (cid:69) ( h ) for all x ∈ G for which ( ) holds, and the sets G C : = { x ∈ G : e ( V ) ∈ ( C , ∞ ) for every V ∈ G ( x ) } , ( ) where e is defined in ( ) , are φ -measurable for any C > .Proof. The fact that G ( x ) is compact is an immediate consequence of Proposition . , the compactness of theGrassmannian in Proposition . , and the convergence result in Proposition . . For any λ , k , r > M λ , k , r ( x , V ) : G × Gr s (cid:69) ( h ) → R as M λ , k , r ( x , V ) : = F k ( r − h T x , r φ , λ C h (cid:120) V ) ,where F k is defined in Definition . . We claim that, for any choice of the parameters, the function M λ , k , r iscontinuous when G × Gr s (cid:69) ( h ) is endowed with respect to the topology induced by the metric d + d G . Indeed,assume { x i } i ∈ N ⊆ G and { V i } ⊆ Gr s (cid:69) ( h ) are two sequences converging to x ∈ G and V ∈ Gr s (cid:69) ( h ) respectively.Thanks to the triangle inequality we havelim sup i → ∞ |M λ , k , r ( x , V ) − M λ , k , r ( x i , V i ) | ≤ lim sup i → ∞ ( |M λ , k , r ( x , V ) − M λ , k , r ( x i , V ) | + |M λ , k , r ( x i , V ) − M λ , k , r ( x i , V i ) | ) ≤ lim sup i → ∞ F k ( r − h T x , r φ , r − h T x i , r φ ) + lim sup i → ∞ F k ( λ C h (cid:120) V , λ C h (cid:120) V i ) ≤ lim sup i → ∞ r − ( h + ) d ( x , x i ) φ ( B ( x , kr + d ( x , x i )))+ lim sup i → ∞ F k ( λ C h (cid:120) V , λ C h (cid:120) V i ) = and the last identitycomes from Proposition . . This in particular implies that the function M ( x , V ) : = sup k > k ∈ Q inf λ > λ ∈ Q lim inf r → r ∈ Q M λ , k , r ( x , V ) k h + ,is Borel measurable.We now claim that for φ -almost every x ∈ G we have that V ∈ G ( x ) if and only if M ( x , V ) =
0. Indeed if V ∈ G ( x ) , there is a λ > { r i } i ∈ N such that lim i → ∞ F k ( r − hi T x , r i φ , λ C h (cid:120) V ) = k >
0, see Proposition . . However, by the scaling properties of F , see Remark . , we can choose an anotherinfinitesimal sequence { s i } i ∈ N ⊆ Q such that r i / s i →
1, and then lim i → ∞ F k ( s − hi T x , s i φ , λ C h (cid:120) V ) = k > M ( x , V ) =
0, then for any j ∈ N there exists a λ j > λ j ∈ Q , and an infinitesimal sequence { r i ( j ) } ⊆ Q such that lim i → ∞ F ( r i ( j ) − h T x , r i ( j ) φ , λ j C h (cid:120) V ) ≤ j .Since 0 < Θ h ∗ ( φ , x ) ≤ Θ h , ∗ ( φ , x ) < ∞ for φ -almost every x ∈ G , we can argue as in the last part of the proof ofProposition . and hence we can assume without loss of generality that λ j converge to some non-zero λ andthat, for every j ∈ N , there exists i j ∈ N such that r i j ( j ) is an infinitesimal sequence and r i j ( j ) − h T x , r ij ( j ) φ (cid:42) λ C h (cid:120) V .This eventually concludes the proof of the claim.Furthermore, since e by Proposition . is lower semicontinuous on Gr s (cid:69) ( h ) , we know that for any C > G × { W ∈ Gr s (cid:69) ( h ) : e ( W ) ≤ C } is closed in G × Gr s (cid:69) ( h ) and in particular, the set M − ( ) ∩ G × { W ∈ Gr s (cid:69) ( h ) : e ( W ) ≤ C } = { ( x , V ) ∈ G × Gr s (cid:69) ( h ) such that M ( x , V ) = e ( V ) ≤ C } ,is Borel. Now, since the projection on the first component of the above set is an analytic set, by the very definitionof analytic sets, and since every analytic set is universally measurable, see for example [ , Section . . ], weget that the set { x ∈ G such that there exists V ∈ Gr s (cid:69) ( h ) with M ( x , V ) = e ( V ) ≤ C } is φ -measurable. Inparticular its complement, that is G C up to φ -null sets - since M ( x , V ) = V ∈ G ( x ) for φ -almostevery x ∈ G - is φ -measurable as well. The interested reader could find this computation in the last inequality of the proof of [ , Proposition . ]. arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 56 Proposition . . Let h ∈ {
1, . . . , Q } , s ∈ S ( h ) , and φ be a P ∗ , (cid:69) h -rectifiable measure supported on a compact set K and forwhich for φ -almost every x ∈ G we have Tan h ( φ , x ) ⊆ { λ S h (cid:120) V : λ > and V ∈ Gr s (cid:69) ( h ) } . ( ) Let us further assume that there exists a constant C > such that φ ( G \ G C ) = , where G C is defined in ( ) . Throughoutthe rest of the statement and the proof we will always assume that C and C are the constants introduced in Definition . in terms of C. Furthermore, let ε ∈ (
0, 5 − ( h + ) C − h ] and µ : = − h − C − h ε .Then, there are ϑ , γ ∈ N , a φ -positive compact subset E of E ( ϑ , γ ) , and a point z ∈ E ∩ G C such that(i) There exists a ρ z > for which φ ( B ( z , C ρ ) \ E ) ≤ µ h + C − h φ ( B ( z , C ρ )) for any < ρ < ρ z ;(ii) There exists an r ∈ (
0, 5 − ( h + ) C − h γ − ] such that for any w ∈ E and any < ρ ≤ C r we can find a V w , ρ ∈ Gr s (cid:69) ( h ) such that e ( V w , ρ ) ≥ C, see ( ) , and . F w ,4 C ρ ( φ , Θ C h (cid:120) w V w , ρ ) ≤ ( − ϑ − C − µ ) ( h + ) · ( C ρ ) h + for some Θ > , . whenever y ∈ B ( w , C ρ ) ∩ w V w , ρ and t ∈ [ µρ , C ρ ] we have φ ( B ( y , t )) ≥ ( − ε )( t / C ρ ) h φ ( B ( w , C ρ )) , . There exists a normal complement L w , ρ of V w , ρ as in Proposition . such that ( − ε ) φ ( B ( w , C ρ ) ∩ wT V w , ρ ( ρ )) ≤ C − h C h ( P ( B (
0, 1 ))) φ ( B ( w , C ρ )) , where T V w , ρ is the cylinder related to the splitting G = V w , ρ · L w , ρ , see Definition . ;(iii) There exists an infinitesimal sequence { ρ i ( z ) } i ∈ N ⊆ (
0, min { r , ρ z } ] such that for any i ∈ N , any w ∈ E and any ρ ∈ ( C ρ i ( z )] we have φ ( B ( w , C ρ )) ≥ ( − ε )( ρ / ρ i ( z )) h φ ( B ( z , C ρ i ( z ))) .Proof. For any positive a , b ∈ R we define F ( a , b ) to be the set of those points in K for which br h ≤ φ ( B ( x , r )) , for any r ∈ ( a ) .One can prove, with the same argument used in the proof of Proposition . , that the sets F ( a , b ) are compact. Asa consequence, this implies that the sets (cid:101) F ( a , b ) : = ∞ (cid:92) p = F ( C a , ( − ε ) b ) \ F ( C a / p , b ) ,are Borel. Since φ is P ∗ h -rectifiable, G can be covered φ -almost all by countably many sets (cid:101) F ( a , b ) . Indeed, φ ( G \∪ a , b ∈ Q + (cid:101) F ( a , b )) = < Θ h ∗ ( φ , x ) < + ∞ holds φ -almost everywhere. In particular thanks to Proposition . we can find a , b ∈ R and ϑ , γ ∈ N such that φ ( (cid:101) F ( a , b ) ∩ E ( ϑ , γ )) >
0. Since (cid:101) F ( a , b ) ∩ E ( ϑ , γ ) is measurable, theremust exist a φ -positive compact subset of (cid:101) F ( a , b ) ∩ E ( ϑ , γ ) that we denote with F . Notice that since φ ( G \ G C ) = F ∩ G C is measurable and φ -positive as well.Let us denote by Gr s , C (cid:69) ( h ) the set { V ∈ Gr s (cid:69) ( h ) such that e ( V ) ≥ C } . Since by the very definition of G C wehave Tan h ( φ , x ) ⊆ M ( h , Gr s , C (cid:69) ( h )) for φ -almost every x ∈ F ∩ G C , we infer that Proposition . together withSeverini-Egoroff theorem, that can be applied since the functions x → d x , kr ( φ , M ( h , Gr s , C (cid:69) ( h ))) are continuous in x for every k , r > . - yield a φ -positive compact subset E of F ∩ G C and an r ≤ − ( h + ) C − h γ − such that d x ,4 C ρ ( φ , M ( h , Gr s , C (cid:69) ( h ))) ≤ ( − ϑ − C − µ ) ( h + ) for any x ∈ E and any 0 < ρ ≤ C r . ( )Let us fix z to be a density point of E with respect to φ , and let us show that E and z satisfy the requirements ofthe proposition. First, by construction E is φ -positive and contained in E ( ϑ , γ ) . Second, since z is a density pointof E , item (i) follows if we choose ρ z small enough. Moreover, the bound ( ) directly implies item (ii. ). Let usprove the remaining items.Since E ⊆ E ( ϑ , γ ) , 4 C r < γ /2 and 4 − ϑ − C − µ ≤ − ( h + ) ϑ , Proposition . (i) implies that for any w ∈ E and any 0 < ρ < C r - choosing σ = − ϑ − C − µ and t = C ρ in Proposition . - there exists a V w , ρ ∈ Gr s , C (cid:69) ( h ) such that φ ( B ( y , r ) ∩ B ( w V , 4 − C − ϑ − µ ρ )) ≥ ( − ( h + ) − C − µ )( r / s ) h φ ( B ( v , s )) , arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 57 whenever y , v ∈ B ( w , 2 C ρ ) ∩ w V w , ρ and ϑ − µρ ≤ r , s ≤ C ρ . Since 2 ( h + ) − C − µ ≤ ε , with the choices s = C ρ and v = w , we finally infer φ ( B ( y , r )) ≥ ( − ε )( r / C ρ ) h φ ( B ( w , C ρ )) ,for any µρ ≤ r ≤ C ρ and any y ∈ B ( w , C ρ ) ∩ w V w , ρ , and this proves item (ii. ). For any w ∈ E and any0 < ρ < C r we choose one normal complement L w , ρ of V w , ρ as in Proposition . , and we denote with P : = P V w , ρ the projection relative to this splitting. Eventually, Proposition . (ii), with the choice k : = C , impliesthat for any 0 < ρ < C r we have φ ( B ( w , C ρ ) ∩ wT V w , ρ ( ρ )) ≤ ( + ( C h + ) ϑ − C − µ ) C − h C h ( P ( B (
0, 1 ))) φ ( B ( w , C ρ )) ≤ ( + ε ) C − h C h ( P ( B (
0, 1 ))) φ ( B ( w , C ρ )) , ( )where the last inequality comes from the fact that ( C h + ) ϑ − C − µ < ε . Hence also item (ii. ) is verified. Inorder to verify item (iii), note that since z ∈ E ⊆ (cid:101) F ( a , b ) on the one hand then there is an infinitesimal sequence { ρ i ( z ) } i ∈ N such that φ ( B ( z , C ρ i ( z )))( C ρ i ( z )) h ≤ b . ( )On the other hand for any w ∈ E , and any 0 < ρ < a we have b ≤ − ε φ ( B ( w , C ρ ))( C ρ ) h . ( )Putting together ( ) and ( ) we finally infer that for any i ∈ N , any w ∈ E and any ρ ∈ ( a ) we have φ ( B ( z , C ρ i ( z ))) ρ i ( z ) h ≤ − ε φ ( B ( w , C ρ )) ρ h ,concluding the proof of item (iii) and thus of the proposition. Proposition . . Assume φ is a P ∗ , (cid:69) h -rectifiable measure supported on a compact set K. Then, there exists a W ∈ Gr (cid:69) ( h ) ,a compact set K (cid:48) (cid:98) W and a Lipschitz function f : K (cid:48) → G such that φ ( f ( K (cid:48) )) > .Proof. Theorem . implies that for φ -almost every x ∈ G the elements of Tan h ( φ , x ) all share the same stratificationvector. Furthermore, thanks to Proposition . , for any s ∈ S ( h ) the set T s : = { x ∈ K : s ( φ , x ) = s } is φ -measurable.Thus, if we prove that for any s ∈ S ( h ) there exists a Lipschitz function as in the thesis of the proposition whoseimage has positive φ (cid:120) T s -measure, the proposition is proved since the sets T s cover φ -almost all K and since thelocality of tangents hold, see Proposition . . Thanks to this argument, we can assume without loss of generalitythat there exists a s ∈ S ( h ) such that for φ -almost every x ∈ K we have s ( φ , x ) = s .Let us further reduce ourselves to the setting in which there exists a constant C > φ ( G \ G C ) = G C is defined in ( ). Thanks to Proposition . , we know that for φ -almost every x ∈ G the set G ( x ) defined in ( ) is compact. Hence, taking item (i) of Proposition . into account, for φ -almost every x ∈ G thereexists a constant C ( x ) > e ( V ) ≥ C ( x ) for every V ∈ G ( x ) . This readily implies that φ ( G \ ∪ n ∈ N G n ) = G n is φ -measurable for every n ∈ N , see Proposition . , we can reduce, with the same argumentused in the previous paragraph, to deal with the case in which there exists C > φ ( G \ G C ) = C : = C ( C ) and C : = C ( C ) be defined as in Definition . , and let (cid:101) ε ≤ − ( h + ) C − h , and (cid:101) µ : = − h − C − h (cid:101) ε . Let E ⊆ K be the compact set and z ∈ E ∩ G C the point yielded by Proposition . with respect to (cid:101) ε , (cid:101) µ . Furthermore let (cid:101) ε ≤ ε ≤ − h − C − h , and µ : = − h − C − h ε such that ( − (cid:101) ε ) ≥ ( − ε ) . We define r : = ρ ( z ) , and r : = ( − ε / h ) r ,where ρ ( z ) is the first term of the sequence { ρ i ( z ) } i ∈ N yielded by item (iii) of Proposition . . arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 58 Let us check that the compact set E ∩ B ( z , C r ) satisfies the hypothesis of Proposition . with respect to thechoiches ε , µ , r . First of all, since r < ρ z , item (i) of Proposition . implies that ( ) holds since (cid:101) µ ≤ µ . Secondly,since r ≤ r , item (ii. ) of Proposition . implies that for any w ∈ E and any 0 < ρ < C r there exists a V w , ρ ∈ Gr s (cid:69) ( h ) such that whenever y ∈ B ( w , C r ) ∩ w V w , ρ and t ∈ [ µρ , C ρ ] we have φ ( B ( y , t )) ≥ ( − (cid:101) ε )( t / C ρ ) h φ ( B ( w , C ρ )) .Furthermore, since r = ρ ( z ) , thanks to item (iii) of Proposition . we finally infer that for any w ∈ E and any0 < ρ < C r we have φ ( B ( y , t )) ≥ ( − (cid:101) ε )( t / C ρ ) h φ ( B ( w , C ρ )) ≥ ( − (cid:101) ε ) ( t / C r ) h φ ( B ( z , C r )) ≥ ( − ε )( t / C r ) h φ ( B ( z , C r )) ,whenever y ∈ B ( w , C r ) ∩ w V w , ρ and t ∈ [ µρ , C ρ ] . The above paragraph shows that the hypotheses of Proposi-tion . are satisfied by z and E ∩ B ( z , C r ) with the choices of r , r , ε , µ as above. Throughout the rest of the proof E will stand for E ∩ B ( z , C r ) , and in order to conclude the argument we willneed to use the other two pieces of information yielded by Proposition . . Indeed, since r < C r , item (ii. ) ofProposition . implies that ( − ε ) φ ( zT V z , r ( r ) ∩ B ( z , C r )) ≤ C h ( P ( B (
0, 1 ))) C − h φ ( B ( z , C r )) , ( )where T is the cylinder related to the splitting G = V z , r · L z , r , and L z , r is one normal complement to V z , r chosenas in item (ii. ) of Proposition . . Furthermore, thanks to item (ii. ) of Proposition . and the fact that r < r weknow that there exists Θ > F z ,4 C r ( φ , Θ C h (cid:120) z V z , r ) ≤ ( − ϑ − C − µ ) h + · ( C r ) h + . ( )The bound ( ) together with Proposition . , that we can apply since 4 C r ≤ γ − , and 2 − ( − ϑ − C − µ ) h + ≤ δ G ,where δ G was introduced in Definition . , imply thatsup w ∈ E ∩ B ( z , C r ) dist (cid:0) w , z V z , r (cid:1) C r ≤ + ( h + ) ϑ ( h + ) ( − ( − ϑ − C − µ ) h + ) h + ≤ C − µ . ( )The above bound shows that the set E inside the ball B ( z , C r ) is very squeezed around the plane V z , r . From nowon we should denote W : = V z , r , P : = P V z , r , L : = L z , r , and T ( · , r ) : = T W ( · , r ) . In order to simplify the notation,since all the statements are invariant up to substituting φ with T z ,1 φ , we can assume that z =
0. Let us recall oncemore that e ( V z , r ) ≥ C from item (ii) of Proposition . .Since it will turn out to be useful later on, we estimate the distance of the points w of E ∩ T ( r ) from 0. Thanksto Proposition . and the fact that w ∈ T ( r ) , we have (cid:107) P W ( w ) (cid:107) ≤ C r . On the other hand, ( ) and ( )imply that (cid:107) P L ( w ) (cid:107) ≤ C dist ( w , W ) ≤ C µ r .This in particular implies that d ( w ) ≤ (cid:107) P W ( w ) (cid:107) + (cid:107) P L ( w ) (cid:107) ≤ C r + C µ r ≤ C r ,showing that E ∩ T ( r ) ⊆ B (
0, 2 C r ) . ( )In the following A , A and A are the sets inside P ( B ( r )) constructed in the statement of Proposition . withrespect to the 0 and the plane W . Now, let (cid:101) A be the set of those u ∈ A for which there exists ρ ( u ) > φ ( B ( C r ) ∩ T ( u , s )) ≤ ( − ε ) ( s / C r ) h C h ( P ( B (
0, 1 ))) φ ( B ( C r )) , ( )for all 0 < s < ρ ( u ) . We claim that (cid:101) A is a Borel set. To prove this, we note that (cid:101) A = (cid:91) k ∈ N { u ∈ A : ( ) holds for any 0 < s < k } = : (cid:91) k ∈ N (cid:101) A k . arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 59 Let us show that (cid:101) A k is a compact set for any k ∈ N , and in order to do this, let us assume { u i } i ∈ N is a sequenceof points of (cid:101) A k . Since (cid:101) A k ⊆ A , and A is compact, we can suppose that, up to a non re-labelled subsequence, u i converges to some u ∈ A . Thus, we have that for every 0 < s < k the following chain of inequality holds φ ( B ( C r ) ∩ T ( u , s )) ≤ lim sup i → ∞ φ ( B ( C r ) ∩ T ( u i , s + d ( u , u i ))) ≤ ( − ε ) C h ( P ( B (
0, 1 )))( s / C r ) h φ ( B ( C r )) .This concludes the proof of the fact that (cid:101) A k is compact and thus (cid:101) A is an F σ set, and thus Borel.Let us notice that, since r < r , by a compactness argument one finds that there exists a (cid:101) s : = (cid:101) s ( r , r ) such thatwhenever u ∈ P ( B ( r )) , then P ( B ( u , (cid:101) s )) ⊆ P ( B ( r )) . The family B : = { P ( B ( u , s )) : u ∈ A \ (cid:101) A , and s ≤ (cid:101) s does not satisfy ( ) } is a fine cover of A \ (cid:101) A by the very definition of (cid:101) A . Thus Proposition . with a routine argument, which wealready employed in Proposition . , implies that B is a S h (cid:120) ( A \ (cid:101) A ) -Vitali relation. Therefore, the set A \ (cid:101) A canbe covered S h -almost all by a sequence of disjointed projected balls { P ( B ( u k , s k )) } k ∈ N such that u k ∈ A \ (cid:101) A and φ ( B ( C r ) ∩ T ( u k , s k )) > ( − ε ) C h ( P ( B (
0, 1 )))( s k / C r ) h φ ( B ( C r )) ,for every k ∈ N . Note that since T ( u k , s k ) = P − ( P ( B ( u k , s k ))) , see Proposition . , we get that { T ( u k , s k ) } k ∈ N isa disjointed family of cylinders. Moreover, from the very definition of (cid:101) s , since u k ∈ P ( B ( r )) and s k ≤ (cid:101) s , we havethat P ( B ( u k , s k )) ⊆ P ( B ( r )) . This implies that φ ( T ( r ) ∩ B ( C r )) ≥ ∑ k ∈ N φ ( B ( C r ) ∩ T ( u k , s k )) > ( − ε ) C h ( P ( B (
0, 1 ))) C − h r − h φ ( B ( C r )) ∑ k ∈ N s hk .Therefore, we have C h ( A \ (cid:101) A ) = ∑ k ∈ N C h ( P ( B ( u k , s k ))) ≤ C h ( P ( B (
0, 1 ))) ∑ k ∈ N s hk < − ( − ε ) − φ ( T ( r ) ∩ B ( C r )) C h r h φ ( B ( C r )) ≤ − ( − ε ) − C h ( P ( B (
0, 1 ))) r h ≤ C h ( P ( B (
0, 1 ))) r h ,where the second inequality on the last line above follows from ( ). Furthermore, from the previous inequalityand from item (iv) of Proposition . we deduce that C h ( (cid:101) A ) = C h ( P ( B ( r ))) − C h ( P ( B ( r )) \ A ) − C h ( A \ (cid:101) A ) > C h ( P ( B (
0, 1 ))) r h − h + C h ε C h ( P ( B (
0, 1 ))) r h − C h ( P ( B (
0, 1 ))) r h ≥ ( − − ) C h ( P ( B (
0, 1 ))) r h > C h ( P ( B (
0, 1 ))) r h .Since (cid:101) A is measurable, we can find a compact set ˆ A ⊆ (cid:101) A and a δ ∈ ( ε r / h ) such that C h ( ˆ A ) > ) holdsfor any u ∈ ˆ A and s ∈ ( δ ) . This can be done by taking an interior approximation with compact sets of (cid:101) A .Thanks to item (v) of Proposition . we know thatˆ A ⊆ A = P ( E ∩ P − ( A )) , ( )and thus for any u ∈ ˆ A we can find a x ∈ E such that P ( x ) = u . We claim that for any x ∈ E for which P ( x ) ∈ ˆ A ,any s < min { r /4, δ / ( + C ) } and any w ∈ V x , s we have (cid:107) P ( w ) (cid:107) > (cid:107) w (cid:107) /2 C . ( )Suppose by contradiction that there are an s < min { r /4, δ / ( + C ) } and a w ∈ V x , s with (cid:107) w (cid:107) = (cid:107) P ( w ) (cid:107) ≤ C . This would imply that for any k =
0, . . . , (cid:98) C /4 (cid:99) − p ∈ B ( s /2 ) we have, byexploiting P ( x ) = u and that P is a homogeneous homomorphism, that d ( P ( x δ ks ( w ) p ) , u ) = d ( δ ks ( P ( w )) P ( p ) , 0 ) ≤(cid:107) δ ks ( P ( w )) (cid:107) + (cid:107) P ( p ) (cid:107)≤ ks (cid:107) P ( w ) (cid:107) + (cid:107) P ( p ) (cid:107) ≤ ks / C + C s ≤ ( + C ) s . ( ) arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 60 Since u ∈ ˆ A ⊆ A ⊆ P ( B ( r )) , and since P ( x ) = u , we conclude that x ∈ T ( r ) . Hence, taking into account that r < r , thanks to the inclusion ( ), we have d ( x δ ks ( w ) p , 0 ) ≤ (cid:107) x (cid:107) + ks + s ≤ C r + ( k + ) s < C r + C r /4 < C r . ( )Putting together ( ) and ( ), we infer that for any k =
0, . . . , (cid:98) C /4 (cid:99) − B ( x δ ks ( w ) , s /2 ) ⊆ T ( u , ( + C ) s ) ∩ B ( C r ) .Furthermore, since x ∈ E , B ( x δ ks ( w ) , s /2 ) are disjoint and contained in B ( x , C s ) , we have by items (ii. ) and (iii)of Proposition . that φ (cid:16) B ( C r ) ∩ T ( u , ( + C ) s ) (cid:17) ≥ (cid:98) C /4 (cid:99)− ∑ k = φ ( B ( x δ ks ( w ) , s /2 )) ≥ ( − ε ) C (cid:16) s /2 C s (cid:17) h φ ( B ( x , C s )) ≥ ( − ε ) C (cid:16) s /2 C r (cid:17) h φ ( B ( C r )) = ( − ε ) C h + (cid:16) sC r (cid:17) h φ ( B ( C r )) . ( )Since by assumption u ∈ ˆ A ⊆ (cid:101) A and ( + C ) s < δ , we infer thanks to ( ) and the definition of ˆ A that ( − ε ) C h + (cid:16) sC r (cid:17) h φ ( B ( C r )) ≤ φ (cid:16) B ( C r ) ∩ T ( u , ( + C ) s ) (cid:17) ≤ ( − ε ) ( + C ) h (cid:16) sC r (cid:17) h C h ( P ( B (
0, 1 ))) φ ( B ( C r )) . ( )The chain of inequalities ( ) is however in contradiction with the choice of C thanks to Remark . , and thusthe claim ( ) is proved.Since P restricted to E ∩ P − ( A ) is surjective on ˆ A as remarked in ( ), thanks to the axiom of choice thereexists a function f : ˆ A → E ∩ P − ( A ) such that P ( f ( u )) = u . We claim that for φ -almost every x ∈ f ( ˆ A ) thereexists a r ( x ) > y ∈ f ( ˆ A ) ∩ B ( x , r ( x )) we have (cid:107) P ( x ) − P ( y ) (cid:107) = (cid:107) P ( x − y ) (cid:107) > C − (cid:107) x − y (cid:107) = C − (cid:107) f ( P ( x )) − f ( P ( y )) (cid:107) , ( )where the last identity comes from the fact that f is bijective on its image and thus the left and right inversemust coincide. In order to prove the latter claim, assume by contradiction that there exists an x ∈ f ( ˆ A ) such thatTan h ( φ , x ) ⊆ M ( h , Gr s (cid:69) ( h )) and a sequence { y i } i ∈ N ⊆ f ( ˆ A ) , with y i → x , such that (cid:107) P ( x − y i ) (cid:107) ≤ C − (cid:107) x − y i (cid:107) , for any i ∈ N . ( )Defined ρ i : = (cid:107) x − y i (cid:107) , thanks to the hypothesis on x and the definitions of y i and ρ i we can assume without lossof generality that . for any i ∈ N we have ρ i ≤ min { r /4, δ / ( + C ) } , . the points g i : = δ ρ i ( x − y i ) converge to some y ∈ ∂ B (
0, 1 ) such that (cid:107) P ( y ) (cid:107) ≤ C − , . ρ − hi T x , ρ i φ (cid:42) λ C h (cid:120) V for some λ > V ∈ Gr s (cid:69) ( h ) .Since C h (cid:120) V ( ∂ B ( p , s )) =
0, see e.g., [ , Lemma . ], for any p ∈ G and any s ≥
0, thanks to [ , Proposition . ] weinfer that λ C h (cid:120) V ( B ( y , ρ )) = lim i → ∞ T x , ρ i φ ( B ( y , ρ )) / ρ hi ≥ lim i → ∞ T x , ρ i φ ( B ( g i , ρ − d ( g i , y ))) / ρ hi ≥ lim i → ∞ φ ( B ( y i , ρ i ρ /2 )) / ρ hi ≥ ϑ − ( ρ /2 ) h > ϑ , γ ∈ N suchthat E ⊆ E ( ϑ , γ ) , since E is provided by Proposition . . The above computation shows that the (contradiction)assumption ( ) implies that at x there is a flat tangent measure whose support V contains an element y ∈ ∂ B (
0, 1 ) such that (cid:107) P ( y ) (cid:107) ≤ C − . Let us prove that if arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 61 (HC) there exists a suitably big i ∈ N such that we can find a q i ∈ V x , ρ i such that d ( y , q i ) ≤ µ ,then we achieve a contradiction with ( ), and thus we prove the claim ( ). Indeed, the claim (HC) above wouldimply thanks to the definition of µ , ( ), Proposition . , and Proposition . , that ( C ) − < ( − µ ) /2 C ≤ ( (cid:107) y (cid:107) − (cid:107) y − q i (cid:107) ) /2 C ≤ (cid:107) q i (cid:107) /2 C < (cid:107) P ( q i ) (cid:107) ≤ (cid:107) P ( y ) (cid:107) + (cid:107) P ( y − q i ) (cid:107) ≤ C − + C µ < C − ,which is a contradiction since C > Q .In this paragraph we prove the claim (HC), which is sufficient to conclude the proof of the claim ( ). Let Θ i be the positive numbers yielded by item (ii. ) of Proposition . with the choices ρ : = ρ i around the point x , andnotice thatlim sup i → ∞ F C ( λ C h (cid:120) V , Θ i C h (cid:120) V x , ρ i ) ≤ lim sup i → ∞ F C (cid:16) T x , ρ i φρ hi , λ C h (cid:120) V (cid:17) + lim sup i → ∞ F C (cid:16) T x , ρ i φρ hi , Θ i C h (cid:120) V x , ρ i (cid:17) = lim sup i → ∞ F C (cid:16) T x , ρ i φρ hi , Θ i C h (cid:120) V x , ρ i (cid:17) = lim sup i → ∞ F x ,4 C ρ i (cid:16) φ , Θ i C h (cid:120) x V x , ρ i (cid:17) ρ h + i ≤ ( ϑ − µ ) ( h + ) ,( )where the first identity in the last line above comes from Proposition . , the second identity from the scalingproperty in Remark . and the last inequality from item (ii. ) of Proposition . and some algebraic computationsthat we omit. Defined g ( w ) : = ( min {
1, 2 − (cid:107) w (cid:107)} ) + by Proposition . for any V (cid:48) ∈ Gr ( h ) we have ˆ gd C h (cid:120) V (cid:48) = h ˆ s h − ( min {
1, 2 − | s |} ) + ds = h (cid:32) ˆ s h − + ˆ s h − ( − s ) ds (cid:33) = h + − h + ( g ) ⊆ B (
0, 4 C ) , thanks to ( ) we infer thatlim sup i → ∞ | λ − Θ i | = lim sup i → ∞ | λ ´ gd C h (cid:120) V − Θ i ´ gd C h (cid:120) V x , ρ i | ´ gd C h (cid:120) V ≤ lim sup i → ∞ ( h + ) F C ( λ C h (cid:120) V , Θ i C h (cid:120) V x , ρ i ) h + − ≤ ( h + )( ϑ − µ ) ( h + ) h + − ≤ ( ϑ − µ ) ( h + ) . ( )Let p ∈ V ∩ B (
0, 1 ) and (cid:96) ( w ) : = ( µ (cid:107) p (cid:107) − (cid:107) p − w (cid:107) ) + . The function (cid:96) is a positive 1-Lipschitz function whosesupport is contained in B ( ( + µ ) (cid:107) p (cid:107) ) and therefore, thanks to Remark . , we deduce thatlim inf i → ∞ λ ˆ (cid:96) ( w ) d C h (cid:120) V x , ρ i ≥ λ ˆ (cid:96) ( w ) d C h (cid:120) V − lim sup i → ∞ λ (cid:12)(cid:12)(cid:12)(cid:12) ˆ (cid:96) ( w ) d C h (cid:120) V − ˆ (cid:96) ( w ) d C h (cid:120) V x , ρ i (cid:12)(cid:12)(cid:12)(cid:12) ≥ λ ˆ (cid:96) ( w ) d C h (cid:120) V − lim sup i → ∞ | λ − Θ i | ˆ (cid:96) ( w ) d C h (cid:120) V x , ρ i − lim sup i → ∞ (cid:12)(cid:12)(cid:12)(cid:12) ˆ (cid:96) ( w ) d λ C h (cid:120) V − ˆ (cid:96) ( w ) d Θ i C h (cid:120) V x , ρ i (cid:12)(cid:12)(cid:12)(cid:12) ≥ λ ˆ (cid:96) ( w ) d C h (cid:120) V − lim sup i → ∞ | λ − Θ i | ˆ (cid:96) ( w ) d C h (cid:120) V x , ρ i − lim sup i → ∞ F ( + µ ) (cid:107) p (cid:107) ( λ C h (cid:120) V , Θ i C h (cid:120) V x , ρ i ) . ( )Let us bound separately the two terms in the last line above. Thanks to the triangle inequality the points q i ∈ V x , ρ i of minimal distance of p from V x , ρ i are contained in B (
0, 2 (cid:107) p (cid:107) ) . This, together with item (iii) of Proposition . ,implies that ˆ (cid:96) ( w ) d C h (cid:120) V x , ρ i ≤ µ (cid:107) p (cid:107)C h (cid:120) V x , ρ i ( B ( q i , ( + µ ) (cid:107) p (cid:107) )) ≤ ( + µ ) h + (cid:107) p (cid:107) h + . ( )On the other hand, thanks to Remark . and the fact that C h (cid:120) V and C h (cid:120) V x , ρ i are invariant under rescaling, weinfer that F ( + µ ) (cid:107) p (cid:107) ( λ C h (cid:120) V , Θ i C h (cid:120) V x , ρ i ) = (cid:18) ( + µ ) (cid:107) p (cid:107) C (cid:19) h + F C ( λ C h (cid:120) V , Θ i C h (cid:120) V x , ρ i ) . ( ) arstrand - mattila rectifiability criterion for co - normal - P ∗ h - rectifiable measures 62 Putting together ( ), ( ), ( ), ( ) and ( ) we finally infer thatlim inf i → ∞ λ ˆ (cid:96) ( w ) d C h (cid:120) V x , ρ i ≥ λ ˆ (cid:96) ( w ) d C h (cid:120) V − ( ϑ − µ ) ( h + ) ( + µ ) h + (cid:107) p (cid:107) h + − (cid:18) ( + µ ) (cid:107) p (cid:107) C (cid:19) h + ( ϑ − µ ) ( h + ) ≥ λ ˆ (cid:96) ( w ) d C h (cid:120) V − ( ϑ − µ ) ( h + ) (cid:107) p (cid:107) h + (cid:0) ( + µ ) h + + (cid:1) . ( )Finally, Proposition . and the fact that x ∈ E ( ϑ , γ ) imply that λ ≥ ϑ − . This together with a simple computationthat we omit, based on Proposition . , shows that λ ˆ (cid:96) ( w ) d C h (cid:120) V = ϑ − ( µ (cid:107) p (cid:107) ) h + / ( h + ) . ( )Putting together ( ) and ( ) we eventually infer thatlim inf i → ∞ λ ˆ (cid:96) ( w ) d C h (cid:120) V x , ρ i ≥ ϑ − ( µ h + / ( h + ) − ( h + ) µ ( h + ) ) (cid:107) p (cid:107) h + > p ∈ B (
0, 1 ) ∩ V we have B ( p , µ (cid:107) p (cid:107) ) ∩ V x , ρ i (cid:54) = ∅ provided i is chosen suitably big. Thus theclaim (HC) is proved taking p = y .Let us conclude the proof of the proposition exploiting the claim ( ) that we have proved. Defined B to be theset of full measure in f ( ˆ A ) on which ( ) holds, we note that since B ( P ( x ) , r ) ⊆ P ( B ( x , r )) , the ( ) implies thefollowing one: for any u ∈ P ( B ) there exists a r ( u ) > (cid:107) f ( u ) − f ( w ) (cid:107) ≤ C (cid:107) u − w (cid:107) , whenever w ∈ ˆ A ∩ B ( u , r ( u )) . ( )Furthermore, note that thanks to the proof of item (v) of Proposition . and recalling that f ( ˆ A ) ⊆ E ∩ P − ( A ) , wededuce that S h (cid:120) f ( ˆ A ) is mutually absolutely continuous with respect to φ and by Proposition . we finally inferthat S h ( ˆ A \ P ( B )) = S h ( P ( f ( ˆ A ) \ B )) = f : ˆ A → f ( ˆ A ) is bijective.We now prove that if r ( u ) is chosen to be the biggest radius for which ( ) holds, then the map u (cid:55)→ r ( u ) is uppersemicontinuous on ˆ A . Indeed, assume { u i } i ∈ N is a sequence in ˆ A such that u i → u ∈ ˆ A and lim sup i → ∞ r ( u i ) = r ≥
0. If r =
0, then the inequality lim sup i → ∞ r ( u i ) ≤ r ( u ) is trivially satisfied. Thus, we can assume that r > < s < r there exists an i ∈ N such that s + d ( u , u i ) < r ( u i ) for any i ≥ i .As a consequence B ( u , s ) ⊆ B ( u i , r ( u i )) and thus for any y ∈ ˆ A ∩ B ( u , s ) and i ≥ i we have (cid:107) f ( u ) − f ( y ) (cid:107) ≤ (cid:107) f ( u ) − f ( u i ) (cid:107) + (cid:107) f ( u i ) − f ( y ) (cid:107) ≤ C (cid:107) u − i u (cid:107) + C (cid:107) u − i y (cid:107) . ( )Sending i to + ∞ , thanks to ( ) we conclude that for any y ∈ B ( u , s ) ∩ ˆ A we have (cid:107) f ( u ) − f ( y ) (cid:107) ≤ C (cid:107) u − y (cid:107) andthus s ≤ r ( u ) . The arbitrariness of s concludes that r is upper semicontinuous and thus for any j ∈ N the sets L j : = { w ∈ ˆ A : r ( w ) ≥ j } ,are Borel. Furthermore, since r ( u ) > P ( B ) , we infer that P ( B ) ⊆ ∪ j ∈ N L j . This, jointly with thefact that S h ( ˆ A ) >
0, and that S h ( ˆ A \ P ( B )) = j ∈ N and compact subset A of L j suchthat S h ( A ) > ( A ) < j .Let us conclude the proof by showing that f is Lipschitz on A and that φ ( f ( A )) >
0. The fact that f ( A ) is φ -positive follows from Proposition . , item (v) of Proposition . and the following computation0 < S h ( A ) = S h ( P ( f ( A ))) ≤ C h S h ( f ( A )) . omparison with other notions of rectifiability 63 On the other hand, for any u , v ∈ A we have d ( u , v ) ≤ j and since u , v ∈ L j then (cid:107) f ( u ) − f ( v ) (cid:107) ≤ C (cid:107) u − v (cid:107) .This eventually concludes the proof of the proposition. Proof of Theorem . . If we prove the result for φ (cid:120) B ( k ) for any k ∈ N , the general case follows taking into accountthe locality of tangents Proposition . and Lebesgue differentiation theorem Proposition . . Therefore, we canassume without loss of generality that φ is supported on a compact set. Let us set F : = {∪ i ∈ N f i ( K i ) : K i is a compact subset of W i with W i ∈ Gr (cid:69) ( h ) and f i : K i → G is Lipschitz } .Let m : = inf F ∈ F { φ ( G \ F ) } . We claim that if m = m = F n ∈ F such that φ ( G \ F n ) < n and then φ ( G \ ∪ n ∈ N F n ) =
0. Let us prove that m =
0. Indeeed, if bycontradiction m >
0, we can take, as before, F (cid:48) n ∈ F such that 0 < φ ( G \ ∪ n ∈ N F (cid:48) n ) ≤ m . Since F (cid:48) : = ∪ n ∈ N F (cid:48) n is Borel,we have, thanks to the locality of tangents Proposition . and Lebesgue differentiation theorem Proposition . ,that φ (cid:120) F (cid:48) is a P ∗ , (cid:69) h -rectifiable measure with compact support. Thus we can apply Proposition . to conclude thatthere exists W ∈ Gr (cid:69) ( h ) , K a compact subset of W and a Lipschitz function f : K → G such that φ (cid:120) F (cid:48) ( f ( K )) > φ ( G \ ( f ( K ) ∪ F (cid:48) )) < m , that is a contradiction with the definition of m .In order to prove the last part of the theorem, let us notice that, thanks to the locality of tangents in Propo-sition . and Lebesgue differentiation theorem in Proposition . , we can reduce on φ (cid:120) E ( ϑ , γ ) , thanks also toProposition . . Moreover, taking into account that S h (cid:120) E ( ϑ , γ ) is mutually absolutely continuous with respect to φ (cid:120) E ( ϑ , γ ) , see Proposition . , we can finally reduce to prove that S h (cid:120) f ( K ) is a P ch -rectifiable measure whenever K is a compact subset of W ∈ Gr (cid:69) ( h ) and f : K → G is a Lipschitz function. The fact that S h (cid:120) f ( K ) is a P ch -rectifiablemeasure follows from the following claim: if K is a compact subset of W ∈ Gr (cid:69) ( h ) and f : K → G is a Lipschitzfunction, then for S h (cid:120) f ( K ) -almost every x ∈ G we have that there exists W ( x ) ∈ Gr ( h ) such that the followingconvergence of measures holds r − h T x , r S h (cid:120) f ( K ) (cid:42) S h (cid:120) W ( x ) , as r goes to 0. ( )Let us finally sketch the proof of ( ). Since W ∈ Gr (cid:69) ( h ) , i.e., it admits a normal complementary subgroup, weget that W is a Carnot subgroup of G , see [ , Remark . ]. Thus we can apply Pansu-Rademacher theorem to f : K ⊆ W → G , see [ , Theorem . . ], to obtain that f is Pansu-differentiable S h -almost everywhere, with Pansudifferential d f , and the area formula holds, see [ , Corollary . . ]. The proof of ( ) with W ( x ) : = D f ( x )( W ) for S h (cid:120) f ( K ) -almost every x is now just a routine task, building on [ , Proposition . . and Proposition . . ],and by using the area formula in [ , Corollary . . ]. We do not give all the details as the proof follows verbatimas in the argument contained in [ , pages - ], with the obvious substitutions taking into account that theauthors in [ ] only deal with Heisenberg groups H n in the case W is horizontal. In this section we provide the proof of Proposition . and Theorem . . The key step for proving the rectifiabilitywith intrinsically differentiable graphs is the following proposition. Proposition . (Hausdorff convergence to tangents) . Let φ be a P h -rectifiable measure. Let K be a compact set suchthat φ ( K ) > . Then for φ -almost every point x ∈ K there exists V ( x ) ∈ Gr ( h ) such that δ r ( x − · K ) → V ( x ) , as r goes to in the sense of Hausdorff convergence on closed balls { B ( k ) } k > . omparison with other notions of rectifiability 64 First of all, by reducing the measure φ to have compact support, e.g., considering the restriction on the ballswith integer radii, and then by using Proposition . , we can assume without loss of generality that K ⊆ E ( ϑ , γ ) for some ϑ , γ ∈ N . In order to prove the Hausdorff convergence to the plane V ( x ) we need to prove two differentthings: first, around almost every point x of K , the points of the set K at decreasingly small scales lies ever closerto the points of x V ( x ) , and this is exactly what comes from the implication ( ), see Proposition . . Secondly, wehave to prove the converse assertion with respect to the previous one, i.e., that the points of x V ( x ) around x atdecreasingly small scales are ever closer to the points of K . For this latter assumption to hold we also need to addto the condition in ( ) the additional control F x , r ( φ (cid:120) K , Θ S h (cid:120) x V ) ≤ δ r h + , see Proposition . . As a consequenceof Proposition . , we can prove Theorem . for measures of the form S h (cid:120) Γ . Finally by the usual reduction to E ( ϑ , γ ) , we can give the proof of Theorem . for arbitrary measures. . C ( G , G (cid:48) ) -rectifiability This subsection is devoted to the proof of Proposition . , i.e., the fact that the spherical Hausdorff measurerestricted to a ( G , G (cid:48) ) -rectifiable set is P -rectifiable. In [ ] the authors give the following definitions of C -submanifold of a Carnot group and rectifiable sets. We first recall the definition of C -function. Definition . ( C -function) . Let G and G (cid:48) be two Carnot groups endowed with left-invariant homogeneousdistances d and d (cid:48) , respectively. Let Ω ⊆ G be open and let f : Ω → G (cid:48) be a function. We say that f is Pansudifferentiable at x ∈ Ω if there exists a homogeneous homomorphism d f x : G → G (cid:48) such thatlim y → x d (cid:48) ( f ( x ) − · f ( y ) , d f x ( x − · y )) d ( x , y ) = f is of class C in Ω if the map x (cid:55)→ d f x is continuous from Ω to the space of homogeneoushomomorphisms from G to G (cid:48) . Definition . ( C -submanifold) . Given an arbitrary Carnot group G , we say that Σ ⊆ G is a C -submanifold of G if there exists a Carnot group G (cid:48) such that for every p ∈ Σ there exists an open neighborhood Ω of p and afunction f ∈ C ( Ω ; G (cid:48) ) such that Σ ∩ Ω = { g ∈ Ω : f ( g ) = } , ( )and d f p : G → G (cid:48) is surjective with K er ( d f p ) complemented. In this case we say that Σ is a C ( G , G (cid:48) ) -submanifold . Definition . (( G , G (cid:48) ) -rectifiable set) . Given two arbitrary Carnot groups G and G (cid:48) of homogeneous dimension Q and Q (cid:48) , respectively, we say that Σ ⊆ G is a ( G , G (cid:48) ) -rectifiable set if there exist countably many subsets Σ i of G that are C ( G , G (cid:48) ) -submanifolds, such that H Q − Q (cid:48) (cid:32) Σ \ + ∞ (cid:91) i = Σ i (cid:33) = ], we prove the following. Proposition . . Let us fix G and G (cid:48) two arbitrary Carnot groups of homogeneous dimensions Q and Q (cid:48) respectively andsuppose Σ ⊆ G is a ( G , G (cid:48) ) -rectifiable set. Then the measure S Q − Q (cid:48) (cid:120) Σ is P cQ − Q (cid:48) -rectifiable.Proof. By [ , Corollary . ] a ( G , G (cid:48) ) -rectifiable set Σ has S Q − Q (cid:48) (cid:120) Σ -almost everyhwere positive and finite density.Thus, by the locality of tangents, see Proposition . , by Lebesgue differentiation theorem in Proposition . , andby the very definitions of ( G , G (cid:48) ) -rectifiable set and C ( G , G (cid:48) ) -submanifold, it suffices to prove the statement when Σ is the zero-level set of a function f ∈ C ( Ω , G (cid:48) ) , with Ω ⊆ G open, and such that for every p ∈ { g ∈ Ω : f ( g ) = } = : Σ the differential d f p : G → G (cid:48) is surjective with Ker ( d f p ) complemented. The distance between two homogeneous homomorphisms is the supremum norm of the two maps restricted to the unit sphere ∂ B (
0, 1 ) . omparison with other notions of rectifiability 65 Fix p ∈ Σ and note that the homogeneous subgroup Ker ( d f p ) , where f is a representation as in ( ), isindependent of the choice of f . This follows for instance from [ , Lemma . , (iii)]. We denote this homogeneoussubgroup with W ( p ) and we call it the tangent subgroup at p to Σ . We first prove thatTan Q − Q (cid:48) (cid:0) S Q − Q (cid:48) (cid:120) Σ , p (cid:1) ⊆ { λ S Q − Q (cid:48) (cid:120) W ( p ) : λ > } , for every p ∈ Σ . ( )Indeed, from [ , Lemma . ], denoting by Σ p , r the set δ r ( p − · Σ ) , we have S Q − Q (cid:48) (cid:120) Σ p , r (cid:42) S Q − Q (cid:48) (cid:120) W ( p ) , for every p ∈ Σ and for r →
0. ( )We claim that this last equality implies that r − ( Q − Q (cid:48) ) i T p , r i (cid:0) S Q − Q (cid:48) (cid:120) Σ (cid:1) (cid:42) S Q − Q (cid:48) (cid:120) W ( p ) , for every infinitesimal sequence r i → ). Indeed, for every measurable set A ⊆ G , we have T p , r i (cid:0) S Q − Q (cid:48) (cid:120) Σ (cid:1) ( A ) = S Q − Q (cid:48) (cid:120) Σ ( p · δ r i ( A )) = S Q − Q (cid:48) (cid:120) ( p − · Σ )( δ r i ( A )) = r Q − Q (cid:48) i S Q − Q (cid:48) (cid:120) Σ p , r i ( A ) ,and thus the claim follows from ( ). In order to conclude the proof, we have to prove that item (i) of Defini-tion . holds. This follows from [ , Corollary . ]. Indeed it is there proved that every ( G , G (cid:48) ) -rectifiable set hasdensity S Q − Q (cid:48) -almost everywhere, that is stronger than item (i) of Definition . . Remark . ( P -rectifiability and ( G , G (cid:48) ) -rectifiable sets) . We remark that the proof above is heavily based on [ ,Lemma . & Corollary . ]. The two latter results in the reference are consequences of the area formula [ ,Theorem . ]. As a consequence the approach in [ ] is, in some sense, reversed with respect to our approach.The authors in [ ] deal with the category of C ( G , G (cid:48) ) -regular submanifolds and prove the area formula relyingupon [ , Proposition . ], that ultimately tells that a Borel regular measure µ with positive and finite Federer’sdensity θ with respect to the spherical Hausdorff measure S h admits a representation µ = θ S h . Then with thisarea formula they are able to prove the results that led to the proof of the above Proposition . . Remark . . From Definition . and Definition . it follows that the tangent subgroup W at a point of a ( G , G (cid:48) ) -rectifiable set is always normal and complemented. Moreover, from [ , Lemma . , (iv)], every complementarysubgroup to W must be a Carnot subgroup of G that in addition is isomorphic to G (cid:48) . This results in a lack ofgenerality of this approach to rectifiability. Let us give here an example where this becomes clear. If we take L anhorizontal subgroup in the first Heisenberg group H , on the one hand S (cid:120) L is P -rectifiable, on the other hand L is not ( H , G (cid:48) ) -rectifiable for any Carnot group G (cid:48) since L is not normal. . Rectifiability with intrinsically differentiable graphs
This subsection is devoted to the proof of Proposition . and Theorem . . Throughout this subsection we let G to be a Carnot group of homogeneous dimension Q and h an arbitrary natural number with 1 ≤ h ≤ Q . Whenever φ is a Radon measure supported on a compact set we freely use the notation E ( ϑ , γ ) introduced in Definition . ,for ϑ , γ ∈ N . We start with some useful definitions and facts. Definition . . For 1 ≤ h ≤ Q and ϑ ∈ N , let us set η ( h ) : = ( h + ) ,and then let us define the constant C = C ( h , ϑ ) : = (cid:32) η ( − η ) h ϑ (cid:33) h + . Proposition . . Let φ be a Radon measure supported on a compact subset of G and let K be a Borel subset of supp φ . Let ϑ , γ and ≤ h ≤ Q be natural numbers. Let x ∈ E ( ϑ , γ ) , < r < γ , and < δ < C . Assume further that there exist Θ > and V ∈ Gr ( h ) such that F x , r ( φ (cid:120) K , Θ C h (cid:120) x V ) + F x , r ( φ , Θ C h (cid:120) x V ) ≤ δ r h + . ( ) Then for any w ∈ B ( x , r /2 ) ∩ x V we have φ ( K ∩ B ( w , δ h + r )) > , and thus in particular K ∩ B ( w , δ h + r ) (cid:54) = ∅ . omparison with other notions of rectifiability 66 Proof.
From the hypothesis we have that F x , r ( φ , Θ C h (cid:120) x V ) ≤ δ r h + . Define g ( x ) : = min { dist ( x , U (
0, 1 ) c ) , η } ,where η is defined in Definition . . From the very definition of the function g and the choice of Θ above wededuce that ϑ − ( − η ) h η r h + − Θ η r h + ≤ η r φ (cid:0) B ( x , ( − η ) r ) (cid:1) − η r Θ C h (cid:120) x V ( B ( x , r )) ≤ ˆ rg ( δ r ( x − z )) d φ ( z ) − Θ ˆ rg ( δ r ( x − z )) d C h (cid:120) x V ( z ) ≤ δ r h + ,where in the first inequality we are using that x ∈ E ( ϑ , γ ) and item (iii) of Proposition . , and in the last inequalitywe are using that rg ( δ r ( x − · )) ∈ Lip + ( B ( x , r )) . Simplifying and rearranging the above chain of inequalities, weinfer that Θ ≥ ϑ − ( − η ) h − δ / η ≥ ( A ) ( ϑ ) − ( − η ) h = ( B ) ( ϑ ) − ( − ( h + )) h ,where (A) comes from the fact that δ < C < (( − η ) h η ) / ( ϑ ) , see Definition . , and (B) comes from thedefinition of η , see Definition . . Since the function h (cid:55)→ ( − ( h + )) h is decreasing and bounded below by e − , we deduce, from the previous inequality, that Θ ≥ ( ϑ e ) .We now claim that for every λ with δ ( h + ) ≤ λ < w ∈ x V ∩ B ( x , r /2 ) we have φ (cid:0) B ( w , λ r ) ∩ K (cid:1) >
0. This will finish the proof. By contradiction assume there is w ∈ x V ∩ B ( x , r /2 ) such that φ (cid:0) B ( w , λ r ) ∩ K (cid:1) = Θ η ( − η ) h λ h + r h + = Θ ηλ r C h (cid:120) x V (cid:0) B ( w , ( − η ) λ r ) (cid:1) ≤ Θ ˆ λ rg ( δ ( λ r ) ( w − z )) d C h (cid:120) x V ( z )= Θ ˆ λ rg ( δ ( λ r ) ( w − z )) d C h (cid:120) x V ( z ) − ˆ λ rg ( δ ( λ r ) ( w − z )) d φ (cid:120) K ( z ) ≤ δ r h + , ( )where the first equality comes from item (iii) of Proposition . , and the last inequality comes from the choice of Θ as in the statement, and the fact that λ rg ( δ ( λ r ) ( w − · )) ∈ Lip + ( B ( w , λ r )) ⊆ Lip + ( B ( x , r )) because λ < w ∈ B ( x , r /2 ) . Thanks to ( ), the choice of λ , and the fact, proved some line above, that 1/ ( e ϑ ) < Θ , we havethat δ h + h + e ϑ η ( − η ) h < Θ λ h + η ( − η ) h ≤ δ , and then δ ( h + ) ≥ η ( − η ) h e ϑ ,which is a contradiction since δ < C = (( η ( − η ) h ) / ( ϑ )) h + , see Definition . . Proof of Proposition . . First of all, by reducing the measure φ to have compact support, e.g., considering therestriction on the balls with integer radii, and then by using Proposition . , we can assume without loss ofgenerality that K ⊆ E ( ϑ , γ ) for some ϑ , γ ∈ N Since φ is a P h -rectifiable measure, by using the locality of tangents with the density ρ ≡ χ K , see Proposition . ,for φ -almost every x ∈ K we have that the following three conditions hold(i) Tan h ( φ , x ) ⊆ { λ S h (cid:120) V ( x ) : λ > } , where V ( x ) ∈ Gr ( h ) ,(ii) 0 < Θ h ∗ ( φ , x ) ≤ Θ h , ∗ ( φ , x ) < + ∞ .(iii) if r i → Θ > r − hi T x , r i φ → Θ C h (cid:120) V ( x ) , then r − hi T x , r i ( φ (cid:120) K ) → Θ C h (cid:120) V ( x ) .From now on let us fix a point x ∈ K for which the three conditions above hold. If we are able to prove theconvergence in the statement for such a point then the proof of the proposition is concluded.Thus, we have to show that for every k > r → d H , G ( δ r ( x − · K ) ∩ B ( k ) , V ( x ) ∩ B ( k )) =
0, ( )where d H , G is the Hausdorff distance between closed subsets in G . For some compatibility with the statementsthat we already proved, we are going to prove ( ) for k = ) for an arbitrary k > omparison with other notions of rectifiability 67 be achieved by changing accordingly the constants in the statements of Proposition . and Proposition . , thatwe are going to crucially use in this proof. We leave this generalization to the reader, as it will be clear from thisproof.Let us fix ε < min { δ G , C } , where δ G is defined in Definition . and C in Definition . , and let us show thatthere exist an r = r ( ε ) and a real function f such that d H , G (cid:16) δ r ( x − · K ) ∩ B (
0, 1/4 ) , V ( x ) ∩ B (
0, 1/4 ) (cid:17) ≤ f ( ε ) , for all 0 < r < r ( ε ) , ( )where f ( ε ) : = max { C ε ( h + ) + f ( ε ) , 3 ε ( h + ) + f ( ε ) } , ( )and where the constant C is defined in Proposition . , and the functions f , f are introduced in ( ) and ( ),respectively. By the definition of f , f , f it follows that f ( ε ) → ε → ), we aredone.In order to reach the proof of ( ) let us add an intermediate step. We claim that there exists an r : = r ( ε ) < γ such that the following holdsfor every 0 < r < r there exists a Θ : = Θ ( r ) for which F x , r ( φ (cid:120) K , Θ C h (cid:120) x V ) + F x , r ( φ , Θ C h (cid:120) x V ) ≤ ε r h + . ( )The conclusion in ( ) follows if we prove thatlim r → inf Θ > F x , r ( φ (cid:120) K , Θ C h (cid:120) x V ) + F x , r ( φ , Θ C h (cid:120) x V ) r h + →
0. ( )We prove ( ) by contradiction. If ( ) was not true, there would exist an (cid:101) ε and an infinitesimal sequence { r i } i ∈ N such that inf Θ > (cid:16) F x , r i ( φ (cid:120) K , Θ C h (cid:120) x V ) + F x , r i ( φ , Θ C h (cid:120) x V ) (cid:17) > (cid:101) ε r h + i , for every i ∈ N . ( )Thus, from items (i) and (ii) above, and from [ , Corollary . ], we conclude that, up to a non re-labelled subse-quence of r i , there exists a Θ ∗ > r − hi T x , r i φ → Θ ∗ C h (cid:120) V ( x ) as r i →
0. Then by exploiting theitem (iii) above we get also that r − hi T x , r i ( φ (cid:120) K ) → Θ ∗ C h (cid:120) V ( x ) as r i →
0. These two conclusions immediately imply,by exploiting Remark . and ( . ), thatlim i → + ∞ r − ( h + ) i (cid:16) F x , r i ( φ (cid:120) K , Θ ∗ C h (cid:120) x V ) + F x , r i ( φ , Θ C h (cid:120) x V ) (cid:17) → ). Thus, the conclusion in ( ) holds. Let us continue the proof of ( ).Taking into account the bound on ε and ( ) we can apply Proposition . , since V ( x ) ∈ Π ε ( x , r ) for all0 < r < r , and Proposition . to obtain, respectively, thatsup p ∈ K ∩ B ( x , r /4 ) dist ( p , x V ( x )) ≤ sup p ∈ E ( ϑ , γ ) ∩ B ( x , r /4 ) dist ( p , x V ( x )) ≤ C r ε ( h + ) , for all 0 < r < r ,for every p ∈ B ( x , r /2 ) ∩ x V ( x ) we have B ( p , ε ( h + ) r ) ∩ K (cid:54) = ∅ , holds for all 0 < r < r . ( )Let us proceed with the proof of ( ). Fix 0 < r < r and note that for any w ∈ δ r ( x − · K ) ∩ B (
0, 1/4 ) there existsa point p ∈ K ∩ B ( x , r /4 ) such that w = : δ r ( x − · p ) . From the first line of ( ) we get that dist ( x − · p , V ( x )) ≤ C r ε ( h + ) and thus there exists a v ∈ V ( x ) such that d ( x − · p , v ) ≤ C r ε ( h + ) . This in particular means that d ( w , δ r v ) ≤ C ε ( h + ) and then, since w ∈ B (
0, 1/4 ) , we get also that δ r v ∈ V ( x ) ∩ B (
0, 1/4 + C ε ( h + ) ) .Thus, we conclude thatdist ( w , V ( x ) ∩ B (
0, 1/4 + C ε ( h + ) )) ≤ C ε ( h + ) , for all w ∈ δ r ( x − · K ) ∩ B (
0, 1/4 ) . ( )Define the following function f ( ε ) : = sup u ∈ V ( x ) ∩ (cid:0) B ( + C ε ( h + ) ) \ U ( ) (cid:1) d ( u , δ − (cid:107) u (cid:107) − u ) , ( ) omparison with other notions of rectifiability 68 and notice that by compactness it is easy to see that f ( ε ) → ε →
0. With the previous definition of f inhands, we can exploit ( ) and conclude thatsup w ∈ δ r ( x − · K ) ∩ B ( ) dist ( w , V ( x ) ∩ B (
0, 1/4 )) ≤ C ε ( h + ) + f ( ε ) . ( )The latter estimate is the first piece of information we need to prove ( ). Let us now estimate dist ( δ r ( x − · K ) ∩ B (
0, 1/4 ) , v ) for any v ∈ V ( x ) ∩ B (
0, 1/4 ) . If u ∈ V ( x ) ∩ (cid:0) B (
0, 1/4 ) \ U (
0, 1/4 − ε ( h + ) ) (cid:1) , then there existsa unique µ = µ ( u ) > δ µ ( u ) u ∈ V ( x ) ∩ ∂ B (
0, 1/4 − ε ( h + ) ) . Let us define f ( ε ) : = sup u ∈ V ( x ) ∩ (cid:0) B ( ) \ U ( − ε ( h + ) ) (cid:1) d ( u , δ µ ( u ) u ) , ( )and by compactness it is easy to see that f ( ε ) → ε →
0. Let us now fix v ∈ V ( x ) ∩ B (
0, 1/4 ) . Then x · δ r v ∈ B ( x , r /4 ) ∩ x V ( x ) ⊆ B ( x , r /2 ) ∩ x V ( x ) . We can use the second line of ( ) to conclude that there exists w ∈ B ( x · δ r v , ε ( h + ) r ) ∩ K . Thus (cid:101) w : = δ r ( x − · w ) ∈ B ( v , ε ( h + ) ) ∩ δ r ( x − · K ) . Now we have two cases • if v was in B (
0, 1/4 − ε ( h + ) ) we would get (cid:101) w ∈ B (
0, 1/4 ) and thendist ( δ r ( x · K ) ∩ B (
0, 1/4 ) , v ) ≤ ε ( h + ) ; ( ) • if instead v ∈ V ( x ) ∩ (cid:0) B (
0, 1/4 ) \ U (
0, 1/4 − ε ( h + ) ) (cid:1) , we denote v (cid:48) : = δ µ ( v ) v the point that we havedefined above and then we still have x · δ r v (cid:48) ∈ B ( x , r /2 ) ∩ x V ( x ) . Thus we can again apply the second lineof ( ) to deduce the existence of w (cid:48) ∈ B ( x · δ r v (cid:48) , ε ( h + ) r ) ∩ K . Then we conclude (cid:101) w (cid:48) : = δ r ( x − · w (cid:48) ) ∈ B ( v (cid:48) , ε ( h + ) ) ∩ δ r ( x − · K ) . Now we can estimate d ( (cid:101) w , (cid:101) w (cid:48) ) = r d ( w , w (cid:48) ) ≤ r (cid:0) d ( w , x · δ r v ) + d ( x · δ r v , x · δ r v (cid:48) ) + d ( x · δ r v (cid:48) , w (cid:48) ) (cid:1) ≤ ε ( h + ) + f ( ε ) . ( )Moreover, since v (cid:48) ∈ ∂ B (
0, 1/4 − ε ( h + ) ) and (cid:101) w (cid:48) ∈ B ( v (cid:48) , ε ( h + ) ) we get that (cid:101) w (cid:48) ∈ B (
0, 1/4 ) ∩ δ r ( x − · K ) .Then by the triangle inequality and ( ) we conclude that, in this second case, d ( (cid:101) w (cid:48) , v ) ≤ ε ( h + ) + f ( ε ) , and then dist ( δ r ( x · K ) ∩ B (
0, 1/4 ) , v ) ≤ ε ( h + ) + f ( ε ) . ( )By joining together the conclusion of the two cases, see ( ) and ( ), we conclude thatsup v ∈ V ( x ) ∩ B ( ) dist ( δ r ( x · K ) ∩ B (
0, 1/4 ) , v ) ≤ ε ( h + ) + f ( ε ) . ( )The equations ( ) and ( ) imply ( ) by the very definition of Hausdorff distance. Thus the proof is concluded.Let us now give the definition of intrinsically differentiable graph. Definition . (Intrinsically differentiable graph) . Let V and L be two complementary subgroups of a Carnotgroup G . Let ϕ : K ⊆ V → L be a continuous function with K compact in V . Let a ∈ K . We say that graph ( ϕ ) isan intrinsically differentiable graph at a · ϕ ( a ) if there exists a homogeneous subgroup V ( a ) such thatfor all k >
0, lim λ → ∞ d H , G (cid:16) δ λ (( a · ϕ ( a )) − · graph ( ϕ )) ∩ B ( k ) , V ( a ) ∩ B ( k ) (cid:17) =
0, ( )where d H , G is the Hausdorff distance between closed subsets of G .We prove now that the support of a P ch -rectifiable measure S h (cid:120) Γ , where Γ is compact, can be written as thecountable union of almost everywhere intrinsically differentiable graphs. omparison with other notions of rectifiability 69 Theorem . . For any ≤ h ≤ Q, there exist a countable subfamily F : = { V k } + ∞ k = of Gr c ( h ) , and L k complementarysubgroups of V k such that the following holds.Let Γ be a compact subset of G such that < S h ( Γ ) < + ∞ , and S h (cid:120) Γ is a P ch -rectifiable measure. Then there arecountably many compact Γ i ’s that are intrinsic graphs of functions ϕ i : P V i ( Γ i ) → L i , and that satisfy the following twoconditions: Γ i are intrinsically differentiable graphs at a · ϕ i ( a ) for S h (cid:120) P V i ( Γ i ) -almost every a ∈ P V i ( Γ i ) , and S h ( Γ \ ∪ + ∞ i = Γ i ) = Proof.
First of all let F ( V , L ) : = ε ( V , L ) , for all ( V , L ) ∈ Sub ( h ) ,where Sub ( h ) is defined in ( ). Given the above defined function F , we construct the family F : = { V k } + ∞ k = and choose L k complementary subgroups of V k as in the statement of Theorem . . Notice that this choice isdependent on the function F that we chose above. We claim that the family for which the statement holds is F .Applying Theorem . with β ≡ S h (cid:120) Γ we get countably many compact sets Γ i ⊆ Γ that are C V i ( F ( V i , L i )) -sets and such that S h ( Γ \ ∪ + ∞ i = Γ i ) = F ( V i , L i ) = ε ( V i , L i ) , we conclude that each Γ i is also the intrinsic graph of a function ϕ i : P V i ( Γ i ) → L i ,see Proposition . . It is left to show that, for every i ∈ N , graph ( ϕ i ) is an intrinsically differentiable graph at a · ϕ i ( a ) for S h (cid:120) P V ( Γ i ) -almost every a ∈ P V i ( Γ i ) .Indeed, since S h (cid:120) Γ is P ch -rectifiable, we can apply Proposition . and, for every i ∈ N , we conclude that δ r ( x − · Γ i ) → V ( x ) , as r goes to 0, for S h (cid:120) Γ i -almost every x ∈ G , where V ( x ) ∈ Gr ( h ) , ( )in the sense of Hausdorff convergence on closed balls { B ( k ) } k > . Moreover, thanks to Proposition . and toLebesgue differentiation theorem in Proposition . , we infer that ( Φ i ) ∗ S h (cid:120) V i is mutually absolutely continuouswith respect to S h (cid:120) Γ i , where Φ i is the graph map of ϕ i . Furthermore, since every point x ∈ Γ i can be writtenas x = a · ϕ i ( a ) , with a ∈ P V i ( Γ i ) , we conclude, from ( ) and latter absolute continuity, that Γ i = graph ( ϕ i ) isan intrinsically differentiable graph at a · ϕ i ( a ) for S h (cid:120) P V ( Γ i ) -almost every a ∈ P V i ( Γ i ) , and this concludes theproof.In the following corollary we provide the proof of Theorem . . Corollary . . For any ≤ h ≤ Q, there exist a countable subfamily F : = { V k } + ∞ k = of Gr c ( h ) , and L k complementarysubgroups of V k such that the following holds.For any P ch -rectifiable measure φ there exist countably many compact sets Γ i ’s that are intrinsic graphs of functions ϕ i : P V i ( Γ i ) → L i , and that satisfy the following conditions: Γ i are intrinsically differentiable graphs at a · ϕ i ( a ) for S h (cid:120) P V i ( Γ i ) -almost every a ∈ P V i ( Γ i ) , and φ ( G \ ∪ + ∞ i = Γ i ) = Proof.
By restricting on closed balls of integer radii we can assume without loss of generality that φ has compactsupport. Let us fix ϑ , γ ∈ N . We can infer this corollary by working on φ (cid:120) E ( ϑ , γ ) , that is mutually absolutelycontinuous with respect to S h (cid:120) E ( ϑ , γ ) , see Proposition . , and by using the previous Theorem . together withProposition . . The resulting strategy is identical to the one in Corollary . so we omit the details. Remark . (Rectifiability with uniformly intrinsically differentiable graphs and C ( G , G (cid:48) ) -surfaces) . By the recentwork of the second named author, see [ , Theorem ], one can show that in an arbitrary Carnot group of homo-geneous dimension Q , the support of a P ∗ Q − -rectifiable measure can be covered by countably many C -regularhypersurfaces. Moreover, it is known that a C -regular hypersurface is characterized, locally, by being the graphof a uniformly intrinsically differentiable function, see [ , Theorem . ]. This means that, in some particular cases,as it is the codimension-one case, we can strenghten the conclusion in Corollary . by obtaining that the maps areuniformly intrinsically differentiable, even if we are asking that the measure is P ∗ Q − -rectifiable, that is a weakercondition than being P Q − -rectifiable. eferences This latter observation gives raise to two questions, that in the co-horizontal case are the same thanks to [ ,Theorem . ], but in general could be different: is it always possible to improve the intrinsic differentiability inCorollary . to some kind of uniform intrinsic differentiability? Is it possible to prove that when a P h -rectifiablemeasure, or even a P ∗ h -rectifiable measure, on G admits only complemented normal subgroups that have onlycomplementary subgroups that are Carnot subgroups , then we can write its support as the countable union of C ( G , G (cid:48) ) -surfaces, see Definition . ? Let us stress that if one answers positively to the second question, thiswould mean, taking into account Proposition . , that whenever they can agree, see Remark . , the two notions of P -rectifiable measure and ( G , G (cid:48) ) -rectifiable set agree. However, we do not address these questions in this paper.In the final part of this section we briefly discuss how the notion of intrinsically differentiable graph in Def-inition . is related to the already available notion of intrinsic differentiability, see [ , Definition . . ] and [ ,Definition . ]. Throughout the rest of this section V and L are two fixed complementary subgroups in a Carnotgroup G .Definition . (Intrinsic translation of a function) . Given a function ϕ : U ⊆ V → L , we define, for every q ∈ G , U q : = { a ∈ V : P V ( q − · a ) ∈ U } ,and ϕ q : U q ⊆ V → L by setting ϕ q ( a ) : = (cid:0) P L ( q − · a ) (cid:1) − · ϕ (cid:0) P V ( q − · a ) (cid:1) . ( ) Definition . (Intrinsically linear function) . The map (cid:96) : V → L is said to be intrinsically linear if graph ( (cid:96) ) is ahomogeneous subgroup of G . Definition . (Intrinsically differentiable function) . Let ϕ : U ⊆ V → L be a function with U Borel in V . Fix adensity point a ∈ D ( U ) of U , let p : = ϕ ( a ) − · a − and denote with ϕ p : U p ⊆ V → L the shifted functionintroduced in Definition . . We say that ϕ is intrinsically differentiable at a if there is an intrinsically linear map d ϕ ϕ a : V → L such that lim b → e , b ∈ U p (cid:107) d ϕ ϕ a [ b ] − · ϕ p ( b ) (cid:107)(cid:107) b (cid:107) =
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