Nowhere differentiable intrinsic Lipschitz graphs
NNOWHERE DIFFERENTIABLE INTRINSIC LIPSCHITZGRAPHS
ANTOINE JULIA, SEBASTIANO NICOLUSSI GOLO, AND DAVIDE VITTONE
Abstract.
We construct intrinsic Lipschitz graphs in Carnot groups with theproperty that, at every point, there exist infinitely many different blow-up lim-its, none of which is a homogeneous subgroup. This provides counterexamplesto a Rademacher theorem for intrinsic Lipschitz graphs.
The notion of Lipschitz submanifolds in sub-Riemannian geometry was intro-duced, at least in the setting of Carnot groups, by B. Franchi, R. Serapioni andF. Serra Cassano in a series of seminal papers [5, 6, 7] through the theory of intrin-sic Lipschitz graphs . One of the main open questions concerns the differentiabilityproperties for such graphs: in this paper we provide examples of intrinsic Lipschitzgraphs of codimension 2 (or higher) that are nowhere differentiable, i.e., that admitno homogeneous tangent subgroup at any point.Recall that a Carnot group G is a connected, simply connected and nilpotentLie group whose Lie algebra is stratified, i.e., it can be decomposed as the directsum ⊕ sj =1 V j of subspaces such that V j +1 = [ V , V j ] for every j = 1 , . . . , s − , [ V , V s ] = { } , V s (cid:54) = { } . We shall identify the group G with its Lie algebra via the exponential map exp : ⊕ sj =1 V j → G , which is a diffeomorphism. In this way, for λ > one can introducethe homogeneous dilations δ λ : G → G as the group automorphisms defined by δ λ ( p ) = λ j p for every p ∈ V j . A subgroup of G is said to be homogeneous if itis dilation-invariant. Assume that a splitting G = WV of G as the product ofhomogeneous and complementary (i.e., such that W ∩ V = { } ) subgroups is fixed;we say that a function φ : W → V intrinsic Lipschitz if there is an open nonemptycone U such that V \ { } ⊂ U and pU ∩ Γ φ = ∅ for all p ∈ Γ φ ,where Γ φ = { wφ ( w ) : w ∈ W } is the intrinsic graph of φ . We say that a set Σ ⊂ G is a blow-up of Γ φ at ˆ p = ˆ wφ ( ˆ w ) if there exists a sequence ( λ n ) n such that λ n → + ∞ and the limit lim n →∞ δ λ n (ˆ p − Γ φ ) = Σ holds with respect to the local Hausdorff convergence. It is worth recalling that, if φ is intrinsic Lipschitz, then every blow-up is automatically the intrinsic Lipschitzgraph of a map W → V . Eventually, we say that φ is intrinsically differentiable Date : January 11, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Sub-Riemannian Geometry, Carnot Groups, Intrinsic Lipschitzgraphs.A.J. has been supported by the Simons Foundation Wave Project. S.N.G. has been supportedby the Academy of Finland (grant 322898 “Sub-Riemannian Geometry via Metric-geometry andLie-group Theory”). D.V. has been supported by FFABR 2017 of MIUR (Italy) and by GNAMPAof INdAM (Italy). All three authors have been supported by the University of Padova STARSProject “Sub-Riemannian Geometry and Geometric Measure Theory Issues: Old and New”. a r X i v : . [ m a t h . M G ] J a n JULIA, NICOLUSSI GOLO, AND VITTONE at ˆ w ∈ W if the blow-up of Γ φ at ˆ p = ˆ wφ ( ˆ w ) is unique and it is a homogeneoussubgroup of G . See [8] for details.We say that a group G along with a splitting WV satisfies an intrinsic Ra-demacher Theorem if all intrinsic Lipschitz maps φ : W → V are intrinsicallydifferentiable almost everywhere (that is, for almost all points of W equipped withits Haar measure). It was proved in [6] that this is the case when V (cid:39) R and G is of step two; other partial results for graphs with codimension 1 ( V (cid:39) R ) arecontained in [4] and [9]. If V is a normal subgroup, the Rademacher Theorem hasbeen proved for general G by G. Antonelli and A. Merlo in [2]. Recently, the thirdnamed author [12] proved that Heisenberg groups (with any splitting) satisfy anintrinsic Rademacher Theorem. The question has been open for a long time if G isthe Engel group (which has step 3) and V (cid:39) R (see [1]). In this paper we prove aresult in the negative direction: namely, we provide examples of intrinsic Lipschitzgraphs that are nowhere intrinsically differentiable. Let us state our main result: Theorem 1.
Let G be a Carnot group with stratification (cid:76) sj =1 V j . Let WV be asplitting of G such that W ∩ V (cid:54)⊂ [ W , W ] and there exists v ∈ V ∩ V such that v (cid:54) = 0 and [ v , W ] = 0 . Then there is an intrinsic Lipschitz function φ : W → V that is nowhere intrinsically differentiable.Moreover, φ can be constructed in such a way that, for every p ∈ Γ φ , the followingproperties hold: (a) there exist infinitely many different blow-ups of Γ φ at p , (b) no blow-up of Γ φ at p is a homogeneous subgroup. The proof of Theorem 1 is postponed in order to first provide some comments.
Remark . The simplest example of a Carnot group where Theorem 1 applies is G = H × R , where H is the first Heisenberg group. As customary, we considergenerators X, Y, T of the Lie algebra of H such that [ X, Y ] = T, [ X, T ] = [
Y, T ] = 0 and fix the exponential coordinates ( x, y, t ) = exp( xX + yY + tT ) . Using coordinates ( x, y, t, r ) on H × R with r ∈ R , we can consider the splitting H × R = WV given bythe vertical subgroup W = { x = r = 0 } of H and the horizontal abelian subgroup V = { y = t = 0 } . Then V ∩ W (cid:54)⊂ [ W , W ] = { } and v = (0 , , , commutes with W . Hence, this splitting of H × R satisfies the conditions of Theorem 1 and it doesnot satisfy an intrinsic Rademacher Theorem.It is worth observing that, in this setting, the map φ : W → V provided inthe proof of Theorem 1 takes the form φ ( y, t ) = (0 , u ( t )) , where u is the -Höldercontinuous function constructed in the Appendix. In particular, the intrinsic graph Γ φ is the set { (0 , y, t, u ( t )) : y, t ∈ R } and it is contained in the Abelian subgroup W × R . One of the properties of u is that the limit lim s → t | u ( t ) − u ( s ) | (cid:112) | t − s | does not exists at any t ∈ R and this is the ultimate reason for the non-differentia-bility of φ .Similar counterexamples can be constructed in any codimension k ≥ : in factone can consider H k − × R = ( R k − x × R k − y × R t ) × R r with splitting WV definedby W = { x = 0 , r = 0 } , V = { y = 0 , t = 0 } . It can be easily checked that the map φ ( y, t ) = (0 , u ( t )) defines an intrinsic Lipschitz graph of codimension k for whichthe properties (a) and (b) in Theorem 1 hold at every point. Remark . The measure µ = H d Γ φ , where d is the Hausdorff dimension of W and H d is the d -dimensional Hausdorff measure, does not have a unique tangent OWHERE DIFFERENTIABLE INTRINSIC LIPSCHITZ GRAPHS 3 measure at any point. Indeed, firstly, any tangent measure of µ is supported on ablow-up of Γ φ . Secondly, by [7, Theorem 3.9], µ and all its dilations are uniformly d -Ahlfors regular, and thus any tangent measure of µ is d -Ahlfors regular. We thenconclude that if µ and µ are two tangent measures of µ supported on differentblow-ups of Γ φ , then they are two distinct measures. Since blow-ups of Γ φ arenot unique, so are tangent measures. Observe also that no tangent measure canbe flat, i.e., supported on a homogeneous subgroup. In particular, Γ φ is purely C H -unrectifiable, i.e., H d (Γ φ ∩ Σ) = 0 for every submanifold Σ of class C H (seee.g. [3, § 2.5 and 6.1]). Remark . If W is a homogeneous subgroup of G with codimension 1, then theconditions of Theorem 1 cannot be met because (cid:76) sj =2 V j = [ W , W ] + [ W , V ] . Actu-ally, intrinsic Lipschitz graphs of codimension 1 are boundaries of sets with finiteperimeter in G (see e.g. [11, Theorem 1.2]), hence at almost every point they possessat least one blow-up which is a homogeneous subgroup of codimension 1, see [1].Therefore, any possible counterexample to the Rademacher Theorem in codimen-sion 1 cannot be as striking as the one provided by Theorem 1, in the sense thatproperty (b) cannot hold on a set with positive measure. Remark . Following the same proof strategy, one can extend Theorem 1 to thecase W ∩ V j (cid:54)⊂ [ W , W ] for some j > and v ∈ V k ∩ V \ { } with k < j and [ v , W ] = 0 , by taking a k/j -Hölder analogue of the function u constructed in theappendix. Proof of Theorem 1.
Let β : W → R be a nonzero linear function such that W ∩ V j ⊂ ker β whenever j (cid:54) = 2 and [ W , W ] ⊂ ker β ; such a β exists because W ∩ V (cid:54)⊂ [ W , W ] .Notice that such a function β is in fact a group morphism W → R .Consider a 1/2-Hölder continuous function u : R → R with the following prop-erties. First, the difference quotients ∆( s, t ) = u ( s ) − u ( t ) sgn ( s − t ) | s − t | / are bounded, namely,(1) | ∆( s, t ) | ≤ for every s, t ∈ R .Second, there exist c > and c > such that, for every t ∈ R and δ ∈ (0 , ,there exist s , s ∈ R such that(2) sgn ( s − t ) = sgn ( s − t ) c δ ≤ | s − t | ≤ δc δ ≤ | s − t | ≤ δ | ∆( s , t ) − ∆( s , t ) | ≥ c . Such a function exists, as we show in the appendix.We can then define φ : W → V as φ ( w ) = u ( β ( w )) v . Notice that the condition [ v , W ] = 0 implies that(3) vw = wv for all w ∈ W and v ∈ R v .Therefore, the intrinsic graph of φ is the set of points wφ ( w ) = w + u ( β ( w )) v for w ∈ W . For instance, one can consider β ( x ) = (cid:104) x, w (cid:105) for some w ∈ ( W ∩ V ) \ [ W , W ] and a scalarproduct on W adapted to the grading (cid:76) sj =1 W ∩ V j of W . JULIA, NICOLUSSI GOLO, AND VITTONE
Claim 1:
The map φ is intrinsic Lipschitz.Fix a homogeneous norm (cid:107) · (cid:107) on G . Notice that, since β ( δ λ x ) = λ β ( x ) for all x ∈ W , there is a constant C such that | β ( x ) | ≤ C (cid:107) x (cid:107) , for all x ∈ W . We checkthat Γ φ has the cone property for the cone U = { wv : w ∈ W , v ∈ V , (cid:107) v (cid:107) > √ C (cid:107) v (cid:107)(cid:107) w (cid:107)} . Given ˆ w, w ∈ W , by (3) we have ( ˆ wφ ( ˆ w )) − ( wφ ( w )) = ( ˆ w − w )( φ ( ˆ w ) − φ ( w )) and (cid:107) φ ( ˆ w ) − φ ( w ) (cid:107) = | u ( β ( w )) − u ( β ( ˆ w )) |(cid:107) v (cid:107) ≤ | β ( w ) − β ( ˆ w ) | / (cid:107) v (cid:107) = | β ( ˆ w − w ) | / (cid:107) v (cid:107) ≤ √ C (cid:107) ˆ w − w (cid:107)(cid:107) v (cid:107) . Thus, ( ˆ wφ ( ˆ w )) − Γ φ ∩ U = ∅ for all ˆ w ∈ W , i.e., Γ φ is an intrinsic Lipschitz graph. Claim 2: for p ∈ Γ φ , none of the blow-ups of Γ φ at p is a homogeneous subgroup.We first observe that, if V ⊂ V ∩ V is the horizontal subgroup generated by v and L : W → V parametrizes a homogeneous subgroup Γ L of G , then L | W ∩ V = 0 .Indeed, the homogeneity of Γ L implies that for every h > and w ∈ W ∩ V onehas L (2 w ) = √ L ( w ) , while the fact that Γ L is a subgroup (plus the fact that V and W commute) gives L (2 w ) = 2 L ( w ) . This proves that L = 0 on W ∩ V .We now prove the claim. Assume by contradiction that there exist ˆ p = ˆ wφ ( ˆ w ) ∈ Γ φ , a map L : W → V such that the intrinsic graph Γ L of L is a homogeneoussubgroup and a sequence ( λ n ) n with λ n → + ∞ , and lim n →∞ δ λ n (ˆ p − Γ φ ) = Γ L . Observe that for every w ∈ W and every nδ λ n (( ˆ wφ ( ˆ w )) − ( wφ ( w ))) = δ λ n ( ˆ w − wφ ( ˆ w ) − φ ( w ))= δ λ n ( ˆ w − w ) (cid:18) u ( β ( w )) − u ( β ( ˆ w ))1 /λ n v (cid:19) . If we set w = ˆ wδ /λ n w (cid:48) , then β ( w ) = β ( ˆ w ) + β ( w (cid:48) ) /λ n . Therefore, the set δ λ n (ˆ p − Γ φ ) is the intrinsic graph of the function from W to V given by φ ˆ p,λ n ( w (cid:48) ) = u ( β ( ˆ w ) + β ( w (cid:48) ) /λ n ) − u ( β ( ˆ w ))1 /λ n v . Since the maps φ ˆ p,λ n take values in V , L is also V -valued and, as we saw above,this implies that L | W ∩ V = 0 .Write ˆ t = β ( ˆ w ) and let w ∈ W ∩ V be such that β ( w ) = 1 ; then for every h ∈ R (4) φ ˆ p,λ n ( hw ) = ( sgn h ) | h | / ∆(ˆ t + h/λ n , ˆ t ) v . By (2) there exists a sequence ( h n ) n such that for every n | h n | ∈ [ c , and (cid:107) φ ˆ p,λ n ( h n w ) (cid:107) ≥ √ c c (cid:107) v (cid:107) / . Up to passing to a subsequence we can also assume that h n → ¯ h with | ¯ h | ∈ [ c , ;since (cid:107) φ ˆ p,λ n ( h n w ) − φ ˆ p,λ n (¯ hw ) (cid:107) = (cid:12)(cid:12)(cid:12)(cid:12) u (ˆ t + h n /λ n ) − u (ˆ t + ¯ h/λ n )1 /λ n (cid:12)(cid:12)(cid:12)(cid:12) (cid:107) v (cid:107)≤ | h n − ¯ h | / (cid:107) v (cid:107) we obtain (cid:107) L (¯ hw ) (cid:107) = lim n (cid:107) φ ˆ p,λ n (¯ hw ) (cid:107) = lim n (cid:107) φ ˆ p,λ n ( h n w ) (cid:107) ≥ √ c c (cid:107) v (cid:107) / . This contradicts the fact that L (¯ hw ) = 0 , and the claim is proved. OWHERE DIFFERENTIABLE INTRINSIC LIPSCHITZ GRAPHS 5
Claim 3: for p ∈ Γ φ , there exist infinitely many different blow-ups of Γ φ at p .Let ˆ p = ˆ wφ ( ˆ w ) ∈ Γ φ be fixed and let ˆ t = β ( ˆ w ) ; as before, fix also w ∈ W ∩ V suchthat β ( w ) = 1 . By (2) we can find infinitesimal sequences ( s n ) n , ( s n ) n such thatsgn ( s n ) = sgn ( s n ) for every n, ∆(ˆ t + s n , ˆ t ) ≥ ∆(ˆ t + s n , ˆ t ) + c . Up to passing to a subsequence, we can assume that there exists σ ∈ { , − } and ∆ , ∆ ∈ R such thatsgn ( s n ) = sgn ( s n ) = σ for every n, ∆(ˆ t + s n , ˆ t ) → ∆ and ∆(ˆ t + s n , ˆ t ) → ∆ as n → ∞ , ∆ ≥ ∆ + c . Due to the continuity of s (cid:55)→ ∆(ˆ t + s, ˆ t ) for s (cid:54) = 0 , given ∆ ∈ (∆ , ∆ ) onecan find an infinitesimal sequence ( s n ) n such that, for every n , sgn ( s n ) = σ and ∆(ˆ t + s n , ˆ t ) = ∆ . Now, as in (4) the set δ | s n | − / (ˆ p − Γ φ ) is the intrinsic graph of amap φ ˆ p, | s n | − / : W → V such that φ ˆ p, | s n | − / ( σw ) = σ ∆(ˆ t + s n , ˆ t ) v = σ ∆ v . Since the family ( φ ˆ p, | s n | − / ) n is uniformly Hölder continuous, up to extractinga subsequence it converges locally uniformly to a map ψ : W → V such that ψ ( σw ) = σ ∆ v . The arbitrariness of ∆ ∈ (∆ , ∆ ) implies that there are infinitelymany different blow-ups at ˆ p , and this concludes the proof. (cid:3) Appendix
We are now going to construct the function u used in the proof of Theorem 1:this function, in a sense, provides a counter-example to a Rademacher property forLipschitz functions from ( R , | · | / ) to ( R , | · | ) . We will use a classical procedureproducing a self-similar function: although these ideas are well-known (see e.g. [10]and the references therein), we prefer to include a detailed construction because wewere not able to find in the literature explicit statements for the precise estimates (6)we need.We construct a function u : [ 0 , → [ 0 , whose difference quotients ∆( s, t ) = u ( s ) − u ( t ) sgn ( s − t ) | s − t | / satisfy(5) | ∆( s, t ) | ≤ for every s, t ∈ [ 0 , .We will construct u in such a way that there exist c > and c > with theproperty that, for every t ∈ [ 0 , and δ ∈ (0 , , one can find s , s ∈ [ 0 , suchthat(6) sgn ( s − t ) = sgn ( s − t ) ,c δ ≤ | s − t | ≤ δ,c δ ≤ | s − t | ≤ δ, | ∆( s , t ) − ∆( s , t ) | ≥ c . We can extend u to R by setting u ( t ) = u ( − t ) for t ∈ [ − , and u ( t + 2 n ) = u ( t ) for all n ∈ Z . The properties (5) and (6) imply the validity of (1) and (2) for theextended u . JULIA, NICOLUSSI GOLO, AND VITTONE
Figure 1.
Four instances of the functions u n defined in (7)The function u is obtained as the limit of a sequence ( u n ) n ∈ N where u ( t ) = t .The function u n +1 is obtained from u n on setting(7) u n +1 ( t ) = u n (cid:0) t (cid:1) if t ∈ (cid:2) , (cid:3) , − u n (cid:0) (cid:0) t − (cid:1)(cid:1) if t ∈ (cid:2) , (cid:3) , + u n (cid:0) (cid:0) t − (cid:1)(cid:1) if t ∈ (cid:2) , (cid:3) . The first few of the functions u , u , u , . . . are plotted in Figure 1. Let us noticethat u n (0) = 0 and u n (1) = 1 for every n , hence u n (4 /
9) = 2 / and u n (5 /
9) = 1 / for every n ≥ .Notice (see Figure 2) that the graph of u n +1 is the union of three affine copiesof the graph of u n , via the following maps (acting on p ∈ R ):(8) A ( p ) = (cid:18) / / (cid:19) p,A / ( p ) = (cid:18) / − / (cid:19) p + (cid:18) / / (cid:19) ,A / ( p ) = (cid:18) / / (cid:19) p + (cid:18) / / (cid:19) . Claim 1:
The functions u n converge uniformly on [ 0 , to a function u forwhich (5) holds.The fact that u n uniformly converge to a continuous function u is a consequenceof the estimate (cid:107) u n +1 − u n (cid:107) C ([ 0 , ≤ (cid:107) u n − u n − (cid:107) C ([ 0 , . OWHERE DIFFERENTIABLE INTRINSIC LIPSCHITZ GRAPHS 7
Figure 2.
Iterated images of the unit square under the affine mapsin (8); red dots are the images of (0 , and (1 , , and they belongto the graph of the limit function u .This estimate follows directly from the definition (7): for instance, for t ∈ [ 0 , / one has | u n +1 ( t ) − u n ( t ) | = 23 | u n (cid:0) t/ (cid:1) − u n − (cid:0) t/ (cid:1) | ≤ (cid:107) u n − u n − (cid:107) C ([ 0 , . Similarly, one can treat the other two cases t ∈ [ 4 / , / and t ∈ [ 5 / , .The bound (5) on the the difference quotients of u follows from the fact that thesame is true for all u n in the sequence, as we are now going to prove by inductionon n . The statement is clearly true for n = 0 . Suppose that u n satisfies | u n ( t ) − u n ( s ) | ≤ | t − s | / for every s, t ∈ [ 0 , ;we will prove that also | u n +1 ( t ) − u n +1 ( s ) | ≤ | t − s | / for every s, t ∈ [ 0 , .We distinguish several cases depending on which intervals ( [ 0 , / , [ 4 / , / or [ 5 / , ) the points s and t belong to. We can suppose that s < t . Case 1: s and t are in the same interval. We can use (7) and the inductionhypothesis to conclude. Case 2: s ∈ [ 0 , / and t ∈ [ 4 / , / . Since ≤ u n ≤ , one sees from thedefinition of u n +1 that max( u n +1 ( s ) , u n +1 ( t )) ≤ / u n +1 (4 / . Thus | u n +1 ( t ) − u n +1 ( s ) | ≤ max( u n +1 (4 / − u n +1 ( t ) , u n +1 (4 / − u n +1 ( s )) ≤ max(( t − / / , (4 / − s ) / ) ≤ ( t − s ) / , where the second inequality follows frow Case 1. Case 3: s ∈ [ 4 / , / and t ∈ [ 5 / , . This is similar to Case 2. Case 4: s ∈ [ 0 , / and t ∈ [ 5 / , . Then either | u n +1 ( t ) − u n +1 ( s ) | ≤ / , andwe are done because | t − s | ≥ / , or | u n +1 ( t ) − u n +1 ( s ) | > / , and then necessarily JULIA, NICOLUSSI GOLO, AND VITTONE u n +1 ( s ) < u n +1 ( t ) (otherwise, ≤ u n +1 ( s ) − u n +1 ( t ) ≤ u n +1 (4 / − u n +1 (5 /
9) =2 / − / / ) and | u n +1 ( t ) − u n +1 ( s ) | = u n +1 ( t ) − u n +1 ( s )= u n +1 ( t ) − u n +1 (5 / − / u n +1 (4 / − u n +1 ( s ) ≤ ( t − / / − / / − s ) / (by Case 1). By squaring the right hand side of the last inequality, we obtain (cid:0) ( t − / / − / / − s ) / (cid:1) = ( t − s ) + 2( t − / / (4 / − s ) / −
23 ( t − / / −
23 (4 / − s ) / = ( t − s ) + ( t − / / (cid:0) (4 / − s ) / − / (cid:1) + (4 / − s ) / (cid:0) ( t − / / − / (cid:1) ≤ t − s, where we used the fact that / − s ≤ / and t − / ≤ / . This is enough toconclude. Claim 2: there exist d > and d > such that, for every t ∈ [ 0 , , one canfind s , s ∈ [ 0 , such that(9) sgn ( s − t ) = sgn ( s − t ) d ≤ | s − t | ≤ d ≤ | s − t | ≤ | ∆( s , t ) − ∆( s , t ) | ≥ d . In fact, we will prove Claim 2 for d = 1 / and d = min (cid:26) (cid:16)(cid:0) − (cid:1) − / − (cid:17) , − , √ (cid:27) . We distinguish several cases.
Case 1: t ∈ [ 0 , / . In this case it suffices to consider s = 5 / and s = 1 ,as we now show. Observe that the distances of s , s from t are both greater than / > d .If u ( t ) ≥ / , by (5) and the equality u (0) = 0 we have t / ≥ u ( t ) , hence t ≥ / ; since u ( t ) ≤ / we obtain ∆(1 , t ) ≥ (cid:113) − and ∆(5 / , t ) ≤ − (cid:113) = 13 , so that ∆(1 , t ) − ∆(5 / , t ) ≥ d .If u ( t ) ≤ / , then (4 / − t ) / ≥ / − / / , hence / − t ≥ / / =(5 / and ∆(1 , t ) ≥ and ∆(5 / , t ) ≤ = 35 and again ∆(1 , t ) − ∆(5 / , t ) ≥ d . Case 2: t ∈ [ 4 / , / . In this case we take s = 5 / and s = 1 . The distancesof s , s from t are both no less than /
18 = d and, since / ≤ u ( t ) ≤ / , onegets ∆(1 , t ) ≥ (cid:113) = 1 √ and ∆(5 / , t ) ≤ . Case 3: t ∈ [ 1 / , . We proved that, if t ∈ [ 0 , / , the claim can beproved on choosing s = 5 / and s = 1 . Therefore, due to the symmetry u ( x ) =1 − u (1 − x ) , when t ∈ [ 1 / , it is enough to take s = 0 and s = 4 / . OWHERE DIFFERENTIABLE INTRINSIC LIPSCHITZ GRAPHS 9
Claim 3: there exist c > and c > such that, for every t ∈ [ 0 , and δ ∈ (0 , , one can find s , s ∈ [ 0 , for which (6) holds.By self-similarity, the graph of u over the interval [ 0 , ∩ [ t − δ, t + δ ] containsthe image of the graph of u over [ 0 , under an affine map L : R → R which is afinite composition L = A j ◦ · · · ◦ A j N of maps ( A j k ) k =1 ,...,N for j k in { , / , / } .Observe that L is an affine map of the form L ( x, y ) = ( L ( x ) , L ( y )) for suitableaffine maps L , L : R → R and it is not restrictive to assume that the length ofthe interval L ([ 0 , , which is contained in [ t − δ, t + δ ] , is at least δ/ : thisimplies that there exists δ/ ≤ c ≤ δ such that | L ( x ) − L ( y ) | = c | x − y | for every x, y ∈ [ 0 , . Let t ∈ [ 0 , be such that L ( t , u ( t )) = ( t, u ( t )) . If s , s ∈ [ 0 , are such that (9) holds, then we havesgn ( L ( s ) − t ) = sgn ( L ( s ) − t ) , d δ ≤ | L ( s ) − t | ≤ δ, d δ ≤ | L ( s ) − t | ≤ δ. Since the maps A j k do not modify the difference quotients, L also has this property,i.e., | ∆( L ( s ) , t ) − ∆( L ( s ) , t ) | = | ∆( s , t ) − ∆( s , t ) | ≥ d . This concludes the proof.
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Départment de Mathématiques d’Orsay, Université Paris-Saclay, 91405,Orsay, France
Email address : [email protected] (Nicolussi Golo) Department of Mathematics and Statistics, 40014 University ofJyväskylä, Finland
Email address : [email protected] (D. Vittone) Dipartimento di Matematica “T.Levi-Civita”, via Trieste 63, 35121Padova, Italy.
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