Nowhere differentiable intrinsic Lipschitz graphs

We construct intrinsic Lipschitz graphs in Carnot groups with the property that, at every point, there exist infinitely many different blow-up limits, none of which is a homogeneous subgroup. This provides counterexamples to a Rademacher theorem for intrinsic Lipschitz graphs.

Recall that a Carnot group G is a connected, simply connected and nilpotent Lie group whose Lie algebra is stratified, i.e., it can be decomposed as the direct sum ⊕ s j=1 V j of subspaces such that V j+1 = [V 1 , V j ] for every j = 1, . . . , s − 1, We shall identify the group G with its Lie algebra via the exponential map exp : ⊕ s j=1 V j → G, which is a diffeomorphism. In this way, for λ > 0 one can introduce the 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 it is dilation-invariant. Assume that a splitting G = WV of G as the product of homogeneous and complementary (i.e., such that W ∩ V = {0}) subgroups is fixed; we say that a function φ : W → V intrinsic Lipschitz if there is an open nonempty cone U such that V \ {0} ⊂ U and 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 atp =ŵφ(ŵ) if there exists a sequence (λ n ) n such that λ n → +∞ and the limit lim n→∞ δ λn (p −1 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 Lipschitz graph of a map W → V. Eventually, we say that φ is intrinsically differentiable atŵ ∈ W if the blow-up of Γ φ atp =ŵφ(ŵ) is unique and it is a homogeneous subgroup of G. See [8] for details.
We say that a group G along with a splitting WV satisfies an intrinsic Rademacher Theorem if all intrinsic Lipschitz maps φ : W → V are intrinsically differentiable almost everywhere (that is, for almost all points of W equipped with its Haar measure). It was proved in [6] that this is the case when V R and G is of step two; other partial results for graphs with codimension 1 (V R) are contained in [4] and [9]. If V is a normal subgroup, the Rademacher Theorem has been proved for general G by G. Antonelli and A. Merlo in [2]. Recently, the third named author [12] proved that Heisenberg groups (with any splitting) satisfy an intrinsic Rademacher Theorem. The question has been open for a long time if G is the Engel group (which has step 3) and V R (see [1]). In this paper we prove a result in the negative direction: namely, we provide examples of intrinsic Lipschitz graphs that are nowhere intrinsically differentiable. Let us state our main result: Moreover, φ can be constructed in such a way that, for every p ∈ Γ φ , the following properties 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. It is worth observing that, in this setting, the map φ : W → V provided in the proof of Theorem 1 takes the form φ(y, t) = (0, u(t)), where u is the 1 2 -Hölder continuous 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 does not exists at any t ∈ R and this is the ultimate reason for the non-differentiability of φ.

Similar counterexamples can be constructed in any codimension
. It can be easily checked that the map φ(y, t) = (0, u(t)) defines an intrinsic Lipschitz graph of codimension k for which the properties (a) and (b) in Theorem 1 hold at every point.
Hausdorff measure, does not have a unique tangent measure at any point. Indeed, firstly, any tangent measure of µ is supported on a blow-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 then conclude that if µ 1 and µ 2 are two tangent measures of µ supported on different blow-ups of Γ φ , then they are two distinct measures. Since blow-ups of Γ φ are not unique, so are tangent measures. Observe also that no tangent measure can be flat, i.e., supported on a homogeneous subgroup. In particular, Γ φ is purely 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 properties. First, the difference quotients Second, there exist c 1 > 0 and c 2 > 0 such that, for every t 0 ∈ R and δ ∈ (0, 1 ], there exist s 1 , s 2 ∈ R such that Such a function exists, as we show in the appendix.
We can then define φ : W → V as Notice that the condition [v 0 , W] = 0 implies that Therefore, the intrinsic graph of φ is the set of points wφ(w) = w + u(β(w))v 0 for w ∈ W.
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 0 ⊂ V ∩ V 1 is the horizontal subgroup generated by v 0 and L : W → V 0 parametrizes a homogeneous subgroup Γ L of G, then L| W∩V2 = 0. Indeed, the homogeneity of Γ L implies that for every h > 0 and w ∈ W ∩ V 2 one has L(2w) = √ 2 L(w), while the fact that Γ L is a subgroup (plus the fact that V 0 and W commute) gives L(2w) = 2L(w). This proves that L = 0 on W ∩ V 2 .
If we set w =ŵδ 1/λn w , then β(w) = β(ŵ) + β(w )/λ 2 n . Therefore, the set δ λn (p −1 Γ φ ) is the intrinsic graph of the function from W to V given by Since the maps φp ,λn take values in V 0 , L is also V 0 -valued and, as we saw above, this implies that L| W∩V2 = 0.

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 for Lipschitz functions from (R, | · | 1/2 ) to (R, | · |). We will use a classical procedure producing 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 we were not able to find in the literature explicit statements for the precise estimates (6) we need.

Claim 1:
The functions u n converge uniformly on [ 0, 1 ] to a function u for which (5) holds.
The fact that u n uniformly converge to a continuous function u is a consequence of the estimate Similarly, one can treat the other two cases t ∈ [ 4/9, 5/9 ] and t ∈ [ 5/9, 1 ].
The bound (5) on the the difference quotients of u follows from the fact that the same is true for all u n in the sequence, as we are now going to prove by induction on n. The statement is clearly true for n = 0. Suppose that u n satisfies |u n (t) − u n (s)| ≤ |t − s| 1/2 for every s, t ∈ [ 0, 1 ]; we will prove that also |u n+1 (t) − u n+1 (s)| ≤ |t − s| 1/2 for every s, t ∈ [ 0, 1 ]. We distinguish several cases depending on which intervals ([ 0, 4/9 ], [ 4/9, 5/9 ] or [ 5/9, 1 ]) 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 induction hypothesis to conclude.