Poincar\'e inequalities for mutually singular measures

Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner,"Inverse limit spaces satisfying a Poincar\'e inequality", we provide a simple example of a metric space $X$ that admits Poincar\'e inequalities for a continuum of mutually singular measures.

In this note we provide a simple example of a metric measure space X that satisfies abstract Poincaré inequalities in the sense of Heinonen-Koskela [HK98] for a 1-parameter family of mutually singular measures. By a recent announcement of M. Csörnyei and P. Jones the classical Rademacher's Theorem is sharp in the sense that if its conclusion holds for the metric measure space (R n , µ), then µ must be absolutely continuous with respect to the Lebesgue measure. From this, using the theory of differentiability spaces (compare [Che99, Sec. 14]), it follows that if µ is a doubling measure on R n such that the metric measure space (R n , µ) admits an abstract Poincaré inequality, then the measure µ is absolutely continuous with respect to the Lebesgue measure. The example presented here shows that for metric measure spaces a similar phenomenon does not hold; in particular, the measure class for which Cheeger's generalization of Rademacher's Theorem holds is not uniquely determined.
Theorem 1.1. There is a compact geodesic metric space X and there is a family of doubling probability measures {µ w } w∈(0,1) defined on X such that: • Each metric measure space (X, µ w ) supports a (1, 1)-Poincaré inequality; • If w = w ′ the measures µ w and µ w ′ are mutually singular.

The Example
The goal of this Section is to prove Theorem 1.1. The metric space X that we consider is the example [LP01, pg. 290], compare also [CK13b, Exa. 1.2]. We briefly recall the construction. We build a sequence of graphs {X n } n≥0 starting with X 0 , which consists of a single edge which we identify with the interval [0, 1]. The graph X n+1 is obtained from X n by subdividing each edge of X n into four equal parts and replacing it by a rescaled copy of the diamond graph ( Figure 1, where we have labelled the edges for future reference). Each edge of X n has length 4 −n and the map which collapses the diamond graph to a segment gives rise to 1-Lipschitz maps π n+1,n : X n+1 → X n . In this way, the graphs {X n } n≥0 fit into an inverse system and, for each pair (k, n) of nonnegative integers satisfying k ≤ n, one has a 1-Lipschitz map π n,k : X n → X k . The maps {π n,k } n≥0,k≥0 satisfy the compatibility relations: (2.1) π k,m • π n,k = π n,m .
Having equipped the graphs X n with the length distance, the space X is the Gromov-Hausdorff limit of the sequence {X n } n≥0 ; by the properties of the inverse system one also gets 1-Lipschitz maps π ∞,n : X → X n which satisfy the compatibility conditions: (2.2) π n,k • π ∞,n = π ∞,k .
We will let Edge(n) and Vertex(n) denote, respectively, the sets of edges and vertices of X n . We now turn to the construction of the family of measures {µ w } w∈(0,1) . For w ∈ (0, 1), the measure µ w is the unique probability measure on X which satisfies the following requirements: (R1): For each nonnegative integer n, one has π ∞,n ♯ (µ w ) = µ w,n where µ w,n is a probability measure on X n ; (R2): The measure µ w,n is a multiple of arclength on each edge of X n and µ 0 is identified with the Lebesgue measure on [0, 1]; (R3): For each edge e n ∈ Edge(n), denoting by {e n+1,i } i=1,··· ,6 ⊂ Edge(n + 1) the edges of X n+1 whose union is π −1 n+1,n (e n ) (labelling as in Figure 1), one has: where H 1 denotes the 1-dimensional Hausdorff measure.
By the main result of [CK13a, Thm. 1.1] each metric measure space (X, µ w ) admits a (1, 1)-Poincaré inequality. Note that the class of spaces considered in [CK13a] is much broader and in this example one can also prove the (1, 1)-Poincaré inequality directly by using pencils of curves similarly as in [Laa00].
We now turn to a probabilistic description of the points in X. Let V ⊂ X be the set of points which project to some vertex: then µ w (V ) = 0 by conditions (R1) and (R2). Let C denote the Cantor set {1, · · · , 6} N and let A : C → X \ V be the map defined as follows: givenp ∈ C, A(p) is the unique point p ∈ X \ V such that, for each n ≥ 1, π n+1 (p) belongs to the edge labelled by the integerp(n) among those in π −1 n+1,n (e n ) (see Figure 1), where e n is the unique edge of X n containing π n (p). We now define the probability measure ν w on {1, · · · , 6} defined as follows: if i = 4, 5; we then let P w denote the probability measure on C which is the product of countably many copies of the measure ν w . We observe that A ♯ P w = µ w and we let T i,n denote the random variable: (2.6) T i,n (p) = 1 ifp(n) = i 0 otherwise.
Lemma 2.7. Let S i,n = k≤n T i,k ; then one has P w -a.s.: Proof. This follows from the Strong Law of Large Numbers as the random variables {T i,n } n≥1 are i.i.d.
We now complete the proof of Theorem 1.1: Lemma 2.9. If w = w ′ the measures µ w and µ w ′ are mutually singular.
Proof. We show that the Radon-Nikodym derivative dµw dµ w ′ is null. We let Reg(w ′ ) denote the set of points of C such that (2.8) holds (for the parameter w ′ ). By an application of measure differentiation, if we consider p ∈ X \ V and let e n (p) ∈ Edge(n) denote the unique edge of X n containing π n (p), then for µ w ′ -a.e. p ∈