Density and Extension of Differentiable Functions on Metric Measure Spaces


 We consider vector valued mappings defined on metric measure spaces with a measurable differentiable structure and study both approximations by nicer mappings and regular extensions of the given mappings when defined on closed subsets. Therefore, we propose a first approach to these problems, largely studied on Euclidean and Banach spaces during the last century, for first order differentiable functions de-fined on these metric measure spaces.


Introduction
In this paper we deal with problems about density of regular mappings and regular extensions of regular mappings on metric measure spaces endowed with a measurable di erentiable structure (MDS for short) in the sense of J. Cheeger [6]. We present here a rst order di erentiability study on these questions following the steps already given in Euclidean spaces as well as in nite dimensional Banach spaces on the same questions.
In any context it is natural to wonder whether a function can be approximated by another one with better properties. This is one of our goals for functions de ned on metric measure spaces with MDS. More precisely, we will study the existence of di erentiable (di erentiable and Lipschitz) approximations for continuous (continuous and Lipschitz) vector valued mappings. There exists a large literature on this subject, mainly given on linear spaces, developed during the last century where a whole collection of related problems are considered, the interested reader may check, for instance, [5,7,11,27] and references therein.
The extension problem for mappings de ned on closed subsets of nite dimensional Banach spaces has been extensively studied. The rst one in dealing with this problem was H. Whitney [29,30], who characterized functions de ned on closed subsets of the real line that can be extended to the whole real line as functions of C k class. Then, G. Gleaser [12] studied the same problem for higher dimensions and C extensions and, nally, C. Fe erman [9,10], in a series of papers, completed the program for nite dimensional spaces and functions de ned on compact subsets of them with C k class extensions.

Measurable Di erentiable Structures
In this section we describe the notion of metric measure space with a measurable di erentiable structure introduced by J. Cheeger [6] and S. Keith [22], as well as the associated notions as that of di erentiable almost everywhere mapping. Metric measure spaces with MDS were also called Lipschitz di erentiability spaces in [4]. The interested reader may also check the survey [23] to learn more about them.
Given a metric space X, the set LIP(X) denotes the set of all Lipschitz real functions on X. We give next the precise de nition of a metric measure space that admits a measurable di erentiable structure.
De nition 2.1. (Cheeger, Keith). Let (X, d, µ) be a metric measure space, and let C ⊂ LIP(X) be a vector space of functions.
(A) A pair (Y , y) is a C -chart if Y ⊂ X is a measurable subset with µ(Y) > and y = (y , . . . , y k ) : X → R k is a function for some k ∈ N ∪ { }, called coordinates on Y, where y i ∈ C for every ≤ i ≤ k. (B) The metric measure space (X, d, µ) has a C -measurable di erentiable structure (C -MDS, for short) if there is a countable collection of C -charts {(Xα , y α )} α∈A , which is called a C -atlas of X, with coordinates y α : X → R k(α) , so that µ(X \ α∈A Xα) = , k = sup α k(α) < ∞, and for every f ∈ C and chart (Xα , y α ) there exists a unique (up to a set of zero measure) measurable function df α : for µ-a.e. x ∈ Xα. Moreover, such a structure is called k-dimensional and it is non-degenerate if k(α) ≥ for all α ∈ A . We will assume we always work with non-degenerate MDS.
(C) We say that the pair (Y , y) is a chart, the countable collection {(Xα , y α )} α∈A is an atlas of X and the metric measure space (X, d, µ) has a measurable di erentiable structure (MDS, for short) whenever (Y , y) is a LIP(X)-chart, {(Xα , y α )} α∈A is a LIP(X)-atlas of X and (X, d, µ) has a LIP(X)-MDS.
In references [6,22,23] the reader may nd examples of metric spaces that can be endowed with a MDS and main facts on them. Next we de ne what we understand by di erentiability at a point of mappings acting on these structures.

De nition 2.2.
Let X be a metric space with a C -MDS and let (Xα , y α ) be a C -chart of X and V a Banach space.
Finally, we can state what we mean by di erentiability almost everywhere of a mapping de ned on metric measure spaces with a MDS.

De nition 2.3.
Let X be a metric space with a C -MDS and let (Xα , y α ) be a C -chart of X and V a Banach : Xα → V} α∈A , ≤m≤k(α) of measurable functions uniquely determined (up to a set of zero measure), such that for almost every x ∈ Xα = . (2.1)

Remark 2.4.
Notice that if we are in the case C = LIP(X), then real-valued Lipschitz functions will be di erentiable almost everywhere by de nition. This fact will be used at several instances regarding the distance function. Also, mappings in the chart are di erentiable almost everywhere and Lipschitz.

Density of di erentiable functions
In this section we study properties of density of di erentiable almost everywhere mappings. We begin by showing that any continuous mapping can be approximated by such mappings. We will achieve our goal by applying standard techniques on partitions of unity. To make the exposition easier, we will work with MDS instead of C -MDS although some comments will be added on this regard.
The functions φn are di erentiable almost everywhere (and Lipschitz). Moreover, it is easy to see that Consider now the function g : X → V de ned as This function is di erentiable almost everywhere because the family {φn} ∞ n= is locally nite and di erentiable almost everywhere. Let us see that g uniformly approximates f . For every where the last inequality is a consequence of the fact that ||f (x) − f (xn)|| < ε whenever φn(x) ≠ .

Remark 3.2.
(i) Let us note that if C ⊂ LIP(X) is a vector space of functions such that (X, d, µ) has a C -MDS, the above theorem holds whenever the functions x → d(x, x ) belong to C for all x ∈ X. (ii) Furthermore, if the functions x → d(x, x ) are di erentiable everywhere but x , then the approximation mapping is di erentiable everywhere on X (assuming X = ∪α Xα). Now we seek approximations by mappings that are di erentiable almost everywhere and Lipschitz. For that we will need the extra hypothesis of working with doubling metric spaces which, in particular, are separable.

De nition 3.3.
A metric space (X, d) is said to be doubling if there is a constant C such that for each r > , every ball contained in X with radius r can be covered by at most C balls of radius r/ . The doubling constant λ(X) if the in mum over all constants C satisfying the doubling condition.
Doubling metric spaces have played a major role in the recent theory of analysis on metric spaces, as reader may check in references [6,22,23] or the monograph by J. Heinonen [13]. Notice also that if (X, d) is doubling, then for each r > , every ball contained in X with radius r can be covered by at most λ(X) n balls of radius r/ n .
The following theorem is needed to prove the next density result.

Theorem 3.4 ([24]
). There exists a universal constant C > such that for every doubling metric space (X, d), for every Y ⊃ X and V Banach spaces, and for every Lipschitz mapping f : Another tool that we will use in this section is the concept of functions that locally depend on nitely many coordinates. This notion was rst de ned on Banach spaces with Schauder basis using the coordinate functionals [25]. Later, a generalization of this notion was considered by some authors using arbitrary continuous linear functionals, see, for instance, [8,18,28].

De nition 3.5.
Let Y and Z be Banach spaces, M ⊂ Y * and G : Y → Z. We say that G locally depends on nitely many coordinates from M (LFC-M, for short) if for each x ∈ Y there are a neighborhood U of x, a nite subset {f , . . . , fn} ⊂ M and a mapping H : A simple example is the sup norm on c , which is LFC-{e * n } ∞ n= away from the origin (where {e * n } ∞ n= are the coordinate functionals in c ). Lemma 3.6. Let (X, d, µ) be a metric measure space with a MDS and let V be a Banach space. Let Φ : X → c be a mapping whose coordinate functions e * n • Φ : X → R are di erentiable almost everywhere, and G : of functionals and a C smooth mapping H : R n → V such that G(y) = H(e * (y), . . . , e * n (y)) for all y ∈ Φ(U) (see [27]). Thus, Now, we can show the main theorem of this section, which gives a su cient condition for the set of di erentiable almost everywhere and Lipschitz vector-valued functions to approximate set of Lipschitz vectorvalued functions.
Theorem 3.7. Let (X, d, µ) be a metric measure space with a MDS where (X, d) is doubling, and let V be a Banach space. Then, for every Lipschitz mapping f : X → V and every ε > , there is a di erentiable almost everywhere and Lipschitz mapping g : X → V such that for all x ∈ X, and where C is the universal constant given by Theorem 3.4. Proof. Notice that since (X, d) is doubling then it is separable. Now, recall that Ahanori proved in [1] that for any ρ > , every separable metric space (X, d) is ( + ρ)-Lipschitz isomorphic to a subset of the Banach space c . Thus, for any ρ > , there is a mapping Φ : X → c such that Later Assouad in [2] and Pelant in [26] re ned this result by showing that every separable metric spaceembeds into c . Finally, Kalton and Lancien [21] constructed a -embedding (resp. -embedding) into c for every separable (resp. proper) metric space. Thus, since our metric space ( We claim that Φ(X) is doubling in c with constant λ(X) . Indeed, for any x ∈ X and r > we have that since Φ − is -Lipschitz and X is doubling. Now, using the fact that Φ is -Lipschitz, which proves the claim.
Let us take g : • by Lemma 3.6, g is di erentiable almost everywhere on X.
In the next proposition we show that subsets of doubling metric spaces are doubling too. We include it in this work as we lack a reference for it. Proof. Let A ⊆ X and let B A (a, r) be a ball in (A, d). We need to show that it can be covered by at most a given number λ(A) of ball in (A, d) with radius at most r/ . Since (X, d) is doubling and B A (a, r) ⊆ B X (a, r), there are at most λ(X) balls in X such that Applying the doubling property again, each ball B X (x i , r/ ) can be covered by, at most, λ(X) balls with center in X and radius r/ . Therefore we have, at most, λ(X) balls of radius r/ which union contains B A (a, r). Taking, for each of these balls, a point in the intersection of A with it (when such intersection is nonempty) we obtain, at most, λ(X) points in A such that the union of balls with center in these points and radius r/ covers B A (a, r). Finally, notice that a doubling metric space is proper (i.e., closed and bounded subsets are compact) if and only if it is complete. Then, using the -embedding into c for proper metric spaces given in [21], we obtain the last result of this section.
for all x ∈ X, and where C is the universal constant given by Theorem 3.4.

Remark 3.11.
(i) Let C ⊂ LIP(X) such that (X, d, µ) has a C -MDS, the main theorem holds whenever the functions x → d(x, x ) belong to C for all x ∈ X. Indeed, following the proof of [27, Proposition 3.1.3, Proposition 3.1.2] (see also [16]) the Lipschitz embedding Φ into c can be chosen with di erentiable almost everywhere coordinate functions and Lipschitz constant less or equal to + r for any r > . (ii) Furthermore, the approximation mapping is di erentiable everywhere on X whenever the functions x → d(x, x ) are di erentiable everywhere but x .

Extension of di erentiable functions
In this section we study the problem of extending di erentiable functions from a metric space with a MDS as di erentiable functions. We begin by obtaining some Lip-derivation inequalities for vector-valued functions similar to the one obtained in [6,14] in the real-valued case. Suppose that (X, d, µ) is a metric measure space with a MDS and V is a Banach space. Given a mapping f : X → V, the point-wise Lipschitz constant, Lipx(f ), at x ∈ X is given by: Proof. Since f : X → V is di erentiable almost everywhere, the function ϕ • f : X → R is di erentiable almost everywhere for any ϕ ∈ V * (actually, they are di erentiable at the points where f is di erentiable). Moreover, it is easy to see that for a.e. x ∈ Xα. Notice that the above inequality is satis ed for every ϕ ∈ V * and any point x ∈ Xα where f is di erentiable. Thus, for any ≤ m ≤ k(α) and any point x ∈ Xα where f is di erentiable, we choose ϕ ∈ V * such that To sum up, inequality (4.1) holds for any x ∈ Xα where f is di erentiable.
To show the main result of this section, we need the following lemma.

Lemma 4.3. Let (X, d, µ) be a metric measure space with a MDS where (X, d) is doubling, let V be a Banach space and A ⊂ X. Then, for every continuous mapping F : X → V such that F | A is Lipschitz, and every ε > ,
there exists a di erentiable almost everywhere mapping G : X → V such that:

is the universal constant given in Theorem 3.4. (iii) In addition, if F is Lipschitz, then there exists a constant C ≥ Cλ(X) , depending only on the doubling constant of X, such that the mapping G can be chosen to be Lipschitz on X and Lip(G) ≤ C Lip(F).
Proof. Assume that the mapping F : X → V is continuous on X and F | A is Lipschitz. By Theorem 3.1 there is a di erentiable almost everywhere mapping h : X → V, such that ||F(x) − h(x)|| < ε for all x ∈ X. Let us apply Corollary 3.9 to F | A to obtain a di erentiable almost everywhere and Lipschitz mapping g : Consider the open sets D = {x ∈ X : ||F(x) − g(x)|| < ε/ } and the closed set There is a di erentiable almost everywhere function u : X → [ , ] such that Indeed, since C ∩ (X \ D) = ∅, the function is continuous on X, p(C) = and p(X \ D) = . Using Theorem 3.1, there is a di erentiable almost everywhere function q : X → R such that |p(x) − q(x)| < . Let us take a C smooth function θ : R → [ , ] such that θ(t) = whenever t ≤ / and θ(t) = whenever t ≥ / . Then, u(x) = θ(q(x)) satis es the desired properties.
Let us de ne G : It is clear that G is a di erentiable almost everywhere mapping. Since u(x) = for all x ∈ X \ D, we deduce that Finally, since u(x) = and G(x) = g(x) for every x ∈ C, we obtain that Lip( To prove the last part, let us now assume that F is Lipschitz on the whole X. Let us apply Theorem 3.7 and Corollary 3.9 to F and F | A to obtain di erentiable almost everywhere mappings g and h from X into V such that We take again the open subsets B, D and the closed subset C as earlier in this proof. Notice that We claim that there is a di erentiable almost everywhere and Lipschitz function u : X → [ , ] such that In fact, let us take a Lipschitz function θ : R → [ , ] such that θ(t) = whenever t ≤ , θ(t) = whenever t ≥ ε and Lip(θ) = /ε . Thus, the function u(x) = θ(dist(x, X \ D)) is Lipschitz, so it is di erentiable almost everywhere, u(C) = , u(X \ D) = and Lip(θ) ≤ /ε . Let us now consider G : X → V as Clearly G is di erentiable almost everywhere on X. We follow the above proof to obtain that Additionally, if y, z ∈ X \ D, then u(y) = , u(z) = , G(y) = h(y), G(z) = h(z), and ||G(y) − G(z)|| = ||h(y) − h(z)|| ≤ Cλ(X) Lip(F)d(y, z). For y, z ∈ D, we have The case z ∈ X \ D and y ∈ D follows the same way as the previous one. We de ne C := + Cλ(X) and, nally, obtain that Lip(G) ≤ C Lip(F).

Equation (2.1) can be also written as
for a.e. x ∈ Xα. This expression inspires the mean value condition we will need to impose in order to obtain our results. This mean value condition was proved to be needed for C extension of vector valued mappings in [20] when working this problem in the context of Banach spaces. We adapt the notion given in [20] to our context in the following de nition.

De nition 4.4.
Let (X, d, µ) be a metric measure space with a MDS, consider {(Xα , y α )} α∈A an atlas on it. Let V be a Banach space and A ⊂ X a measurable subset with µ(A) > . We say that a di erentiable almost everywhere mapping f : A → V satis es the mean value condition on A if for every α, almost every x ∈ A ∩ Xα and every ε > , there is an open ball B(x, r) in X such that for every z, y ∈ A ∩ Xα ∩ B(x, r).
The following theorem is the main result in this section. It is rather an approximation result than an extension one as it shows that continuous extensions can be approached by smooth mappings in a precise way that can be useful to obtain the extensions result we are seeking. How to apply this result will be show later.

assume that f is Lipschitz on A and F is a Lipschitz extension of f to X. Then the function G can be chosen to be Lipschitz on X and Lip(G) ≤ C Lip(F) for a certain constant C.
Proof. Assume that F is a continuous extension of f . Since X is a separable metric space, A ⊂ X is a closed subspace and f satis es the mean value condition on for x ∈ X. Notice that Tn satis es the following properties: (1) Tn is di erentiable almost everywhere on X,  Xα ∩ B(xn , rn), Thus, Lip((Tn − F)| A∩Xα∩B(xn ,rn) ) ≤ ε C . By Theorem 3.1 there is a di erentiable almost everywhere mapping F : X → V such that ||F(x)−F (x)|| < ε for every x ∈ X. Let us de ne Ln := max{Lip(φn), } for every n ≥ . Now, for every n ∈ N we apply Lemma 4.3 to Tn − F on A ∩ Xα ∩ B(xn , rn) to obtain a di erentiable almost everywhere mapping δn : X −→ V so that for every x ∈ X and Lip(δn | A∩Xα ∩B(xn ,rn ) ) ≤ ε .
Let us de ne The mapping G is di erentiable almost everywhere since {φn} ∞ n= is locally nitely nonzero. For every x ∈ X Let us prove that Lip(f − G | A ) < ε. In order to simplify the notation let us write Let us now consider the case when F is a Lipschitz extension of f on X. In this case, we can assume that f is not constant (otherwise the assertion is trivial) and thus Lip(F) ≥ Lip(f ) > . Let us x < ε < Lip(F). If Besides, by applying Corollary 3.9, we select a di erentiable almost everywhere mapping F : Similarly to the rst case, the de nition of G is The proofs that G is di erentiable almost everywhere, ||G(x) − F(x)|| < ε for all x ∈ X and Lip(f − G | A ) < ε follow along the same lines. To show that G is Lipschitz, notice that, from the fact that the mappings {φn} are a partition of unity, We close the paper by showing how this result can be applied to obtain smooth extensions of smooth mappings. We begin with a convergence theorem. Theorem 4.6. Let (X, d, µ) be a metric measure space with a MDS and V a Banach space. If {fn} is a sequence of di erentiable almost everywhere functions from X to V and f : X → V is such that (i) Lip x (fn − f ) → for almost every x ∈ X, and (ii) there exists a function g : X → V such that df α n (x) → g(x) for almost every x ∈ X, then, f is di erentiable for almost every x ∈ X and df α (x) = g(x).
Proof. We have to show that Lip x (f − g(x)y α ) = for almost every x ∈ X, that is, for ε > there exists δ > such that Let us x x ∈ X so that (i) and (ii) hold. From (i), there exists n ∈ N such that for n ≥ n we have that Lip x (fn − f ) < ε/ . Therefore, there exists δ > such that From (ii) and the fact the mappings y α are Lipschitz by de nition, there exists n ∈ N such that for n ≥ n we have that Let us take now N = max{n , n }, since f N is di erentiable almost everywhere, there exists δ > , we can choose it so that δ < δ , such that which completes the proof.
We obtain the following corollary. Proof. We will apply Theorem 4.6 for functions n j= f j . We need the function g(x) = ∞ n= df α n (x). This function is well de ned due to Proposition 4.2 and (ii). Also, (ii) in Theorem 4.6 follows from Proposition 4.2 and (ii) in this corollary. Therefore, f is di erentiable almost everywhere.
We go next with the nal result of this paper where we provide existence of smooth extensions. We will require of a very strong use of the mean valued condition. Of course, the mean valued condition as required in the next statement can be weakened by imposing it only to the mappings that are needed in the proof, however we impose it on all di erentiable mappings so it makes the exposition easier. Theorem 4.8. Let (X, d, µ) be a metric measure space with a MDS where (X, d) is doubling, let V be a Banach space and A ⊂ X a closed subset with µ(A) > . Let us supposed that any di erentiable almost everywhere mapping from X to V satis es the mean value condition on A. Then, given a di erentiable almost everywhere and Lipschitz mapping from A to V, there is a di erentiable almost everywhere mapping F : X → V such that Moreover, if the mapping f is Lipschitz, then the di erentiable almost everywhere extension F : X → V can be chosen to be Lipschitz with Lip(F) ≤ C Lip(f ), where C ≥ is a constant that only depends on X.
Proof. By Theorem 4.5, there exists a di erentiable almost everywhere and Lipschitz g : X → V such that Let us take f − g | A : A → V, since we are assuming all di erentiable functions satisfy the mean value condition, we can apply Theorem 4.5, for globally de ned Lipschitz functions on the whole X, again and so there exists a di erentiable almost everywhere and Lipschitz g : X → V such that (i) ||f (x) − g (x) − g (x)|| < / en A.
Proceeding in this way, for all n ≥ there exists a di erentiable almost everywhere and Lipschitz mapping gn : X → V such that (i) ||f (x) − n j= g j (x)|| < / n in A. (ii) Lip(f − ( n j= g j ) | A ) < / n . (iii) Lip(gn) ≤ C/ n− . We de ne now g(x) = ∞ n= gn(x) for x ∈ X. Mapping g veri es: (i) g(x) = f (x) for x ∈ A. (ii) Lip(g) ≤ ∞ n= Lip(gn) ≤ Lip(g ) + ∞ n= C/ n− ≤ C(Lip(f ) + ). (iii) For x ∈ X, take a ∈ A, and so ||g(x)|| ≤ ||g(x) − g(a)|| + ||f (a)|| ≤ Lip(g)d(x, a) + ||f (a)|| < ∞. Finally, from Corollary 4.7, g is di erentiable almost everywhere and the theorem is proved. Remark 4.9. As nal remark we point out that the mean value condition was needed in the studies in Banach spaces because they were looking for C extensions. Since being of class C has not been considered at all in this work, the remaining question is not if the strong assumptions on the mean value condition may be weakened but rather if it can be completely dropped.