The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson’s inequalities

We show that the algebra of cylinder functions in the Wasserstein Sobolev space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{1,q}(\mathcal {P}_p(X,\textsf{d}), W_{p, \textsf{d}}, \mathfrak {m})$$\end{document}H1,q(Pp(X,d),Wp,d,m) generated by a finite and positive Borel measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {m}$$\end{document}m on the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p,\textsf{d})$$\end{document}(p,d)-Wasserstein space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal {P}_p(X, \textsf{d}), W_{p, \textsf{d}})$$\end{document}(Pp(X,d),Wp,d) on a complete and separable metric space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X,\textsf{d})$$\end{document}(X,d) is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {B}$$\end{document}B, then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {B}$$\end{document}B is reflexive (resp. if the dual of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {B}$$\end{document}B is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson’s type inequalities in the Wasserstein Sobolev space.


Introduction
The study of Sobolev spaces on metric measure spaces is a well established area of interest in metric geometry, we refer to [18,19,22,25] for a general treatment of the subject.It is thus important to provide examples (as concrete as possible) of such Sobolev spaces built on relevant metric spaces; the study carried out in [12] goes precisely in this direction: the authors analyze the properties of the 2-Sobolev space on the 2-Wasserstein space [29,30,2,24] of probability measures on a separable Hilbert space H. Let us also mention that, besides being interesting from a purely theoretical point of view, the study of Sobolev spaces on spaces of probability measures has also important applications to functional analysis over spaces of probability measures.Indeed, problems such as evolutionary games or Kolmogorov equations in nonlinear filtering have been treated so far by classical notions of solutions, lacking weak formulations.In this regard, providing a functional analytic framework in which to set those problems may be very relevant.We refer to the introduction of [12] for a more detailed list of applications and references.In this work we generalize the results in [12] considering a general complete and separable metric space (X, d), instead of a Hilbert space, and we treat general exponents p, q both in the order of the Wasserstein distance and in the Sobolev space (instead of p = q = 2).Before entering into the details of the present work, let us briefly recall the definition of metric Sobolev space and the results of [12] that are the starting point for our analysis.Among the possible approaches to Sobolev spaces on metric measure spaces (for example the Newtonian one [26,5]), here we focus on the approach, strictly related to the ideas of Cheeger [7], that is contained in the work of Ambrosio, Gigli and Savaré [4] where they define the following concept of q-relaxed (q ∈ (1, +∞)) gradient: given a metric measure space (X, d, m), we say that G ∈ L q (X, m) is a q-relaxed gradient of f ∈ L 0 (X, m) if there exist a sequence (f n ) n ⊂ Lip b (X, d) and G ∈ L q (X, m) such that (1)  The q-Cheeger energy of f ∈ L 0 (X, m) is then defined as where |Df | ⋆,q is the minimal relaxed gradient of f i.e. the element of minimal norm in the set of relaxed gradients of f .It is well known that the q-Cheeger energy can be also characterized as the relaxation of the so called pre-q-Cheeger energy in the sense that 2) The Sobolev space à la Cheeger H 1,q (X, d, m) is thus the vector space of functions f ∈ L q (X, m) with finite Cheeger energy endowed with the norm which makes it a Banach space.A remarkable result [4] is the so called strong approximation property by Lipschitz functions: if f ∈ H 1,q (X, d, m) then there exists a sequence ( (1.3)In [12] the authors provide a general criterion for the validity of the same property where, instead of the whole Lip b (X, d), the approximating sequence can be taken from a suitable subalgebra A ⊂ Lip b (X, d) satisfying ½ ∈ A , for every x 0 , x 1 ∈ X there exists f ∈ A : f (x 0 ) = f (x 1 ), where ½ : X → R is the constant function equal to 1.This is equivalent to say that A is dense in energy in the Sobolev space.In particular (see Theorem 2.9 below or [12,Theorem 2.12] for the proof) it is proven that an equivalent condition is that for every y ∈ X it holds |Dd y | ⋆,q,A ≤ 1 m-a.e. in X, (1.4) where d y (x) := d(x, y) and |D • | ⋆,q,A is a suitably A -adapted notion of minimal relaxed gradient (cf.Definition 2.1).The second part of [12] is devoted to apply the criterion in (1.4) to the L 2 -Kantorovich-Rubinstein-Wasserstein (in brief, Wasserstein) space on a separable Hilbert space H, denoted by (P 2 (H), W 2 ), with the algebra C P(H), C 1 b (H) of cylinder functions (cf.Definition 4.1), which is the algebra of functions of the form where ψ ∈ C 1 b (R N ) and φ n ∈ C 1 b (H) for n = 1, . . ., N ∈ N. The density of C P(H), C 1 b (H) in H 1,2 (P 2 (H), W 2 , m) (here m is a finite and positive Borel measure on P 2 (H)) is particularly interesting since cylinder functions come already with some structure; in particular to any function F as in (1.5) we can associate a notion of gradient given by DF (µ, x) := (1.6) It is proven in [12,Proposition 4.9] that, for every F ∈ C P(H), C 1 b (H) , we have so that the pre-Cheeger energy satisfies the parallelogram identity (cf.[12,Subsection 4.2]) and thus forces the Cheeger energy to be a quadratic form.This amounts to say that the Sobolev space H 1,2 (P 2 (H), W 2 , m) is a Hilbert space, a crucial property in the theory of metric Sobolev spaces [21,27,28,4,14].
The aim of this work is, following the approach of [12], to study the properties of the more general class of Wasserstein Sobolev spaces H 1,q (P p (B, d • ), W p,d • , m) (p, q ∈ (1, +∞)), where (B, • ) is a separable Banach space, (P p (B, d • ), W p,d • ) is the (p, d • )-Wasserstein space on it and m is positive and finite Borel measure on P p (B, d • ).First of all, in Proposition 4.7 we provide a generalization of the representation in (1.7): if F is a cylinder function in P(B) as in (1.5), then where p ′ = p/(p − 1) is the conjugate exponent of p, • * is the dual norm in B * and DF is adapted from (1.6) in the obvious way.Notice that, differently from the representation (1.7), the right hand side of (1.8) doesn't coincide with the q-th power of a L q norm on a suitable space but it is rather the q-th power of the norm in the L q -direct integral of a family of Banach spaces (see Subsection 5 for more details).Approaching first the case in which B = R d and • is a sufficiently regular norm on R d (cf.(3.3)), we are able to prove the density of C P(R d ), C 1 b (R d ) in the corresponding Sobolev space (first part of Theorem 4.15) adapting the techniques of [12] (in particular using (1.4) and the representation in (1.8)) to this more general situation: this requires more sophisticated results (which may be interesting by themselves) in terms of the properties of Kantorovich potentials which are no longer convex if p = 2 (see Section 3.1).The density of cylinder functions is then extended to the Sobolev-Wasserstien space corresponding to an arbitrary norm on R d by an approximation procedure and then to an arbitrary separable Banach space (B, • ) via a standard embedding technique in ℓ ∞ (N) and finite dimensional projections (second part of Theorem 4.15 and Corollary 4.19, respectively).The precise statement of the result is the following.Theorem 1.1.Let (B, • ) be a separable Banach space; then the algebra of cylinder functions Let us also mention that we are able to extend the density result (Theorem 4.18) to an arbitrary complete and separable metric space (X, d) where, instead of using C 1 b functions (which of course are not available) to generate cylinder functions on P(X), we consider a sequence (φ for every x, y ∈ X, and we use the smallest unital algebra containing (φ k ) k .
In the same spirit of the Hilbertianity result in [12], we are led to study which properties of the Banach space B are transferred to the Sobolev space, thanks to the density of cylinder functions provided by the above Theorem 1.1.This is the case of uniform convexity and the validity of some Clarkson's type inequalities.The argument we adopt for such properties is similar: since the pre-Cheeger energy of a cylinder function as in (1.8) corresponds to the q-th power of a norm in a suitable Banach space, it is sufficient to prove that such norm enjoys the uniform convexity (resp.the validity of Clarkson's type inequalities) to obtain that the pre-Cheeger satisfies the same property; thanks to the density of cylinder functions such a property is extended to the Cheeger energy and thus to the whole Sobolev norm (Theorems 5.4 and 5.10).For what concerns reflexivity the argument is somehow standard: again using the representation of the pre-Cheeger energy in (1.8), we can isometrically embed the Sobolev space into a reflexive Banach space (Theorem 5.2).The precise statement regarding reflexivity and uniform convexity is reported below (we refer to Subsection 5.1 for the results related to Clarkson's type inequalities).
Theorem 1.2.Let (B, • ) be a separable Banach space.If B is reflexive (resp.its dual is uniformly convex), then the Sobolev space Plan of the paper.In Section 2 we summarize the construction of metric Sobolev spaces depending on a subalgebra of Lipschitz and bounded functions A and we report a few results of [12] concerning the density in energy of A .The beginning of Section 3 contains the general framework for Wasserstein spaces we are going to work with and Subsection 3.1 presents some compactness results for Kantorovich potentials in a specific geometric situation: these results combine the ideas of [11,13] to provide useful Lipschitz estimates on the potentials.Section 4 contains the core of our density results: after stating the definition of cylinder functions in the framework of a metric space (X, d), in Subsection 4.1, we study the asymptotic Lipschitz constant of a cylinder function (Proposition 4.7); in Subsection 4.2 we prove the density result when the base space is R d endowed with an arbitrary norm (Theorem 4.15); in Subsection 4.3 we generalize this result to a complete and separable metric space (X, d) (Theorem 4.18).Finally, in Section 5 we prove that relevant properties of the underlying Banach space pass to the Sobolev space: first we treat reflexivity and uniform convexity (Theorems 5.2, 5.4) and then we study Clarkson's type inequalities in Subsection 5.1 (see in particular Theorem 5.10).
Acknowledgments.The author warmly thanks G. Savaré for proposing the problem and for his many valuable suggestions.The author is grateful to E. Pasqualetto for his comments on a first draft of the present paper.The author also gratefully acknowledges the support of the Institute for Advanced Study of the Technical University of Munich, funded by the German Excellence Initiative.The author was also partially financed by the Austrian Science Fund (FWF) project F65.The authors is grateful to the anonymous reviewer for their valuable comments.

Metric Sobolev spaces and density of subalgebras
In this section we describe the general metric setting and we list a few results of [12] which are the starting point of our analysis.For this whole section, we fix a separable metric space (X, d), a finite and positive Borel measure m on (X, d) (the triple (X, d, m) is called a Polish metricmeasure space), an exponent q ∈ (1, +∞) and a unital and separating subalgebra A ⊂ Lip b (X, d) i.e. satisfying where Lip b (X, d) is the space of real valued and bounded d-Lipschitz functions on X and ½ : ) is another complete and separable metric space and f : X → Y is a Borel function, we define the finite (with the same total mass of m) and Borel measure f ♯ m on (Y, d Y ) as We define the d-asymptotic Lipschitz constant of f ∈ Lip b (X, d) as where B d (x, r) ⊂ X denotes the d-open ball centered at x with radius r > 0 and, for A ⊂ X, the quantity Lip(f, A, d) is defined as We denote by L 0 (X, m) the space of real valued and Borel measurable functions on X, identified up to equality m-a.e.and, analogously, by L r (X, m) the usual Lebesgue spaces of real valued, r-summable and Borel measurable functions, identified up to equality m-a.e., r ∈ [1, +∞].We endow L 0 (X, m) with the topology of the convergence in m-measure, while L r (X, m) is endowed with the usual norm, r ∈ [1, +∞].When dealing with vector-valued or extended real-valued functions we also specify the codomain in the notation for L r -spaces, i.e. we write L r (X, µ; Y ), where Y is a (subset of a) Banach space.The following is the definition of relaxed gradient we adopt [4,3,25] (see also [26,5] for a different approach).
Definition 2.1 ((q, A )-relaxed gradient).We say that G ∈ L q (X, m) is a (q, A )-relaxed gradient of f ∈ L 0 (X, m) if there exist a sequence (f n ) n∈N ∈ A and G ∈ L q (X, m) such that: (1) The next result is a simple but important property of relaxed gradients [4,3,25].
Theorem 2.2.If f ∈ L 0 (X, m) has a (q, A )-relaxed gradient then there exists a unique element of minimal L q (X, m)-norm in Thanks to Theorem 2.2 the next definition is well posed.
Definition 2.3 (Minimal relaxed gradient).Let f ∈ L 0 (X, m) be such that f has a (q, A )relaxed gradient.Its relaxed gradient with minimal L q (X, m)-norm is denoted by |Df | ⋆,q,A and called minimal (q, A )-relaxed gradient of f .Definition 2.4 (Cheeger energy and Sobolev space).We call D 1,q (X, d, m; A ) the set of functions in L 0 (X, m) with a (q, A )-relaxed gradient and we define the (q, A )-Cheeger energy as with CE q,A (f ) := +∞ if f ∈ L 0 (X, m) \ D 1,q (X, d, m; A ).The Sobolev space H 1,q (X, d, m; A ) is defined as L q (X, m) ∩ D 1,q (X, d, m; A ).The Sobolev norm of f ∈ H 1,q (X, d, m; A ) is defined as f q H 1,q (X,d,m;A ) := f q L q (X,m) + CE q,A (f ).In the next theorem we collect a few properties of relaxed gradients and Sobolev spaces that will be useful (for a more comprehensive list and references to the proofs, see [12]).
(1) The set is convex and it is closed with respect to to the product topology of the convergence in mmeasure and the weak convergence in L q (X, m).In particular, the restriction S r := S ∩ L r (X, m) × L q (X, m) is weakly closed in L r (X, m) × L q (X, m) for every r ∈ (1, +∞).
(2) (Strong approximation) If f ∈ D 1,q (X, d, m; A ) takes values in a closed (possibly unbounded) interval I ⊂ R then there exists a sequence f n ∈ A with values in I such that If moreover f ∈ L r (X, m) for some r ∈ [1, +∞) then we can also find a sequence as in (2.5) converging strongly to f in L r (X, m).
Remark 2.6 (Pre-Cheeger energy and its relaxation).It is well known (see e.g.[25,Corollary 3.1.7])that for every r ∈ {0} ∪ [1, +∞) it holds that where the pre-Cheeger energy pCE q : Lip b (X, d) → [0, +∞) is defined as (2.9) The main property we are interested in investigating is the density of the subalgebra A in the metric Sobolev space.
Definition 2.7 (Density in energy of a subalgebra of Lipschitz functions).We say that A ⊂ Lip b (X, d) is dense in q-energy in D 1,q (X, d, m) if for every f ∈ D 1,q (X, d, m) there exists a sequence (f n ) n∈N satisfying (2.10) Remark 2.8.It is not difficult to see that Definition 2.7 is equivalent to the equality D 1,q (X, d, m; A ) = D 1,q (X, d, m) with equal minimal relaxed gradients, the equality of the Sobolev spaces H 1,q (X, d, m; A ) = H 1,q (X, d, m) with equal norms and the following strong approximation property: for every f ∈ H 1,q (X, d, m) there exists a sequence (f n ) n∈N satisfying The following characterization of the density of a subalgebra of Lipschitz and bounded functions is proven in [12].
Theorem 2.9.Let Y ⊂ X be a dense subset.Then for every y ∈ Y it holds d y ∈ D 1,q (X, d, m; A ), Dd y ⋆,q,A ≤ 1 (2.12) if and only if A is dense in q-energy according to Definition 2.7, where the function d y : X → [0, +∞) is defined as d y (x) := d(x, y), x ∈ X.

Wasserstein spaces
We devote the first part of this section to a few general properties of Wasserstein spaces (see [29,30,24,2] for a general review of Optimal Transport).The second part of the section will treat properties of Kantorovich potentials in particular geometric situations.We fix an exponent p ∈ (1, +∞) (recall that p ′ := p/(p − 1)) and we remark that all vector spaces (and thus all Banach spaces) are real vector spaces.If (X, d) is a metric space, we denote by P(X) the space of Borel probability measures on X and by P p (X, d) the set < +∞ for some (and hence for all) x 0 ∈ X .
Given µ, ν ∈ P(X) the set of transport plans between µ and ν is denoted by Γ(µ, ν) and defined as Γ(µ, ν) := γ ∈ P(X × X) : π 1 ♯ γ = µ, π 2 ♯ γ = ν , where π i (x 1 , x 2 ) = x i for every (x 1 , x 2 ) ∈ X × X and ♯ denotes the push forward operator as in (2.2).The (p, d)-Wasserstein distance W p,d between µ, ν ∈ P p (X, d) is defined as It is well known that the infimum above is attained in a non-empty and convex set Γ o,p,d (µ, ν) ⊂ Γ(µ, ν) and that the (p, d)-Wasserstein space (P p (X, d), W p,d ) is complete and separable, if (X, d) is complete and separable.The Kantorovich duality for the Wasserstein distance states that for every µ, ν ∈ P p (X, d), It is easy to check that for every f ∈ Lip(X, d) and µ, ν ∈ P p (X, d) we have in fact, choosing γ ∈ Γ o,p,d (µ, ν) and setting L := Lip(f, X), we have In this subsection we study some stability properties for Kantorovich potentials in finite dimensional real Banach spaces for the cost induced by the p-th power of the norm.We thus fix a dimension d ∈ N and a norm • on R d such that the unit • -ball B • (0, 1) := {x ∈ R d : x < 1} is strictly convex and has C where • * is the dual norm of • .Since h is strictly convex, h * is differentiable and we denote by j p ′ the gradient 1 of h * which satisfies the equality where •, • is the standard scalar product on R d .The reason for restricting our analysis to norms satisfying condition (3.3) is that the cost function c(x, y) induced by h through the formula is compatible with the frameworks studied in [13,11] whose results we will often use.More specifically, the function h as above, satisfies hypotheses (H1), (H2) and (H3) in [13]: while (H1) and (H3) are obvious, let us just mention that the smoothness of the unit • -sphere gives that the unit • -ball satisfies the so called ε-ball condition for some ε > 0 (see e.g.[9, Definition 1.1, Theorem 1.9]) which in turn implies (H2).Also notice that, being the unit • -sphere smooth, Finally let us set The next theorem uses the results of Gangbo and McCann [13, Sections 3 and 4] and Figalli and Gigli [11] to collect various useful properties of the optimal potentials realizing the supremum in (3.1) when the support of one of the measures is a closed ball.This result plays the same role of [12, Theorem 3.2] for the case p = q = 2 and • equal to the Euclidean norm.
satisfies also There exists a unique pair as above satisfying conditions (i), (ii) and the additional conditions Such a unique pair is denoted by (Φ(ν, µ), Φ * (ν, µ)).Finally, the function ψ := Φ * (ν, µ) satisfies the following estimates: there exists a constant K p,R , depending only on p and R, such that Proof.Let γ ∈ Γ o,p (ν, µ) be fixed and let c be as in (3.5).We divide the proof in several steps.
) be locally Lipschitz functions satisfying (i) and (ii).Then the maps t, s defined as are Optimal Transport maps from ν to µ and from µ to ν, respectively.In particular φ and ψ satisfy (3.7).
Proof of claim 1.The proof of this statement is classical and thus omitted (see e.g.[30, Theorem 2.12], [24, Theorem 1.17] or [2, Theorem 6.2.4]).By the proof of [2, Theorem 6.14] and [2, Theorem 6.15], we have that there exists a c-concave where u c is the c-transform of u defined as and Recall that c-concavity of u means that there exists some proper v : then ψ is real valued and locally Lipschitz, it is finite on π 2 (supp γ), ψ ∈ L 1 (R d , µ) and Proof of claim 2. Since u(x) + u c (y) = c(x, y) for every (x, y) ∈ supp γ, then for µ-a.e.y ∈ R d there exists some x ∈ B • (0, R) such that u(x)+u c (y) = c(x, y).This gives that u c = ψ µ-a.e. in R d and proves that ψ ∈ L 1 (R d , µ) and (3.14).Since ψ ≥ u c everywhere, we also have that ψ is finite in π 2 (supp γ).Let us prove that ψ is real valued.Since v is proper, there exists some To prove that ψ is locally Lipschitz, it is enough to observe that, for every M > 0 and every where If we define φ as the restriction of u to B • (0, R), then we have that φ is real valued and locally Lipschitz.
Proof of claim 3. Let us denote the effective domain of u by and its interior by Ω.Since u is c-concave, by [11, Step 2 in the proof of Theorem 1], we know that for every point x ∈ D(u) \ Ω there exists some v x ∈ R d \ {0} and ℓ x > 0 such that the interior of the set does not intersect D(u).Thus we have that int(D(u)) ⊂ Ω. Moreover u is locally Lipschitz in Ω by [11, Step 3 in the proof of Theorem 1].It is then enough to prove that B ) and they satisfy points (i), (ii), (iii), (iv) of the statement.
Proof of claim 4. The only nontrivial fact to be checked is (3.8).Let us define the modified cost function c : and the function w : It is thus clear that By [30,Prop. 5.8], we have that w c = w ccc that reduces precisely to ) satisfying points (i), (ii), (iii) and (iv).By claim 1. the maps t, t 0 defined as are Optimal Transport maps from ν to µ.Since ϕ and ϕ 0 are (restrictions of) c-concave functions and both t and t 0 push ν to µ, by [13,Theorem 4.4] we get that t = t 0 ν-a.e. in B • (0, R) and this gives in particular (recall that h is everywhere differentiable) that ∇ϕ 0 = ∇ϕ L d -a.e. in B • (0, R).From this and point (iii) we get uniqueness.Claim 6.Let ψ := Φ * (ν, µ).Then there exists a constant K p,R depending only on p and R such that (3.10), (3.11), (3.12) and (3.13) hold.
Proof of claim 6. (3.10) is a consequence of the elementary inequality and of the fact that (3.11), (3.12) directly follow by (3.10).Finally, (3.13) follows by the inequality that holds for a suitable constant K p > 0 that depends solely on p.In fact, using (iii), we have that ϕ(x) ≤ 1 p x p for every x ∈ R d so that Remark 3.2.There exists a constant D p,R , depending only on p and R such that, if ψ : R d → R is a function satisfying (3.10), then In fact, by (3.10), if we take any γ ∈ Γ o,p (µ ′ , µ ′′ ), we have 1/p ′ so that (3.16) follows.
In the following results we collect some properties of sequences of pairs of potentials as in Theorem 3.1.The aim is to prove that, under suitable conditions, such sequences converge to optimal potentials.In the next Propositions we use the notation Proof.The proof is strongly based on [11] (see also the similar approach in [13, Proposition C.3]) and it is divided in two claims.For the whole proof, we extend ϕ n to R d simply setting Let us suppose by contradiction that there exist some x ∈ B • (0, R) and a sequence ( we get that x − y n → +∞ which in particular gives that x n − y n → +∞.Let us define now the curves γ n : [0, Since x n − y n → +∞, we can assume that x n − y n ≥ 1 for every n ∈ N and define where we have used the elementary inequality (a − b) p ≤ a p − pa p−1 b p for every 0 ≤ a ≤ b.
Letting n → +∞, we obtain the claim.Up to passing to a (unrelabeled) subsequence, we can thus assume that ϕ n (x) ≤ −n for every x ∈ C n , n ∈ N.
Since x ∈ B • (0, R) there exists some ε > 0 such that B • (x, ε) ⊂ B • (0, R).Let N ∈ N be such that x n − x < ε/2 for every n ≥ N and let δ := ε 3 ; it is not difficult to check that the truncated sets Then for every n ≥ N we have where where | • | is the Euclidean norm on R d .Passing to the limit as n → +∞ gives that I = ∞, a contradiction.Claim 2. For every K ⊂⊂ B • (0, R) there exists M K > 0 such that for every x ∈ K and for every n ∈ N, so that the sequence ϕ n is locally (w.r.t. and where we have used claim 1 to ensure that the supremum and the infimum are finite.Let us consider (x, y, n) ∈ K × R d × N as above.To prove that C K is the sought constant, it is enough to consider the case in which x − y ≥ 1.If we define γ : [0, x − y ] → R d as the curve given by we have, arguing as in (3.17), that If x ∈ K and n ∈ N are fixed, we can find a minimizing sequence (y k ) k ⊂ R d , in the sense that so that, by the result we have just proven, (y k ) k is bounded and thus converges, up to a (unrelabeled) subsequence to some ȳ ∈ B • (x, C K ).Passing the above inequality to the limit as k → +∞, we get that This proves that ȳ is a minimizer and, since it belongs to B • (0, M K ), we get the second claim.
Proposition 3.4.Let R, I > 0 and let (ϕ n ) n , (ψ n ) n be sequences of functions such that for every x ∈ B • (0, R), every y ∈ R d and every n ∈ N. Then there exist a subsequence j → n(j) and two locally Lipschitz functions ϕ : B • (0, R) → R and ψ : R d → R such that ϕ n(j) → ϕ locally uniformly in B • (0, R), ψ n(j) → ψ locally uniformly in R d and ) where ∂ c ψ is the c-superdifferential operator of ψ, whose graph is given by Proof.The proof is divided in three claims, the first of which is adapted from [ bounded and Lipschitz.Hence we can apply Ascoli-Arzelà theorem and obtain the existence of a subsequence j → n(j) and two locally Lipschitz functions ϕ : B • (0, R) and ψ : R d → R such that ϕ n(j) → ϕ locally uniformly in B • (0, R) and ψ n(j) → ψ locally uniformly in R d .Claim 1.For every K > 0 there exists a constant C K > 0 such that where ∂ c ψ n is the c-superdifferential operator of ψ n (which is defined analogously to ∂ c ψ adapting (3.20) in the obvious way).Moreover ∂ c ψ n (y) = ∅ for every y ∈ R d and every n ∈ N.
Proof of claim 1.Let us start from the last part of the claim; let y ∈ R d and n ∈ N be fixed.By hypothesis, we can find a sequence (x j ) j ⊂ B • (0, R) such that for every j ∈ N it holds This gives that Up to a (unrelabeled) subsequence, there exists some x ∈ R d such that x j → x and we get passing to the limit the above inequality that Let us come to the first part of the claim.Let K > 0 be fixed and observe that, arguing as in Theorem 3.1 (see in particular (3.11)), the sequence ψ n is locally (w.r.t.y ∈ R d ) uniformly (w.r.t.n ∈ N) bounded, so that there exists some We claim that there exists a constant D K such that, whenever (x, λ, u, n then x ≤ D K .Suppose by contradiction that we can find a sequence ( and x k → +∞, we may assume that v k > 1 for every k ∈ N. Let us also define ξ We can thus evaluate (3.22) and (3.23) written for (x k , λ k , u k , n k ) with z = z k obtaining that We thus deduce that 1 Let now w k be the unique element of the subdifferential of h at the point Combining (3.24) with (3.26) we get where for the last inequality we have used (3.25).Since ξ k ≥ 1 2 and v k → +∞, we get a contradiction, proving our first claim.We set C K := max{K, D K } and we prove that Proof of claim 2. Let y ∈ R d be fixed; observe that, if (x j ) j is any sequence such that x j ∈ ∂ c ψ n(j) (y) for every j ∈ N (there exists at least one such a sequence by claim 1), then by claim 1 ∂ c ψ n(j) (y) are uniformly (w.r.t j ∈ N) bounded so that we can extract a subsequence k → j k such that x j k → x for some x ∈ R d .By the very definition of c-superdifferential, we have that for every k ∈ N it holds Passing to the limit as k → +∞, we get that x ∈ ∂ c ψ(y), so that ∂ c ψ(y) = ∅.This proves in particular that from any sequence of points (x j ) j such that x j ∈ ∂ c ψ n(j) (y) for every j ∈ N we can extract a subsequence converging to an element of ∂ c ψ(y).Let us come now to the proof of the limit.We prove that from any (unrelabeled) subsequence of n(j) we can extract a further subsequence such that we have the convergence above.For every j ∈ N we can find some Reasoning as above, we find a subsequence k → j k such that Then we have This concludes the proof of the claim.Claim 3. We have that ∇ψ n(j) → ∇ψ L d -a.e. in R d .Proof of claim 3. Let A := y ∈ R d : ψ and ψ n(j) are differentiable at y for every j ∈ N ; notice that A has full L d measure.Let y ∈ A; since ψ is differentiable at y we know by [13, Lemma 3.1] that any x ∈ ∂ c ψ(y) will satisfy ∇ψ(y) p ′ * = ∇h(x − y) p ′ * = x − y p so that ∂ c ψ(y) is contained in B • (0, C) for some C > 0. On the other hand, claim 1 implies that there exists some constant D > 0 such that ∂ c ψ n(j) (y) ⊂ B • (0, D) for every j ∈ N. Let us define M > 0 as the uniform modulus of continuity of the map x → ∇h(x − y) in the compact set B • (0, C + D).By claim 2, for every ε > 0 we can find This means that for every j ≥ J ε and every x j ∈ ∂ c ψ n(j) (y) there exists some z j ∈ ∂ c ψ(y) such that x j − z j < ε/M .Using again [13, Lemma 3.1] and the fact that both ψ n(j) and ψ are differentiable at y, we have that ∇ψ n(j) (y) = ∇h(x j − y) and ∇ψ(y) = ∇h(z j − y) for every j ≥ J ε .Then for every j ≥ J ε .This concludes the proof of the claim.
4. The Wasserstein Sobolev space H 1,q (P p (X, d), W p,d , m) The aim of this section is to study the q-Sobolev space on the (p, d)-Wasserstein space on a separable and complete metric space (X, d).In particular we will show at the end of this section that the algebra of cylinder functions generated by a sufficiently rich algebra of Lipschitz and bounded functions on (X, d) is dense in q-energy in the Sobolev space H 1,q (P p (X, d), W p,d , m).For the whole section p, q ∈ (1, +∞) are fixed exponents.

Cylinder functions on (X, d) and their differential in Banach spaces
Let (X, d) be a complete and separable metric space.We extend the definition of cylinder function in [12] to this more general setting.To every φ ∈ Lip b (X, d) we can associate the functional L φ on P(X) Definition 4.1 (E -cylinder functions).Let E ⊂ Lip b (X, d) be an algebra of functions; we say that a function We denote the set of such functions by C P(X), E .
Remark 4.2.Since for every φ ∈ E N the range of L φ is always contained in the bounded set [−M, M ] N where M := max i=1,...,d φ i ∞ , also functions In particular every function of the form L φ , φ ∈ E , belongs to C P(X), E .
We use the notation D X for the Borel set We introduce now the definition of differential of a cylinder function, still following [12], in case We will also denote by DF ψ,φ [µ] the function x → DF ψ,φ (µ, x) and we will set where We define the curves µ, ν : [0, 1] → P p (B, d • ) as where x t : B × B → B is the map defined as x t (x 0 , x 1 ) := (1 − t)x 0 + tx 1 , for every (x 0 , x 1 ) ∈ B × B and t for every s ∈ [0, 1].This is a simple consequence of the chain rule and the regularity of φ. ).We will be then able to remove the subscript in DF ψ,φ and denote it simply by D p F .
The following Lemma is proved in [12] and will be useful in the proof of Proposition 4.7.The following proposition corresponds to [12, Proposition 4.9] and the proof is quite similar but, because of a few differences, we still report it here.
Let (µ t n ) t∈[0,1] be the curves defined as in (4.6) for plans , where the first equality comes from (4.8).Dividing both sides by W p,d • (µ ′ n , µ ′′ n ), we obtain Observe that µ n → (i B , i B ) ♯ µ in P(B × B) so that µ t n → µ in P(B) for every t ∈ [0, 1].We can pass to the limit as n → +∞ the above inequality using the dominated convergence Theorem and Lemma 4.6 with To prove the other inequality we consider a countable dense subset E := {x n } n of the unit sphere in B and, for every ε > 0, the maps N ε : B * → N and j p ′ ,ε : B * → B defined as It is not difficult to check that j p ′ ,ε is measurable: if we define the sets then {U n } n∈N is a countable measurable partition of B * and j p ′ ,ε can be written as j p ′ ,ε (x * ) = x * p ′ /p * x n if x ∈ U n , so that j p ′ ,ε is measurable.Moreover it is obvious that

Thus lip
.
Passing to lim ε↓0 we obtain the sought inequality and this concludes the proof.
where Recall that for a bounded Lipschitz function F : P p (R d ) → R the pre-Cheeger energy (cf.(2.9)) associated to m is defined by q dm(µ).
q/p ′ dm(µ), (4.15)where • * is the dual norm induced by • .Notice that (pCE q ) 1/q is not simply the L q -L p ′ mixed norm of D p F in a Bochner space, since the measures µ w.r.t. the inner norm is computed varies, but rather the norm in the direct L q (P p (R d ), m)-integral of the Banach spaces L p ′ (B, µ; (B * , • * )) (see e.g.[17,10] or Section 5).We adopt the notation such that F n and |DF n | ⋆,q,A are uniformly bounded in every bounded set of P p (R d ) and let F, G be Borel functions in L q (P p (R d ), m), G nonnegative.If ) be such that supp κ = B(0, 1) (here B(0, 1) is the unit d-dimensional Euclidean ball) , κ(x) ≥ 0 for every x ∈ R d , κ(x) > 0 for every x ∈ B(0, 1), R d κdL d = 1 and κ(−x) = κ(x) for every x ∈ R d .Let us define, for every 0 < ε < 1, the standard mollifiers and it is easy to check that, if we set then we have Definition 4.12.Let 0 < ε < 1 and ν ∈ P p (R d ).We define the continuous functions The proof of the following proposition follows the one of [12, Proposition 4.17] but since the exponent p may be different from 2, the estimates are more complicated and thus reported in full.
Proposition 4.13.Let 0 < ε < 1, δ, R > 0, let ν ∈ P r p (R d ) be such that supp ν = B • (0, R) and ν B • (0, R) ≥ δL d B • (0, R), and let ζ : R → R be a C 1 nondecreasing function whose derivative has compact support.Then Proof.Let G := {µ h } h∈N be a dense and countable set in P p (R d ) and let us set, for every h ∈ N, Notice that, by (3.11), By the definition of ϕ h and ϕ * h we have, for every µ ∈ P p (R d ) and h ∈ N, that Clearly, if µ ∈ G, this is an equality.By (3.16) there exists a constant D p,R such that, for every h ∈ N and µ, µ where we used (4.19) and (4.21).We hence deduce that for every Choosing µ ′ ∈ G and passing to the limit as k → +∞ we get from (4.23) that Using the density of G and the continuity of . ., k}.Proof of claim 2. For every h ∈ N, (3.13) and (3.12) (also using (3.15)) yield where A p,R,ε > 0 is a constant depending only on p, R, ε.Since the map ℓ h : Lemma 4.11 and the above estimates yield Since H k can be written as we can apply Theorem 2.5 (4) and conclude the proof of the second claim.Claim 3.For every R > 0 there exists a constant C > 0 independent of h such that Proof of Claim 3. It is sufficient to use (4.24).Claim 4. Let (h n ) n ⊂ N be an increasing sequence and let It is then enough to prove that Let us set φ n := ϕ hn and ψ n := ϕ * hn and let us extract a (unrelabeled) subsequence such that the lim sup in the statement of the claim is achieved as a limit.Since By the convergence of R d u hn dµ ε to F ε ν (µ), we can find N ∈ N such that where the last inequality comes from (3.11).Combining the above inequality with (4.27) we get that there exists some I > 0 such that We can thus apply Proposition 3.4 and obtain the existence of a subsequence j → n(j) and two locally Lipschitz functions φ and ψ such that φ n(j) → φ locally uniformly in B • (0, R), ψ n(j) → ψ locally uniformly in R d and (3.19) holds.Being the inequality satisfied for every j ∈ N, we can pass to the limit and obtain point (i) in Theorem 3.1 for the pair (φ, ψ).Since φ n (x) ≤ 1 p x p for every x ∈ R d we get by Fatou's lemma that This in particular gives that φ ∈ L 1 (B • (0, R), ν).By (3.11) we have that |ψ(y)| ≤ K p,R (1+ y p ) so that ψ ∈ L 1 (R d , µ) and we can also apply the dominated convergence theorem and obtain that Combining (4.29) with (4.30) and using that lim n R d u hn dµ ε = F ε ν (µ), we get point (ii) in Theorem 3.1 for the pair (φ, ψ), so that, by the first part of Theorem 3.1, we conclude that Then, since by (3.12) there exists a constant K p,R > 0 such that e. y ∈ R d , and for every j ∈ N, we can use the dominated convergence theorem to conclude that This concludes the proof of the fourth claim.Claim 5.It holds Proof of claim 5. Let B ⊂ P p (R d ) be defined as where A k h is the full m-measure subset of B k h where claim 2 holds.Notice that B has full m-measure.Let µ ∈ B be fixed and let us pick an increasing sequence k → h k such that By claim 1 we know that G n (µ) → F ε ν (µ) so that we can apply claim 4 and conclude that By claim 2, the right hand side is greater than lim sup k |DH k | p ′ ⋆,q,A (µ); this concludes the proof of the fifth claim.Eventually, we observe that by Claim 1 ) is uniformly bounded.We can then combine the expression of Claim 2, the uniform estimate of Claim 3, the limit of Claim 5 with Lemma 4.10 to get (4.22).
Precisely as in [12,Corollary 4.18] we get the following corollary (its proof can be easily adapted and thus omitted). (4.32) The following theorem provides the main result of this subsection.Recall the notation for Wasserstein spaces as in Section 3 and the definition of density in q-energy as in Definition 2.7.
Theorem 4.15.Let d ∈ N, let p, q ∈ (1, +∞) be (not necessarily conjugate) fixed exponents and let • be any norm on R d .Then the algebra , where d • is the distance induced by the norm • .
Proof.Let us first suppose that • is a norm as in (3.3).Then the density result follows by Corollary 4.14 and Theorem 2.9, since the set of ν ∈ P r p (R d ) such that there exist R, δ > 0 for which supp(ν) = B • (0, R) and ν B • (0, R) ≥ δL d B • (0, R) is dense in P p (R d ).This can be seen defining for every ε ∈ (0, 1) the measures 3) and such that For example we can take as • k the norm whose unit ball is given by the set where | • | is the Euclidean norm on R d and η > 0 is a constant such that |x| ≤ η x for every x ∈ R d .Notice that by construction C k is strictly convex, it contains the • -unit ball and it satisfies the 1/k-ball condition so that its boundary is C m) by the first part of the proof, we have that Passing to the lim inf n we get that which gives the desired density, since this entails equality of the Cheeger energies and thus of the minimal relaxed gradients (see also Remark 2.8). 3 Here we are adding a subscript both in the notation for the Cheeger and the pre-Cheeger energy to specify the distance w.r.t. which they are computed.
Arguing precisely as in [12,Proposition 4.19], it is not difficult to see that we can obtain the density result also for smaller algebras of functions on R d .
Proposition 4.16.Let d ∈ N, p, q ∈ (1, +∞) and let • be any norm on R d .Let E ⊂ C 1 b (R d ) be an algebra of functions such that for every Then the algebra C P(R d ), E is dense in q-energy in D where C > 0 is any positive constant.

The density result in (X, d)
In this subsection we prove the density in energy of suitable cylinder functions for an arbitrary separable metric space (X, d).We consider on (X, d) a sequence Φ : and we define Φ(d We will have to embed (X, d) into ℓ ∞ (N) and, for this reason, we need to fix some related notation.
We consider the family of maps E := {e i } i∈N ⊂ (ℓ ∞ (N)) * , where for every i ∈ N, e i is defined as We consider the collection of maps (4.39) On P(ℓ ∞ (N)), we consider the algebra of cylinder functions  and let E ⊂ Lip b (X, d) be the smallest unital subalgebra of functions on X containing {φ k } k∈N .Finally, let p, q ∈ (1, +∞) be (not necessarily conjugate) fixed exponents and let m be a positive and finite Borel measure on P p (X, d); then the algebra of cylinder functions generated by E , C P(X), E , is dense in q-energy in D 1,q (P p (X, d), W p,d , m).

.44)
We split the proof in two steps.
Step 1: it is sufficient to prove that, for every h ∈ N, the function where πh : In fact, using the continuity property of the Wasserstein distance, it is clear that for every so that it is enough to apply Theorem 2.5( 1)-( 3) to obtain (4.44).
Step 2: Let h ∈ N be fixed and let us denote by W p,h the Wasserstein distance on P Thus, if we define the function Fh : We also introduce the measure m h which is the push-forward of m through the (1-Lipschitz) map P h : P , m h ), we can find a sequence of cylinder functions Fh,n ∈

.48)
We thus consider the functions F h,n ∈ A ′ defined as in (4.41) by M) satisfying (4.34) for d g .

Reflexivity, uniform convexity and Clarkson's inequalities
The aim of this section is to study how some properties of a Banach space (or its dual) pass to the Wasserstein Sobolev space built on it.In the whole section (B, • ) is a fixed separable Banach space and m is a positive and finite Borel measure on P p (B, d • ).
To study the properties of the pre-Cheeger energy as in (4.15) we will use the concept of L q -direct integral of Banach spaces ( [10,17]) (see also the related notion of Banach bundle [23]), that we introduce here briefly in the form that it is best suited to our needs.Let V be the vector space of functions It is not difficult to check that, for every f ∈ V , the map M f : The direct integral of (V µ ) µ with respect to m is the vector space of Bochner measurable functions G : Proof.The completeness property can be found in [10, Proposition 3.2] (see also [17] where the notion of direct integral was introduced for the first time and let us define G ⊂ X × Y as the closure in X × Y (endowed with the norm (x, y) F n → |DF | ⋆,q in X.
By Proposition 4.7, we have that D p F n Y is uniformly bounded so that, by the reflexivity of Y , we can find a subsequence of (F n k ) k such that This gives that F ∈ π X (G) and that G Y ≤ CE 1/q q (F ).Moreover, if we consider F ∈ H 1,q (P p (B, d • ), W p,d • , m) and G ∈ D m F , then there exists (F n ) n ⊂ C P(B), C 1 b (B) such that (F n , D p F ) → (F, G) in X × Y so that G Y = lim n D p F n Y = lim n pCE 1/q q (F n ) ≥ CE 1/q q (F ) so that min G∈DmF G Y = CE 1/q q (F ), F ∈ H 1,q (P p (B, d • ), W p,d • , m).
We define G 0 := G/({0} × D m 0) with the quotient norm • Y,0 : elements of G 0 are equivalence classes [F ] = {(F, G) : G ∈ D m F } and the quotient norm is simply given by The linear isometry is thus given by ι : H 1,q (P p (B, d • ), W p,d • , m) → G 0 defined as Since closed subspaces of reflexive spaces are reflexive and quotients of reflexive spaces are reflexive, G 0 is reflexive and this concludes the proof.
The next result [31,Lemma 2] provides a quantitative version of the uniform convexity property that will pass from the pre-Cheeger energy to the Cheeger energy.Proof.By the uniform convexity of Pp(B,d • ) V µ dm(µ) L q provided by Proposition 5.1 and using Proposition 5.3, we have that for every t ∈ (1, +∞) there exists a continuous and strictly increasing function g t : [0, 2] → [0, +∞) such that pCE q 1 2 (F + G) t/q + pCE 1/q q (F ) ∨ pCE 1/q q (G) t g t pCE 1/q q (F − G) pCE 1/q q (F ) ∨ pCE 1/q q (G) ≤ 1 2 pCE t/q q (F ) + 1 2 pCE t/q q (G) for every F, G ∈ C P(B), where we are able to identify the norm in the direct integral and the q-th root of the pre-Cheeger energy thanks to Proposition 4.7.If F, G ∈ H By the lower semicontinuity of the Cheeger energy w.r.t. the L q convergence and the continuity and monotonicity of g t , we obtain that (5.1) holds for CE q instead of pCE q .By Proposition 5.3 we obtain that CE 1/q q is uniformly convex, in the sense that, for every ε > 0 there exists a δ(ε) > 0 such that, whenever F, G ∈ H 1,q (P p (B, d • ), W p,d • , m) are such that CE 1/q q (F ) = CE 1/q q (G) = 1 and CE 1/q q (F − G) ≥ ε, then CE 1/q q (F + G) < 2(1 − δ(ε)).Since the L q (P p (B, d • , m)-norm is uniformly convex, the q-sum (i.e. the q-th root of the sum of the q-powers) of it with CE 1/q q is uniformly convex (see e.g.[8,Theorem 1]).This means that the Sobolev norm is uniformly convex and concludes the proof.It is immediate to check that, for every q ∈ (1, +∞), the Sobolev spaces H 1,q (B, d • , m) and H 1,q (P 2 (B, d • ), W 2,d • , D ♯ m) are isomorphic, see also [12,Section 5.2].Notice that the choice p = 2 is irrelevant, since D(B) ⊂ P p (B, d • ) for every p ∈ [1, +∞].As a consequence of this observation and Theorems 5.2 and 5.4 we obtain the following corollary.
We remark that the statement regarding reflexivity in the above corollary was already known (see e.g.[25,Corollary 5.3.11])while the part concerning uniform convexity is new, at least to our knowledge.

Clarkson's type inequalities
In this subsection we show how one can deduce Clarkson's type inequalities for the Sobolev norm starting from analogous inequalities that are satisfied for the dual norm in the separable Banach space B. The following inequality was introduced for the first time by [6] and it is a generalization of the classical Clarkson's inequalities [8].Definition 5.6.Let V be a real vector space and let r, s ∈ (1, +∞).We say that a functional J : V → [0, +∞) satisfies the (r, s)-Boas inequality ((r, s)-(BI)) in V if (J(u + v) r + J(u − v) r ) 1/r ≤ 2 1/s ′ (J(u) s + J(v) s ) 1/s for every u, v ∈ V.
Proof.Let u, v ∈ V ; let us set Since J 1 and J 2 satisfy the (r, s)-(BI) in V , we have Then 2 1/s ′ (J(u) s + J(v) s ) 1/s = 2 1/s ′ (a q + c q ) s/q + (b q + d q ) s/q 1/s = 2 1/s ′ ((a s ) q/s + (c s ) q/s ) s/q + ((b s ) q/s + (d s ) q/s ) s/q 1/s ≥ 2 1/s ′ (a s + b s ) q/s + (c s + d s ) q/s 1/q ≥ (a r 0 + b r 0 ) q/r + (c r 0 + d r 0 ) q/r 1/q = (a q 0 ) r/q + (b q 0 ) r/q q/r + (c q 0 ) r/q + (d q 0 ) r/q q/r 1/q ≥ (a q 0 + c q 0 ) r/q + (b q 0 + d q 0 ) r/q 1/r = (J(u + v) r + J(u − v) r ) 1/r , where we have used the triangular inequality for the α-norm in R 2 first with α = q/s (from the second to the third line) and then with α = r/q (from the fifth to the sixth line), and the inequalities (5.2) and (5.3) to pass from the third to the fourth line.

20 ,
Proposition 13.14].We consider the complete and separable metric space (R d , d • ), where d • is the distance induced by • , and the corresponding (p, d • )-Wasserstein space.To simplify the notation, in this subsection, we will simply write P p (R d ), W p and Γ o,p , omitting the dependence on d • .Moreover we denote by P r p (R d ) the subset of P p (R d ) of probability measures that are absolutely continuous w.r.t. the d-dimensional Lebesgue measure L d and we set
and satisfies(3.22)and(3.23)by the very definition of ∂ c ψ n so that by the above claim we get that x ≤ D K ≤ C K and of course y ≤ K ≤ C K .This concludes the proof of the claim.Claim 2. We have that (X, d) = (B, d • ), where (B, • ) is a Banach space and d • is the distance induced by • .In this case we denote by C 1 b (B) the space of bounded, continuously differentiable and Lipschitz functions on B. Definition 4.3.Let F ∈ C P(B), C 1 b (B) ; then, given N ∈ N, ψ ∈ C 1 b (R N ) and φ ∈ (C 1 b (B)) N such that F = ψ • L φ , we define the Wasserstein differential of F conditioned to (ψ, φ), DF ψ,φ : D B → B * , as DF ψ,φ (µ, x) := N n=1 ∂ n ψ (L φ (µ)) ∇φ n (x), (µ, x) ∈ D B .(4.4)

Remark 4 . 5 .
It is not difficult to check that DF ψ,φ is continuous in P(B) × B (4.10) with respect to the natural product (narrow and norm) topology.In principle DF ψ,φ may depend on the choice of N ∈ N, ψ ∈ C 1 b (R N ) and φ ∈ (C 1 b (B)) N used to represent F .In Proposition 4.7 we show that for every µ ∈ P p (B, d • ) the function DF ψ,φ [µ] is uniquely characterized in supp(µ) so that DF ψ,φ is uniquely characterized by F in D p B := D B ∩ (P p (B, d • ) × B

Lemma 4 . 6 .
Let Y be a Polish space and let G : P(Y ) × Y → [0, +∞) be a bounded and continuous function.If (µ n ) n∈N is a sequence in P(Y ) narrowly converging (i.e.converging in duality with continuous and bounded functions on Y ) to µ as n → +∞, then lim n→∞ Y G(µ n , y)dµ n (y) = Y G(µ, y)dµ(y).

5 . 4 . 2 .
and D p is as in Remark 4.The density result in (R d , • ) for any normIn this whole subsection (apart from Theorem 4.15) we focus again on a finite dimensional Banach space with a sufficiently regular norm.To this aim we fix a dimension d ∈ N and a norm• on R d satisfying (3.3).As we did in Subsection 3.1, we work on the complete and separable metric space (R d , d • ), where d • is the distance induced by • , and the corresponding (p, d • )-Wasserstein space.To simplify the notation, as we did in Subsection 3.1, in this subsection, we will simply write P p (R d ), W p and Γ o,p , omitting the dependence on d • .For the rest of this subsection m is a positive and finite Borel measure on P p (R d ).

(4. 14 )
Thanks to Proposition 4.7, if F is a cylinder function in C P(R d ), C 1 b (R d ) , we have a nice equivalent expression ) and we devote this subsection to the proof of Theorem 4.15.The following Lemmas are the obvious adaptation of [12, Lemma 4.14, Lemma 4.15] and their proofs are the same, and thus omitted.

Lemma 4 . 10 .
Let F n be a sequence of functions in D1,q

Proof.
It is enough to consider the functions of the form φ k (x) := x * k , x , x ∈ B, where {x * k } k∈N is a subset of unit sphere in B * separating the points in B. Notice that each φ k belongs to C 1 b (B).Corollary 4.20.Let (M, g) be a complete Riemannian manifold and let m be a positive and finite Borel measure on P p (M, d g ), where d g is the Riemannian distance induced by g.Then the algebra C P Given a separable Banach space (B, • ) and a positive and finite Borel measure m on B, we can consider the measure D ♯ m on P(B, d • ), where D : B → P(B, d • ) is defined as D(x) := δ x , x ∈ B.
1,1(see [9, Definition 1.1 and Theorem 1.8]).Also by construction we have that• k ↑ • as k → +∞.Let us denote by W p,k the (p, d • k )-Wasserstein distance on P p (R d ) = P p (R d , d • ) = P p (R d , d • k ).We have that W p,k (µ, ν) ↑ W p,d • (µ,ν) for every µ, ν ∈ P p (R d ) and, being the norms • and • k equivalent, then the distances W p,k and W p,d • induce the same topology on P p (R d ).Let F ∈ Lip b (P p (R d ), W p,d • ); by [22, Proposition 3.3] (see also [1, Theorem 9.1]), we can find a subsequence (k n ) n such that, for every n ∈ N, there exists F n ∈ Lip b (P p (R d ), W p,kn ) such that 3 1,p(P p (R d , d • ), W p,d • , m).
.40) It is also easy to check that a function F belongs to A ′ if and only if there exists d ∈ N and F ∈ C P(R d ), P d,Φ(d) such that Corollary 4.19.Let (B, • ) be a separable Banach space and let m be a positive and finite Borel measure on P p (B, d • ).Then the algebra C .47) yields that F h,n → F h in m-measure as n → ∞.On the other hand, (4.43) and Remark 4.8 yieldlip W p,d • ′ F h,n (µ) ≤ lip W p,h Fh,n (P h (µ))so that, up to a unrelabeled subsequence, we havelip W p,d • ′ F h,n ⇀ G h ≤ g h • P h in L q (P p (B, d • ), m)and g h • P h ≤ 1 m-a.e. in P p (ι(X), d • ′ ).By Theorem 2.5(1)-(3), we obtain (4.45), concluding the proof.
where ∼ µ is the equivalence relation of the equality µ-a.e..We say that s :P p (B, d • ) → µ V µ (i.e.s(µ) ∈ V µ for every µ ∈ P p (B, d • ))is a simple function if there exist a finite measurable partition of P p (B, d • ), {A k } N k=1 , and values {f k [17, (B * , • * ) is uniformly convex, then we can use [15, Theorem 2.2], together with the fact that L q (P p (B, d • ), m) is uniformly convex and that the modulus of convexity of V µ is larger than the one of L p ′ (B, µ; (B * , • )), being V µ just a subspace of the latter, and the fact that the modulus of convexity of the Bochner space L p ′ (B, µ; (B * , • )) does not depend on µ but only on p ′ and on the modulus of convexity of the uniformly convex space (B * , • * ).If B is reflexive, then also B * and thus L p ′ (B, µ; (B * , • * )) is reflexive so that we can apply[17,  Theorem 6.19], also noting that, obviously, L q for some N ∈ N; notice that we are looking at D p F as the map sending µ ∈ P p (B, d• ) to D p [µ] : B → B * .Of course D p F [µ] ∈ V µ for every µ ∈ P p (B,d • ) and it is uniformly bounded.Moreover it is Bochner measurable since, in order to approximate it with simple functions, it is enough to approximate every term of the form ∂ψ n ( B φ 1 dµ, . . ., B φ N dµ) with R-valued simple function (s n k ) k on P p (B, d • ) (which is possible since this map is measurable) so that (µ)∇φ n , µ ∈ P p (B, d • ), is a sequence of simple functions approximating D p F .Theorem 5.2.Let (B, • ) be a separable reflexive Banach space, let p, q ∈ (1, +∞) be (not necessarily conjugate) fixed exponents and let m be a positive and finite Borel measure on P p (B, d • ).Then the Sobolev space H 1,q (P p (B, d • ), W p,d • , m) is reflexive.Proof.It is sufficient to provide a linear isometry ι from H 1,q (P p (B, d • ), W p,d • , m) into a reflexive Banach space.For simplicity let us denote by by Corollary 4.19, we can find a sequence (F n) n ⊂ C P(B), C 1 b (B) such that F n → F, lip W p,d • Proposition 5.3.Let (W , • W ) be a Banach space.Then (W , • W ) is uniformly convex if and only if for every t ∈ (1, +∞) there exists a strictly increasing and continuous functiong t : [0, 2] → [0, +∞) such that W ∨ y W ) t g t x − y W x W ∨ y W )Theorem 5.4.Let (B, • ) be a separable Banach space such that (B * , • * ) is uniformly convex, let p, q ∈ (1, +∞) be (not necessarily conjugate) fixed exponents and let m be a positive and finite Borel measure on P p (B, d • ).Then the Sobolev space H 1,q (P p (B, d • ), W p,d • , m) is uniformly convex.
t W + ( x W for every x, y ∈ W . 1,q (P p (B, d • ), W p,d • , m), thanks to Corollary 4.19, we can find sequences (F n ) n , (G n ) n ⊂ C P(B), C 1 b (B) such that