Yet another proof of the density in energy of Lipschitz functions

We provide a new, short proof of the density in energy of Lipschitz functions into the metric Sobolev space defined by using plans with barycenter (and thus, a fortiori, into the Newtonian-Sobolev space). Our result covers first-order Sobolev spaces of exponent $p\in(1,\infty)$, defined over a complete and separable metric space endowed with a boundedly-finite Borel measure. Our proof is based on a completely smooth analysis: first we reduce the problem to the Banach space setting, where we consider smooth functions instead of Lipschitz ones, then we rely on classical tools in convex analysis and on the superposition principle for normal $1$-currents. Along the way, we obtain a new proof of the density in energy of smooth cylindrical functions in Sobolev spaces defined over a separable Banach space endowed with a finite Borel measure.

1. Introduction 1.1.General overview.Sobolev calculus on metric measure spaces has been a field of intense research activity for almost three decades.In this paper, we focus on two approaches: the Sobolev space H 1,p (X, µ) obtained via relaxation and the Sobolev space W 1,p (X, µ) defined using plans with barycenter.More specifically, fix a metric measure space (X, d, µ), i.e. (X, d) is a complete separable metric space endowed with a boundedly-finite Borel measure µ ≥ 0, and p ∈ (1, ∞).Then: • By H 1,p (X, µ) we mean the Sobolev space via relaxation of Lipschitz functions, which was introduced by Ambrosio-Gigli-Savaré [3] as a variant of Cheeger's approach [6]; see Definition 2.4.We denote by |Df | H the minimal relaxed slope of f ∈ H 1,p (X, µ).• By W 1,p (X, µ) we mean the Sobolev space defined using plans with barycenter, which was introduced in [23] after [1,3,2]; see Definition 2.8.The notion of plan with barycenter we consider is essentially taken from [23].We denote by |Df | W the minimal weak upper gradient of f ∈ W 1,p (X, µ), while B q (X, µ) is the space of plans π with barycenter Bar(π) in L q (µ), where q stands for the conjugate exponent of p; see Definition 2.6.
In this paper, we provide a new proof of the equivalence between H 1,p (X, µ) and W 1,p (X, µ), i.e.
It is worth underlining that we do not make any additional assumptions on (X, d, µ).In particular, we are not assuming that µ is doubling nor the validity of a Poincaré inequality; in the doubling-Poincaré framework, the equivalence was proved in [6,24].We can rephrase (1.1) as follows: Lipschitz functions are dense in energy in W 1,p (X, µ).
The result stated in (1.1) or variants of it were obtained earlier in the literature: • The first proof was obtained by Ambrosio-Gigli-Savaré in [2], where another class of plans (called test plans) was used; see Remark 3.6 for a comparison with the notion of W 1,p (X, µ) we consider in this paper.The proof in [2] is based on the metric Hopf-Lax semigroup.• Savaré proved in [23] that (1.1) holds by using the von Neumann min-max principle and two representations of the dual Cheeger energy.This approach is the closest to ours.• Eriksson-Bique proved in [10] a variant of (1.1) via a more direct approximation technique.
On the one hand, the result in [10] is (a priori) weaker, since it shows the identification between H 1,p (X, µ) and the Newtonian-Sobolev space N 1,p (X, µ); see the relative discussion in Section 1.3.On the other hand, [10] covers also the case of the exponent p = 1.
Compared to the previous arguments, the novelty of our proof of (1.1) is that it relies on a purely smooth analysis.More precisely, since we can embed (X, d) isometrically into a Banach space, we can reduce the problem to the case where X = B itself is a Banach space.In this framework, we argue by using only smooth functions, their Fréchet differentials, classical tools in convex analysis, and normal 1-currents.Neither Lipschitz functions nor other metric tools are actually needed.
1.2.The proof strategy.Up to a localisation argument and a Kuratowski embedding, we can reduce ourselves to addressing the problem in the case where µ is finite and X = B is a separable Banach space.We then consider the algebra Cyl(B) of cylindrical functions (Definition 2.2).The advantage of working with cylindrical functions is that they are both smooth (of class C ∞ ) and strongly dense in L p (µ).We define the space H 1,p cyl (B, µ) in analogy with H 1,p (B, µ), but using cylindrical functions in the relaxation procedure instead of Lipschitz functions.We denote the corresponding minimal relaxed slope by |Df | H,cyl .Therefore, the new goal is to prove that It is easy to show that H In order to prove (1.3), we apply well-known results in convex analysis about Fenchel conjugates; see (the proof of) Theorem 3.3.Our arguments are strongly inspired by some ideas contained in Bouchitté-Buttazzo-Seppecher's paper [5], where Sobolev spaces on weighted Euclidean spaces were introduced.Roughly speaking, we consider the densely-defined unbounded linear operator d : L p (µ) → L p (µ; B * ) with domain D(d) = Cyl(B), which assigns to each function f ∈ Cyl(B) the µ-a.e.equivalence class df of its Fréchet differential.To prove the property (1.3) amounts to showing that sc , where sc − F denotes the weak lower semicontinuous envelope of the functional F : L p (µ) → [0, +∞] given by and F (f ) := +∞ otherwise (some extra care is needed when µ is not fully supported).In order to achieve this goal, we need to prove the following statement: given any L ∈ D(d * ), there exists a plan π ∈ B q (B, µ) such that ∂π = (d * L)µ (see (2.1)) and Bar(π) L q (µ) ≤ L L p (µ;B * ) * .Here, we denote by d * : L p (µ; B * ) * → L q (µ) the adjoint operator of d.This is the content of Proposition 3.1, whose proof is based on Smirnov's superposition principle for normal 1-currents [25].
1.3.Some additional comments.With (1.2), we recover a result by Savaré [23], which states that cylindrical functions are dense in energy in W 1,p (B, µ); see also the paper [12].Whereas Savaré obtains (1.2) as a consequence of the density in energy of Lipschitz functions, our proof goes in the opposite direction: we prove directly (1.2), then we obtain (1.1) as a corollary.
We now consider the Newtonian-Sobolev space N 1,p (X, µ) introduced by Shanmugalingam [24] (see also [15]) and the associated notion of minimal weak upper gradient |Df | N .It is not too difficult to show that H 1,p (X, µ) , in other words that Lipschitz functions are dense in energy in the Newtonian-Sobolev space; cf. with Remark 3.5.
We also mention that it seems that our proof strategy cannot be used to prove the identification between the spaces H 1,1 and W 1,1 .Nevertheless, we do believe that it can be adapted to show the equivalence of the different notions of functions of bounded variation, as well as to study various notions of Sobolev spaces of exponent p = ∞.These questions will be addressed in future works.
Acknowledgements.The authors thank Luigi Ambrosio, Giuseppe Buttazzo, Toni Ikonen, and Giacomo Del Nin for the useful discussions on the topics of this paper.The second named author has been supported by the MIUR-PRIN 202244A7YL project "Gradient Flows and Non-Smooth Geometric Structures with Applications to Optimization and Machine Learning".

Preliminaries
Given p ∈ [1, ∞), we tacitly denote by q := p p−1 ∈ (1, ∞] its conjugate exponent, and vice versa. .We denote by M(X) the set of (finite) signed Borel measures on X and M + (X) := {µ ∈ M(X) : µ ≥ 0}.For any µ ∈ M(X), we denote by µ + , µ − ∈ M + (X) the positive part and the negative part of µ, respectively.Recall that µ = µ + − µ − .The total variation measure of µ ∈ M(X) is defined as . We endow M(X) with the weak topology, i.e. with the coarsest topology such that M(X) whose support spt(f ) is bounded and we define LIP bs (X) := LIP(X) ∩ C bs (X).By Lip(f ) we mean the Lipschitz constant of f ∈ LIP(X), while its asymptotic slope lip a (f ) : X → [0, +∞) is given by Let us now focus on the space C([0, 1]; X) of curves.The evaluation maps e ± : C([0, 1]; X) → X are the 1-Lipschitz maps given by e + (γ) := γ 1 and e − (γ where L 1 stands for the restriction of the one-dimensional Lebesgue measure to [0, 1].The length of γ is defined as ℓ(γ) := By a plan on X we mean any measure π ∈ M + (C([0, 1]; X)).We define the boundary of π as (2.1) Moreover, we define the Borel measure π ≥ 0 on X as One can also readily prove that, given any function f ∈ C b (X) such that f ≥ 0, it holds that 2.2.Banach spaces and 1-currents.Let B be a Banach space.For any function ).When V is a Euclidean space, these are the 1-currents in the sense of Federer-Fleming [11].The elements of M 1 (V) can be identified with the V-valued Borel measures on V, thus we can consider the total variation measure We say that T is acyclic if its unique cycle is the null current.Then the following result holds (see e.g.[18, Proposition 3.8]): for any T ∈ M 1 (V), there exists a cycle C of T such that T − C is acyclic.The following result states that acyclic normal 1-currents are superpositions of curves: Theorem 2.1 (Superposition principle).Let V be a finite-dimensional Banach space.Then for every acyclic current T ∈ N 1 (V) there exists π ∈ M + (C([0, 1]; V)) concentrated on non-constant Lipschitz curves of constant speed such that (e + ) # π = (∂T ) + , (e − ) # π = (∂T ) − , and T = π .
Proof.Since all norms on a finite-dimensional vector space are equivalent and the Euclidean norm is strictly convex, one can deduce the statement from Smirnov's results in [25].Alternatively, one can argue as follows: the metric 1-currents on V can be identified with the V-valued Borel measures on V (see [20,Lemma A.3]), thus the statement follows from [18,Lemma 5.4].
We will focus on a distinguished class of smooth functions: the algebra of cylindrical functions.

Definition 2.2 (Cylindrical function). Let B be a Banach space. Then we say that
, and some bounded linear map p : B → V. We denote by Cyl(B) the space of cylindrical functions.
Given Banach spaces B, V with V finite-dimensional and a linear 1-Lipschitz operator p : B → V, we define the pullback operator p * : where p adj : V * → B * stands for the adjoint of p, which is a linear 1-Lipschitz operator.Hence, ) , and that p * (df ) = d(f • p) thanks to the chain rule for Fréchet differentials.Given any µ ∈ M + (B) and p ∈ [1, ∞), the µ-a.e.equivalence class [p * ω] µ of p * ω belongs to the Lebesgue-Bochner space L p (µ; B * ), which consists of all L p (µ)-integrable maps from B to B * in the sense of Bochner [9] 2.3.Metric Sobolev spaces.By a metric measure space (X, d, µ) we mean a complete and separable metric space (X, d) together with a boundedly-finite Borel measure µ ≥ 0 on X, where "boundedly-finite" means that µ(B) < +∞ whenever B ⊆ X is a bounded Borel set.Given any exponent p ∈ [1, ∞], we denote by (L p (µ), • L p (µ) ) the p-Lebesgue space on (X, d, µ).For any measurable function f : X → R, we denote by [f ] µ its equivalence class up to µ-a.e.equality.If μ is a boundedly-finite Borel measure on X such that µ ≤ μ, then we denote by ext μ : L p (µ) → L p (μ) the unique map satisfying [ext μ(f )] µ = f and ext μ(f ) = 0 μ-a.e. on dµ dμ = 0 for every f ∈ L p (µ).
Remark 2.3.Let (X, d, µ) be a metric measure space with spt(µ) = X and let Indeed, if two continuous functions agree µ-a.e. on X, then they agree everywhere on spt(µ).

2.3.1.
Sobolev spaces via relaxation.The first notion of metric Sobolev space we recall is based on a relaxation procedure.The next definition, taken from [2], is a variant of Cheeger's one [6].
Definition 2.4 (Sobolev space via relaxation of Lipschitz functions).Let (X, d, µ) be a metric measure space and p ∈ (1, ∞).We define the Cheeger energy functional Ch : Then we define H 1,p (X, µ) On a weighted Banach space, one can give a similar definition using cylindrical functions instead: Definition 2.5 (Sobolev space via relaxation of cylindrical functions).Let B be a separable Banach space and µ ∈ M + (B).Let p ∈ (1, ∞) be a given exponent.We define the cylindrical Cheeger energy functional Ch cyl : L p (µ) → [0, +∞] as Then we define Definition 2.5 is a particular instance of the notion of metric Sobolev space via relaxation introduced in [23], because Cyl(B) is a unital separating subalgebra of LIP b (B) [ We define B q (X, µ) as the set of all π ∈ M + (C([0, 1]; X)) concentrated on LIP([0, 1]; X) such that: i) π has barycenter in L q (µ), i.e. there exists a (unique) function Bar(π) ∈ L q (µ) such that ii) It holds that (e ± ) # π ≪ µ and In Corollary 2.10 and Proposition 3.1, we will need the following technical result about plans.
Remark 2.11.Let (X, d, µ) be a metric measure space and p ∈ (1, ∞).Let (Ω n ) n be a sequence of open sets in X such that Ω n ⊆ Ω n+1 for all n ∈ N and X = n∈N Ω n .Let f ∈ L p (µ) be such that f n ∈ H 1,p (X, µ| Ωn ) for every n ∈ N and s := sup n |Df n | p H dµ| Ωn < +∞, where f n := [f ] µ|Ω n .Then it holds that f ∈ H 1,p (X, µ) and |Df | p H dµ = s.This property can be proved by combining the locality of minimal relaxed slopes with a cut-off argument, see e.g.[6, Proposition 2.17].

Main results
Let B be a separable Banach space, µ ∈ M + (B) a measure satisfying spt(µ) = B, and p ∈ [1, ∞).In view of Remark 2.3, we can identify Cyl(B) with a subspace of L p (µ), thus the Fréchet differential induces an unbounded linear operator d : Proof.By [13, Proposition 1.2.13], there exists a unique L ∞ (µ)-linear map ℓ : the µ-a.e.sense for every ω ∈ L p (µ; B * ), for some |L| ∈ L q (µ) + satisfying |L| L q (µ) = L L p (µ;B * ) * .Since B can be embedded linearly and isometrically into ℓ ∞ via a Kuratowski embedding and ℓ ∞ has the metric approximation property (see e.g.[18, Lemma 5.7]), we have that B is the subspace of a Banach space B having the metric approximation property.Given that µ is concentrated on a σ-compact set, we can find a sequence (p n ) n of finite-rank 1-Lipschitz linear operators pn : B → B such that lim n pn (x) − x B = 0 holds for µ-a.e.x ∈ B. Now let us fix a separable closed subspace B of B containing B ∪ n∈N pn (B).
For any n ∈ N, we denote by V n the finite-dimensional Banach space pn (B) ⊆ B and we define the operator p n : B → V n as p n := pn | B .We define the 1-current , where p * n ω is given by (2.3).We claim that T n ∈ N 1 (V n ) and To prove the first property in (3.1), fix any open set Ω ⊆ V n and an element ω ∈ C ∞ c (V n ; V * n ) satisfying spt(ω) ⊆ Ω and ω(x) V * n ≤ 1 for every x ∈ V n .Recalling (2.4), we can estimate To prove the second property in (3.1), notice that for every given function whence it follows that T n is normal and ) is acyclic.Using Theorem 2.1, we obtain a plan π n ∈ M + (C([0, 1]; B)), concentrated on the set Γ of non-constant Lipschitz curves in V n of constant speed, such that Tn = π n and (e ± ) # π n = (∂ Tn ) ± .Now we follow the proof of [18,Lemma 4.11].Since π n ≤ T n and (e ± ) # π n ≤ |∂T n | for every n ∈ N, the sequences ( π n ) n , ((e ± ) # π n ) n ⊆ M + ( B) are tight, thus we can find compact subsets (K j ) j of B such that π n ( B \ K j ) ≤ 4 −j and ((e + ) # π n )( B \ K j ) ≤ 2 −j for all j, n ∈ N. We also define where Γk := {γ ∈ Γ : and 2 j π n ({γ ∈ Γ : ℓ(γ) > 2 j }) ≤ π n ( B) ≤ |L| dµ =: m for every n ∈ N, we deduce that for all j, n ∈ N. We now show that Γ j is a precompact subset of C([0, 1]; B).Fix (γ i ) i ⊆ Γ j .Then: • Suppose lim i ℓ(γ i ) = 0. Since ((γ i ) 1 ) i ⊆ K j and K j is compact, x := lim i (γ i ) 1 ∈ K j exists, up to subsequence.Hence, (γ i ) i converges uniformly to the curve constantly in x. • Suppose lim i ℓ(γ i ) > 0, so that c := inf i ℓ(γ i ) > 0 up to subsequence.Since Lip(γ i ) ≤ 2 j and L 1 (( All in all, we proved that each Γ j is precompact, thus (3. for every f ∈ C bs ( B) thanks to (2.2), (3.1), and the dominated convergence theorem.This implies that π has barycenter in L q (µ) and Bar( π Therefore, it holds π ∈ B q ( B, µ).Also, Thanks to the last part of Lemma 2.7, we then conclude that π := π| LIP([0,1];B) ∈ B q (B, µ) verifies the statement.Remark 3.2.Alternatively, in the proof of Proposition 3.1 we could have used metric 1-currents in the sense of Ambrosio-Kirchheim [4] and Paolini-Stepanov's metric version of the superposition principle [18,19].We opted for the proof we presented for two reasons: first, the current T we want to associate to L is defined on cylindrical functions, but it is not obvious how to extend it to Lipschitz functions; second, extending T to Lipschitz functions is de facto unnecessary.
With Proposition 3.1 at disposal, we prove the equivalence result for weighted Banach spaces: and F (f ) := +∞ otherwise.Consider its Fenchel conjugate and its double Fenchel conjugate, i.e.

2. 1 .
Metric and measure spaces.Given metric spaces (X, d X ), (Y, d Y ), we denote by C(X; Y) the space of continuous maps from X to Y.We endow its subset C b (X; Y) consisting of bounded elements with the distance d C b (X;Y) (ϕ, ψ) := sup x∈X d Y (ϕ(x), ψ(x)).If Y = B is a Banach space, C b (X; B) is a vector space and d C b (X;B) is induced by the supremum norm • C b (X;B) for every i ∈ N and k > j, the sequence (γ i ) i has a uniformly converging subsequence by the Arzelà-Ascoli theorem [18, Proposition 2.1].