Characterisation of upper gradients on the weighted Euclidean space and applications

In the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all Lipschitz functions.


Introduction
In this paper we study first-order Sobolev spaces on the Euclidean space R n equipped with an arbitrary Radon measure µ ≥ 0. This theory has been initiated in the late nineties, with the pioneering work [9] by G. Bouchitté, G. Buttazzo, and P. Seppecher. The motivations and applications were numerous, in the fields of calculus of variations [9], shape optimisation [10], optimal transport problems with gradient penalisation [33], amongst many others. Compared to Allard's theory of varifolds [3] or to Federer-Fleming's theory of currents [21], the usage of measures in optimisation problems presents two main advantages: it allows to model objects made of parts having different Hausdorff dimension (such as multijunctions), and it rests on a solid functional-analytic machinery. About the former feature, we just mention that the aim of [9] was to represent low-dimensional elastic structures (such as membrane and beams) in an intrinsic way, as opposed to the more classical idea of first 'fattening' the structure under consideration and then passing to the limit in the vanishing thickness parameter (via Γ-convergence methods, for instance). With regard to the latter feature, let us briefly explain which is the analytic framework the theory of Sobolev spaces on weighted R n relies upon.
The key idea introduced by [9] was to define a suitable 'tangent distribution' associated with the measure µ, namely, a µ-a.e. defined measurable subbundle of T R n ∼ = R n × R n ; see Definition 1.21. In the approach adopted in [9], the tangent fibers are identified by looking at vector fields whose distributional divergence belongs to L 2 (µ) (see (1.24) for the precise definition we are referring to). A different (but similar in spirit) notion was studied by Fragalà-Mantegazza [22]; we do not investigate it in this paper. For a complete account on this technique via the distributional divergence, we refer to the survey [11] and the references therein. An alternative way to select the tangent fibers was proposed by Zhikov in [40,41], where the strategy was to perform a relaxation at the level of gradients of smooth functions. We introduce a useful generalisation -called G-structure -of Zhikov's concept in Definition 2.8. Later on, J. Louet studied in his PhD thesis [32] the relation between the above two approaches, but their complete equivalence was not known; we will obtain it as a byproduct of Theorem 2. 16. Once the tangent distribution is given, the Sobolev space is defined by first projecting the gradients of smooth functions on the tangent fibers (obtaining the tangential gradient with respect to µ) and then passing to the closure. The resulting energy functional is lower semicontinuous, or equivalently the associated notion of weak gradient yields a closed linear operator. It is worth to recall that other geometric and measure-theoretic notions of tangent space to a measure are studied in the literature -for instance, Preiss' notion of 'tangent measure' [36] or Simon's notion of 'approximate tangent space' [39]. However, these are not the correct objects to look at in order to define a Sobolev space: besides the fact that they not always exist, a noteworthy problem is that the consequent tangential gradient may well be not closable (since the geometric fibers are typically bigger than the analytic ones).
In the present paper we start our investigation of the Sobolev space on weighted R n from a rather different viewpoint. More precisely, we regard it as a special case of the more general theory of Sobolev spaces over a metric measure space (X, d, µ). In this respect, the first definition was given by P. Haj lasz in [29], but we will not consider it here because of its 'non-local' nature. At a later time, several other notions (which eventually turned out to be equivalent) have been proposed by J. Cheeger [13], N. Shanmugalingam [38], L. Ambrosio, N. Gigli, and G. Savaré [6], and S. Di Marino [17]. It will be convenient for us to work with the approach W 1,2 (X, µ) based on the concept of test plan, introduced in [6]; see Definition 1.3. The common feature of all the above approaches is the following: in lack of an underlying Banach structure, the weakly differentiable functions f on a metric measure space are detected by estimating the entity of their variation, rather than the variation itself. In other words, one obtains the 'modulus of the weak differential' |D µ f | instead of the weak differential D µ f .
Let us focus our attention on the case in which µ is a Radon measure on R n . Contrarily to what was discussed in the first part of this introduction, we now have a Sobolev space W 1,2 (R n , µ) at our disposal, but not (a priori) a notion of tangent fiber. Still, a tangent distribution can be recovered by appealing to results available in the literature, as we are going to describe: • N. Gigli built in [24] an abstract tensor calculus for metric measure spaces (X, d, µ), which is based upon the notion of L 2 (µ)-normed L ∞ (µ)-module. In particular, the Sobolev space gives rise to a natural notion of tangent module L 2 µ (T X), whose elements should be regarded as the 'synthetic' vector fields over (X, d, µ). See Definition 1.10.
• In the framework of the weighted Euclidean space, N. Gigli and the second named author proved in [27] that the tangent module L 2 µ (T R n ) can be isometrically embedded into the space L 2 (R n , R n ; µ) of all L 2 (µ)-maps from R n to itself. See Theorem 1.16.
• The first and second named authors proved in [34] that (locally finitely-generated) L 2 (µ)-normed L ∞ (µ)-modules can be always represented as the spaces of sections of a measurable Banach bundle. In the specific case of weighted R n , this grants that the tangent module L 2 µ (T R n ) is canonically associated with a distribution T µ in R n , that we will call the tangent distribution. See Definition 2.4.
One of the main achievements of the present paper is Theorem 2.16, where we prove that the tangent distribution T µ -and accordingly the Sobolev space W 1,2 (R n , µ) -is consistent both with the notion obtained via divergence by Bouchitté-Buttazzo-Seppecher [9] and with the one via vectorial relaxation by Zhikov [40,41]. Moreover, by building on top of this equivalence result, we will identify the minimal object |D µ f | (called the minimal weak upper gradient) associated with any compactly-supported Lipschitz function f on R n ; see Theorem 2.20. The case n = 1 was previously investigated by S. Di Marino and G. Speight in [20].
In order to establish the above-mentioned characterisation of the weak gradient of Lipschitz functions, we will need to study the interaction between the Sobolev calculus on weighted R n and the Alberti-Marchese differentiation theorem [1], which says -roughly speaking -that there exists a maximal distribution V µ in R n along which all Lipschitz functions are µ-a.e. differentiable (in the sense of Fréchet); cf. Theorem 2.1. This kind of investigation has been initiated by the first and second named authors together with S. Di Marino in [19], where it is proven that the absolute value of the Alberti-Marchese gradient is a weak upper gradient (see Theorem 2.3). By using the machinery discussed so far, we show (in Corollary 2.17) that for µ-a.e. x ∈ R n .
However, in general 'Sobolev calculus' and 'Lipschitz calculus' are not equivalent, thus one cannot expect the equality T µ = V µ to hold for all measures µ. Indeed, the Alberti-Marchese distribution just depends on the negligible sets of µ, while the Sobolev space W 1,2 (R n , µ) -and thus, a fortiori, the tangent distribution T µ -strongly depends on the measure µ itself. An example of a measure µ on R for which T µ = V µ will be described in Remark 2.18.
We are now in a position to state Theorem 2.20: given any f ∈ LIP c (R n ), it holds that where we denote by pr Tµ : V µ → T µ the natural projection operator, while ∇ AM f stands for the Alberti-Marchese gradient of f (that is a measurable section of the distribution V µ ).
In the last part of the paper -namely, in Section 3 -we shall provide a few applications (for the moment, only at a theoretical level) of our main Theorems 2.16 and 2.20: • Section 3.1: By combining our techniques with a deep result by G. De Philippis and F. Rindler [15] about Radon measures on R n , we prove that for µ s -a.e. point x ∈ R n the tangent fiber T µ (x) cannot coincide with the whole R n , where µ s stands for the singular part of µ with respect to the Lebesgue measure L n ; see Theorem 3.6. • Section 3.2: The tangent distribution T µ admits a geometric interpretation, in terms of the initial velocities of suitably chosen test plans on (R n , d Eucl , µ); see Theorem 3.16. • Section 3.3: Sobolev spaces over the weighted Euclidean space satisfy the expected tensorisation property; see Theorem 3.21.
We wish to point out that in the whole paper we just stick to the case p = 2, but mostly for a matter of practicality. The main reason is that many of the tools we will use -those concerning the theory of normed modules -are explicitly written in the literature only for the case p = 2. However, we expect that our results have appropriate counterparts for every p ∈ (1, ∞).
Finally, we conclude this introduction by mentioning that also second-order Sobolev spaces on weighted Euclidean spaces (for suitable Radon measures) have been studied, e.g., in [12]. It would be definitely interesting to understand whether even these second-order spaces admit an equivalent reformulation in the language of metric measure spaces. Yet another interesting problem would be to study the space BV(R n , µ) of functions of bounded variation.
Observe that the identity γ t = |γ t | is satisfied for L 1 -a.e. t ∈ [0, 1]. is indicated with LIP(X). The subfamily of those Lipschitz functions having compact support (resp. bounded support) is denoted by LIP c (X) (resp. LIP bs (X)). Given any f ∈ LIP(X), we define its local Lipschitz constant as , whenever x ∈ X is an accumulation point, (1.6) and lip(f )(x) := 0 elsewhere.

Sobolev space via test plans.
We recall here the definition of Sobolev space in the metric measure setting and its main properties. The approach we are going to describe has been proposed in [6,5]. To begin with, let us recall the important notion of test plan: Definition 1.1 (Test plan [6,5]). Let (X, d, µ) be a metric measure space. Then we say that a Borel probability measure π on C([0, 1], X), d ∞ is a test plan on (X, d, µ) provided the following properties are satisfied: i) There exists a compression constant Comp(π) > 0 such that where (e t ) * π stands for the pushforward measure of π under the evaluation map e t . ii) The measure π is concentrated on AC 2 ([0, 1], X) and has finite kinetic energy, i.e., (1.7) Then the map Const X is an isometry and the measure π := Const X * ν is a test plan on (X, d, µ) for every ν ∈ P(X) satisfying ν ≤ Cµ for some constant C > 0.
The notion of test plan plays an essential role in the definition of Sobolev space: Definition 1.3 (Sobolev space via test plans [6,5]). Let (X, d, µ) be a metric measure space. Fix f ∈ L 2 (µ). Then a function G ∈ L 2 (µ) is said to be a weak upper gradient of f provided for any test plan π on (X, d, µ) the following property is satisfied: for π-a.e. γ it holds that f • γ ∈ W 1,1 (0, 1) and We define the Sobolev space W 1,2 (X, µ) as the family of all those functions f ∈ L 2 (µ) that admit a weak upper gradient. Given any f ∈ W 1,2 (X, µ), we denote by |D µ f | the minimal weak upper gradient of f , where minimality is intended in the µ-a.e. sense.
The original notion of Sobolev space W 1,2 (X, µ) via test plans has been introduced in [6], but its equivalent reformulation we presented above has been established in [23,Appendix B]. We chose the unusual notation W 1,2 (X, µ), where the distance d does not appear (even though it plays a role in the definition), for a matter of practicality, since in all the cases we shall consider, the distance -differently from the measure -will always remain fixed.
1.1.4. Energy functionals. Throughout the whole paper, we will consider several different energy functionals E : L 2 (µ) → [0, +∞] over a given metric measure space (X, d, µ). Let us fix some notation. The finiteness domain of E is given by The functional E is said to satisfy the parallelogram rule if it holds that Moreover, we say that the functional E is lower semicontinuous provided The lower semicontinuous envelopeẼ : where the infimum is taken among all sequences (f n ) n ⊆ L 2 (µ) such that f n → f in L 2 (µ). It holds thatẼ is the greatest lower semicontinuous functional which is dominated by E.
The most important energy functional we will consider is the so-called Cheeger energy: Definition 1.4 (Cheeger energy). Let (X, d, µ) be a metric measure space. Then we define The functional E Ch : L 2 (µ) → [0, +∞] is called the Cheeger energy associated with (X, d, µ).
Another energy functional to take into account is the following one: (1.10) In view of (1.8), we know that E Ch ≤ E lip . Actually, E Ch is the lower semicontinuous envelope of E lip , as granted by the following important result.
1.1.6. Laplacian and heat flow. Let (X, d, µ) be an infinitesimally Hilbertian space. Given Since h is uniquely determined, we denote it by ∆ µ f and call it the Laplacian of f . It holds that D(∆ µ ) is a linear subspace of W 1,2 (X, µ) and ∆ µ : The heat flow {P t } t≥0 on (X, d, µ) is defined as follows: for any given function f ∈ L 2 (µ), we have that [0, +∞) ∋ t → P t f ∈ L 2 (µ) is the unique continuous curve satisfying P 0 f = f , which is absolutely continuous on (0, +∞), such that P t f ∈ D(∆ µ ) holds for all t > 0 and d dt P t f = ∆ µ P t f, for L 1 -a.e. t > 0. (1.12) Given any function f ∈ W 1,2 (X, µ), it holds that P t f W 1,2 (X,µ) ≤ f W 1,2 (X,µ) , for every t > 0. (1.13) The above properties are ensured by the classical Komura-Brezis theory of gradient flows.
1.2. Differential structure of metric measure spaces. A first-order differential calculus on metric measure spaces has been developed in [24,25]. Let us briefly recall the key concepts. (All inequalities are intended in the µ-a.e. sense.) We say that M , | · | , or just M , is an L 2 (µ)-normed L ∞ (µ)-module provided the norm v M := |v| L 2 (µ) on M is complete.
By a morphism ϕ : M → N between two given L 2 (µ)-normed L ∞ (µ)-modules M , N we mean an L ∞ (µ)-linear and continuous map. The dual module M * of M is defined as the space of all L ∞ (µ)-linear and continuous maps from M to L 1 (µ). It holds that M * has a natural L 2 (µ)-normed L ∞ (µ)-module structure, the pointwise norm |L| of L ∈ M * being defined as the minimal function G ∈ L 2 (µ), where minimality is intended in the µ-a.e. sense, such that the inequality L(v) ≤ G|v| is satisfied µ-a.e. on X for every element v ∈ M .
Clearly, M is a Hilbert module if and only if it is Hilbert when viewed as a Banach space.
The resulting mapping ·, · : is an isometric isomorphism. We call R M the Riesz isomorphism associated with M . Let (X, d, µ) be a metric measure space. Let M be an L 2 (µ)-normed L ∞ (µ)-module and let E ⊆ X be a Borel set such that µ(E) > 0. Then: We say that M has dimension n ∈ N on E provided it admits a local basis v 1 , . . . , v n ∈ M on E, i.e., the elements v 1 , . . . , v n are independent on E and {v 1 , . . . , v n } generates M on E.
Let (X, d X , µ), (Y, d Y , ν) be metric measure spaces. Let ϕ : X → Y be a given Borel map. Then we say that ϕ is a map of bounded compression provided ϕ * µ ≤ Cν for some C > 0.
Moreover, given two is a linear operator, such that the following properties are satisfied: is a Hilbert module and the tangent module is defined as It holds that a given metric measure space (X, d, µ) is infinitesimally Hilbertian if and only if its associated modules L 2 µ (T * X) and L 2 µ (T X) are Hilbert.
Given a test plan π on a metric measure space (X, d, µ), it holds that for every t ∈ [0, 1] the evaluation map e t is a map of bounded compression between C([0, 1], X), π and (X, µ). This allows us to consider the pullback modules e * t L 2 µ (T * X) and e * t L 2 µ (T X).
Proposition 1.12 (Velocity of a test plan [24, Theorem 2.3.18]). Let (X, d, µ) be a metric measure space such that the module L 2 µ (T X) is separable. Let π be a test plan on (X, d, µ).
The uniquely determined function h will be denoted by div µ (v) and called the abstract divergence of v. It can be readily checked that a function f ∈ W 1,2 (X, µ) belongs to D(∆ µ ) if and only if ∇ µ f ∈ D(div µ ). In this case, it also holds that div µ (∇ µ f ) = ∆ µ f .
Let f ∈ LIP bs (X) and v ∈ D(div µ ) be given. Then it holds that f v ∈ D(div µ ) and In other words, we say that the abstract divergence satisfies the Leibniz rule.
Proof. First of all, fix f ∈ W 1,2 (X, µ) and consider P t f ∈ D(∆ µ ) for every t > 0. Since the family {P t f } t>0 ⊆ W 1,2 (X, µ) is bounded by (1.13) and W 1,2 (X, µ) is reflexive, there exists a sequence t n ց 0 such that P tn f ⇀ f weakly in W 1,2 (X, µ). By Banach-Saks theorem, we have that (possibly passing to a not relabelled subsequence) the sequence (f n ) n ⊆ D(∆ µ ) given by f n := 1 n n i=1 P t i f satisfies f n → f with respect to the strong topology of W 1,2 (X, µ). In order to prove the last part of the statement, fix v ∈ L 2 µ (T X) and ε > 0. We can find Thanks to the first part of the statement and the fact that boundedly-supported Lipschitz functions are weakly * dense in L ∞ (µ), there are f 1 , . . . , f n ∈ D(∆ µ ) and g 1 , . . . , g n ∈ LIP bs (X) such that Consequently, we conclude that the vector field w : . Concrete 1-forms and vector fields on weighted R n . Let (X, d, µ) be a metric measure space and B, · a separable Banach space. Then we denote by L 2 (X, B; µ) the family of all Borel maps v : X → B such that´ v(x) 2 dµ(x) < +∞, considered up to µ-a.e. equality.
It holds that L 2 (X, B; µ) is an L 2 (µ)-normed L ∞ (µ)-module when endowed with the natural pointwise operations and the following pointwise norm: given any v ∈ L 2 (X, B; µ), we define Moreover, it holds (assuming µ = 0) that L 2 (X, B; µ) is Hilbert if and only if B is Hilbert.
We denote by d Eucl the Euclidean distance d Eucl (x, y) := |x − y| on R n . Given any t ∈ [0, 1], we define the mapping Der t : (1.18) Standard arguments show that Der t is Borel. Given a non-negative Radon measure µ on R n and a test plan π on (R n , d Eucl , µ), we define the space B π as Observe that B π is a separable Hilbert space.
Proposition 1.14. Let µ be a Radon measure on R n . Let π be a test plan on (R n , d Eucl , µ).
Then it holds that (the equivalence classes up to L 1 -a.e. equality of ) the mappings where the derivative is intended with respect to the strong topology of B π .
Proof. First of all, let us observe that thus Der t ∈ B π for L 1 -a.e. t ∈ [0, 1] and Der ∈ L 2 ([0, 1], B π ; L 1 ); we omit the standard proof of the fact that Der is Borel. Moreover, for every s, t ∈ [0, 1] with s < t it holds Der r dr (γ), for π-a.e. γ, so that (e t − e 0 ) − (e s − e 0 ) = e t − e s =´t s Der r dr ∈ B π , whence the statement follows.
Given any non-negative Radon measure µ on R n , we will refer to the metric measure space (R n , d Eucl , µ) as a weighted Euclidean space. The rest of this paper is devoted to the study of the Sobolev space and the differential structure associated with (R n , d Eucl , µ). We will refer to the elements of the Hilbert module L 2 (R n , R n ; µ) as the concrete vector fields on (R n , d Eucl , µ). Given any f ∈ C ∞ c (R n ), we denote by ∇f ∈ L 2 (R n , R n ; µ) the (equivalence class of the) 'strong' gradient of f , i.e., for any x ∈ R n we characterise ∇f (x) ∈ R n as the unique vector satisfying The Hilbert module L 2 (R n , (R n ) * ; µ) is the dual module of L 2 (R n , R n ; µ) and its elements are said to be the concrete 1-forms on (R n , d Eucl , µ). The 'strong' differential of a given function f ∈ C ∞ c (R n ) will be denoted by df ∈ L 2 (R n , (R n ) * ; µ).
The relation between abstract and concrete vector fields on the weighted Euclidean space has been investigated in [27], where the following results have been proven: Theorem 1.15 (Density in energy of smooth functions). Let µ be a non-negative Radon measure on R n . Let f ∈ W 1,2 (R n , µ) be given. Then there exists a sequence The proof of the above result was obtained by combining a standard convolution argument with (a stronger variant of) Theorem 1.5. As a consequence, the following statement holds: Theorem 1.16 (The isometric embedding ι µ ). Let µ ≥ 0 be a Radon measure on R n . Then there exists a unique morphism P µ : Calling ι µ : L 2 µ (T R n ) → L 2 (R n , R n ; µ) the adjoint of P µ , i.e., the unique morphism satisfying Remark 1.17. Given any Radon measure µ ≥ 0 on R n and any vector field v ∈ L 2 µ (T R n ), it holds that ι µ (v) can be characterised as the unique element of L 2 (R n , R n ; µ) such that This readily follows from the fact that df : f ∈ C ∞ c (R n ) generates L 2 (R n , (R n ) * ; µ) and that ι µ : As it was observed in [27], it immediately follows from Theorem 1.16 that Euclidean spaces are universally infinitesimally Hilbertian, in the following sense. Theorem 1.18 (Infinitesimal Hilbertianity of weighted R n ). Let µ ≥ 0 be a Radon measure on R n . Then the metric measure space (R d , d Eucl , µ) is infinitesimally Hilbertian.
We point out that other two different proofs of Theorem 1.18 are known: it directly follows from [18, Theorem 1.1], and it is one of the main achievements of [19]; in Section 2.1, we will briefly describe the strategy of the latter approach. Let us now recall an important consequence of Theorems 1.15 and 1.18. For the reader's usefulness, we also provide its proof.
Proof. Thanks to Theorem 1.15, we can find a sequence ( . By using Proposition 1.11 (and the Riesz isomorphism), we obtain that v = ∇ µ f . Moreover, where the first inequality is granted by the weak convergence , which shows the validity of (1.23c).

1.2.5.
Divergence of concrete vector fields. Let µ ≥ 0 be a Radon measure on R n . Then we denote by D(div µ ) the space of all those vector fields v ∈ L 2 (R n , R n ; µ) whose distributional divergence belongs to L 2 (µ). Namely, there exists a function div µ (v) ∈ L 2 (µ) such that Observe that div µ satisfies the Leibniz rule, i.e., it holds that f v ∈ D(div µ ) and Lemma 1.20 (Relation between div µ and div µ ). Let µ ≥ 0 be a Radon measure on R n . Then for any vector field v ∈ L 2 In this case, it holds that div µ (v) = div µ ι µ (v) in the µ-a.e. sense.
Proof. On the one hand, suppose v ∈ D(div µ ). Then for any f ∈ C ∞ c (R n ) it holds that This shows that v ∈ D(div µ ) and div µ (v) = div µ ι µ (v) holds µ-a.e. on R n , as required.
1.3. Distributions on the Euclidean space. We denote by Gr(R n ) the Grassmannian of R n , i.e., the family of all linear subspaces of R n . We endow Gr(R n ) with the distance i.e., d Gr(R n ) (V, W ) is the Hausdorff distance in R n between the closed unit balls of V and W . It holds that Gr(R n ), d Gr(R n ) is a compact metric space; see, for instance, [2].
Given any Radon measure µ on R n , we denote by D n (µ) the family of all distributions on R n , considered up to µ-a.e. equality. Given any V ∈ D n (µ), we define Γ(V ) ⊆ L 2 (R n , R n ; µ) as Moreover, we define a partial order on D n (µ) in the following way: given any V, W ∈ D n (µ), It can be readily checked that Γ(V ) is an L 2 (µ)-normed L ∞ (µ)-submodule of L 2 (R n , R n ; µ). Proposition 1.22. Let µ ≥ 0 be a Radon measure on R n . Then the mapping V → Γ(V ) is a bijection between D n (µ) and the family of L 2 (µ)-normed L ∞ (µ)-submodules of L 2 (R n , R n ; µ). Moreover, the map Γ is order-preserving, i.e., one has V ≤ W if and only if Γ(V ) ⊆ Γ(W ).
Proof. The only non-trivial fact to check is that the mapping Γ is surjective. To this aim, let us fix an L 2 (µ)-normed L ∞ (µ)-submodule M of L 2 (R n , R n ; µ). Also, take any countable, for µ-a.e. x ∈ R n . (1.26) The resulting map V : R n → Gr(R n ) is Borel. Indeed, for every W ∈ Gr(R n ) we have that holds for µ-a.e. x ∈ R n , where (w j ) j is any dense sequence in the closed unit ball of W , thus accordingly x → d Gr(R n ) V (x), W is µ-a.e. equivalent to a Borel function. Then, V ∈ D n (µ).
Let us now prove that M = Γ(V ). Given that v i ∈ Γ(V ) for every i ∈ N by construction and M = cl {v i : i ∈ N}, we deduce that M ⊆ Γ(V ). Conversely, fix any v ∈ Γ(V ). By dominated convergence theorem we see that the sequence (w j ) j ⊆ Γ(V ), given by w j : , for every i ∈ N and µ-a.e. x ∈ E j i . (1.27) By exploiting the inequality in (1.27), we thus obtain that Remark 1.23. The statement of Proposition 1.22 is a particular instance of a more general result proven in [34], concerning the representation of a certain class of normed modules as spaces of sections of a measurable Banach bundle. Nevertheless, in the special case under consideration (i.e., only submodules of L 2 (R n , R n ; µ) are taken into account) the argument is simpler than the original one in [34], so we opted for providing an easier proof.
Lemma 1.24. Let µ be a Radon measure on R n . Let V be a linear subspace of L 2 (R n , R n ; µ) such that gv ∈ V holds for every g ∈ C ∞ c (R n ) and v ∈ V . Given a dense sequence e. x ∈ R n . Then V (x) x∈R n is a family of linear subspaces of R n , which are µ-a.e. independent of (v i ) i . Moreover, it holds that In particular, the map V : R n → Gr(R n ) is a distribution on R n , the Banach space cl V is an L 2 (µ)-normed L ∞ (µ)-submodule of L 2 (R n , R n ; µ), and Γ(V ) = cl V .
Proof. The first part of the statement follows, e.g., from [12, Lemma A.1]. The fact that V is a distribution on R n can be proved by arguing exactly as in the proof of Proposition 1.22, whence the remaining claims immediately follow. Remark 1.25 (Orthogonal projection). Let µ ≥ 0 be a Radon measure on R n . Let V ∈ D n (µ) be given. We define the orthogonal projection mapping pr V : L 2 (R n , R n ; µ) → Γ(V ) as where π V (x) : R n → V (x) is the standard orthogonal projection. Clearly, the mapping pr V is a surjective, 1-Lipschitz morphism of L 2 (µ)-normed L ∞ (µ)-modules.
Remark 1.26 (Orthogonal complement, II). Given any Radon measure µ on R n and any distribution V ∈ D n (µ), we define the orthogonal complement V ⊥ ∈ D n (µ) of V as Moreover, observe that Γ(V ⊥ ) = Γ(V ) ⊥ , where Γ(V ) ⊥ is defined as in Remark 1.7.

2.
Characterisation of the Sobolev space on weighted Euclidean spaces 2.1. Alberti-Marchese distribution. In our investigation of the Sobolev space associated with a weighted Euclidean space, a key role is played by the following result, whose statement can be roughly summed up in this way: given a Radon measure µ on R n , there is a 'maximal' distribution V µ on R n along which all Lipschitz functions are µ-a.e. (Fréchet) differentiable.
Theorem 2.1 (Alberti-Marchese distribution [1]). Let µ ≥ 0 be a Radon measure on R n . Then there exists a unique distribution V µ ∈ D n (µ) such that the following properties hold: i) Every function f ∈ LIP c (R n ) is µ-a.e. differentiable with respect to V µ , i.e., there exists a vector field ii) There exists a function f 0 ∈ LIP(R n ) such that for µ-a.e. x ∈ R n it holds that f 0 is not differentiable at x with respect to any direction v ∈ R n \ V µ (x).
We call V µ the Alberti-Marchese distribution associated with µ.
In [1] the object V µ is called the 'decomposability bundle' of µ. Here, we chose the term 'distribution' in order to be consistent with our Definition 1.21. Moreover, Theorem 2.1 was actually proven under the additional assumption of µ being a finite measure, whence the case of a possibly infinite Radon measure follows by arguing as in [19,Remark 1.6].

Remark 2.2. It follows from Rademacher theorem that
In particular, if µ ≪ L n , then V µ (x) = R n holds for µ-a.e. x ∈ R n .
We shall refer to ∇ AM as the Alberti-Marchese gradient operator. It readily follows from (2.1) that the element ∇ AM f is uniquely determined (up to µ-a.e. equality). Moreover, for µ-a.e. x ∈ R n , are satisfied for every f, g ∈ LIP c (R n ). Let us also recall that it holds that i) Let π be a test plan on (R n , d Eucl , µ). Then for π-a.e. curve γ it holds thaṫ ii) Let f ∈ LIP c (R n ) be given. Then |∇ AM f | ∈ L 2 (µ) is a weak upper gradient of f .
iii) Let f ∈ W 1,2 (R n , µ) be given. Then there exists a sequence (f i ) i ⊆ LIP c (R n ) such that f i → f and |∇ AM f i | → |D µ f | in the strong topology of L 2 (µ).
As we already mentioned in the paragraph below Theorem 1.18, the universal infinitesimal Hilbertianity of R n was obtained in [19, Theorem 2.3] as a consequence of Theorem 2.3. The argument was the following: the Cheeger energy E Ch is the lower semicontinuous envelope of the Alberti-Marchese energy functional E AM : L 2 (µ) → [0, +∞], given by which is clearly 2-homogeneous by construction, and satisfies the parallelogram rule by (2.2). Consequently, the Cheeger energy associated with (R n , d Eucl , µ) satisfies the parallelogram rule, thus yielding the sought conclusion.

2.2.
Identification of the tangent module. Let µ be a given Radon measure on R n . We know from Theorem 1.16 that the tangent module L 2 µ (T R n ) can be canonically seen as a submodule of L 2 (R n , R n ; µ), whence (by Proposition 1.22) we have a natural notion of tangent distribution T µ . In this section, we provide some alternative characterisations of T µ , thus showing (as described in the introduction) that our approach is equivalent to the ones introduced in [9] and [40,41]. Some of the proofs that we will carry out are inspired by [33].

Tangent distribution.
We introduce the notion of tangent distribution on (R n , d Eucl , µ): Definition 2.4 (Tangent distribution). Let µ be a Radon measure on R n . Then we define the tangent distribution T µ as the unique element of D n (µ) such that

Remark 2.5.
It is straightforward to check that the module L 2 µ (T R n ) has dimension k on a given Borel set E ⊆ R n with µ(E) > 0 if and only if dim T µ (x) = k for µ-a.e. x ∈ E.
The following result shows that 'test plans are tangent to the distribution T µ ', in a sense. Lemma 2.6. Let µ be a Radon measure on R n . Let π be a given test plan on (R n , d Eucl , µ). Then for π-a.e. curve γ it holds thaṫ Proof. Let π be a given test plan on (R n , d Eucl , µ). Given any t ∈ [0, 1], consider the pullback morphisms e * t P µ : e * t L 2 (R n , (R n ) * ; µ) → e * t L 2 µ (T * R n ) and e * t ι µ : e * t L 2 µ (T R n ) → e * t L 2 (R n , R n ; µ) as in Theorem 1.9. The spaces e * t L 2 µ (T R n ) and e * t L 2 (R n , R n ; µ) can be identified with the dual modules of e * t L 2 µ (T * R n ) and e * t L 2 (R n , (R n ) * ; µ), respectively, as a consequence of the separability of L 2 (R n , R n ; µ) (which can be readily checked) and of its subspace ι µ L 2 µ (T R n ) ; cf. [24, Theorem 1.6.7]. Since ι µ is the adjoint of P µ , it holds that e * t ι µ is the adjoint of e * t P µ , thus in particular for any element z ∈ e * t L 2 µ (T R n ) we have that (e * t P µ )(e * t df ) (z) = (e * t df ) (e * t ι µ )(z) π-a.e., for every f ∈ C ∞ c (R n ). (2.7) Moreover, the morphism e * whence e * t ι µ is an isometry as we know that e * t v : v ∈ L 2 µ (T R n ) generates e * t L 2 µ (T R n ). One can readily check that e * t L 2 (R n , R n ; µ) can be identified with the space B π , the pullback map e * t : L 2 (R n , R n ; µ) → B π being given by e * t v := v • e t for every v ∈ L 2 (R n , R n ; µ). An analogous statement holds for e * t L 2 (R n , (R n ) * ; µ). Observe that (e * t ι µ ) e * t L 2 µ (T R n ) = z ∈ B π z(γ) ∈ T µ (γ t ), for π-a.e. γ . (2.8) Let us consider, for L 1 -a.e. t ∈ [0, 1], the velocity π ′ t ∈ e * t L 2 µ (T R n ) of π as in Proposition 1.12. We deduce from (1.15) that for any given function for π-a.e. γ. (2.9) For L 1 -a.e. t ∈ [0, 1], consider the mapping Der t ∈ B π defined in (1.18). We claim that Given any function f ∈ C ∞ c (R n ), we have that for π-a.e. curve γ it holds that . Since the identity in (2.7) actually characterises e * t ι µ , we deduce that the claim (2.10) holds. In particular, we have that for (π ⊗ L 1 )-a.e. (γ, t).
Thanks to Fubini theorem, we finally conclude that the sought property (2.6) is satisfied.
Clearly, in order to identify the minimal weak upper gradient of a given Sobolev function, it is sufficient to look at the directions that are selected by the test plans. The following result makes this claim precise.
Lemma 2.7. Let µ be a Radon measure on R n . Let V ∈ D n (µ) satisfy the following property: given any test plan π on (R n , d Eucl , µ), it holds thatγ t ∈ V (γ t ) for (π ⊗ L 1 )-a.e. (γ, t). Then for any function f ∈ C ∞ c (R n ) we have that pr V (∇f ) is a weak upper gradient of f .
Proof. Fix any test plan π on (R n , d Eucl , µ). Then for π-a.e. curve γ it holds that . By arbitrariness of π, we conclude that pr V (∇f ) is a weak upper gradient of f .

2.2.2.
An axiomatic notion of weak gradient. Another possible way to define the tangent fibers is via the vectorial relaxation procedure proposed by Zhikov in [40,41] and studied by Louet in [33]. Below we introduce a generalisation of such approach, tailored for our purposes.
Definition 2.8 (G-structure). Let µ ≥ 0 be a Radon measure on R n . Then by G-structure on (R n , d Eucl , µ) we mean a couple (V,∇) satisfying the following list of axioms: A1. V is a linear subspace of W 1,2 (R n , µ) containing C ∞ c (R n ). A2.∇ : V → L 2 (R n , R n ; µ) is a linear operator. A3. |∇f | is a weak upper gradient of f for any f ∈ V, with |∇f | ∈ L ∞ (µ) if f ∈ C ∞ c (R n ).
A4.∇ satisfies the Leibniz rule, i.e., if f ∈ V and g ∈ C ∞ c (R n ), then f g ∈ V and ∇(f g) = f∇g + g∇f, in the µ-a.e. sense.
A5. Calling E G : L 2 (µ) → [0, +∞] the energy functional it holds that E Ch is the lower semicontinuous envelope of E G .
The term 'G-structure' is somehow inspired by the notion of D-structure, which has been proposed by V. Gol'dshtein and M. Troyanov in the paper [28]. Therein, they developed an axiomatic theory of Sobolev spaces on general metric measure spaces. In our setting, thanks to the presence of an underlying linear structure, the axiomatisation can be formulated in terms of 'gradients' rather than 'moduli of the gradients'. Remark 2.9 (Density in energy). Observe that axiom A5 is equivalent to requiring that the elements of V are dense in energy in W 1,2 (R n , µ), i.e., for every f ∈ W 1,2 (R n , µ) there exists a sequence (f i ) i ⊆ V such that f i → f and |∇f i | → |D µ f | strongly in L 2 (µ).

Example 2.10 (Examples of G-structures). Let us describe two examples of G-structures
on (R n , d Eucl , µ) that will play a fundamental role in the forthcoming discussion: The axioms defining a G-structure are satisfied both in a) and in b), as a consequence of the results contained in Sections 1.2 and 2.1, respectively.
Much like in the case of Sobolev spaces via test plans and minimal weak upper gradients, any G-structure naturally comes with a unique minimal object, called the minimal G-gradient: Definition 2.11 (G-gradient). Let µ ≥ 0 be a Radon measure on R n and (V,∇) a G-structure on (R n , d Eucl , µ). Fix f ∈ L 2 (µ). Then we say that f admits a G-gradient v ∈ L 2 (R n , R n ; µ) provided there exists a sequence (f i ) i ⊆ V such that We denote by G(f ) the closed affine subspace of L 2 (R n , R n ; µ) made of all G-gradients of f . The (unique) element of G(f ) of minimal norm is called the minimal G-gradient of f .
Observe that∇f ∈ G(f ) for every f ∈ V, as one can see by taking f i := f for every i ∈ N.
Remark 2.12. Note that the space G(0) is closed under multiplication by C ∞ c (R n )-functions: given any g ∈ C ∞ c (R n ) and v ∈ G(0), it holds that gv ∈ G(0).
and∇(gf i ) = g∇f i + f i∇ g → gv in L 2 (R n , R n ; µ). In particular, we deduce from Lemma 1.24 that the space G(0) is an L 2 (µ)-normed L ∞ (µ)-submodule of L 2 (R n , R n ; µ). Definition 2.13. Let µ be a Radon measure on R n and (V,∇) a G-structure on (R n , d Eucl , µ). Then we define W G as the unique element of D n (µ) such that Notice that the previous definition is meaningful as a consequence of Remark 2.12.

2.2.3.
Alternative characterisations of the tangent distribution. The following two results show that G-structures can be used to provide an alternative notion of Sobolev space, which turns out to be fully equivalent to the approach via test plans.
Theorem 2.14 (Alternative characterisation of W 1,2 ). Let µ ≥ 0 be a Radon measure on R n and let (V,∇) be a G-structure on (R n , d Eucl , µ). Then Moreover, for every f ∈ W 1,2 (R n , µ) it holds that the minimal weak upper gradient |D µ f | coincides (in the µ-a.e. sense) with the pointwise norm of the minimal G-gradient of f .
Proof. First of all, let us fix any function f ∈ W 1,2 (R n , µ). We claim that G(f ) = ∅ and that there exists an element v ∈ G(f ) such that |v| L 2 (µ) ≤ |D µ f | L 2 (µ) . In order to prove it, , whose existence is observed in Remark 2.9. Up to a not relabelled subsequence, it holds that∇f i ⇀ v weakly in L 2 (R n , R n ; µ) for some vector field v ∈ L 2 (R n , R n ; µ). By Banach-Saks theorem we know that (up to taking a further subsequence) it holds that the functions g i := 1 i i j=1 f j ∈ V satisfy g i → f in L 2 (µ) and∇g i → v strongly in L 2 (R n , R n ; µ), which yields v ∈ G(f ). It also holds that |v| L 2 (µ) = lim i |∇g i | L 2 (µ) ≤ lim i Conversely, let us suppose that f ∈ L 2 (µ) satisfies G(f ) = ∅. Fix an element v ∈ G(f ). Pick any sequence (f i ) i ⊆ V such that f i → f in L 2 (µ) and∇f i → v in L 2 (R n , R n ; µ). In particular, |∇f i | → |v| in L 2 (µ). Since |∇f i | is a weak upper gradient of f i for every i ∈ N, we deduce from Proposition 1.11 that f ∈ W 1,2 (R n , µ) and |D µ f | ≤ |v| holds µ-a.e. in R n . All in all, the proof of the statement is finally achieved. Proof. We claim that for any element v ∈ G(f ) it holds that pr W ⊥ G (v) belongs to G(f ) and is independent of v. First, recall that∇f ∈ G(f ). Since Lemma 1.24 yields pr W G (∇f ) ∈ G(0), All in all, the claim is proven. Now the first part of the statement readily follows: given any v ∈ G(f ), it holds that Therefore, we finally conclude that pr W ⊥ G (∇f ) is the minimal G-gradient of f . The last part of the statement now follows from Theorem 2.14, thus the proof is complete.
We are now ready to state and prove the main result of this section. It says that the tangent distribution T µ can be expressed either in terms of the domain of the distributional divergence div µ , or of the G µ -structure. We point out that, to the best of our knowledge, the equivalence between these two approaches (namely, items ii) and iii) of the following result) was previously not known; one of the two implications is proved in [33, end of Section 1].
Theorem 2.16 (Alternative characterisations of T µ ). Let µ ≥ 0 be a Radon measure on R n . Then the tangent distribution T µ can be equivalently characterised in the following ways: i) T µ is the unique minimal element of D n (µ) with the property that for any test plan π where W µ := W Gµ stands for the distribution on R n associated with the G µ -structure C ∞ c (R n ), ∇ , which is described in item a) of Example 2.10. In items i) and ii), minimality has to be intended with respect to the partial order ≤ on D n (µ).
Proof. We subdivide the proof into several steps: Step 1. First of all, we claim that T µ ≤ W ⊥ µ . This would follow from the inclusions Indeed, by using (2.13), (2.5), (2.12), and Remark 1.26, we deduce that Γ(T µ ) ⊆ Γ(W ⊥ µ ), whence T µ ≤ W ⊥ µ by the last part of the statement of Proposition 1.22. To prove the first inclusion in (2.13), recall that cl D(div µ ) = L 2 µ (T R n ) by Lemma 1.13 and notice that To prove the second inclusion in (2.13), it clearly suffices to show that D(div µ ) ⊆ G µ (0) ⊥ . To this aim, fix v ∈ D(div µ ) and w ∈ G µ (0). Choose any By arbitrariness of v and w, we conclude that D(div µ ) ⊆ G µ (0) ⊥ , so that (2.13) is proven.
Step 2. Let V ∈ D n (µ) be a distribution on R n such that for any test plan π on (R n , d Eucl , µ) it holds thatγ t ∈ V (γ t ) for (π ⊗ L 1 )-a.e. (γ, t). Then we claim that W ⊥ µ ≤ V .

First, from
Step 1 and Lemma 2.6 we know that W ⊥ µ ∩ V ∈ D n (µ) satisfies the same property as V , i.e., for any test plan π one hasγ t ∈ W µ (γ t ) ⊥ ∩ V (γ t ) for (π ⊗ L 1 )-a.e. (γ, t). Let W ∈ D n (µ) be defined so that W (x) is the orthogonal complement of W µ (x) ⊥ ∩ V (x) in W µ (x) ⊥ for µ-a.e. point x ∈ R n . Given any function f ∈ C ∞ c (R n ), it holds that pr W ⊥ µ ∩V (∇f ) is a weak upper gradient of f by Lemma 2.7, while pr W ⊥ µ (∇f ) is the minimal weak upper gradient of f by Proposition 2.15. This implies that pr W ⊥ µ ∩V (∇f ) = pr W ⊥ µ (∇f ) holds µ-a.e. in R n for every f ∈ C ∞ c (R n ), thus accordingly we might conclude that pr W (∇f ) Given that ∇f : f ∈ C ∞ c (R n ) generates L 2 (R n , R n ; µ) on R n , we deduce that the image of pr W coincides with {0}, thus necessarily W = {0}. This means that W µ (x) ⊥ ∩V (x) = W µ (x) ⊥ for µ-a.e. point x ∈ R n , which grants that W ⊥ µ ≤ V . Hence, the claim is proven.
Step 3. By Lemma 2.6 we know that T µ satisfies the property in item i), whence by Steps 1 and 2 we see that T µ = W ⊥ µ is the (unique) minimal distribution on R n having this property, proving items iii) and i). Moreover, notice that ι µ L 2 µ (T R n ) = cl D(div µ ) = G µ (0) ⊥ follows from (2.13) and the identity T µ = W ⊥ µ , thus item ii) is proven as well.
Note that by combining Lemma 1.20 with item ii) of the previous theorem, we obtain that ι µ D(div µ ) = D(div µ ), for every Radon measure µ ≥ 0 on R n .
As another immediate consequence of Theorem 2.16, we also see that the tangent distribution is always contained in the Alberti-Marchese distribution: Corollary 2.17. Let µ ≥ 0 be a Radon measure on R n . Then it holds that (2.14) Proof. Combine item i) of Theorem 2.3 with item i) of Theorem 2.16.
This means that ι µ (∇ µ f ) ∈ G µ (f ). Given that ι µ (∇ µ f ) = |D µ f | holds µ-a.e. in R n , we infer from Theorem 2.14 that ι µ (∇ µ f ) is the minimal G µ -gradient of f . The proof is complete. Theorem 2.20 (Minimal weak upper gradient of Lipschitz functions). Let µ ≥ 0 be a Radon measure on R n . Then it holds that for every f ∈ LIP c (R n ).
Proof. Consider the G µ -structure and the G AM -structure, which were defined in items a) and b) of Example 2.10, respectively. For brevity, we call W AM := W G AM . First, we prove that In order to show one inclusion, fix v ∈ G AM (0) ⊥ ∩ Γ(V µ ) and w ∈ G µ (0). Let us pick any sequence (f i ) i ⊆ C ∞ c (R n ) satisfying f i → 0 in L 2 (µ) and ∇f i → w in L 2 (R n , R n ; µ). The latter convergence, together with (2.11), yields ∇ AM f i = pr Vµ (∇f i ) → pr Vµ (w) in L 2 (R n , R n ; µ), which gives pr Vµ (w) ∈ G AM (0). Since v ∈ G AM (0) ⊥ ∩ Γ(V µ ), we get that v · w = v · pr Vµ (w) = 0 holds µ-a.e., which implies To prove the converse inclusion, let us consider the orthogonal complement in the µ-a.e. sense. (2.17) By applying Proposition 2.15 to the G µ -structure and the G AM -structure, we obtain that respectively. By plugging (2.18) into (2.17), we deduce that pr Z (∇f ) = pr Z (∇ AM f ) = 0 holds µ-a.e. for all f ∈ C ∞ c (R n ). Since ∇f : f ∈ C ∞ c (R n ) generates L 2 (R n , R n ; µ) on R n , we conclude that Z = {0}, which means that the identity in (2.16) is verified.
Now fix any function f ∈ LIP c (R n ). We know that pr W ⊥

AM
(∇ AM f ) is the minimal weak upper gradient of f by Proposition 2.15. Since it also holds that we finally conclude that pr Tµ (∇ AM f ) is the minimal weak upper gradient of f .
It readily follows from Theorem 2.20 that those measures µ on R n for which minimal weak upper gradient and local Lipschitz constant always coincide can be explicitly characterised in terms of the tangent distribution T µ , as the next result shows.
Proof. Suppose i) holds. To prove ii), we argue by contradiction: suppose there exists a Borel set E ⊆ R n such that µ(E) > 0 and T µ (x) = R n for µ-a.e. x ∈ E. This means that for µ-a.e. point x ∈ E there exists a vector v ∈ Q n such that v / ∈ T µ (x), in other words Hence, there exist a vector v ∈ Q n and a Borel set F ⊆ E such that µ(F ) > 0 and v / ∈ T µ (x) for µ-a.e. x ∈ F . Choose a radius r > 0 such that µ F ∩ B r (0) > 0 and a function f ∈ C ∞ c (R n ) satisfying ∇f (x) = v for every x ∈ B r (0). By using Proposition 2.19, we thus deduce that This leads to a contradiction with i), whence accordingly ii) is proven. Conversely, suppose ii) holds. A fortiori, we have that V µ (x) = R n for µ-a.e. x ∈ R n (recall Corollary 2.17), so that any given function f ∈ LIP c (R n ) is µ-a.e. differentiable and thus |∇ AM f | = lip(f ) in the µ-a.e. sense. Finally, by using Theorem 2.20 we obtain that |D µ f | = pr Tµ (∇ AM f ) = |∇ AM f | = lip(f ), holds µ-a.e. on R n , proving the validity of i).

Some applications
3.1. Tangent fibers on the singular part. In the structure theory of Radon measures on Euclidean spaces, a breakthrough is represented by the celebrated paper [15] by G. De Philippis and F. Rindler. A consequence of their main result is reported in Theorem 3.1.
In this section, we will combine the results by De Philippis-Rindler with our knowledge of the tangent distribution, in order to prove that for any Radon measure µ = ρL n + µ s on R n (where µ s ⊥ L n ) it holds that T µ (x) = R n for µ s -a.e. point x ∈ R n ; see Theorem 3.6. This gives a positive answer to a variant of a question raised by I. Fragalà and C. Mantegazza in [22,Remark 4.4]; the original problem was posed in terms of a different notion of tangent fiber. However, by adapting our arguments one can solve also their original open problem. We point out that neither the kind of results we will prove in this section, nor the techniques we will use, are really new. See, e.g., [14,31,27] for similar statements and arguments.
3.1.1. Reminder on Euclidean 1-currents. Recall that a 1-current T on R n is a linear and continuous real-valued functional defined on the space of smooth, compactly-supported 1forms on R n . Its total mass M(T) is given by the supremum of T(ω) among all smooth, compactly-supported 1-forms ω on R n that satisfy |ω| ≤ 1 on all R n . If M(T) is finite, then T is an R n -valued Radon measure on R n , whence by using the Radon-Nikodým theorem one can find a finite, non-negative Borel measure T on R n and a vector field T ∈ L 1 (R n , R n ; T ), with T(x) = 1 for T -a.e. point x ∈ R n , such that T = T T . The boundary ∂T of T is the 0-current (i.e., the generalised function) on R n which is defined as ∂T(f ) := T(df ) for all f ∈ C ∞ c (R n ). A 1-current T on R n is said to be normal provided M(T), M(∂T) < +∞, where M(∂T) := sup ∂T(f ) : f ∈ C ∞ c (R n ), |f | ≤ 1 on R n . When the total mass M(∂T) is finite, the 0-current ∂T can be canonically identified with a (finite) signed measure on R n .
The following deep result, concerning the structure of normal 1-currents in the Euclidean space, has been proven by G. De Philippis and F. Rindler in the paper [15].
Theorem 3.1. Let µ ≥ 0 be a Radon measure on R n and let T 1 , . . . , T n be normal 1-currents in R n such that µ ≪ T i for every i = 1, . . . , n. Suppose that T 1 (x), . . . , T n (x) ∈ R n are linearly independent, for µ-a.e. x ∈ R n .
Then it holds that µ ≪ L n .
As pointed out in [15], Theorem 3.1 has -amongst many others -the following consequence: Theorem 3.2 (Weak converse of Rademacher theorem). Let µ be a Radon measure on R n such that every function f ∈ LIP(R n ) is µ-a.e. differentiable. Then it holds that µ ≪ L n .
In turn, the weak converse of Rademacher theorem readily implies that the Alberti-Marchese distribution has full rank if and only if the measure under consideration is absolutely continuous with respect to the Lebesgue measure: Corollary 3.3. Let µ be a given Radon measure on R n . Then it holds that V µ (x) = R n , for µ-a.e. x ∈ R n ⇐⇒ µ ≪ L n .
Proof. If V µ (x) = R n for µ-a.e. x ∈ R n , then every Lipschitz function is µ-a.e. differentiable, whence µ ≪ L n by Theorem 3.2. The converse implication is observed in Remark 2.2.
Example 3.4 (Vector fields with divergence as normal 1-currents). Let µ be a finite Borel measure on R n and v ∈ D(div µ ). Let us associate to v the 1-current I(v) on R n , defined as I(v)(ω) :=ˆω(v) dµ, for every smooth, compactly-supported 1-form ω on R n . (3.1) Then we claim that I(v) is a normal 1-current and that it satisfies − − → Indeed, the fact that M I(v) < +∞, and the explicit formulae for − − → I(v) and I(v) , are immediate consequences of (3.1), while it readily follows from the identity that the 0-current ∂I(v) has finite total mass and satisfies ∂I(v) = −div µ (v)µ.

3.1.2.
The dimension drops on the singular part. As a first step, we show that a given Radon measure on R n must be absolutely continuous with respect to the Lebesgue measure L n if restricted to any Borel set where the tangent module has maximal dimension.
Proposition 3.5. Let µ be a finite Borel measure on R n . Suppose L 2 µ (T R n ) has dimension equal to n on a Borel set E ⊆ R n . Then µ| E ≪ L n .
Proof. Fix a countable dense subset C of D(div µ ). Given that Γ(T µ ) = cl D(div µ ) by item ii) of Theorem 2.16 and div µ satisfies the Leibniz rule, we know from Lemma 1.24 that T µ (x) coincides with cl v(x) : v ∈ C for µ-a.e. x ∈ R n . In particular, Remark 2.5 grants that: For µ-a.e. x ∈ E, there exist v 1 , . . . , v n ∈ C : span v 1 (x), . . . , v n (x) = R n . (3.2) Consider the family (S k ) k∈N of all those subsets of C made exactly of n elements. Given any k ∈ N, we denote by E k the set of all points x ∈ E such that v 1 (x), . . . , v n (x) ∈ R n are linearly independent, where {v 1 , . . . , v n } = S k . Then (3.2) grants that the Borel sets E k satisfy µ E \ k E k = 0. Now fix any k ∈ N and call S k = {v 1 , . . . , v n }. Thanks to Example 3.4, the 1-currents I(v 1 ), . . . , I(v n ) are normal and satisfy µ| E k ≪ I(v i ) for all i = 1, . . . , n. Therefore, we conclude from Theorem 3.1 that µ| E k ≪ L n for all k ∈ N, thus µ| E ≪ L n .
It is now easy to prove, as an immediate consequence of Proposition 3.5, that the tangent fibers cannot have dimension n on the singular part of the measure µ under consideration. Theorem 3.6 (Tangent fibers on the singular part). Let µ be a finite Borel measure on R n , with Lebesgue decomposition µ = ρL n + µ s . Then it holds that dim T µ (x) < n, for µ s -a.e. x ∈ R n .

(3.3)
Proof. Fix a Borel set B ⊆ R n such that L n (B) = µ s (R n \B) = 0. We argue by contradiction: suppose there is a Borel set E ⊆ B such that µ s (E) > 0 and dim T µ (x) = n for µ s -a.e. x ∈ E.
In particular, µ(E) > 0 and dim T µ (x) = n for µ-a.e. x ∈ E. As observed in Remark 2.5, this means that the tangent module L 2 µ (T R n ) has dimension n on E. Therefore, Proposition 3.5 grants that µ s | E = µ| E ≪ L n . This leads to a contradiction, as L n (E) = 0 but µ s (E) > 0.
Remark 3.7. Actually, Theorem 3.6 holds for any non-negative Radon measure µ on R n . Indeed, given anyx ∈ spt(µ) and r > 0, it can be readily deduced from [23, Proposition 2.6] that T µr (x) = T µ (x) is satisfied for µ-a.e. x ∈ B r (x), where we set µ r := µ| Br(x) . Moreover, notice that (µ r ) s = µ s | Br(x) . Therefore, by applying Theorem 3.6 to the measures (µ k ) k∈N we deduce that µ itself satisfies (3.3), thus showing that in the statement of Theorem 3.6 the finiteness assumption on µ can be dropped.
Remark 3.8 (Weighted real line). As already mentioned in the introduction, the Sobolev space on weighted R has been fully understood by S. Di Marino and G. Speight in [20]. More specifically, they completely characterised the minimal weak upper gradient of any Lipschitz function f ∈ W 1,2 (R, µ), where µ is a given Radon measure on R; see [20, Theorem 2]. We point out that our results imply a part (but not the whole) of their statement: Theorem 2.20 grants that |D µ f |(x) ∈ 0, lip(f )(x) is satisfied for µ-a.e. x ∈ R, while Theorem 3.6 ensures that T µ (x) = {0} and thus |D µ f |(x) = 0 hold for µ s -a.e. x ∈ R.
It is worth to isolate the following statement, which might be seen as a special case of Theorem 3.6 (or, alternatively, of Corollary 3.3). Corollary 3.9. Let µ ≥ 0 be a Radon measure on R n such that for every f ∈ LIP c (R n ). (3.4) Then it holds that µ ≪ L n .
Proof. By Corollary 2.21, we know that (3.4) is equivalent to T µ (x) = R n for µ-a.e. x ∈ R n . Therefore, it follows from Theorem 3.6 that µ s = 0, which exactly means that µ ≪ L n . Alternatively, one can argue as follows: since T µ (x) = R n for µ-a.e. x ∈ R n , we know a fortiori that V µ (x) = R n for µ-a.e. x ∈ R n , thus accordingly µ ≪ L n by Corollary 3.3.
Remark 3.10. Suppose that µ is a Radon measure on R n such that the resulting metric measure space (R n , d Eucl , µ) is doubling and supports a weak (1, 2)-Poincaré inequality, in the sense of [30]. Then the property in (3.4) is satisfied, as proven by J. Cheeger in [13]. Therefore, it follows from Corollary 3.9 that the measure µ must be absolutely continuous with respect to L n . This fact was already proven by A. Schioppa in [37]. See also [14].

3.2.
A geometric characterisation of the tangent distribution. The aim of this section is to show that the tangent distribution T µ associated with a given Radon measure µ on R n admits a 'geometric' characterisation in terms of the velocity of test plans, somehow refining Theorem 2.16. More precisely, we will prove that there exists a sequence (π i ) i of test plans on (R n , d Eucl , µ) having the following property: T µ is obtained as the closure of the velocities of the plans π i at time 0, in a suitable sense; see Theorem 3.16 for the correct statement. In order to achieve this goal, a key tool is given by the notion of test plan representing a gradient, which has been defined and proven to exist (in high generality) by N. Gigli in [23].
3.2.1. Reminder on test plans representing a gradient. First of all, let us report the notion of test plan representing the gradient of a Sobolev function; recall the definition (1.5) of KE t . Definition 3.11 (Test plan representing a gradient [23]). Let (X, d, µ) be a metric measure space. Let f ∈ W 1,2 (X, µ) be given. Then a test plan π on (X, d, µ) is said to represent the gradient of the function f provided it satisfies the following property: Test plans representing a gradient exist under mild assumptions, as the next result shows.
In lack of an appropriate reference, we provide a quick proof of the following elementary continuity result. To do so, we use the well-known density of LIP c (R n , R n ) in L 2 (R n , R n ; µ). Lemma 3.13. Let µ ≥ 0 be a Radon measure on R n . Let π be a test plan on (R n , d Eucl , µ). Then for every v ∈ L 2 (R n , R n ; µ) it holds that Proof. Fix any v ∈ L 2 (R n , R n ; µ). Choose compactly-supported Lipschitz maps v i : R n → R n such that v i → v in L 2 (R n , R n ; µ). Given any t ∈ [0, 1], we have lim s→t´| v i •e s −v i •e t | 2 dπ = 0 by dominated convergence theorem, so [0, 1] ∋ t → v i • e t ∈ B π is continuous. Moreover, the curves t → v i • e t uniformly converge to t → v • e t as i → ∞. Indeed, it holds that Therefore, the curve [0, 1] ∋ t → v • e t ∈ B π is continuous as well, as required.
As one might expect, if a test plan π represents the gradient of a Sobolev function f , then for any other Sobolev function g we have, roughly speaking, that the derivative at t = 0 of the map t → g • e t ∈ L 1 (π) coincides with the scalar product ∇ µ g, ∇ µ f • e 0 . This claim is made precise by the ensuing result, which has been proven in [35,Corollary 2.4].
Proof. Given that C ∞ c (R n ) is strongly dense in W 1,2 (R n , µ) by Corollary 1.19, we can find a countable Q-linear subspace (f i ) i of C ∞ c (R n ) that is dense in W 1,2 (R n , µ). In particular, the family V := k j=1 g j ∇ µ f i j : k ∈ N, (g j ) k j=1 ⊆ L ∞ (µ), (i j ) k j=1 ⊆ N is dense in L 2 µ (T R n ), thus the linear space ι µ (V ) is dense in Γ(T µ ). By using Lemma 1.24, we can deduce that for µ-a.e. x ∈ R n . (3.12) It is straightforward to check that one can find a Borel probability measure ν on R n such that´|x| 2 dν(x) < +∞ and µ ≪ ν ≤ Cµ for some C > 0. Given any i ∈ N, we know from Theorem 3.12 that there exists a test plan π i on (R n , d Eucl , µ) representing the gradient of f i and satisfying (e 0 ) * π i = ν. Theorem 3.15 grants that D π i exists as in (3.6). Also, it holds Im e 0 ,π i (D π i ) = Im e 0 ,π i ι µ (∇ µ f i ) By taking (3.12) into account, we eventually obtain (3.11), as desired.
3.3. Tensorisation of the Cheeger energy on weighted Euclidean spaces. In the framework of Sobolev calculus on metric measure spaces, a surprisingly difficult problem is the following: given two metric measure spaces (X, d X , µ) and (Y, d Y , ν), is the Sobolev space on the product space (X × Y, d X×Y , µ ⊗ ν) the tensorisation of W 1,2 (X, µ) and W 1,2 (Y, ν)?
The precise statement would read as follows: given any function f ∈ W 1,2 (X × Y, µ ⊗ ν), it holds for (µ ⊗ ν)-a.e. (x, y) ∈ X × Y that f (y) ∈ W 1,2 (X, µ), f (x) ∈ W 1,2 (Y, ν), and where we set f (y) (x) = f (x) (y) := f (x, y). (Here, Fubini theorem plays a role.) A positive answer to the above question is known only in some particular circumstances. About the spaces having such tensorisation property, this is the current state of the art: a) L. Ambrosio, N. Gigli, and G. Savaré proved in [7] that RCD(K, ∞) spaces, for any given K ∈ R, have the tensorisation property. b) L. Ambrosio, A. Pinamonti, and G. Speight proved in [8] the tensorisation property on doubling metric measure spaces supporting a weak (1, 2)-Poincaré inequality. c) N. Gigli and B.-X. Han showed in [26] that the Sobolev space tensorises as soon as one of the two factors is a closed real interval I ⊆ R.
To the best of our knowledge, these are all the cases that have been studied so far. The aim of this section is to prove that weighted Euclidean spaces have the tensorisation property (cf. Theorem 3.21), and we do so by first showing that the fibers of the tangent distribution 'tensorise' as well (cf. Proposition 3.19). Notice that the family of all weighted Euclidean spaces is not contained in any of the classes of spaces described in items a), b), and c) above.
Therefore, by arbitrariness of π n we can finally conclude that T µ (x) ⊆ p n T µ⊗ν (x, y) holds for (µ ⊗ ν)-a.e. (x, y) ∈ R n+m , whence the proof of the statement is complete.
Remark 3.22. Proposition 3.19 and Theorem 3.21 are verified even when µ and ν are (not necessarily finite) Radon measures, by taking into account [23, Proposition 2.6], which says that the Sobolev space can be 'localised' in a suitable sense. We omit the details.