Exact integrability conditions for cotangent vector fields

In Quantum Hydro-Dynamics the following problem is relevant: let $(\sqrt{\rho},\Lambda) \in H^1(\R^d) \times L^2(\R^d,\R^d)$ be a finite energy hydrodynamics state, i.e. $\Lambda = 0$ when $\rho = 0$ and \begin{equation*} E = \int_{\R^d} \frac{1}{2} \big| \nabla \sqrt{\rho} \big|^2 + \frac{1}{2} \Lambda^2 \mathcal L^d<\infty. \end{equation*} The question is under which conditions there exists a wave function $\psi \in H^1(\R^d,\C)$ such that \begin{equation*} \sqrt{\rho} = |\psi|, \quad J = \sqrt{\rho} \Lambda = \Im \big( \bar \psi \nabla \psi). \end{equation*} The second equation gives for $\psi = \sqrt{\rho} w$ smooth, $|w| = 1$, that $i \Lambda = \sqrt{\rho} \bar w \nabla w$. Interpreting $\rho \mathcal L^d$ as a measure in the metric space $\R^d$, this question can be stated in generality as follows: given metric measure space $(X,d,\mu)$ and a cotangent vector field $v \in L^2(T^* X)$, is there a function $w \in H^1(\mu,\mathbb S^1)$ such that \begin{equation*} dw = i w v. \end{equation*} %dw = i w v$? We show that under some assumptions on the metric measure space $(X,d,\mu)$ (conditions which are verified on Riemann manifolds with the measure $\mu = \rho \mathrm{Vol}$ or more generally on non-branching $MCP(K,N)$), we show that the necessary and sufficient conditions for the existence of $w$ is that (in the case of differentiable manifold) \begin{equation*} \int v(\gamma(t)) \cdot \dot \gamma (t) dt \in 2\pi \Z \end{equation*} for $\pi$-a.e. $\gamma$, where $\pi$ is a test plan supported on closed curves. This condition generalizes the conditions that the vorticity is quantized. We also give a representation of every possible solution. In particular, we deduce that the wave function $\psi = \sqrt{\rho} w$ is in $W^{1,2}(X)$ whenever $\sqrt{\rho} \in W^{1,2}(X)$.

Apart from the 1d case, where a weak solution to (1.2) can be constructed and then the wave function ψ is recovered (see [AMZ21]), in the general case one first obtains the solution to (1.3) and then shows that the associated hydrodynamics quantities (1.1) satisfy (1.2); for an overview of the mathematical theory for QHD we refer to [AM].
An implication that (ρ, J) are derived from a wave function ψ is that which mimics the fact that curl ∇θ = 0. Being this condition a local condition, it is not sufficient for the existence of a wave function ψ, as one can see with the elementary example in R 2 ρ(x) ∈ C 1 c (R 2 ), ρ(0) = 0, J(x) = ρ(x) αx ⊥ |x| 2 , α / ∈ Z.
(1.5) Indeed (1.4) is verified and being v = αx ⊥ /|x| 2 , curl v = 2παδ 0 , for α / ∈ N it is impossible to solve where Dw is the differential of the function w.
In [AMZ] the authors present some results which extend the 1d case recontruction of the wave function ψ to the case when x ∈ R d and ρ, J are radially symmetric, and more interestingly when x ∈ R 2 , the mass ρ is continuous and the vacuum set {ρ = 0} is given by isolated points {x 1 , x 2 , . . .}, and the velocity field v = J/ρ is a distribution such that (1.6) In this case the local function w(x) given by Dw = ivw (which is obtained by the above curl-free condition) can be extended to a global one because of the integer circuitation condition (1.6): indeed, considering a single singular point x 1 = 0 curl v = 2πkδ 0 , one deduce that for all test function φ − ˆ∇⊥ φ • vL 2 = 2πkφ(0).
With more generality, one can see that the condition (1.6) is equivalent to the following.Let Γ(R 2 ) be the metric space of Lipschitz curves γ : [0, 1] → R 2 with the uniform topology C 0 , and define a test plan π as a probability measure such that its evaluation e(t) π at time t satisfies ˆφ(x)(e(t) π)(dx) = ˆφ(γ(t))π(dγ) ≤ C ˆφ(x)dx, for all φ ≥ 0 continuous function.The formula (1.8) is then rewritten as follows: for π-a.e.γ such that γ(0) = γ(1) (i.e. it is a closed path) it holds ˆ1 0 v(γ(t)) • γ(t)dt = 2π j k j Wi(γ, x j ) ∈ 2πZ, (1.9) where Wi(γ, x) is the winding number of γ around z / ∈ γ([0, 1]).Being the set {ρ = 0} locally finite (or more generally of H 1 -measure 0), one can prove that π-a.e.γ is not passing through {ρ = 0}, so that for π-a.e.γ the winding number Wi(γ, x i ) is defined for all x i .The above formula is actually equivalent to the weak formulation (1.7) because for sufficiently regular functions the boundary of almost all level sets is the union of closed curves.
Formula (1.9) is meaningful in any dimension, and actually even in metric measure spaces, and gives rise to the following questions.
Problem.Let (X, d, µ) be a metric measure space (a Polish space with µ ∈ P(X) Borel probability measure) and let v ∈ L 2 (T * X) be a cotangent vector field such that, if π is a test plan supported on the set of closed curves, then ˆ1 0 v(γ(t)) • γ(t)dt ∈ 2πZ for π-a.e.γ. (1.10) (1) Is there a function w ∈ L ∞ (X, S 1 ) such that its differential satisfies Dw = iwv ?(1.11) (2) Assuming moreover that m is the ambient measure in (X, d) and (1.12) does the function f w belong to W 1,2 (X, C)?
Above and in the following we will use the identification {z ∈ C, |z| = 1} = S 1 , this should not generate confusion.In the case X = R d (or a differentiable manifold) the objects considered in the previous statement are standard, while in Section 3 we recall their definitions for generic metric measure spaces following [GP20].
The assumption that v ∈ L 2 (T * X) is the analog of the requirement Λ = √ ρv ∈ L 2 (R d ) in the Euclidean setting.We will call the condition (1.10) the integer circuitation condition.It is obvious that the opposite is trivial, i.e. v = −iDw/w has integer circuitation.It turns out that in general metric spaces the answer is no, simple because the space may have too few curves and the measure µ may be too singular (Example 6.3).But in more regular spaces the answer is positive: Theorem 1.1.Assume that (X, d, m) has the measure contraction property M CP (K, N ) (and if branching the ambient measure m is the Hausdorff measure H N ).Then, for every f ∈ W 1,2 (X, R) and cotangent vector field v ∈ L 2 (T * X) w.r.t. the measure µ = |f | 2 m/ f 2 2 , there exists w ∈ L ∞ (X, S 1 ) such that Dw = iwv and f w ∈ W 1,2 (X, C). (1.13) Moreover, there exists a partition of {|f | > 0} into at most countably many sets Ω i , m(Ω i ) > 0, such that for every function w satisfying Dw = iwv it holds w (x) w(x) = c i ∈ S 1 constant for m-a.e.x ∈ Ω i .
The second part of the statement is the same as the 1d case, where in every interval such that {|f | > 0} the function w is determined up to a constant.Moreover, every Riemann manifold satisfies the assumptions of the theorem.
Before giving an overview of the proof, some comments are in order.The actual result proved here is more general: the idea is to decompose the space X into regions {Ω α } α in which a.e. points x, y ∈ Ω α can be connected by test plans, and in these regions to construct the function w by integrating along curves.The technical part is to remove curves γ which are not "good" for v: either the integral along γ is not defined or the circuitation (1.13) is not integer if γ is closed.The non technical part is that such a decomposition does not need to be sufficiently regular to decompose the metric measure space (X, d, m) into metric measure spaces (Ω α , d, m α ).Example 6.3 indeed uses the well known construction of the Vitali set to show that the decomposition above does not exist in general.From a measure theoretical point of view, one needs to assume that the partition {Ω α } α admits a strongly consistent disintegration, i.e. one can write µ = ˆµα ω(dα) with µ α ∈ P(Ω α ). (1.14) This condition is certainly fullfilled if the partition {Ω α } α is countable, and this countability property is true under the assumptions of Theorem 1.1.It is not clear to us if such a decomposition into these measure-theoretic indecomposable components Ω α has some interests on its own.Within these setting, the function v is not locally integrable, so curl v is meaningless even distributionally.On the other hand, as in the example of (1.5), the study of curl Λ does not yield any condition into the existence of the function w.It is not clear if there exists some curl-like functional which allows to write the integer circuitation condition in a more PDE point of view, for example identifying the quantum vortex structure on which the rotational part of v is concentrated and has 2πZ-density, as in (1.6).
Another question is whether the integer circuitation assumption (1.10) is preserved by the flow of (1.2).A tautological argument is that since a solution to the QHD system is constructed by means of a wave function ψ solving Shcrödinger equation, a posteriori the integer circuitation condition must be compatible with the QHD formulation.Similarly for the study of the identity (1.4), it can be shown that the integer circuitation condition is compatible for smooth solutions of (1.2).Indeed one rewrites the system (1.2) as so that both ρ and v are transported by the same vector field v. Hence any curve γ at time t is transported back to a curve γ at time 0 (being the flow Ẋ = v(t, X) locally smooth), and defining Hence the fact that the l.h.s. of the equation for v in (1.15) is a gradient yields that the integer circuitation condition is preserved.The last two remarks (i.e. if one can give a meaning to curl v and if the computation above can be performed for nonsmooth solutions) are related to the fundamental question of regularity of the QHD solution, which is echoed by the regularity of the flow of the transport equation with vector field v.As far as we know, these questions are completely open in the general case and they are deeply connected to the proof of existence of a solution by purely fluid dynamics techniques.
As a last observation, a completely similar analysis can be done for the question: given v ∈ L 1 (T * X), there exists a function w such that Dw = v, i.e. the form is exact?
The results and the proof are exactly the same, by replacing the integer circuitation condition with the circuitation free condition: if π is a test plan supported on the set of closed curves, then ˆ1 0 v(γ(t)) • γ(t)dt = 0 for π-a.e.γ. (1.16) 1.1.Sketch of the proof and structure of the paper.In a nutshell, the idea of constructing w is the same as for the smooth case: given a value w(x) as some point x, find w(x) by solving the ODE (1.17) The integer circuitation condition should take care of the possibility of having multiple curves connecting x to x.
The problem is that all definitions are given in terms of measures, and then one has to consider only the good path, i.e. the ones for which the circuitation condition holds.This difficulty occurs also for X = R d , because of the density f ∈ W 1,2 .Given a closed curve γ such that γ(t) = γ(s) for some 0 < s < t < 1, it is not even true in general that the restricted closed curve R s,t γ = γ [s,t] is of integer circuitation, even if γ is.The first results (Proposition 3.10 and Theorem 3.12) state that under the integer circuitation condition (1.10) every test plan is concentrated on a set of curves A π such that every closed curve obtained by patching pieces of finitely many curves of A π has integer circuitation.In particular, restrictions of curves satisfy (1.10).We call the curves obtained by piecing together parts of other curves subcurve or subloops if closed.
The argument in R d (and in differentiable manifolds) is much easier than the general case in MMS: the case X = R 2 is particularly straightforward (Section 4).Indeed, one can use test plans to move mass along the i-axis, i = 1, 2, and then construct a finite dimensional map from R 4 into the set of curves Γ such that every point in some Cantor-like neighborhood is connected with another one by a specific curve moving first along the 1-direction and then along the 2-direction (Lemma 4.1).The integer circuitation condition and the result of the previous part imply that we can remove a set of measure 0 in R 4 and the remaining curves have integer circuitation together with the subcurves (Corollary 4.2).Since by W 1,2 -regularity of the function f one can locally assume that f > 1/2 on sufficiently many of these segments, the function w can be constructed on this set (which is of positive L 2 -measure, Lemma 4.3).This procedure can be repeated in every Lebesgue point of f and Df , and the different w (which are defined up to a constant) can be related with each other by a test plan (if it exists) concentrated on curves with initial points in one set and end points in the other set, making them compatible (Lemma 4.5).The same analysis can be done in R d , d ≥ 3, at the expense of a more complicated selection procedure for the curves.Since we will address the problem in general spaces, we will leave this extension to the reader which wants to avoid some measure theoretic arguments used in the remaining parts.
The observation above is that locally one can construct suitable paths which have integer circuitation, where f > 0. In particular these sets cannot be further divided by just removing a set of measure 0 trajectories of the above plan.The idea is then to find this minimal partition (see Section 5 for the abstract theory, and Section 5.1 for the application to out case).Given a set of curves {γ α } α , define the minimal (by set inclusion) equivalence relation E by requiring that γ α (t)Eγ α (s) for all s, t ∈ [0, 1], γ α (Definition 5.7).If we have a test plan π, for every set of curves A π , π(A π ) = 1, one has clearly different equivalence relations E Aπ .However the space X is separable, and we use this fact to deduce that there is a set Āπ such that the equivalence relation E Āπ is minimal: this means that if M Āπ is the family of µ-measurable sets which are saturated w.r.t.E Āπ , then for all other sets A π where π is concentrated there exists a µ-conegligible set where the σ-algebra generated by E Aπ is a subalgebra of M E Āπ (Proposition 5.8).This does not means that every atom of M Āπ coincides with an equivalence class of E Āπ : the situation is completely equivalent to the Vitali set of [0, 1], where E = [0, 1]/Q and then the E-saturated L 1 -measurable sets have Lebesgue measure 0 or 1 (see Example 6.3).
Notwithstanding the counterexample right above, the next step we perform is to find a test plan π such that the maximal measure algebra M π is the minimal among all measure algebras M π (Proposition 5.9).The reason for this analysis is the following: first, being M π maximal, up to a negligible set the space X cannot be further divided into smaller atoms, i.e. the atoms of M π are the minimal components used by the test plan π.Being M π minimal among the maximal measure algebras, we deduce that the all others test plans use curves which are contained into atoms of M π (up to a negligible set as usual, see Corollary 5.11).These components are in some sense the minimal connected-by-curves components of (X, µ), in the sense that one cannot find a further decomposition such that all test plans have trajectories concentrated on the atoms of this new refined decomposition.
Unfortunately finding the minimal-maximal σ-algebra is not sufficient: indeed the problem is that in general the components of the minimal-maximal decomposition do not coincide with the equivalence classes of the generating relation (as observed above, and as Example 6.3 shows).This condition seems essential and it is assumed in the construction of w (Assumption 6.1).In the case of sufficiently regular MMS (in particular M CP (K, N )-spaces where the ambient measure is f H N with f ∈ W 1,2 (X, H N )), it is satisfied (Example 7.2).Under the assumption that the equivalence classes coincide with the atoms of the measure algebra M π , the construction is now fairly simple, and follows the line for the R 2 case: for the test plan π take the set A π such that every subloop is of integer circuitation, choose a Borel section and a Borel function ω on the quotient space and for every x compute w by solving the ODE (1.17) with initial data ω(y) along a path connecting y to x, where y is the point in Borel section in the equivalence class of x (Proposition 6.4).It is important to note that the existence of such a path follows because we assume that the atoms of the measure algebra coincide with the equivalence classes, or equivalently that the disintegration is strongly consistent (1.14).The fact that Āπ is of free circuitation gives that the value w(x) is independent of the particular path chosen.The same argument yields that any other solution w is such that w /w is constant on the equivalence class (Theorem 6.6).
The final analysis of the case when µ = f m, with f ∈ W 1,2 (X, m) is at this point a corollary, once it is assumed that the space (X, d, m) has a test plan connecting all points at lest locally (a Democratic test plan, as defined in [LV07], see Assumption 7.1): indeed in this case the equivalence classes have positive measure, and then the disintegration is strongly consistent.Metric measure spaces enough regular satisfy this assumption (Example 7.2).For the existence of w solving (1.11), the only thing to verify is that there exists a test plan π satisfying the assumptions of the previous sections, and this is true in these spaces (Lemma 7.3).Finally, to check that f w ∈ W 1,2 (X, m) it is enough to prove that that it holds along test plans for m, and this is fairly easy once one have a Borel way of passing from test plans for m to test plans for µ (Lemma 7.6 and Proposition 7.7).

Notation
The real numbers are denoted by R, the rational numbers by Q, the natural numbers by N and the integers by Z.The unitary circle in R 2 is denoted by S 1 .The cardinality of a set A is denoted by A.
A generic constant (which may vary from line to line) is denoted with const.or C, eventually with an index or accent.
Let X be a separable complete metric space with distance d.A point in X will be denoted by x.The balls of radius r centered at x is denoted as B X r (x), and more generally the r neighborhood of a set A as B X r (A).The family of compact subsets of a metric space X is denoted by K(X), and we will use the Hausdorff distance d H (K 1 , K 2 ).Countable intersection of open sets are G δ -sets, while countable union of closed sets are F σ -sets.The projection onto X is written as p X , or sometimes p i if it is the i-component of j X j .The Borel σ-algebra is B(X), the projection of closed sets are the analytic sets Σ 1 1 , and the σ-algebra generated by them is σ(Σ 1 1 ).In case there is no ambiguity, the geodesic connecting x to y is −−→ [x, y]: this notation will be used mainly in R d .
A generic function from X to Y will be denoted by f : X → Y , and we will call f a Borel function when it is a Borel function whose domain of definition Dom(f ) is a Borel set.The graph of a function f : X → Y is denoted by and its range by f (X) = f (x), x ∈ X .For a (multi-valued) function f : X → Y we will denote f (x) both as the image of f (x) or as the section Graph f ∩ {x} × Y .In general this will not generate confusion.
The space of Lipschitz functions Lip([0, 1]; X) from [0, 1] into X is denoted by Γ: we will refer to it as the set of Lipschitz curves on X.Its subset of closed Lipschitz curves (or closed loops or loops for shortness) will be written as Γ c .An element of both sets will be written as γ.
It is known that Γ is a separable metric space with the C 0 -topology [Sri98, Theorem 2.4.3].
The evaluation map e(t) = e(t, γ) is the map The restriction operator R s,t (γ) is defined as the restriction of the curve The time inversion of a curve is denoted by R(γ The metric derivative of γ is denoted by |γ|.The length of a curve γ ∈ Γ is defined as Given a set A ⊂ Γ, we say that γ ∈ Γ is a sub-curve of A if there are finitely many curves The set of all sub-curves of A is denoted by Γ A .The Lebesgue measure in R d is denoted with L d and the d-dimensional Hausdorff measure by H d .Let P(X) be the space of probability measures over X and m,µ elements of P(X): in the second part of our analysis µ = f m (and m will be used as ambient probability for the space W 1,2 (X)), but in the first part we will consider (X, d, µ) as a given metric measure space.The topology on P(X) is the narrow topology.We denote the standard integral spaces by L p (µ).The push forward of a measure µ by a map f is written by f µ.The disintegration of a probability according to a map f : X → Y will be denoted as We will also write = µ M, where M is the σ-algebra made of µ-measurable sets saturated w.r.t. the equivalence relation If E is an equivalence relation, we will write xEx if x, x are equivalent, and the equivalence class for x as Ex or E(x).We will not distinguish between E and its graph in the product space.We will often use the letters α, β, . . .for the elements of the quotient space X/E.The quotient map is considered either as a section of X (i.e. a set S containing a single point for every equivalence class) or as a map e : X → [0, 1] whose level sets are the equivalence classes.

Settings and basic computations
In this section we introduce the setting for our problem: it is based on the analysis on metric measure space of [Gig,GP20], with elementary slight variations.The set of µ-admissible test plans is denoted by Adm(µ), and C is called the compressibility constant of π.
The above formula (3.1) is equivalent to It is clear that the above definition is invariant if we reparameterize the curves γ.In the following we will often consider the parametrization of γ by length, so that |γ| = 1: this is to avoid to write the factor |γ| in many equations.In that case the interval of integration is [0, L(γ)], and we will often not specify the interval of integration if clear from the context.
Remark 3.2.For the analysis in this work we do not require the conditions see for example [GP20, Definition 2.1.1].These assumptions will be analyzed in Proposition 3.9: here we can just say that since the integer circuitation condition (1.10) is required for π-a.e.γ and is independent on the parametrization of γ, it is natural to consider test plans as in Definition 3.1.
Remark 3.3.An admissible test plan π is given by the constant curves: define the map and set so that it is admissible.In particular the measure defined in (3.3) when f ≡ 1 is denoted with π.It may happens that these are the only test plans: for example, if (X, d) is totally disconnected (or zero dimensional [Sri98, Section 2.2]), the only Lipschitz curves γ : [0, 1] → X are the constant ones.

Tangent and cotangent bundles.
Here we recall the definition of tangent and cotangent bundles in a metric measure space X, following the approach of [Gig,GP20].
Let S 1 (X) be the set of functions f : X → R with the following property: there exists g ∈ L 1 (X) such that for all test plan π where C is the compressibility constant of π (see (3.1)) and e : [0, 1] × Γ → X is the evaluation map.The essential infimum of all g's is called the minimal upper gradient |Df |.
Let v be an element of the cotangent bundle L 1 (T * X), which can be defined as the where the equivalence relation ∼ is w.r.t. the above norm, and |Df | is the minimal upper gradient (see [GP20, Theorem 4.1.1]).We can associate to a function f ∈ S 1 (X) the cotangent vector field Df = (X, f ), so that The above norm definition gives that the pointwise norm is If π is a test plan, then for every simple element v = i 1 I Ωi Df i we can define Hence, being the elements of the form (3.4) dense in L 1 (T * X) and v → Λ γ (t, v) linear, passing to the limit we obtain that for every v ∈ L 1 (T * X) there exists a unique where is linear and continuous.
Remark 3.4.The function Λ γ (•, v) depends on π.Since we are using it only with a test plan which is clear from the context, we will not add the dependence w.r.t.π in what follows.
We will need the following property of Λ γ (t, v): we write the proof for completeness.
Proof.By (3.7) we have that if , so that up to subsequences it converges L 1 × π-a.e..In particular, let v n → v be a sequence of cotangent vector fields in L 1 (T * X) such that the statement is true for each v n , and let Clearly the set It is thus sufficient to prove the statement is true for v of the form (3.4), and in particular for a single f .Define the set of transversal self-intersections It is fairly easy to see that for every γ N n (γ) is finite for every γ: For a given f ∈ S 1 (X), consider the conegligible set Note that if γ(s) = γ(t), s, t ∈ M (γ), then there are sequences s n → s, t n → t, s n , t n = s, t respectively such that γ(s n ) = γ(t n ): hence by the formula (3.5) we deduce that The complement N = M c satisfies the statement.
be the corresponding L 1 -function given by (3.5): being v fixed from now on, for shortness we will not write the dependence w.r.t.v.We restate the assumption of integer circuitation (1.10) here below.
Assumption 3.7.For all test plans π concentrated on Γ c it holds i.e. v (or Λ) has integer circuitation.
We define so that we can rephrase the above assumption as I c has full π-measure.Note that as far as we know I c depends on π, but this should not create confusion being π fixed in the construction of the function w satisfying (1.11) (see Section 6).
An elementary example of an integer circuitation cotangent vector field is given by with π-a.e.γ, with π test plan.
Remark 3.8.Another possible condition is to ask that In this case we are looking for functions whose differential along γ is Λ γ , i.e. v = Df is exact: we will say that every admissible π has 0 circuitation or it is circuitation free.The analysis of this situation is completely similar to the case we are considering in the following, so the same constructions apply.
For completeness we show that we can restrict the set of test plans to a more tame one.
Proposition 3.9.Assume that Assumption 3.7 holds only for test plans which satisfy Then Assumption 3.7 holds for all test plans.
Proof.Let π ∈ Adm(µ) be a test plan supported on Γ c as in Definition 3.1.Since Assumption 3.7 is invariant for countable unions of test plans, we can assume that π is concentrated on curves with length L(γ) ∈ ( L, L + 1), L fixed, and we thus consider the pameterization In this case Condition (3.1) implies Hence for all δt > 0 the measure π defined by is supported on Γ c and satisfies L ≤ L(γ) ≤ L + 1 and where for simplicity we assume that 0 ≤ t ≤ L − δ.The assumptions on the proposition thus imply that Assumption 3.7 holds for π -a.e.γ, and by Corollary 3.6 so that we conclude that π satisfies Assumption 3.7.
3.4.Intersection properties of curves.By definition, any test plan π supported on Γ c is concentrated on the set of curves such that ˆL(γ) However a priori nothing is said about properties of intersecting curve: for example, given two curves γ, γ such that γ(s) = γ (s ), γ(t) = γ (t ), s ≤ t, s ≤ t , (3.9) one can construct a closed curve γ by setting (3.10) i.e. moving from γ(s) to γ(t) = γ (t ) along γ(τ ) and then returning to γ (s ) = γ(s) along γ(−τ ).We will use the above construction (i.e. a new closed curve obtained by piecing together parts of other curves) to show that we can also require by removing a π-negligible set of trajectories, where γ is the curve constructed above.We first start with self-intersecting curves.
Proposition 3.10.Let Λ be of integer circuitation.Every Proof.The proof is given in 3 steps.
Step 1: construction of a countable Borel set of loops.The following argument is the same as in [Sri98, Proposition 5.2.7]: we give it for completeness.The set is closed, and moreover By construction, the graphs of the functions L k,m are dense in F (γ) for each γ, i.e.
By reordering, we will denote the functions L m,k and domains A m,k with L n , A n respectively.
Step 2: negligibility of the set of bad loops.If π ∈ P(Γ c ) is a test plan, for each map L n (γ) = (s n , t n ) define the transport plan Being a restriction, clearly π n ∈ Adm(µ).In particular, it follows by Assumption 3.7 that Step 3: conclusion.The previous point gives that π ∈ Adm(µ) is concentrated on a set of curves A π such that for all n ∈ N it holds R sn,tn (γ) ∈ I c .Let now (s n , t n ), (s, t) ∈ F (γ), with (s n , t n ) → (s, t).Then 2πZ which gives that every subcurve of γ ∈ A π belongs to I c , i.e. it is of integer circuitation.
As a consequence of the previous proposition, the next lemma shows that we can construct the function w along π-a.e.curve.
Lemma 3.11.Let γ be a Lipschitz curve such that if γ(s) = γ(t) then R s,t γ ∈ I c .Then there exists a function Proof.Define the function w γ (t) = e i ´t 0 Λγ |γ|L 1 .
If γ(t) = γ(s), then by the assumption R s,t γ ∈ I c it holds w γ (t) = w γ (s)e i ´t s Λγ |γ|L 1 = w γ (s), so that the function w γ can be actually defined on Graph(γ) by We next generalize Proposition 3.10 to closed subcurves of the set of curves where π is concentrated: Theorem 3.12.Let Λ be of integer circuitation.Every π ∈ Adm(µ) is concentrated on a set of curves A π such that A π (n) ⊂ I c for all n, i.e. every subloop of A π belongs to I c .
Proof.The proof is given in two steps.The first one illustrates the procedure in a simple case, while the second is the general case.
Step 1: the case of two curves.Consider the set (Γ c ) 2 = Γ c × Γ c and its closed subset In particular the multifunction F (γ 1 , γ 2 ) with graph is Borel measurable and F (γ 1 , γ 2 ) is closed, so that by [Sri98, Theorem 5.2.1] there is a Borel selection We thus can define the curve (3.11) Clearly the map (γ 1 , γ 2 ) → γ (γ 1 , γ 2 ) is Borel.Up to reparametrization, the curve γ (γ 1 , γ 2 ) is the closed curve obtained by moving along the first curve and then along the second one.
Consider now the set of positive measures where π ∈ Adm(µ) is the test plan in the statement.We can define the new test plan It is fairly easy to see that π ∈ Adm(µ) because and since γ ∈ Γ c then π is supported on Γ c .Proposition 3.10 implies that the set In our case, we have observed that the l.h.s. is 0 when B = C(2), and then there are set B 1 , B 2 negligible w.r.t.π such that for every γ 1 , γ 2 ∈ Γ c \ (B 1 ∪ B 2 ) every subloop of the curve γ (γ 1 , γ 2 ) has integer circuitation.This proves the theorem for n = 2.
Step 2: the general case.Define the closed set Starting with γ 1 , as in the previous step we can join it to some curve γ i , i = 1, by formula (3.11): call this curve γ2 .Using the assumption that ∪ i Graph γ i is connected, we can repeat the process by adding to γ2 a new curve γ j , j = 1, i, as in (3.11), obtaining a new curve γ3 .Proceeding by adding curves, after n steps we obtain a single curve γ: it is fairly easy to see that since each merging operation is Borel, so the map (γ 1 , . . ., γ n ) → γ(γ 1 , . . ., γ n ) is Borel too.As in the previous step, for every we can construct the test plan π = γ(γ 1 , . . ., γ n ) ξ, so that we deduce that the set In our case the left side is 0, so that we can take the sets B i = {f i > 0} to deduce that there are π-negligible sets B i such that In particular in the complement of Bn = ∪ i B i it holds that every subloop of γ(γ 1 , . . ., γ n ) belongs to I c .Removing the negligible set ∪ n Bn we obtain the set A π of the statement.
Given a sequence of test plan π n , n ∈ N, it is immediate to see that where C n is the constant for π n in (3.1) (the constant in front the sum is for normalization).Hence the previous theorem gives the following statement.
Corollary 3.13.If {π n } n∈N ⊂ Adm(µ) is a countable family of µ-test plans, then there exist Borel sets A πn such that π n (A πn ) = 1 and it holds As a preliminary analysis, we perform the construction of a function w : R 2 → S 1 such that where v ∈ R d is a (cotangent) vector field defined f -a.e.having integer circuitation, and f ∈ W 1,2 (R 2 , L d ): the integer circuitation assumption is for π-a.e.γ, with π-test plan concentrated on Γ c (R 2 ) for the measure µ = f 2 L 2 / f 2 2 .With some additional technicalities, a similar analysis can be adapted to the d-dimensional case: the advantage in the 2d-case is that the construction of the test plans for the measure µ = f 2 L 2 is particularly easy.
Consider a Lebesgue point for f and Df : let us assume for definiteness By standard density argument, for the directions e 1 , e 2 along the coordinate axis there are segments In particular it follows that |f | ≥ 1/2 H 1 -a.e. on all the segments { k,y , y ∈ K k , k = 1, 2}.Consider the map R 4 (y, y ) = (y 1 , y 2 , y 1 , y 2 ) → γ (y,y ) (t) ∈ Γ c where where −−→ [x, y] is the geodesic connecting x, y: γ (y,y ) is the oriented boundary of the rectangle with opposite corners y, y .Lemma 4.1.For all g ∈ L ∞ (R 4 , L 4 ), g 1 = 1 with bounded support, the measure π = (γ (y,y ) ) (gL 4 ) is a test plan in (R 2 , L 2 ).
Proof.Since γ (y,y ) is the union of 4 curves, it is enough to prove the statement for a single part, for example − −−−−−− → [y, (y 1 , y 2 )]: in this case ˆφ (1 − t)y 1 + ty 1 , y 2 g(y, y )dydy = ˆφ(z 1 , y 2 )g(z, y )(( where Using the same ideas of the first part of the proof of Theorem 3.12 applied to the measures , we deduce the following corollary. Corollary 4.2.There is a L 2 -conegligible set I 0 ⊂ K 1 × K 2 such that for every y, y ∈ I 0 the curve γ (y,y ) has integer circuitation.
Using Fubini, we obtain that for L 2 -a.e.y = (y 1 , y 2 ) ∈ I 0 the set has full measure.Define for one of these y w(y, y ) = e i ´y Proof.The proof is a consequence that the statement holds for y ∈ I 0 such that (y 1 , y 2 ) ∈ I 0 because setting y = (y 1 , y 2 ) ˆy and the set y ∈ I 0 , (y 1 , y 2 ) ∈ I 0 × y ∈ I 0 , (y 1 , y 2 ) ∈ I 0 has full measure due to the assumption on (4.3) and on I 0 .
By repeating the construction, we obtain the following corollary.
Corollary 4.4.The is a countable family of sets K 1,n × K 2,n and of functions w n such that and w n satisfies Lemma 4.3 in every K 1,n × K 2,n .
Note that we do not require the sets K 1,n × K 2,n to be disjoint.
Proof.Since by restricting the balls we can require that a standard covering argument as in the proof of [Bog, Theorem 5.5.1]gives the statement.
The above lemma defines the function w in a set of positive Lebesgue measure, with the property that every two points are connected with a path where |f | ≥ 1/2.This is the hardest step, since the remaining part needs only to show that this w is compatible with all test plans and it can be extended to the whole {f = 0}.
Proof.For π-a.e.γ the set γ such that γγ,γ is of integer circuitation has full π-measure.For every such a γ, define c ij as the inverse of the ratio of (4.4): Since by (4.4) the ratio is the same for π-a.e.γ, the proof is complete.
Starting from the function w 1 defined in K 1,1 ×K 2,1 , we can then find constants c 1j such that the above lemma holds for all sets K 1,j × K 2,j for which there is a test plan connecting K 1,1 × K 2,1 to K 1,j × K 2,j .If at this step we cover {|f | > 0} up to a L 2 -negligible set, we are done, otherwise we take the next K 1,i × K 2,i and connect to some others subsets K 1,j × K 2,j : clearly these sets cannot belong to the previous family, otherwise there is a test plan from K 1,1 × K 2,1 to K 1,j × K 2,j and then to K 1,i × K 2,i .
We this conclude with the following statement.
Proposition 4.6.There exists a function w such that for all test plans π w.r.t. the measure |f | 2 L 2 it holds w(γ(1)) = w(γ(0))e i ´v• γ π-a.e.γ.Moreover all these functions depend at most on countably many multiplicative constants c j .
Proof.We are left in proving the second part, which follows from the following observations: (1) every solution w satisfies w w j = constant in every K 1,j × K 2,j : indeed it is constant on the curves used in Lemma 4.3; (2) the relation R ) ⇐⇒ ∃π plan connecting them as in Lemma 4.5 is an equivalence relation; (3) the construction of w presented before the statement of this proposition is defined up to a multiplicative constant in each equivalence class.
This concludes the proof.
Once the function w is constructed, it follows from the chain rule in one dimension that for all test plans w.r.
for π-a.e.γ.In particular, by elementary arguments, for L 1 -a.e.y 2 it holds y 1 → (f w)(y 1 , y 2 ) is a.c. and has derivative w∂ x1 f + iwv 1 , and this means that Since |f w| = |f | we thus conclude with the following result.
Proposition 4.7.For every w given by Proposition 4.6 the function f w is in W 1,2 (R 2 ) and has differential D(f w) = wDf + iwf v.

Construction of the minimal equivalence relation
In this section we construct a decomposition X = ∪ α X α into disjoint sets with the following properties: • each test plan is concentrated on a set of curves γ such that γ([0, 1]) belongs to a singe elements Z α up to a H 1 -negligible est; • there exists a test plan π generating the partition {X α } α up to a µ-negligible set.
The first property means that every test plan move mass only inside the equivalence classes X α : this can be interpreted by saying that the X α are the connected components from the point of view of the MMS (X, µ), in the sense that no mass transfer can occur between different equivalence classes.For example, the MMS is such that the segment connecting [−1, 1] has L 2 -measure 0, so the no test plans π can charge mass on it.Hence it is not seen by any Sobolev function f ∈ W 1,2 (X): from their point of view the two balls are completely disjoint.
The second property instead says that no smaller partitions can be constructed if they need to satisfy the first property: even by changing the set where the plan π is concentrated, the partition {X α } α remains the same in a set of full µ-measure.Returning to the example (5.1), it is elementary to see that the two balls cannot be further subdivided if we require the above first point to hold (recall Section 4 and see also Example 7.2).
In the first part of this section we take an abstract point of view as follow.
(1) The (complete) σ-algebra M of µ is separable, i.e. countably generated: identifying elements of M up to negligible sets, we obtain a separable measure algebra which we still denote by (M, µ) or M since the measure µ is fixed here.The function is a distance on M, which makes M a complete and separable metric space.(2) Each equivalence relation E on X generates the sub σ-algebra M E made of all saturated measurable sets, M E = Ω ∈ M : EΩ = Ω . (5.3) The corresponding measure algebra M E is clearly a measure sub-algebra of M. In particular it is separable and closed.
The statements below can be deduced easily from the properties of measure algebras, see [Fre02a,Chapter 3].Consider a family of measure sub-algebras {M α } α , M α ⊂ M, closed under countable union: for every sequence M αi there exists M ᾱ such that i M αi ⊂ M ᾱ.
The following result shows that there is a maximal element M ᾱ.
The proof is just the observation that every subset of a separable metric space has a countable dense subset.
Proof.Consider the measure sub-algebra generated by the union of all M α : it corresponds to the closure of the algebra generated by the algebra ∪ α M α w.r.t. the distance (5.2).In particular, being countably generated, it coincides with the measure sub-algebra generated by countably many elements Ω n ∈ ∪ α M α .
Let the sub-algebras M αn be such that Ω n ∈ M αn , and let M ᾱ be measure sub-algebra containing all of them.This is the measure sub-algebra of the statement.
A symmetric result considers the case where the family {M α } α is closed under countable intersection: for every sequence {M αn } n there exists M ᾱ such that (5.4) Proposition 5.2.There exists The proof follows by observing that every union of open balls is obtained by considering only countably many balls.
Proof.First of all, for all Ω / ∈ α M α there is α Ω such that Ω / ∈ M αΩ , and then there exists r Ω > 0 such that the set By separability, we need only a countable family of balls Hence we deduce that if which is the statement by (5.4).
In terms of σ-algebras, the two statements above can be rewritten as follows.Consider a family of µ-complete σ-algebras M α ⊂ M, where M is the µ-completion of a separable σ-algebra.
Corollary 5.3.The following holds: (1) if {M α } α is closed under countable union, then there exists an element M ᾱ such that ∀α, Ω ∈ M α ∃Ω ∈ M ᾱ µ(Ω∆Ω ) = 0 ; (2) if {M α } α is closed under countable intersection, then there exists an element M ᾱ such that The same statement can be rewritten in terms of equivalence relations, by reversing the intersection and union.
5.1.Application to test plans.The statement of Proposition 5.1 in our setting will be used as follows.
Definition 5.5.Let {Ω α } α be a family of subsets of a set X.The equivalence relation generated by Lemma 5.6.The equivalence relation generated by {Ω α } α is the following: Proof.Immediate from the definition.
We apply the above construction to our case, the sets Ω α being the range of the curves {γ ∈ A π }.
Definition 5.7.We will say that two points x, x in X are equivalent according to the set A π ⊂ Γ and we will write xE Aπ x if they are equivalent according to Definition 5.5 where the sets Ω α are the family of sets {γ([0, 1]), γ ∈ A π }.Equivalently, there exists a finite set {γ i } i ⊂ A π of curves and a path γ ∈ Γ such that Graph γ ⊂ i Graph γ i and γ(0) = x, γ(1) = x .
For every test plan π ∈ Adm(µ) consider the family of sets A π satisfying Theorem 3.12 (i.e.every closed subloop of A π is of integer cicuitation), and let E Aπ , M E Aπ be the corresponding equivalence relation and σ-algebra, respectively (see (5.3)).
In particular, for every subset A π ⊂ Āπ such that π(A π ) = 1 it holds as measure algebras. (5.5) We will denote M E Āπ as M π , and the corresponding equivalence relation by E π = E Āπ .We stress that the only unique object is M π considered as a complete σ-subalgebra of M or as a measure algebra.
Proof.We need only to verify that the family {E Aπ } Aπ is closed under countable intersection: this follows from the implications and We next apply Proposition 5.2 to sequences of test plans.
Proof.Consider a sequence of minimal equivalence relations {E πi } i , and define the test plan where C i is the compressibility constant (3.1) (as before we normalized the sum of measures in order to be a probability).Let Āπ , Āπi be a set of curves satisfying Proposition 5.8 above for π, π i respectively.
If we consider the sets Âπi = Āπ ∩ Āπi , then π i ( Âπi ) = 1 because π i π, and then by maximality of M E Āπ i we conclude that by (5.5) that We thus deduce from E Âπ i ⊂ E Āπ that M π ⊂ M πi for all i, which shows that we are in the situation of Point (2) of Corollary 5.3: the minimal measure algebra M π satisfies the statement.
Let Āπ be a σ-compact set where π is concentrated such that E Āπ generates the maximal measure algebra M π , which by the previous proposition is the smallest among all maximal measure algebras M π .
Let {Ω n } n be a countable family of E Āπ -saturated sets generating M π up to negligible sets, and define the equivalence relation Ē = n (5.6) By construction E Āπ ⊂ Ē, and in general the inclusion is strict.The equivalence relation Ē is made of the atoms of the σ-algebra generated by the {Ω n } n , whose completion coincides with M π .
Remark 5.10.Consider another set Ā π generating M π , and let {Ω n } n be E Ā π -saturated sets generating M π .Since M π is unique and it is the completion of the σ-algebra generated by Ē, then there are Ēsaturated sets we deduce that the equivalence relation obtained by (5.6) using the family {Ω n } n is equal to the one constructed through the sets {Ω n } n up to a µ-negligible set.Indeed the equivalence relation generated by {Ω n \ N } n contains Ē since the sets Ω n are Ē-saturated.Hence Reversing the argument we obtain that there exists a µ-negligible set N such that and then Let π ∈ Adm(µ) and Āπ generating the maximal M π as in Proposition 5.8.By minimality of M π , for every Ω n there is a set Ω π n ∈ M π such that µ Ω n ∆Ω π n = 0. (5.7) In particular, following the reasoning of the previous remark, there exists a µ-negligible set N ⊂ X such that if Ē is given by (5.6) then (5.8) Since γ ∈ Āπ takes values in a single class of E Āπ , we conclude that Corollary 5.11.Every π ∈ Adm(µ) is concentrated on a family of trajectories A π such that Proof.Since we know that γ(t)E Āπ γ(s) for s, t ∈ [0, L(γ)] by construction, then we deduce from (5.8) that γ(s) so that we need only to remove the negligible set of curves γ such that L 1 (γ −1 (N )) > 0.
Remark 5.12.The presence of the negligible set N is natural: take for example the space X = S 1 , µ = L 1 , and the equivalence relation The measure algebra is the trivial one, but any curve passing through 0 sees N = {0}.

Construction of the function w
In the previous section we have constructed an equivalence relation E Āπ for the test plan π such that the measure algebra M π generated by the saturated sets M π satisfies the following properties: (1) Āπ satisfies Theorem 3.12, (2) it is the maximal measure algebra among all measure algebra generated by sets where π is concentrated, (3) all others test plans π are concentrated on a set of curves Āπ generating a measure algebra which contains M π .Nothing is said about the strong consistency of the disintegration w.r.t.E Āπ : if then in general the conditional probability µ α is not concentrated on the corresponding equivalence class E Āπ α : this implies that Ē ⊇ E Āπ .Assumption 6.1.We assume that the minimal disintegration w.r.t.E Āπ is strongly consistent for every Āπ generating M π , i.e.
We recall that this is equivalent to each of the following conditions ([BC09] or [Fre02b, Section 452]): (1) the equivalence relation E Āπ is essentially countably generated, and in particular we can assume that Ē = E Āπ up to a µ-negligible set, where Ē is given by (5.6); (2) there exists a µ-measurable map f : X → [0, 1] whole level sets are the equivalence classes: Moreover, strong consistency implies that, up to a µ-negligible set of equivalence classes of E Āπ , there exists a Borel section S, i.e. a Borel set in X such that it contains exactly one point of each equivalence class of E Āπ .We collect the previous observations in the following statement.
Lemma 6.2.Under Assumption 6.1, there exists a test plan π concentrated on a set of trajectories Āπ satisfying Theorem 3.12 such that: (1) the equivalence relation Ē = Ē Āπ generated by Āπ admits a Borel section S and has the following equivalent properties (up to a µ-negligible saturated set): (a) the disintegration (6.1) is consistent, (b) the atoms of the separable σ-algebra M π are the equivalence classes of Ē; (2) every two points of an equivalence class are connected by finitely many curves in Āπ .Assumption 6.1 is necessary to prove that the integer circuitation condition of Λ implies the existence of the function w: indeed if the disintegration is not consistent, then there are counterexamples as the following one.
Figure 1.The Cantor set K together with a piece of the orbit o, and the evolution of the point (y, θ) in the angular coordinates.
Example 6.3.Consider the Cantor 1/3-set and let V : [0, 1] → [0, 1] be the Vitali function which is defined up to the countably many points belonging to the boundary Z of the intervals n−1 i=1 s i 3 −i + 3 −n (1, 2), n ∈ N, s i ∈ {0, 2}.For definiteness we can assume that those points are removed from K, in which case the set K \ Z is a G δ -set, i.e. still a Polish space.
If we consider the torus T 2 , then the values of the angles θ(y) on a single orbit corresponds to the trajectory of the points n → y + nα mod 1, θ(y) 2π + nβ mod 1 .
We can slightly change the space X and the test plan π in order to see that the assumption that the disintegration is strongly consistent for all Āπ is essential.Set X = X ∪ [0, 1] × {0}, i.e. we add the segment containing K.This segment is negligible for all test plans, since we still keep the measure supported on X.To have a set of curves with integer circuitation, just remove a point to every orbit o(t, x), for example assume that Āπ = (t, 0, 0 It is fairly easy to see that Āπ satisfies Theorem 3.12, and the equivalence class is X \ ∪ x {o(1/2, x)}.However removing the negligible curve (t, 0, 0) the measure algebra M is not the minimal one M π .The fact that the last possibility is ruled out by Assumption 6.1 is essential in the proof of Theorem 6.6.
We now prove the existence of a function w using the curves of Āπ .Proposition 6.4.There exists a Borel function w such that for all sub-curves γ of Āπ it holds w(γ(1)) = w(γ(0))e i ´1 0 Λγ (t) |γ|(t)dt .
Proof.Let S be a Borel section of Ē, and let Γ Āπ, S be the family of subcurves of Āπ passing through S. We will assume that S has a point in each equivalence class, by removing a µ-negligible set of equivalence classes.
For every curve γ ∈ Γ Āπ, S let s(γ) = max t : γ(t) ∈ S .Notice that by the assumption that Āπ generates Ē, every point x ∈ X is connected to the unique point S ∩ Ex by a subcurve γ ∈ Γ Āπ, S .Being {(t, γ), γ(t) ∈ S} a Borel set, it follows that s is Borel.
Define the function is a sub-loop of Āπ .This shows that w(t, γ) is the graph of a Borel function w defined on X.
Finally, let γ ∈ Γ Āπ be a subcurve of Āπ .Let γ 0 , γ 1 ∈ Γ Āπ, S be subcurves connecting γ(0), γ(1) to a point S, respectively: such curves exists because γ takes values in a single equivalence class, and each point in an equivalence class is connected to S by a subcurve.The subloop made by these 3 curves has integer circuitation, and then w(γ(1)) = w(γ(0))e where we have used that the curve obtained by joining γ, γ 0 , γ 1 is a subloop of Āπ .
The next proposition shows the importance that M π is the maximal measure algebra generated by π.
Observe that in this prof we do not need the assumption on the strong consistency of the disintegration, Assumption 6.1: this is due to the fact that the function w is µ-measurable.
Proof.Consider the test plan π generating the minimal equivalence relation Ē.
Assumption (6.5) implies that t → w(γ(t)) is equal to a constant h(γ) for L 1 -a.e.t ∈ [0, 1] for π-a.e.γ.Define the set The same analysis of Step 1 of Theorem 3.12 implies that there is a π-negligible set N ⊂ Γ such that Indeed, for every measure 0 < ξ ∈ Π ≤ (π, π) concentrated on W the test plan obtained by (3.11) would not satisfy the assumption of the proposition.
Hence, t → w(γ(t)) is constant in each subcurve γ of Γ \ N for L 1 -a.e.t.In particular, it is constant in each equivalence class of the equivalence relation E Āπ\N generated by Āπ \ N .
This equivalence relation E Āπ\N is smaller than Ē, but by the maximality of M Ē it generates the same measure algebra M Ē = M π .If w is not constant in each equivalence class of Ē up to a µ-negligible set N ⊂ X, then it follows that taking f as in (6.2) for the equivalence relation Ē and defining we obtain a Borel map such that its equivalence relation Ē = (f , f ) −1 (id) is smaller than Ē and the measure algebra M Ē strictly contains M π , which contradicts the maximality of the latter.
The last result gives the general formula for every solution to the differential equation Dw = iwv or more precisely rewritten in (6.6) below.
In particular the map is bijective on the set of solutions.
Proof.Given a function ω ∈ L ∞ (ζ, S 1 ), the following extension of formula (6.3) constructs the candidate function for the first part of the statement: where γ ∈ Γ Āπ, S is a subcurve of Āπ connecting y to x.
We need to prove that the integral formula (6.6) holds for all π.Let π be a test plan: by removing a negligible set of curves, we can assume that (π + π)/2 is concentrated on a set of integer circuitation curves, together with its subloops (Corollary 3.13).Being the equivalence relation Ē maximal in the sense of Proposition 5.9, we can apply Corollary 5.11: the curves γ on which π is concentrated are such that (1) γ(t) belongs to the same equivalence class of Ē up to a L 1 -negligible subset of [0, 1]; (2) the function w(γ(t); ω) given by formula (6.7) is defined L 1 -a.e.Since the disintegration is strongly consistent, for every γ as above there is a subcurve curve γ of the set where π is concentrated connecting γ(s) to γ(t) (by Point (1) above and Assumption 6.1).Being π + π of integer circuitation, we deduce as in (6.4) that (6.6) holds.
Finally we show that every solution w is computed in this way, i.e. the second part of the statement.By construction, along π-a.e.
so that we can apply Proposition 6.5.
7. The W 1,2 -case In this section we apply the above analysis to the case µ = |f | 2 m/ f 2 2 , where m is now the ambient measure on X and f ∈ W 1,2 (X, m).We still need some assumptions on the structure of the MMS (X, d, m), as explained in the introduction.
Assumption 7.1.For m-a.e. point x ∈ X there exist two sequence of numbers r n , R n , r n ≤ R n , converging to 0 and a test plan π n ∈ P(Γ) such that for some constant and (e(0), e(1) Up to the restriction that the curves γ used by π n are localized inside the ball B Rn (x), the test plans π n are the Democratic plans of [LV07].
Example 7.2.The above condition holds for example for which is equivalent to m by coarea formula, being In the case of branching spaces, the same condition holds if (locally, being all computations done in a small ball) and for every x ∈ X, 0 < r < 1, there is a transport plan π x,r ∈ Geo(X) concentrated on geodesics such that e(0) π x,r = δ x , e(1) π Being π x,r ∈ P(Geo(X)), i.e. it is concentrated on the set of geodesic, it follows that L(γ) ≤ 2r, and thus e(t) π y,x,r is concentrated on B 2r (x) for y ∈ B r (x).The measure contraction property (the last estimate of (7.2)) gives that m(B 2r (x)) ≤ C2 N m(B r (x)).Moreover by construction (e(0), e(1)) m = 1 m(B r (x)) 2 m × m Br(x)×Br(x) .
Finally, consider a ball B r (x) and compute where we have used the last estimate in (7.2) and the simple observation e(t) πy,x,r (B r (x)) ≤ 1 y ∈ B r+2tr (x), 0 otherwise.
Using again the measure contraction property one obtains and then for t ≥ r/r and using (7.1) For t ≤ r/r we have m(B r+2tr (x)) ≤ C4 N m(B r (x)), and thus we obtain for some constant C .This shows that e(t) π ≤ C m/m(B r (x)).
Lemma 7.3.There exists a test plan π satisfying Assumption 7.1 and such that the equivalence relation Ē given by (5.6) is made of countably many equivalence classes of positive measure.
Proof.Take a point x as in Assumption 7. with c n , c nn chosen as in (3.12) to have a probability measure.
We next consider a function f ∈ W 1,2 (X, m) and define probability measure We assume that we are given the cotangent vector field v ∈ L 2 (T * X) on the space (X, µ), and we denote with Λ γ (t) the corresponding function as in (3.6).
Proposition , so that the end points of the previous trajectories cover a subset of B δ (x) with m-measure larger than 7m(B rn (x))/8.The set of x such that |f (x) − 1| > 1/4 has measure smaller than 16 2 m(B rn (x)), so that there is a set of positive m-measure such that |f (x) − 1| > 1/4 and the trajectories starting from x verifying (7.4) have end points in a set with measure > 7m(B rn (x))/8.In particular the trajectories starting from each of these points must intersect, and since Corollary 7.5.The disintegration w.r.t. the equivalence classes of E Āπ for the measure µ is strongly consistent for every Āπ , hence that there exists a solution w.
This corollary, together with Example 7.2, proves the existence part of (1.13) of Theorem 1.1.
Proof.We just observe that the disintegration with equivalence classes of positive measure is certainly strongly consistent, corresponding to a countable partition of the ambient space up to a negligible set.
Let now π be a test plan for (X, m): we can assume that π is concentrated on a family of curves A π such that f • γ ∈ H 1 ([0, L(γ)]) for all γ ∈ A π .In particular, f • γ is continuous and for every n The next lemma assures that the dependence w.r.t.k, γ of t ± k,n is Borel.Lemma 7.6.There exists a family of countable maps of Borel regularity γ → t − k,n (γ), t + k,n (γ) such that (7.5) holds.
(7.8) Moreover any other solution w satisfies w(x) w (x) = ω(Ex), where E is the equivalence relation constructed in Section 5 in the space (X, µ) with µ = f 2 m/ f 2 2 .In particular there are at most (S 1 ) N -different solutions.Together with Proposition 7.4, the above statement completes the proof of Theorem 1.1.
Proof.Because of Theorem 6.6, we are left to prove (7.8): this follows from (7.6) and the observation that every cotangent vector field is uniquely defined by its behavior on test plans as in (3.5).

Lemma 4. 3 .
For every (y , y ) ∈ (K 1 × K 2 ) 2 \ N , L 4 (N ) = 0, it holds w(y, y ) = w(y, y )e 1, where M i are the d × d-minors of the matrix [(1 − t)id tid].In particular, it holds in a Riemann manifold by a slight variation of the above computation.More generally, it holds in non-branching metric measure space under the Measure Contraction Property M CP (K, N ) [Oht07, Definition 2.1], just by adapting the proof of [LV07, Theorem 3.1].
with non integer circuitation is negligible for every ξ ∈ Π ≤ (π, . . ., π).We now use [Kel84, Lemma 1.8, Theorems 2.14 and 2.21]: they state that if B is Borel (or analytic) then sup ξ∈Π ≤ (π,...,π) 1, fix n and let A πn be a set of curves where π n is concentrated: by assumption m-a.e. point x 1 ∈ B rn (x) is connected by a curve to m-a.e. point x 2 ∈ B rn (x).By the transitivity of the equivalence relation, all these points are in the same equivalence class, i.e.B rn (x) is in a single equivalence class up to a m-negligible set.It is clear that by removing a negligible set of trajectories we changes the equivalence class only in a m-negligible set.Let {Ω n } n = {B rn (x n )} n be the family of balls as above covering m-a.e.x ∈ X.Let π nn be a test plan (if it exists) such that e(0) π nn m Br n (xn) , e(1) π nn m Br n (x n ) .n c n π n + nn c nn π nn , 7.4.There exists a test plan π (w.r.t. the measure µ) satisfying Assumption 6.1 such that its equivalence classes are of positive measure.Proof.As in the previous proof, it is enough to prove the statement locally by showing that nearby to a Lebesgue point of |f |, |Df | > 0 there exists a plan π generating an equivalence class of positive measure essentially invariant w.r.t. the choice of the sets A π .Let x ∈ X be a Lebesgue point for |f | and |Df | (being |Df | the upper gradient), i.e. for definiteness Consider the ball B rn (x), with R n ≤ r, and the test plan π n defined in Assumption 7.1: we have ˆ ˆ1For all other points, there exists a family of geodesics of π n,x -measure greater than 7/8 such that 0 |Df |(γ(t)) |γ|(t)dt π n (dγ) = ˆL(γ)π n (dγ) + ˆ ˆ1 0 |Df |(γ(t)) − 1 |γ|(t)dt π n (dγ) L(γ) ≤ 2R n , e(t) π n ≤ C xm/m(B rn (x)) rn (x)) Begin A π,k Borel, the functions t± k are Borel with domain A π,k .It is clear that with this procedure we have { f > 0} = Being the last set a Borel set, the functions t ± k are again Borel.By definingγ k,n = γ (t − k,n (γ),t + k,n (γ)) , π k,n = (γ k,n) π, we have that π k,n is a test plan for the measure µ = f 2 m/ f 2 2 , with compressibility constant 2 2n C/ f 2 2 , C being the compressibility constant of π.Along π k,n -a.e.curve, by elementary 1d computations we ∈ S 1 , for π n,k -a.e.γ we deduce γ) > 0} and define the function t+k : A π,k → [0, 1] as the function whose graph is γ : t+ k (γ) ≥ t = s k ≤s ≤t A π, .In the same way the function t−k : A π,k → [0, 1] is given by γ : t− k (γ) ≤ t =The only step left is to have that every interval of { f (•, γ) > 0} is counted once.Set t ± 1,n (γ) = t± 1 (γ), and in general t ± k,n (γ) = t± k (γ) if k = min j : s j ∈ ( t− k (γ), t+ k (γ)) .