# Exact integrability conditions for cotangent vector fields

Stefano Bianchini  S. Bianchini: S.I.S.S.A., via Bonomea 265, 34136 Trieste,
Italy<EMAIL_ADDRESS>
###### Abstract.

In Quantum Hydro-Dynamics the following problem is relevant: let
$(\sqrt{\rho},\Lambda)\in H^{1}(\mathbb{R}^{d})\times
L^{2}(\mathbb{R}^{d},\mathbb{R}^{d})$ be a finite energy hydrodynamics state,
i.e. $\Lambda=0$ when $\rho=0$ and

$E=\int_{\mathbb{R}^{d}}\frac{1}{2}\big{|}\nabla\sqrt{\rho}\big{|}^{2}+\frac{1}{2}\Lambda^{2}\mathcal{L}^{d}<\infty.$

The question is under which conditions there exists a wave function $\psi\in
H^{1}(\mathbb{R}^{d},\mathbb{C})$ such that

$\sqrt{\rho}=|\psi|,\quad
J=\sqrt{\rho}\Lambda=\Im\big{(}\bar{\psi}\nabla\psi).$

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
$\mathbb{R}^{d}$, 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

$dw=iwv.$

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)

$\int v(\gamma(t))\cdot\dot{\gamma}(t)dt\in 2\pi\mathbb{Z}$

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 $\sqrt{\rho}\in W^{1,2}(X)$.

###### Key words and phrases:

Wave functions, exact forms, quantum hydro-dynamics

###### 2020 Mathematics Subject Classification:

35Q40

I wish to thank Paolo Antonelli and Piero Marcati for introducing me to the
beautiful subject of QHD

Preprint SISSA 18/2021/MATE

###### Contents

  1. 1 Introduction
    1. 1.1 Sketch of the proof and structure of the paper
  2. 2 Notation
  3. 3 Settings and basic computations
    1. 3.1 Test plans
    2. 3.2 Tangent and cotangent bundles
    3. 3.3 The integer circuitation condition
    4. 3.4 Intersection properties of curves
  4. 4 A simple case: $x\in\mathbb{R}^{2}$, $\mu=f^{2}\mathcal{L}^{2}/\|f\|_{2}^{2}$ with $f\in W^{1,2}$
  5. 5 Construction of the minimal equivalence relation
    1. 5.1 Application to test plans
  6. 6 Construction of the function $w$
  7. 7 The $W^{1,2}$-case

##  1\. Introduction

This paper is devoted to the following question arising in Quantum Hydro-
Dynamics (QHD): assume that $(\sqrt{\rho},\Lambda)\in
H^{1}(\mathbb{R}^{d})\times L^{2}(\mathbb{R}^{d},\mathbb{R}^{d})$ are such
that $\Lambda=0$ when $\rho=0$. The question is under which conditions there
exists a wave function $\psi\in H^{1}(\mathbb{R}^{d},\mathbb{C})$ such that

$\sqrt{\rho}=|\psi|,\quad
J=\sqrt{\rho}\Lambda=\Im\big{(}\bar{\psi}\nabla\psi).$ (1.1)

The second equation gives for $\psi=\sqrt{\rho}w$ smooth, $|w|=1$, that
$i\Lambda=\sqrt{\rho}\bar{w}\nabla w$. The interest for the above question
arises because $\rho,J$ are obtained as solutions to the QHD system

$\begin{cases}\partial_{t}\rho+\operatorname{div}J=0,\\\
{\displaystyle\partial_{t}J+\operatorname{div}\big{(}\Lambda\otimes\Lambda+p(\rho)\mathrm{id}\big{)}=\operatorname{div}\bigg{(}\frac{\nabla^{2}\rho}{4}-\nabla\sqrt{\rho}\otimes\nabla\sqrt{\rho}\bigg{)},}\end{cases}$
(1.2)

which is the system of PDE satisfied by the hydrodynamics quantities in (1.1)
if $\psi\in C(\mathbb{R},H^{1}(\mathbb{R}^{d},\mathbb{C}))$ solves the
Schrödinger equation

$i\partial_{t}\psi=-\frac{1}{2}\Delta\psi+f^{\prime}(|\psi|^{2})\psi,$ (1.3)

where

$f(\rho)=\rho\int_{\rho*}^{\rho}\frac{p(s)-p(\rho*)}{s^{2}}ds,\quad\rho^{*}\text{
reference density}.$

Apart from the 1d case, where a weak solution to (1.2) can be constructed and
then the wave function $\psi$ 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 $(\rho,J)$ are derived from a wave function $\psi$ is that

$\mathrm{curl}\,J=2\nabla\sqrt{\rho}\wedge\Lambda,$ (1.4)

which mimics the fact that $\mathrm{curl}\,\nabla\theta=0$. Being this
condition a local condition, it is not sufficient for the existence of a wave
function $\psi$, as one can see with the elementary example in
$\mathbb{R}^{2}$

$\rho(x)\in C^{1}_{c}(\mathbb{R}^{2}),\rho(0)=0,\qquad
J(x)=\rho(x)\frac{\alpha x^{\perp}}{|x|^{2}},\ \alpha\notin\mathbb{Z}.$ (1.5)

Indeed (1.4) is verified and being $v=\alpha x^{\perp}/|x|^{2}$,
$\mathrm{curl}\,v=2\pi\alpha\delta_{0}$, for $\alpha\notin\mathbb{N}$ it is
impossible to solve

$Dw=iwv,$

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 $\psi$ to the case when
$x\in\mathbb{R}^{d}$ and $\rho,J$ are radially symmetric, and more
interestingly when $x\in\mathbb{R}^{2}$, the mass $\rho$ is continuous and the
vacuum set $\\{\rho=0\\}$ is given by isolated points
$\\{x_{1},x_{2},\dots\\}$, and the velocity field $v=J/\rho$ is a distribution
such that

$\mathrm{curl}\,v=2\pi\sum_{j}k_{j}\delta_{x_{j}},\qquad k_{j}\in\mathbb{Z}.$
(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 $x_{1}=0$

$\mathrm{curl}\,v=2\pi k\delta_{0},$

one deduce that for all test function $\phi$

$-\int\nabla^{\perp}\phi\cdot v\mathcal{L}^{2}=2\pi k\phi(0).$

In particular (by considering radial function $\psi$, for example) one deduce
that for $\mathcal{L}^{1}$-a.e. $r$ it holds

$\int_{0}^{1}v(\gamma_{r}(t))\cdot\dot{\gamma}_{r}(t)dt=2\pi
k,\quad\gamma_{r}(t)=r(\cos(2\pi t),\sin(2\pi t)).$ (1.7)

Hence, the local functions $w$ can be extended globally, since after a
rotation around $x_{1}$ the value

$w(\gamma_{r}(1))=w(\gamma_{r}(0))e^{i\int_{0}^{1}v(\gamma_{r}(t))\cdot\dot{\gamma}_{r}(t)dt}=w(\gamma_{r}(0))$
(1.8)

is the same, and indeed the counterexample (1.5) does not satisfy exactly
(1.6) about the point $x=0$.

With more generality, one can see that the condition (1.6) is equivalent to
the following. Let $\Gamma(\mathbb{R}^{2})$ be the metric space of Lipschitz
curves $\gamma:[0,1]\mapsto\mathbb{R}^{2}$ with the uniform topology $C_{0}$,
and define a _test plan $\pi$_ as a probability measure such that its
evaluation $e(t)_{\sharp}\pi$ at time $t$ satisfies

$\int\phi(x)(e(t)_{\sharp}\pi)(dx)=\int\phi(\gamma(t))\pi(d\gamma)\leq
C\int\phi(x)dx,$

for all $\phi\geq 0$ continuous function. The formula (1.8) is then rewritten
as follows: for $\pi$-a.e. $\gamma$ such that $\gamma(0)=\gamma(1)$ (i.e. it
is a closed path) it holds

$\int_{0}^{1}v(\gamma(t))\cdot\dot{\gamma}(t)dt=2\pi\sum_{j}k_{j}\mathrm{Wi}(\gamma,x_{j})\in
2\pi\mathbb{Z},$ (1.9)

where $\mathrm{Wi}(\gamma,x)$ is the _winding number_ of $\gamma$ around
$z\notin\gamma([0,1])$. Being the set $\\{\rho=0\\}$ locally finite (or more
generally of $\mathcal{H}^{1}$-measure $0$), one can prove that $\pi$-a.e.
$\gamma$ is not passing through $\\{\rho=0\\}$, so that for $\pi$-a.e.
$\gamma$ the winding number $\mathrm{Wi}(\gamma,x_{i})$ 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 $(X,d,\mu)$ be a metric measure space (a Polish space with
$\mu\in\mathcal{P}(X)$ Borel probability measure) and let $v\in L^{2}(T^{*}X)$
be a cotangent vector field such that, if $\pi$ is a test plan supported on
the set of closed curves, then

$\int_{0}^{1}v(\gamma(t))\cdot\dot{\gamma}(t)dt\in
2\pi\mathbb{Z}\qquad\text{for $\pi$-a.e. $\gamma$.}$ (1.10)

  1. (1)

Is there a function $w\in L^{\infty}(X,\mathbb{S}^{1})$ such that its
differential satisfies

$Dw=iwv\,?$ (1.11)

  2. (2)

Assuming moreover that $m$ is the ambient measure in $(X,d)$ and

$\mu=\frac{|f|^{2}m}{\|f\|_{2}^{2}}\quad\text{with}\ f\in
W^{1,2}(X,\mathbb{R}),$ (1.12)

does the function $fw$ belong to $W^{1,2}(X,\mathbb{C})$?

Above and in the following we will use the identification
$\\{z\in\mathbb{C},|z|=1\\}=\mathbb{S}^{1}$, this should not generate
confusion. In the case $X=\mathbb{R}^{d}$ (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 $v\in L^{2}(T^{*}X)$ is the analog of the requirement
$\Lambda=\sqrt{\rho}v\in L^{2}(\mathbb{R}^{d})$ 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 $\mu$ 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 $MCP(K,N)$ (and if
branching the ambient measure $m$ is the Hausdorff measure $\mathcal{H}^{N}$).
Then, for every $f\in W^{1,2}(X,\mathbb{R})$ and cotangent vector field $v\in
L^{2}(T^{*}X)$ w.r.t. the measure $\mu=|f|^{2}m/\|f\|_{2}^{2}$, there exists
$w\in L^{\infty}(X,\mathbb{S}^{1})$ such that

$Dw=iwv\quad\text{and}\quad fw\in W^{1,2}(X,\mathbb{C}).$ (1.13)

Moreover, there exists a partition of $\\{|f|>0\\}$ into at most countably
many sets $\Omega_{i}$, $m(\Omega_{i})>0$,

$\\{|f|>0\\}=\bigcup_{i\in\mathbb{N}}\Omega_{i},$

such that for every function $w^{\prime}$ satisfying $Dw=iwv$ it holds

$\frac{w^{\prime}(x)}{w(x)}=c_{i}\in\mathbb{S}^{1}\quad\text{constant for
$m$-a.e. $x\in\Omega_{i}$}.$

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 $\\{\Omega_{\alpha}\\}_{\alpha}$ in which a.e. points
$x,y\in\Omega_{\alpha}$ 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 $\gamma$ which are not ”good” for $v$: either the integral
along $\gamma$ is not defined or the circuitation (1.13) is not integer if
$\gamma$ 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 $(\Omega_{\alpha},d,m_{\alpha})$. 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
$\\{\Omega_{\alpha}\\}_{\alpha}$ admits a _strongly consistent disintegration_
, i.e. one can write

$\mu=\int\mu_{\alpha}\omega(d\alpha)\quad\text{with}\quad\mu_{\alpha}\in\mathcal{P}(\Omega_{\alpha}).$
(1.14)

This condition is certainly fullfilled if the partition
$\\{\Omega_{\alpha}\\}_{\alpha}$ 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
$\Omega_{\alpha}$ has some interests on its own.

Within these setting, the function $v$ is not locally integrable, so
$\mathrm{curl}\,v$ is meaningless even distributionally. On the other hand, as
in the example of (1.5), the study of $\mathrm{curl}\,\Lambda$ does not yield
any condition into the existence of the function $w$. It is not clear if there
exists some $\mathrm{curl}$-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 $2\pi\mathbb{Z}$-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 $\psi$
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

$\begin{cases}\partial_{t}\rho+v\cdot\nabla\rho=0,\\\
{\displaystyle\partial_{t}v+v\cdot\nabla
v=\nabla\bigg{(}\frac{\Delta\sqrt{\rho}}{2\sqrt{\rho}}-f^{\prime}(\rho)\bigg{)},}\end{cases}$
(1.15)

so that both $\rho$ and $v$ are transported by the same vector field $v$.
Hence any curve $\gamma$ at time $t$ is transported back to a curve
$\gamma^{\prime}$ at time $0$ (being the flow $\dot{X}=v(t,X)$ locally
smooth), and defining

$\gamma_{t}(\alpha)=X(t,\gamma(\alpha))$

one has

$\begin{split}\frac{d}{dt}\int
v(t,\gamma_{t}(\alpha))\cdot\dot{\gamma}_{t}(\alpha)d\alpha\bigg{|}_{t=0}&=\int\nabla\bigg{(}-\frac{|v_{0}|^{2}}{2}+\frac{\Delta\sqrt{\rho_{0}}}{2\sqrt{\rho_{0}}}-f^{\prime}(\rho_{0})\bigg{)}\cdot\dot{\gamma}(\alpha)d\alpha\\\
&\quad+\int v_{0}(\gamma(\alpha))\cdot\nabla
v_{0}(\gamma(\alpha))\dot{\gamma}(\alpha)d\alpha\\\
&=\int\nabla\bigg{(}\frac{\Delta\sqrt{\rho_{0}}}{2\sqrt{\rho_{0}}}-f^{\prime}(\rho_{0})\bigg{)}\cdot\dot{\gamma}(\alpha)d\alpha=0.\end{split}$

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 $\mathrm{curl}\,v$ 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 $v\in L^{1}(T^{*}X)$, 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 $\pi$ is a
test plan supported on the set of closed curves, then

$\int_{0}^{1}v(\gamma(t))\cdot\dot{\gamma}(t)dt=0\qquad\text{for $\pi$-a.e.
$\gamma$.}$ (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 $w(\bar{x})$ as some point $\bar{x}$, find $w(x)$ by
solving the ODE

$\frac{d}{dt}w(\gamma(t))=iw(\gamma(t))v(\gamma(t))\cdot\dot{\gamma}(t),\qquad
w(\gamma(0))=w(\bar{x}),\ \gamma(0)=\bar{x},\gamma(1)=x.$ (1.17)

The integer circuitation condition should take care of the possibility of
having multiple curves connecting $\bar{x}$ 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
$X=\mathbb{R}^{d}$, because of the density $f\in W^{1,2}$. Given a closed
curve $\gamma$ such that $\gamma(t)=\gamma(s)$ for some $0<s<t<1$, it is not
even true in general that the restricted closed curve
$R_{s,t}\gamma=\gamma\llcorner_{[s,t]}$ is of integer circuitation, even if
$\gamma$ 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 $A_{\pi}$ such that every closed curve
obtained by patching pieces of finitely many curves of $A_{\pi}$ 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 $\mathbb{R}^{d}$ (and in differentiable manifolds) is much
easier than the general case in MMS: the case $X=\mathbb{R}^{2}$ 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 $\mathbb{R}^{4}$ into the set of curves $\Gamma$ 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
$\mathbb{R}^{4}$ 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 $\mathcal{L}^{2}$-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 $\mathbb{R}^{d}$, $d\geq 3$, 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 $\\{\gamma_{\alpha}\\}_{\alpha}$, define the minimal (by
set inclusion) equivalence relation $E$ by requiring that
$\gamma_{\alpha}(t)E\gamma_{\alpha}(s)$ for all $s,t\in[0,1]$,
$\gamma_{\alpha}$ (Definition 5.7). If we have a test plan $\pi$, for every
set of curves $A_{\pi}$, $\pi(A_{\pi})=1$, one has clearly different
equivalence relations $E^{A_{\pi}}$. However the space $X$ is separable, and
we use this fact to deduce that there is a set $\bar{A}_{\pi}$ such that the
equivalence relation $E^{\bar{A}_{\pi}}$ is minimal: this means that if
$\mathcal{M}^{\bar{A}_{\pi}}$ is the family of $\mu$-measurable sets which are
saturated w.r.t. $E^{\bar{A}_{\pi}}$, then for all other sets $A_{\pi}$ where
$\pi$ is concentrated there exists a $\mu$-conegligible set where the
$\sigma$-algebra generated by $E^{A_{\pi}}$ is a subalgebra of
$\mathcal{M}^{E^{\bar{A}_{\pi}}}$ (Proposition 5.8). This does not means that
every atom of $\mathcal{M}^{\bar{A}_{\pi}}$ coincides with an equivalence
class of $E^{\bar{A}_{\pi}}$: the situation is completely equivalent to the
Vitali set of $[0,1]$, where $E=[0,1]/\mathbb{Q}$ and then the $E$-saturated
$\mathcal{L}^{1}$-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 $\bar{\pi}$ such that the maximal measure algebra
$\mathcal{M}^{\bar{\pi}}$ is the minimal among all measure algebras
$\mathcal{M}^{\pi}$ (Proposition 5.9). The reason for this analysis is the
following: first, being $\mathcal{M}^{\bar{\pi}}$ maximal, up to a negligible
set the space $X$ cannot be further divided into smaller atoms, i.e. the atoms
of $\mathcal{M}^{\bar{\pi}}$ are the minimal components used by the test plan
$\bar{\pi}$. Being $\mathcal{M}^{\bar{\pi}}$ minimal among the maximal measure
algebras, we deduce that the all others test plans use curves which are
contained into atoms of $\mathcal{M}^{\bar{\pi}}$ (up to a negligible set as
usual, see Corollary 5.11). These components are in some sense the minimal
connected-by-curves components of $(X,\mu)$, 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 $\sigma$-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 $MCP(K,N)$-spaces where
the ambient measure is $f\mathcal{H}^{N}$ with $f\in
W^{1,2}(X,\mathcal{H}^{N})$), it is satisfied (Example 7.2). Under the
assumption that the equivalence classes coincide with the atoms of the measure
algebra $\mathcal{M}^{\bar{\pi}}$, the construction is now fairly simple, and
follows the line for the $\mathbb{R}^{2}$ case: for the test plan $\bar{\pi}$
take the set $A^{\bar{\pi}}$ such that every subloop is of integer
circuitation, choose a Borel section and a Borel function $\omega$ on the
quotient space and for every $x$ compute $w$ by solving the ODE (1.17) with
initial data $\omega(y)$ 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 $\bar{A}^{\bar{\pi}}$ 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 $w^{\prime}$ is such that
$w^{\prime}/w$ is constant on the equivalence class (Theorem 6.6).

The final analysis of the case when $\mu=fm$, with $f\in W^{1,2}(X,m)$ 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 $\bar{\pi}$ satisfying the assumptions
of the previous sections, and this is true in these spaces (Lemma 7.3).
Finally, to check that $fw\in W^{1,2}(X,m)$ 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 $\mu$ (Lemma 7.6 and
Proposition 7.7).

##  2\. Notation

The real numbers are denoted by $\mathbb{R}$, the rational numbers by
$\mathbb{Q}$, the natural numbers by $\mathbb{N}$ and the integers by
$\mathbb{Z}$. The unitary circle in $\mathbb{R}^{2}$ is denoted by
$\mathbb{S}^{1}$. The cardinality of a set $A$ is denoted by $\sharp A$.

A generic constant (which may vary from line to line) is denoted with
$\mathrm{const.}$ 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 $\mathcal{K}(X)$, and we will use the Hausdorff distance
$d_{H}(K_{1},K_{2})$. Countable intersection of open sets are
$G_{\delta}$-sets, while countable union of closed sets are $F_{\sigma}$-sets.
The projection onto $X$ is written as $\mathtt{p}_{X}$, or sometimes
$\mathtt{p}_{i}$ if it is the $i$-component of $\prod_{j}X_{j}$. The Borel
$\sigma$-algebra is $\mathcal{B}(X)$, the projection of closed sets are the
analytic sets $\Sigma^{1}_{1}$, and the $\sigma$-algebra generated by them is
$\sigma(\Sigma^{1}_{1})$.

In case there is no ambiguity, the geodesic connecting $x$ to $y$ is →[x,y]:
this notation will be used mainly in $\mathbb{R}^{d}$.

A generic function from $X$ to $Y$ will be denoted by
$\lx@glossaries@gls@link{main}{genfunc}{\leavevmode{\color[rgb]{0,0,1}f}}:X\to
Y$, and we will call $f$ a Borel function when it is a Borel function whose
domain of definition $\operatorname{Dom}(f)$ is a Borel set. The graph of a
function $f:X\to Y$ is denoted by

$\lx@glossaries@gls@link{main}{Graphf}{\leavevmode{\color[rgb]{0,0,1}\operatorname{Graph}(f)}}=\big{\\{}(x,f(x)),x\in
X\big{\\}}\subset X\times Y,$

and its range by

$\lx@glossaries@gls@link{main}{Rangef}{\leavevmode{\color[rgb]{0,0,1}f(X)}}=\big{\\{}f(x),x\in
X\big{\\}}.$

For a (multi-valued) function $f:X\to Y$ we will denote $f(x)$ both as the
image of $f(x)$ or as the section $\operatorname{Graph}f\cap\\{x\\}\times Y$.
In general this will not generate confusion.

The space of Lipschitz functions $\mathrm{Lip}([0,1];X)$ from $[0,1]$ into $X$
is denoted by $\Gamma$: 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 $\Gamma_{c}$. An element of both sets will be
written as $\gamma$. Explicitly

$\Gamma=\operatorname{Lip}([0,1];X),\quad\Gamma_{c}=\Big{\\{}\gamma\in\operatorname{Lip}([0,1];X),\gamma(0)=\gamma(1)\Big{\\}}.$

It is known that $\Gamma$ is a separable metric space with the
$C^{0}$-topology [Sri98, Theorem 2.4.3].

The evaluation map
$\lx@glossaries@gls@link{main}{evaluation}{\leavevmode{\color[rgb]{0,0,1}e(t)}}=e(t,\gamma)$
is the map

$\Gamma\ni\gamma\mapsto e(t,\gamma)=\gamma(t)\in X.$

The restriction operator $R_{s,t}(\gamma)$ is defined as the restriction of
the curve $\gamma$ to the interval $[s,t]\subset[0,1]$,

$R_{s,t}(\gamma)(\tau)=\gamma\big{(}(1-\tau)s+\tau t\big{)},\quad 0\leq s\leq
t\leq 1.$

The time inversion of a curve is denoted by $R(\gamma)(t)=\gamma(1-t)$.

The metric derivative of $\gamma$ is denoted by $\dot{|\gamma|}$. The length
of a curve $\gamma\in\Gamma$ is defined as

$\lx@glossaries@gls@link{main}{Lengthgam}{\leavevmode{\color[rgb]{0,0,1}L(\gamma)}}=\int_{0}^{1}\dot{|\gamma|}(t)dt.$

Given a set $A\subset\Gamma$, we say that $\gamma\in\Gamma$ is a sub-curve of
$A$ if there are finitely many curves $\gamma_{i}\in A$ such that

$\operatorname{Graph}\gamma\subset\bigcup_{i}\operatorname{Graph}\gamma_{i}.$

The set of all sub-curves of $A$ is denoted by $\Gamma_{A}$.

The Lebesgue measure in $\mathbb{R}^{d}$ is denoted with $\mathcal{L}^{d}$ and
the $d$-dimensional Hausdorff measure by $\mathcal{H}^{d}$. Let
$\mathcal{P}(X)$ be the space of probability measures over $X$ and $m$,$\mu$
elements of $\mathcal{P}(X)$: in the second part of our analysis $\mu=fm$ (and
$m$ will be used as ambient probability for the space $W^{1,2}(X)$), but in
the first part we will consider $(X,d,\mu)$ as a given metric measure space.
The topology on $\mathcal{P}(X)$ is the narrow topology. We denote the
standard integral spaces by $L^{p}(\mu)$. The push forward of a measure $\mu$
by a map $f$ is written by $f_{\sharp}\mu$. The disintegration of a
probability according to a map $f:X\to Y$ will be denoted as

$\mu=\int\lx@glossaries@gls@link{main}{muy}{\leavevmode{\color[rgb]{0,0,1}\mu_{y}}}\varpi,\quad\lx@glossaries@gls@link{main}{varpi}{\leavevmode{\color[rgb]{0,0,1}\varpi}}=f_{\sharp}\mu.$

We will also write $\varpi=\mu\llcorner\mathcal{M}$, where $\mathcal{M}$ is
the $\sigma$-algebra made of $\mu$-measurable sets saturated w.r.t. the
equivalence relation $Ex=f^{-1}(f(x))$.

If $E$ is an equivalence relation, we will write $xEx^{\prime}$ if
$x,x^{\prime}$ 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 $\alpha,\beta,\dots$ 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
$\lx@glossaries@gls@link{main}{quotinetmap}{\leavevmode{\color[rgb]{0,0,1}\mathtt{e}}}:X\to[0,1]$
whose level sets are the equivalence classes.

##  3\. 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.

###  3.1. Test plans

######  Definition 3.1.

A measure
$\lx@glossaries@gls@link{main}{pi}{\leavevmode{\color[rgb]{0,0,1}\pi}}\in\mathcal{P}(\Gamma)$
is a _$\mu$ -admissible test plan_ (or _test plan_ when the reference measure
$\mu$ is clear from the context) if for some constant $C$ it holds

$\int_{0}^{1}e(t)_{\sharp}(\dot{|\gamma|}\eta)dt\leq C\mu.$ (3.1)

The set of $\mu$-admissible test plans is denoted by $\mathrm{Adm(\mu)}$, and
$C$ is called the _compressibility constant_ of $\pi$.

The above formula (3.1) is equivalent to

$\int\bigg{[}\int_{0}^{1}1\\!\\!\mathrm{I}_{\gamma(t)\in
A}(t,\gamma)\dot{|\gamma|}(t)dt\bigg{]}\eta(d\gamma)\leq C\mu(A).$

It is clear that the above definition is invariant if we reparameterize the
curves $\gamma$. In the following we will often consider the parametrization
of $\gamma$ by length, so that $\dot{|\gamma|}=1$: this is to avoid to write
the factor $\dot{|\gamma|}$ in many equations. In that case the interval of
integration is $[0,L(\gamma)]$, 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

$\int\bigg{[}\int_{0}^{1}\big{(}\dot{|\gamma|}(t)\big{)}^{2}dt\bigg{]}\pi(d\gamma)<\infty,\quad
e(t)_{\sharp}\pi\leq C\mu,$ (3.2)

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 $\pi$-a.e. $\gamma$ and is independent on the
parametrization of $\gamma$, it is natural to consider test plans as in
Definition 3.1.

######  Remark 3.3.

An admissible test plan $\pi$ is given by the constant curves: define the map

$X\ni x\mapsto\gamma_{x}\in\Gamma_{c},\qquad\gamma_{x}(t)=x\quad\forall
t\in[0,1],$

and set

$\pi=(\gamma_{x})_{\sharp}(f(x)\mu(dx))$ (3.3)

for every $f\in L^{\infty}(\mu)$, $\|f\|_{1}=1$. It is immediate from the
definition to see that

$\pi(\Gamma)=1,\quad e(t)_{\sharp}\pi=f\mu,$

so that it is admissible. In particular the measure defined in (3.3) when
$f\equiv 1$ is denoted with $\hat{\pi}$.

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 $\gamma:[0,1]\to X$ are the constant ones.

It is immediate to see that if $\pi\in\text{{Adm}}(\mu)$ and
$\gamma\mapsto(s(\gamma),t(\gamma))$ is a Borel function, then
$(R_{s,t})_{\sharp}\pi,R_{\sharp}\pi\in\text{{Adm}}(\mu)$ because

$e(\tau)_{\sharp}\big{(}(R_{s,t})_{\sharp}\pi\big{)}=\gamma((1-\tau)s(\gamma)+\tau
t(\gamma))_{\sharp}\pi,\quad
e(\tau)_{\sharp}\big{(}R_{\sharp}\pi\big{)}=e(1-\tau)_{\sharp}\pi.$

###  3.2. 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 $f:X\to\mathbb{R}$ with the following
property: there exists $g\in L^{1}(X)$ such that for all test plan $\pi$

$f\circ\gamma\ \text{is a.c. $\pi$-a.e. $\gamma$}\quad\wedge\quad
e_{\sharp}\bigg{(}\bigg{|}\frac{d}{dt}f(\gamma(t))\bigg{|}\mathcal{L}^{1}\llcorner_{[0,1]}\times\pi\bigg{)}\leq
Cg\mu,$

where $C$ is the compressibility constant of $\pi$ (see (3.1)) and
$e:[0,1]\times\Gamma\to X$ 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 $L^{1}(\mu)$-normed $L^{\infty}(\mu)$-modulus closure of the
set

$\bigcup_{n\in\mathbb{N}}\Big{\\{}\\{(\Omega_{i},f_{i})\\}_{i=1}^{n},\\{\Omega_{i}\\}_{i=1}^{n}\
\text{partition of}\ X,\\{f_{i}\\}_{i=1}^{n}\subset S^{1}(X)\Big{\\}}/\sim$

w.r.t. the norm

$\big{\|}\\{(\Omega_{i},f_{i})\\}_{i=1}^{n}\big{\|}=\sum_{i}\int_{\Omega_{i}}|Df_{i}|\mu,$

where the equivalence relation $\sim$ 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 $f\in S^{1}(X)$ the cotangent vector field
$\lx@glossaries@gls@link{main}{df}{\leavevmode{\color[rgb]{0,0,1}Df}}=(X,f)$,
so that

$\\{(\Omega_{i},f_{i})\\}_{i=1}^{n}=\sum_{i=1}^{n}1\\!\\!\mathrm{I}_{\Omega_{i}}Df_{i}.$
(3.4)

The tangent bundle $L^{\infty}(TX)$ is defined as the dual
$L^{\infty}(\mu)$-normed $L^{\infty}(\mu)$-modulus, i.e. the space of linear
functional $L:L^{1}(T^{*}X)\to L^{1}(\mu)$ such that for $f\in
L^{\infty}(\mu)$, $v\in L^{1}(T^{*}X)$

$L(fv)=fL(v),\quad\|L\|=\sup_{\|v\|\leq 1}\int|L(v)|\mu<\infty.$

The above norm definition gives that the pointwise norm is

$|L|=\mathrm{ess\text{-}sup}\big{\\{}L(v),\|v\|\leq 1\big{\\}}\in
L^{\infty}(\mu).$

If $\pi$ is a test plan, then for every simple element
$v=\sum_{i}1\\!\\!\mathrm{I}_{\Omega_{i}}Df_{i}$ we can define

$\Lambda_{\gamma}(t,v)=\sum_{i}1\\!\\!\mathrm{I}_{\Omega_{i}}(\gamma(t))\frac{d}{dt}f_{i}(\gamma)(t)\quad(\dot{|\gamma|}=1),$
(3.5)

which satisfies by (3.1)

$\int\bigg{[}\int_{0}^{L(\gamma)}|\Lambda_{\gamma}(t,v)|dt\bigg{]}\eta(d\gamma)\leq\sum_{i}C\int_{\Omega_{i}}|Df_{i}|\mu=C\|v\|_{L^{1}(T^{*}X)}.$

Hence, being the elements of the form (3.4) dense in $L^{1}(T^{*}X)$ and
$v\mapsto\Lambda_{\gamma}(t,v)$ linear, passing to the limit we obtain that
for every $v\in L^{1}(T^{*}X)$ there exists a unique
$L^{1}(\mathcal{L}^{1}\times\pi)$-function $\Lambda_{\gamma}(t,v)$ such that

$\lx@glossaries@gls@link{main}{Lambda}{\leavevmode{\color[rgb]{0,0,1}\Lambda_{\gamma}(t,v)}}=\lim_{v_{n}\to
v}\Lambda_{\gamma}(t,v_{n}),$ (3.6)

where $v_{n}=\sum_{i}1\\!\\!\mathrm{I}_{\Omega_{i}}Df_{i}$ is converging to
$v$ in $L^{1}(T^{*}X)$, and it satisfies

$\|\Lambda_{\cdot}(\cdot,v)\|_{L^{1}(\mathcal{L}^{1}\otimes\pi)}\leq
C\|v\|_{L^{1}(T^{*}(X))},\quad\text{$C$ compressibility constant}.$ (3.7)

Hence $L^{1}(T^{*}X)\ni v\mapsto\Lambda_{\gamma}(t,v)\in
L^{1}(\mathcal{L}^{1}\times\pi)$ is linear and continuous.

######  Remark 3.4.

The function $\Lambda_{\gamma}(\cdot,v)$ depends on $\pi$. 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. $\pi$ in what follows.

We will need the following property of $\Lambda_{\gamma}(t,v)$: we write the
proof for completeness.

######  Lemma 3.5.

There exists a $\mathcal{L}^{1}\times\pi$-negligble set $N$ such that

$\gamma(t)=\gamma(s)\quad\Rightarrow\quad\Lambda_{\gamma}(t,v)=\Lambda_{\gamma}(s,v)$

for all $s,t\in[0,L(\gamma)]\setminus N(\gamma)$.

In particular, the function $\Lambda_{\gamma}$ can be defined as an
$L^{1}$-function of $\mathcal{H}^{1}\llcorner_{\gamma(0,L(\gamma))}$.

###### Proof.

By (3.7) we have that if $v_{n}\to v$ in $L^{1}(T^{*}X)$ then
$\Lambda_{\gamma}(t,v_{n})\to\Lambda_{\gamma}(t,v)$ in
$L^{1}(\mathcal{L}^{1}\times\pi)$, so that up to subsequences it converges
$\mathcal{L}^{1}\times\pi$-a.e.. In particular, let $v_{n}\to v$ be a sequence
of cotangent vector fields in $L^{1}(T^{*}X)$ such that the statement is true
for each $v_{n}$, and let $N_{n}\subset[0,1]\times\Gamma$ be the corresponding
$\mathcal{L}^{1}\times\pi$-negligible set such that

$\forall s,t\notin N_{n}(\gamma)\ \Big{(}\gamma(t)=\gamma(s)\ \Rightarrow\
\Lambda_{\gamma}(s,v_{n})=\Lambda_{\gamma}(t,v_{n})\Big{)}.$

Clearly the set $N=\cup_{n}N_{n}$ is negligible. Since
$\Lambda_{\gamma}(t,v_{n})\to\Lambda_{\gamma}(t,v)$ up to a subsequence in a
$\mathcal{L}^{1}\times\pi$-conegligble set $A\subset\mathbb{R}\times\Gamma$,
we obtain that for all $(t,\gamma),(s,\gamma)\in A\setminus N$ it holds

$\gamma(t)=\gamma(s)\ \Rightarrow\
\Lambda_{\gamma}(s,v)=\Lambda_{\gamma}(t,v).$

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

$\begin{split}N^{\prime}&=\bigcup_{n}N^{\prime}_{n}\\\
&=\bigcup_{n\in\mathbb{N}}\bigg{\\{}(s,\gamma):\exists
t\in[0,L(\gamma)]\,\Big{(}\gamma(s+2^{-n}(-1,1))\cap\gamma(t+2^{-n}(-1,1))=\\{\gamma(s)\\}=\\{\gamma(t)\\}\Big{)}\bigg{\\}}.\end{split}$

It is fairly easy to see that for every $\gamma$ $N^{\prime}_{n}(\gamma)$ is
finite for every $\gamma$: indeed if $s_{n},t_{n}\in[0,L(\gamma)]$ are a
sequence of points in the definition of $N^{\prime}_{n}(\gamma)$, then it is
clear that $|s_{n}-s_{m}|\vee|t_{n}-t_{m}|\geq 2^{-n}$. Hence
$N^{\prime}(\gamma)$ is countable and $N^{\prime}$ is
$\mathcal{L}^{1}\times\pi$-negligible.

For a given $f\in S^{1}(X)$, consider the conegligible set

$M=\Big{\\{}(t,\gamma)\notin N^{\prime}:\dot{|\gamma|}(t)\ \text{exists and
}f\circ\gamma\ \text{differentiable in $t$}\Big{\\}}.$

Note that if $\gamma(s)=\gamma(t)$, $s,t\in M(\gamma)$, then there are
sequences $s_{n}\to s,t_{n}\to t$, $s_{n},t_{n}\not=s,t$ respectively such
that $\gamma(s_{n})=\gamma(t_{n})$: hence by the formula (3.5) we deduce that

$\frac{d}{dt}f\circ\gamma(t)=\frac{d}{dt}f\circ\gamma(s).$

The complement $N=M^{c}$ satisfies the statement. ∎

In particular we deduce that

######  Corollary 3.6.

For $\pi$-a.e. $\gamma$ the function $\Lambda_{\gamma}$ depends only on
$\gamma(t)$ for $\mathcal{L}^{1}$-a.e. $t$, i.e. can be written as a function
on $\gamma([0,1])$ defined $\mathcal{H}^{1}$-a.e. or equivalently it is a
measurable function for the pull-back $\sigma$-algebra
$e^{-1}(\gamma)(\mathcal{B}(X))$.

###  3.3. The integer circuitation condition

Let $v\in L^{1}(T^{*}X)$ and $\Lambda_{\gamma}(t)=\Lambda(t,v)\in
L^{1}(\mathcal{L}^{1}\times\pi)$ 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 $\pi$ concentrated on $\Gamma_{c}$ it holds

$\int_{0}^{1}\Lambda_{\gamma}(t)\dot{|\gamma|}(t)dt\in
2\pi\mathbb{Z}\quad\pi\text{-a.e.}\ \gamma,$ (3.8)

i.e. $v$ (or $\Lambda$) has _integer circuitation_.

We define

$\lx@glossaries@gls@link{main}{InteCirc}{\leavevmode{\color[rgb]{0,0,1}\mathcal{I}_{c}}}=\bigg{\\{}\gamma\in\Gamma_{c},\int_{0}^{1}\Lambda_{\gamma}(t)\dot{|\gamma|}(t)dt\in
2\pi\mathbb{Z}\bigg{\\}},$

so that we can rephrase the above assumption as $\mathcal{I}_{c}$ has full
$\pi$-measure. Note that as far as we know $\mathcal{I}_{c}$ depends on $\pi$,
but this should not create confusion being $\pi$ 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

$v=-i\frac{Df}{f},$

where $f\in L^{0}(\mu,\mathbb{S}^{1})$ with $|Df|\in L^{1}$: in this case

$\Lambda_{\gamma}(t)=-\frac{i}{f}\frac{d}{dt}f\circ\gamma(t)\quad(\dot{|\gamma|}=1),$

with $\pi$-a.e. $\gamma$, with $\pi$ test plan.

######  Remark 3.8.

Another possible condition is to ask that

$\int_{0}^{1}\Lambda_{\gamma}(t)\dot{|\gamma|}(t)dt=0\quad\pi\text{-a.e.}\
\gamma.$

In this case we are looking for functions whose differential along $\gamma$ is
$\Lambda_{\gamma}$, i.e. $v=Df$ is exact: we will say that every admissible
$\pi$ 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

$L(\gamma)\in L^{\infty}(\pi),\quad e(t)_{\sharp}\pi\leq C\mu,$

Then Assumption 3.7 holds for all test plans.

###### Proof.

Let $\pi\in\text{{Adm}}(\mu)$ be a test plan supported on $\Gamma_{c}$ as in
Definition 3.1.

Since Assumption 3.7 is invariant for countable unions of test plans, we can
assume that $\pi$ is concentrated on curves with length
$L(\gamma)\in(\bar{L},\bar{L}+1)$, $\bar{L}$ fixed, and we thus consider the
pameterization

$L(\gamma\llcorner_{[0,t]})=L(\gamma)t.$

In this case Condition (3.1) implies

$\int_{0}^{1}e(t)_{\sharp}\pi\leq\frac{C}{\bar{L}}\mu=C^{\prime}\mu.$

Hence for all $\delta t>0$ the measure $\pi^{\prime}$ defined by

$\pi^{\prime}=\fint_{0}^{\delta}\big{[}(\gamma^{\epsilon})_{\sharp}\pi\big{]}d\epsilon,\quad\gamma^{\epsilon}(t)=\gamma(\epsilon+t\mod
L(\gamma)),$

is supported on $\Gamma_{c}$ and satisfies $\bar{L}\leq
L(\gamma)\leq\bar{L}+1$ and

$e(t)_{\sharp}\pi^{\prime}=\fint_{t}^{\delta+t}\big{[}e(s)_{\sharp}\pi\big{]}ds\leq\frac{C^{\prime}}{\delta}\mu,$

where for simplicity we assume that $0\leq t\leq\bar{L}-\delta$. The
assumptions on the proposition thus imply that Assumption 3.7 holds for
$\pi^{\prime}$-a.e. $\gamma$, and by Corollary 3.6

$\Lambda_{\gamma^{\epsilon}}(t)=\Lambda_{\gamma}(t+\epsilon\mod L(\gamma)),$

so that we conclude that $\pi$ satisfies Assumption 3.7. ∎

###  3.4. Intersection properties of curves

By definition, any test plan $\pi$ supported on $\Gamma_{c}$ is concentrated
on the set of curves such that

$\int_{0}^{L(\gamma)}\Lambda_{\gamma}(t)dt\in 2\pi\mathbb{Z}.$

However a priori nothing is said about properties of intersecting curve: for
example, given two curves $\gamma,\gamma^{\prime}$ such that

$\gamma(s)=\gamma^{\prime}(s^{\prime}),\
\gamma(t)=\gamma^{\prime}(t^{\prime}),\quad s\leq t,s^{\prime}\leq
t^{\prime},$ (3.9)

one can construct a closed curve $\gamma^{\prime\prime}$ by setting

$\gamma^{\prime\prime}(\tau)=\begin{cases}R_{s,t}(\gamma)(2\tau)&\tau\in[0,1/2],\\\
R_{1-t^{\prime},1-s^{\prime}}(R(\gamma^{\prime}))(2\tau-1)&\tau\in(1/2,1],\end{cases}$
(3.10)

i.e. moving from $\gamma(s)$ to $\gamma(t)=\gamma^{\prime}(t^{\prime})$ along
$\gamma(\tau)$ and then returning to $\gamma^{\prime}(s^{\prime})=\gamma(s)$
along $\gamma(-\tau)$. 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

$\int_{0}^{1}\Lambda(\gamma^{\prime\prime})(\tau)\dot{|\gamma^{\prime\prime}|}(\tau)d\tau\in
2\pi\mathbb{Z}$

by removing a $\pi$-negligible set of trajectories, where
$\gamma^{\prime\prime}$ is the curve constructed above.

We first start with self-intersecting curves.

######  Proposition 3.10.

Let $\Lambda$ be of integer circuitation. Every $\pi\in\text{{Adm}}(\mu)$
supported on $\Gamma_{c}$ is concentrated on a set of closed curves $A_{\pi}$
such that if $\gamma(s)=\gamma(t)$ then $R_{s,t}(\gamma)$ has integer
circuitation.

###### 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

$F=\big{\\{}(\gamma,s,t):\gamma(s)=\gamma(t)\big{\\}}\subset\Gamma_{c}\times[0,1]^{2}$

is closed, and moreover

$\Big{\\{}\gamma:\gamma(s)=\gamma(t)\ \text{for some}\
(s,t)\in\bar{B}^{2}_{r}((\bar{s},\bar{t}))\Big{\\}}$

is closed in $\Gamma$. Hence $F:\Gamma_{c}\to[0,1]^{2}$ is a Borel measurable
multifunction with $F(\gamma)$ closed. For every
$\bar{B}^{2}_{2^{-k}}((\bar{s}_{m},\bar{t}_{m}))$, $s_{m},t_{m}\in\mathbb{Q}$
and $k\in\mathbb{N}$, there is a Borel selection [Sri98, Theorem 5.2.1]

$L_{k,m}:A_{k,m}=F^{-1}(\bar{B}^{2}_{2^{-k}}((\bar{s}_{m},\bar{t}_{m})))\to[0,1]^{2},\quad
L_{k,m}(\gamma)=(s_{k,m},t_{k,m})\in
F(\gamma)\cap\bar{B}^{2}_{2^{-k}}((\bar{s}_{m},\bar{t}_{m})),$

By construction, the graphs of the functions $L_{k,m}$ are dense in
$F(\gamma)$ for each $\gamma$, i.e.

$F(\gamma)=\operatorname{clos}\big{\\{}L_{k,m}(\gamma),\gamma\in
A_{k,m}\big{\\}}.$

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
$\pi\in\mathcal{P}(\Gamma_{c})$ is a test plan, for each map
$L_{n}(\gamma)=(s_{n},t_{n})$ define the transport plan

$\pi_{n}=(R_{s_{n},t_{n}})_{\sharp}(\pi\llcorner_{A_{n}}),\quad
A_{n}=\operatorname{Dom}(L_{n}).$

Being a restriction, clearly $\pi_{n}\in\text{{Adm}}(\mu)$. In particular, it
follows by Assumption 3.7 that

$\pi_{n}\big{(}\\{\gamma\notin\mathcal{I}_{c}\\}\big{)}=\pi\big{(}\big{\\{}\gamma\in
A_{n}:R_{s_{n},t_{n}}(\gamma)\notin\mathcal{I}_{c}\big{\\}}\big{)}=0.$

By $\sigma$-additivity

$\pi\big{(}\big{\\{}\gamma\in A_{n}:\exists
n\big{(}R_{s_{n},t_{n}}(\gamma)\notin\mathcal{I}_{c}\big{)}\big{\\}}\big{)}=0.$

Step 3: conclusion. The previous point gives that $\pi\in\text{{Adm}}(\mu)$ is
concentrated on a set of curves $A_{\pi}$ such that for all $n\in\mathbb{N}$
it holds $R_{s_{n},t_{n}}(\gamma)\in\mathcal{I}_{c}$. Let now
$(s_{n},t_{n}),(\bar{s},\bar{t})\in F(\gamma)$, with
$(s_{n},t_{n})\to(\bar{s},\bar{t})$. Then

$2\pi\mathbb{Z}\ni\int_{s_{n}}^{t_{n}}\Lambda_{\gamma}(\tau)\dot{|\gamma|}(\tau)d\tau\to\int_{\bar{s}}^{\bar{t}}\Lambda_{\gamma}(\tau)\dot{|\gamma|}(\tau)d\tau,$

which gives that every subcurve of $\gamma\in A_{\pi}$ belongs to
$\mathcal{I}_{c}$, 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 $\pi$-a.e. curve.

######  Lemma 3.11.

Let $\gamma$ be a Lipschitz curve such that if $\gamma(s)=\gamma(t)$ then
$R_{s,t}\gamma\in\mathcal{I}_{c}$. Then there exists a function
$w_{\gamma}:\gamma([0,1])\to\mathbb{S}^{1}$ such that

$\frac{d}{dt}w_{\gamma}(\gamma(t))=i\Lambda_{\gamma}(t)\dot{|\gamma|}(t)w(\gamma(t)).$

###### Proof.

Define the function

$w_{\gamma}(t)=e^{i\int_{0}^{t}\Lambda_{\gamma}\dot{|\gamma|}\mathcal{L}^{1}}.$

If $\gamma(t)=\gamma(s)$, then by the assumption
$R_{s,t}\gamma\in\mathcal{I}_{c}$ it holds

$w_{\gamma}(t)=w_{\gamma}(s)e^{i\int_{s}^{t}\Lambda_{\gamma}\dot{|\gamma|}\mathcal{L}^{1}}=w_{\gamma}(s),$

so that the function $w_{\gamma}$ can be actually defined on
$\operatorname{Graph}(\gamma)$ by

$w_{\gamma}(x)=w_{\gamma}(t),\quad x=\gamma(t).\qed$

We next generalize Proposition 3.10 to closed subcurves of the set of curves
where $\pi$ is concentrated: given a set $A\subset\Gamma_{c}$ define

$\lx@glossaries@gls@link{main}{An}{\leavevmode{\color[rgb]{0,0,1}A(n)}}=\bigg{\\{}\gamma\in\Gamma_{c}:\exists\gamma_{1},\dots,\gamma_{n}\in
A\bigg{(}\operatorname{Graph}\gamma\subset\bigcup_{i}\operatorname{Graph}\gamma_{i}\bigg{)}\bigg{\\}}.$

######  Theorem 3.12.

Let $\Lambda$ be of integer circuitation. Every $\pi\in\text{{Adm}}(\mu)$ is
concentrated on a set of curves $A_{\pi}$ such that
$A_{\pi}(n)\subset\mathcal{I}_{c}$ for all $n$, i.e. every subloop of
$A_{\pi}$ belongs to $\mathcal{I}_{c}$.

###### 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
$(\Gamma_{c})^{2}=\Gamma_{c}\times\Gamma_{c}$ and its closed subset

$\mathcal{C}(2)=\Big{\\{}(\gamma_{1},\gamma_{2})\in(\Gamma_{c})^{2}:\gamma_{1}([0,1])\cap\gamma_{2}([0,1])\not=\emptyset\Big{\\}}.$

In particular the multifunction $F(\gamma_{1},\gamma_{2})$ with graph

$F=\Big{\\{}(\gamma_{1},\gamma_{2},t_{1},t_{2})\in\mathcal{C}(2)\times[0,1]^{2}:\gamma_{1}(t_{1})=\gamma_{2}(t_{2})\Big{\\}}$

is Borel measurable and $F(\gamma_{1},\gamma_{2})$ is closed, so that by
[Sri98, Theorem 5.2.1] there is a Borel selection

$L:\mathcal{C}(2)\to[0,1]^{2},\quad L(\gamma_{1},\gamma_{2})=(t_{1},t_{2})\
\text{such that $\gamma_{1}(t_{1})=\gamma_{2}(t_{2})$}.$

We thus can define the curve

$\gamma^{\prime}(\gamma_{1},\gamma_{2};t)=\begin{cases}\gamma_{1}(2t)&t\in[0,t_{1}/2]\\\
\gamma_{2}(t_{2}+2t-t_{1}\mod 1)&t\in(t_{1}/2,t_{1}/2+1/2),\\\
\gamma_{1}(2t-1)&t\in[t_{1}/2+1/2,1].\end{cases}$ (3.11)

Clearly the map
$(\gamma_{1},\gamma_{2})\to\gamma^{\prime}(\gamma_{1},\gamma_{2})$ is Borel.
Up to reparametrization, the curve $\gamma^{\prime}(\gamma_{1},\gamma_{2})$ is
the closed curve obtained by moving along the first curve and then along the
second one.

Consider now the set of positive measures

$\Pi^{\leq}(\pi,\pi)=\Big{\\{}\xi\in\mathcal{M}^{+}(\mathcal{C}(2)):(\mathtt{p}_{1})_{\sharp}\xi\leq\pi,(\mathtt{p}_{2})_{\sharp}\xi\leq\pi\Big{\\}},$

where $\pi\in\text{{Adm}}(\mu)$ is the test plan in the statement. We can
define the new test plan

$\pi^{\prime}=(\gamma^{\prime})_{\sharp}\xi,\quad\xi\in\Pi^{\leq}(\pi,\pi).$

It is fairly easy to see that $\pi^{\prime}\in\text{{Adm}}(\mu)$ because

$\begin{split}\int\bigg{[}\int_{0}^{1}\phi(\gamma(t))\dot{|\gamma|}(t)dt\bigg{]}\pi^{\prime}(d\gamma)&=\sum_{i=1,2}\int\bigg{[}\int_{0}^{1}\phi(\gamma(t))\dot{|\gamma|}(t)dt\bigg{]}(\mathtt{p}_{i})_{\sharp}\xi(d\gamma)\\\
&\leq
2\int\bigg{[}\int_{0}^{1}\phi(\gamma(t))\dot{|\gamma|}(t)dt\bigg{]}\pi(d\gamma),\end{split}$

and since $\gamma^{\prime}\in\Gamma_{c}$ then $\pi^{\prime}$ is supported on
$\Gamma_{c}$.

Proposition 3.10 implies that the set

$\tilde{\mathcal{C}}(2)=\Big{\\{}(\gamma_{1},\gamma_{2})\in\mathcal{C}(2):\gamma^{\prime}(\gamma_{1},\gamma_{2})\
\text{has a self-intersection with non integer circuitation}\Big{\\}}$

is negligible for every $\xi\in\Pi^{\leq}(\pi,\pi)$. We recall now the
following result [Kel84, Theorem 2.19 and Proposition 3.5]: if $B$ is Borel
(or analytic)

$\sup_{\xi\in\Pi^{\leq}(\pi,\pi)}\xi(B)=\min\Big{\\{}\pi(B_{1})+\pi(B_{2}):\,B\subset(B_{1}\times\Gamma)\cup(\Gamma\times
B_{2})\Big{\\}}.$

In our case, we have observed that the l.h.s. is $0$ when
$B=\tilde{\mathcal{C}}(2)$, and then there are set $B_{1}$, $B_{2}$ negligible
w.r.t. $\pi$ such that for every
$\gamma_{1},\gamma_{2}\in\Gamma_{c}\setminus(B_{1}\cup B_{2})$ every subloop
of the curve $\gamma^{\prime}(\gamma_{1},\gamma_{2})$ has integer
circuitation.

This proves the theorem for $n=2$.

Step 2: the general case. Define the closed set

$\lx@glossaries@gls@link{main}{calCn}{\leavevmode{\color[rgb]{0,0,1}\mathcal{C}(n)}}=\bigg{\\{}(\gamma_{1},\dots,\gamma_{n})\in(\Gamma_{c})^{n}:\bigcup_{i}\operatorname{Graph}\gamma_{i}\
\text{is connected}\bigg{\\}}.$

Starting with $\gamma_{1}$, as in the previous step we can join it to some
curve $\gamma_{i}$, $i\not=1$, by formula (3.11): call this curve
$\tilde{\gamma}_{2}$. Using the assumption that
$\cup_{i}\operatorname{Graph}\gamma_{i}$ is connected, we can repeat the
process by adding to $\tilde{\gamma}_{2}$ a new curve $\gamma_{j}$,
$j\not=1,i$, as in (3.11), obtaining a new curve $\tilde{\gamma}_{3}$.
Proceeding by adding curves, after $n$ steps we obtain a single curve
$\tilde{\gamma}$: it is fairly easy to see that since each merging operation
is Borel, so the map
$(\gamma_{1},\dots,\gamma_{n})\to\tilde{\gamma}(\gamma_{1},\dots,\gamma_{n})$
is Borel too.

As in the previous step, for every

$\xi\in\Pi^{\leq}\big{(}\underbrace{\pi,\dots,\pi}_{n\
\text{times}}\big{)}=\big{\\{}\xi\in\mathcal{M}^{+}(\mathcal{C}(n)),(\mathtt{p}_{i})_{\sharp}\xi\leq\pi\big{\\}}$

we can construct the test plan

$\pi^{\prime}=\big{(}\tilde{\gamma}(\gamma_{1},\dots,\gamma_{n})\big{)}_{\sharp}\xi,$

so that we deduce that the set

$\tilde{\mathcal{C}}(n)=\Big{\\{}(\gamma_{1},\dots,\gamma_{n})\in\mathcal{C}(n):\gamma^{\prime}\
\text{has a self intersection with non integer circuitation}\Big{\\}}$

is negligible for every $\xi\in\Pi^{\leq}(\pi,\dots,\pi)$. We now use [Kel84,
Lemma 1.8, Theorems 2.14 and 2.21]: they state that if $B$ is Borel (or
analytic) then

$\sup_{\xi\in\Pi^{\leq}(\pi,\dots,\pi)}\xi(B)=\min\bigg{\\{}\sum_{i=1}^{n}\int
f_{i}\pi:\,1\\!\\!\mathrm{I}_{B}(\gamma_{1},\dots,\gamma_{n})\leq\sum_{i=1}^{n}f_{i}(\gamma_{i}),0\leq
f_{i}\leq 1\bigg{\\}}.$

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 $\pi$-negligible sets $B_{i}$
such that

$\tilde{\mathcal{C}}(n)\subset\bigcup_{i}(\mathtt{p}_{i})^{-1}(B_{i}),\quad\text{where}\
\mathtt{p}_{i}(\gamma,\dots,\gamma_{n})=\gamma_{i}.$

In particular in the complement of $\tilde{B}_{n}=\cup_{i}B_{i}$ it holds that
every subloop of $\tilde{\gamma}(\gamma_{1},\dots,\gamma_{n})$ belongs to
$\mathcal{I}_{c}$. Removing the negligible set $\cup_{n}\tilde{B}_{n}$ we
obtain the set $A_{\pi}$ of the statement. ∎

Given a sequence of test plan $\pi_{n}$, $n\in\mathbb{N}$, it is immediate to
see that

$\frac{1}{\sum_{n}\frac{2^{-n}}{C_{n}}}\sum_{n}\frac{2^{-n}}{C_{n}}\pi_{n}\in\text{{Adm}}(\mu),$
(3.12)

where $C_{n}$ is the constant for $\pi_{n}$ in (3.1) (the constant in front
the sum is for normalization). Hence the previous theorem gives the following
statement.

######  Corollary 3.13.

If $\\{\pi_{n}\\}_{n\in\mathbb{N}}\subset\text{{Adm}}(\mu)$ is a countable
family of $\mu$-test plans, then there exist Borel sets $A_{\pi_{n}}$ such
that $\pi_{n}(A_{\pi_{n}})=1$ and it holds

$\bigg{\\{}\gamma\in\Gamma_{c}:\exists\gamma_{1}\in
A_{\pi_{1}},\dots,\gamma_{n}\in
A_{\pi_{n}}\bigg{(}\operatorname{Graph}\gamma\subset\bigcup_{i}\operatorname{Graph}\gamma_{i}\bigg{)}\bigg{\\}}\subset\mathcal{I}_{c}.$

##  4\. A simple case: $x\in\mathbb{R}^{2}$,
$\mu=f^{2}\mathcal{L}^{2}/\|f\|_{2}^{2}$ with $f\in W^{1,2}$

As a preliminary analysis, we perform the construction of a function
$w:\mathbb{R}^{2}\mapsto\mathbb{S}^{1}$ such that

$D(fw)=Dfw+fv,$

where $v\in\mathbb{R}^{d}$ is a (cotangent) vector field defined $f$-a.e.
having integer circuitation, and $f\in
W^{1,2}(\mathbb{R}^{2},\mathcal{L}^{d})$: the integer circuitation assumption
is

$\int_{0}^{1}v(\gamma(t))\cdot\dot{\gamma}(t)dt\in 2\pi\mathbb{Z}$

for $\pi$-a.e. $\gamma$, with $\pi$-test plan concentrated on
$\Gamma_{c}(\mathbb{R}^{2})$ for the measure
$\mu=f^{2}\mathcal{L}^{2}/\|f\|_{2}^{2}$. 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
$\mu=f^{2}\mathcal{L}^{2}$ is particularly easy.

Consider a Lebesgue point for $f$ and $Df$: let us assume for definiteness

$\fint_{B^{2}_{r}(0)}|f-1|^{2}\mathcal{L}^{d},\fint_{B^{2}_{r}(0)}|Df-1|^{2}<\epsilon^{2},\quad\epsilon=o(r).$
(4.1)

By standard density argument, for the directions
$\mathbf{e}_{1},\mathbf{e}_{2}$ along the coordinate axis there are segments

$\ell_{k,y}=\big{\\{}x\in\mathbb{R}^{2}:\mathbf{e}_{k}\cdot
x\in[-r,r],\mathtt{p}_{\mathbf{e}_{k}^{\perp}}x=y\big{\\}},\quad y\in
K_{k}\subset[-r,r]\ \text{compact},$

such that

$\mathcal{L}^{1}(K_{k})\geq
r,\quad\int_{\ell_{k,y}}|f-1|^{2}\mathcal{H}^{1},\int_{\ell_{k,y}}|\partial_{x_{k}}f-1|^{2}\mathcal{H}^{1}<C\epsilon^{2}r.$

In particular it follows that $|f|\geq 1/2$ $\mathcal{H}^{1}$-a.e. on all the
segments $\\{\ell_{k,y},y\in K_{k},k=1,2\\}$.

Consider the map
$\mathbb{R}^{4}\ni(y,y^{\prime})=(y_{1},y_{2},y_{1}^{\prime},y_{2}^{\prime})\mapsto\gamma_{(y,y^{\prime})}(t)\in\Gamma_{c}$
where

$\gamma_{(y,y^{\prime})}=\overrightarrow{[y,(y_{1}^{\prime},y_{2})]}\cup\overrightarrow{[(y_{1}^{\prime},y_{2}),y^{\prime}]}\cup\overrightarrow{[y^{\prime},(y_{1},y_{2}^{\prime})]}\cup\overrightarrow{[(y_{1},y_{2}^{\prime}),y]},$
(4.2)

where $\overrightarrow{[x,y]}$ is the geodesic connecting $x,y$:
$\gamma_{(y,y^{\prime})}$ is the oriented boundary of the rectangle with
opposite corners $y,y^{\prime}$.

######  Lemma 4.1.

For all $g\in L^{\infty}(\mathbb{R}^{4},\mathcal{L}^{4})$, $\|g\|_{1}=1$ with
bounded support, the measure
$\pi=(\gamma_{(y,y^{\prime})})_{\sharp}(g\mathcal{L}^{4})$ is a test plan in
$(\mathbb{R}^{2},\mathcal{L}^{2})$.

###### Proof.

Since $\gamma_{(y,y^{\prime})}$ is the union of 4 curves, it is enough to
prove the statement for a single part, for example
$\overrightarrow{[y,(y_{1}^{\prime},y_{2})]}$: in this case

$\begin{split}\int\phi\big{(}(1-t)y_{1}+ty_{1}^{\prime},y_{2}\big{)}g(y,y^{\prime})dydy^{\prime}&=\int\phi(z_{1},y_{2})g(z,y^{\prime})((1-t)^{2}+t^{2})dzdy^{\prime}\\\
&\leq
C\|g\|_{\infty}\big{(}\operatorname{diam}(\operatorname{supp}g)\big{)}^{2}\int\phi(z_{1},y_{2})dz_{1}dy_{2},\end{split}$

where

$\left(\begin{array}[]{c}z_{1}\\\
z_{1}^{\prime}\end{array}\right)=\left[\begin{array}[]{cc}1-t&t\\\
-t&1-t\end{array}\right]\left(\begin{array}[]{c}y_{1}\\\
y_{1}^{\prime}\end{array}\right),$

which shows that the evaluation of $\pi$ is $\leq C(g)\mathcal{L}^{2}$. ∎

Using the same ideas of the first part of the proof of Theorem 3.12 applied to
the measures

$\xi\in\Pi^{\leq}\big{(}\mathcal{L}^{2}\llcorner_{K_{1}\times
K_{2}},\mathcal{L}^{2}\llcorner_{K_{1}\times K_{2}}\big{)},$

we deduce the following corollary.

######  Corollary 4.2.

There is a $\mathcal{L}^{2}$-conegligible set $I_{0}\subset K_{1}\times K_{2}$
such that for every $y,y^{\prime}\in I_{0}$ the curve
$\gamma_{(y,y^{\prime})}$ has integer circuitation.

Using Fubini, we obtain that for $\mathcal{L}^{2}$-a.e. $y=(y_{1},y_{2})\in
I_{0}$ the set

$\big{\\{}y^{\prime}\in I_{0}:p_{2}(y^{\prime})=y_{2}\big{\\}}\subset K_{2},$
(4.3)

has full measure. Define for one of these $y$

$w(y,y^{\prime})=e^{i\int_{y_{1}}^{y_{1}^{\prime}}v_{1}(t,y_{2})dt+i\int_{y_{2}}^{y_{2}^{\prime}}v_{2}(y_{1}^{\prime},t)dt},\quad
y^{\prime}\in K_{1}\times K_{2},$

i.e. the integral along the curve
$\overrightarrow{[y,(y_{1}^{\prime},y_{2})]}\cup\overrightarrow{[(y_{1}^{\prime},y_{2}),y^{\prime}]}$.
The function $w(y,y^{\prime})$ is defined on $\mathcal{L}^{2}$-a.e. point of
$K_{1}\times K_{2}$.

######  Lemma 4.3.

For every $(y^{\prime},y^{\prime\prime})\in(K_{1}\times K_{2})^{2}\setminus
N$, $\mathcal{L}^{4}(N)=0$, it holds

$w(y,y^{\prime\prime})=w(y,y^{\prime})e^{i\int_{y^{\prime}_{1}}^{y_{1}^{\prime\prime}}v_{1}(t,y_{2}^{\prime})dt+i\int_{y^{\prime}_{2}}^{y_{2}^{\prime\prime}}v_{2}(y_{1}^{\prime\prime},t)dt}.$

###### Proof.

The proof is a consequence that the statement holds for $y^{\prime\prime}\in
I_{0}$ such that $(y^{\prime\prime}_{1},y_{2})\in I_{0}$ because setting
$y^{\prime\prime\prime}=(y^{\prime\prime}_{1},y_{2})$

$\begin{split}&\int_{y_{1}}^{y_{1}^{\prime}}v_{1}(t,y_{2})dt+\int_{y_{2}}^{y_{2}^{\prime}}v_{2}(y_{1}^{\prime},t)dt+\int_{y^{\prime}_{1}}^{y_{1}^{\prime\prime}}v_{1}(t,y_{2}^{\prime})dt+\int_{y^{\prime}_{2}}^{y_{2}^{\prime\prime}}v_{2}(y_{1}^{\prime\prime},t)dt\\\
&\quad=\int_{y_{1}}^{y_{1}^{\prime\prime}}v_{1}(t,y_{2})dt+\int_{y_{2}}^{y_{2}^{\prime\prime}}v_{2}(y_{1}^{\prime\prime},t)dt+\int_{\gamma_{(y^{\prime},y^{\prime\prime\prime})}}v_{1}\cdot\dot{\gamma}_{(y^{\prime},y^{\prime\prime\prime})}\\\
\
&\quad\in\int_{y_{1}}^{y_{1}^{\prime\prime}}v_{1}(t,y_{2})dt+\int_{y_{2}}^{y_{2}^{\prime\prime}}v_{2}(y_{1}^{\prime\prime},t)dt+2\pi\mathbb{Z},\end{split}$

and the set

$\big{\\{}y^{\prime}\in I_{0},(y^{\prime}_{1},y_{2})\in
I_{0}\big{\\}}\times\big{\\{}y^{\prime\prime}\in
I_{0},(y^{\prime\prime}_{1},y_{2})\in I_{0}\big{\\}}$

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 $K_{1,n}\times K_{2,n}$ and of functions
$w_{n}$ such that

$\bigcup_{n\in\mathbb{N}}K_{1,n}\times K_{2,n}\quad\text{has full
$|f|^{2}\mathcal{L}^{2}$-measure}$

and $w_{n}$ satisfies Lemma 4.3 in every $K_{1,n}\times K_{2,n}$.

Note that we do not require the sets $K_{1,n}\times K_{2,n}$ to be disjoint.

###### Proof.

Since by restricting the balls we can require that

$\frac{\mathcal{L}^{2}(K_{1,n}\times K_{2,n})}{(2r)^{2}}\to 1,$

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 $|f|\geq 1/2$. 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 $\\{f\not=0\\}$.

Consider a test plan $\pi$ such that

$e(0)_{\sharp}\pi\leq C\mathcal{L}^{2}\llcorner_{K_{1,i}\times K_{2,i}},\quad
e(1)_{\sharp}\pi\leq C\mathcal{L}^{2}\llcorner_{K_{1,j}\times K_{2,j}}.$

For every $\gamma,\gamma^{\prime}$ such that

$\gamma(0),\gamma^{\prime}(0)\in K_{1,i}\times
K_{2,i},\quad\gamma(1),\gamma^{\prime}(1)\in K_{1,j}\times K_{2,j},$

let $\hat{\gamma}_{\gamma,\gamma^{\prime}}$ be the closed curve obtained by
joining $\gamma(0),\gamma^{\prime}(0)$ and $\gamma(1),\gamma^{\prime}(1)$ with
the curves used in Lemma 4.3 to compute $w$. Define

$\tilde{\pi}=(\hat{\gamma}_{\gamma,\gamma})_{\sharp}\big{(}\pi(d\gamma)\times\pi(d\gamma^{\prime})\big{)},$

which is a transference plan due to Lemma 4.1. The condition of integer
circuitation yields that by removing a $\tilde{\pi}$-negligible set of
trajectories the curves $\hat{\gamma}_{\gamma,\gamma}$ have integer
circuitation. For all these curves it follows that

$\begin{split}\frac{w_{j}(\gamma(1))}{w_{i}(\gamma(0))e^{i\int_{0}^{1}v\cdot\dot{\gamma}\mathcal{L}^{1}}}&=\frac{w_{j}(\gamma^{\prime}(1))e^{i\int
v\cdot\dot{\overrightarrow{[\gamma^{\prime}(1),\gamma(1)]}}}}{w_{i}(\gamma^{\prime}(0))e^{i\int
v\cdot\dot{\overrightarrow{[\gamma^{\prime}(0),\gamma(0)]})}+i\int_{0}^{1}v\cdot\dot{\gamma}\mathcal{L}^{1}}}=\frac{w_{j}(\gamma^{\prime}(1))}{w_{i}(\gamma^{\prime}(0))e^{i\int_{0}^{1}v\cdot\dot{\gamma}^{\prime}\mathcal{L}^{1}}}.\end{split}$
(4.4)

The above computation yields the following lemma.

######  Lemma 4.5.

There exists a constant $c_{ij}$ such that for every test plan $\pi$ with

$e(0)_{\sharp}\pi\leq C\mathcal{L}^{2}\llcorner_{K_{1,i}\times K_{2,i}},\quad
e(1)_{\sharp}\pi\leq C\mathcal{L}^{2}\llcorner_{K_{1,j}\times K_{2,j}},$

it holds for $\pi$-a.e. $\gamma$

$c_{ij}w_{j}(\gamma(1))=w_{i}(\gamma(0))e^{i\int v\cdot\dot{\gamma}}.$

###### Proof.

For $\pi$-a.e. $\gamma$ the set $\gamma^{\prime}$ such that
$\hat{\gamma}_{\gamma,\gamma^{\prime}}$ is of integer circuitation has full
$\pi$-measure. For every such a $\gamma$, define $c_{ij}$ as the inverse of
the ratio of (4.4):

$c_{ij}=\frac{w_{i}(\gamma(0))e^{i\int_{0}^{1}v\cdot\dot{\gamma}\mathcal{L}^{1}}}{w_{j}(\gamma(1))}.$

Since by (4.4) the ratio is the same for $\pi$-a.e. $\gamma$, the proof is
complete. ∎

Starting from the function $w_{1}$ defined in $K_{1,1}\times K_{2,1}$, we can
then find constants $c_{1j}$ such that the above lemma holds for all sets
$K_{1,j}\times K_{2,j}$ for which there is a test plan connecting
$K_{1,1}\times K_{2,1}$ to $K_{1,j}\times K_{2,j}$. If at this step we cover
$\\{|f|>0\\}$ up to a $\mathcal{L}^{2}$-negligible set, we are done, otherwise
we take the next $K_{1,i}\times K_{2,i}$ and connect to some others subsets
$K_{1,j^{\prime}}\times K_{2,j^{\prime}}$: clearly these sets cannot belong to
the previous family, otherwise there is a test plan from $K_{1,1}\times
K_{2,1}$ to $K_{1,j^{\prime}}\times K_{2,j^{\prime}}$ and then to
$K_{1,i}\times K_{2,i}$.

We this conclude with the following statement.

######  Proposition 4.6.

There exists a function $w$ such that for all test plans $\pi$ w.r.t. the
measure $|f|^{2}\mathcal{L}^{2}$ it holds

$w(\gamma(1))=w(\gamma(0))e^{i\int v\cdot\dot{\gamma}}\quad\text{$\pi$-a.e.
$\gamma$}.$

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. (1)

every solution $\tilde{w}$ satisfies

$\frac{\tilde{w}}{w_{j}}=\ \text{constant in every $K_{1,j}\times K_{2,j}$}:$

indeed it is constant on the curves used in Lemma 4.3;

  2. (2)

the relation $R$

$(K_{1,j}\times K_{2,j})R(K_{1,i}\times
K_{2,i})\quad\Longleftrightarrow\quad\exists\pi\ \text{plan connecting them as
in Lemma \ref{Lem:single_constat}}$

is an equivalence relation;

  3. (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.t. $|f|^{2}\mathcal{L}^{2}$ it holds

$\frac{d}{dt}(fw)\circ\gamma=\frac{df\circ\gamma}{dt}w+f\frac{dw\circ\gamma}{dt}=wDf\cdot\dot{\gamma}+iwfv\cdot\gamma,$

for $\pi$-a.e. $\gamma$. In particular, by elementary arguments, for
$\mathcal{L}^{1}$-a.e. $y_{2}$ it holds $y_{1}\mapsto(fw)(y_{1},y_{2})$ is
a.c. and has derivative $w\partial_{x_{1}}f+iwv_{1}$, and this means that

$\|D(fw)\|_{2}\leq\|Df\|_{2}+\int|v|^{2}|f|^{2}\mathcal{L}^{2}<\infty,$

Since $|fw|=|f|$ we thus conclude with the following result.

######  Proposition 4.7.

For every $w$ given by Proposition 4.6 the function $fw$ is in
$W^{1,2}(\mathbb{R}^{2})$ and has differential $D(fw)=wDf+iwfv$.

##  5\. Construction of the minimal equivalence relation

In this section we construct a decomposition $X=\cup_{\alpha}X_{\alpha}$ into
disjoint sets with the following properties:

  * •

each test plan is concentrated on a set of curves $\gamma$ such that
$\gamma([0,1])$ belongs to a singe elements $Z_{\alpha}$ up to a
$\mathcal{H}^{1}$-negligible est;

  * •

there exists a test plan $\bar{\pi}$ generating the partition
$\\{X_{\alpha}\\}_{\alpha}$ up to a $\mu$-negligible set.

The first property means that every test plan move mass only inside the
equivalence classes $X_{\alpha}$: this can be interpreted by saying that the
$X_{\alpha}$ are the connected components from the point of view of the MMS
$(X,\mu)$, in the sense that no mass transfer can occur between different
equivalence classes.For example, the MMS

$B_{1}((-2,0))\cup B_{1}((2,0))\cup[-1,1]\times\\{0\\}\subset\mathbb{R}^{2}$
(5.1)

is such that the segment connecting $[-1,1]$ has $\mathcal{L}^{2}$-measure
$0$, so the no test plans $\pi$ can charge mass on it. Hence it is not seen by
any Sobolev function $f\in W^{1,2}(X)$: 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 $\bar{\pi}$ is concentrated, the partition $\\{X_{\alpha}\\}_{\alpha}$
remains the same in a set of full $\mu$-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. (1)

The (complete) $\sigma$-algebra $\mathcal{M}$ of $\mu$ is separable, i.e.
countably generated: identifying elements of $\mathcal{M}$ up to negligible
sets, we obtain a separable measure algebra which we still denote by
$(\mathcal{M},\mu)$ or $\mathcal{M}$ since the measure $\mu$ is fixed here.
The function

$d(\Omega,\Omega^{\prime})=\mu(\Omega\Delta\Omega^{\prime})$ (5.2)

is a distance on $\mathcal{M}$, which makes $\mathcal{M}$ a complete and
separable metric space.

  2. (2)

Each equivalence relation $E$ on $X$ generates the sub $\sigma$-algebra
$\mathcal{M}^{E}$ made of all saturated measurable sets,

$\lx@glossaries@gls@link{main}{calME}{\leavevmode{\color[rgb]{0,0,1}\mathcal{M}^{E}}}=\big{\\{}\Omega\in\mathcal{M}:E\Omega=\Omega\big{\\}}.$
(5.3)

The corresponding measure algebra $\mathcal{M}^{E}$ is clearly a measure sub-
algebra of $\mathcal{M}$. 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
$\\{\mathcal{M}^{\alpha}\\}_{\alpha}$,
$\mathcal{M}^{\alpha}\subset\mathcal{M}$, closed under countable union: for
every sequence $\mathcal{M}^{\alpha_{i}}$ there exists
$\mathcal{M}^{\bar{\alpha}}$ such that

$\bigcup_{i}\mathcal{M}^{\alpha_{i}}\subset\mathcal{M}^{\bar{\alpha}}.$

The following result shows that there is a maximal element
$\mathcal{M}^{\bar{\alpha}}$.

######  Proposition 5.1.

There exists
$\mathcal{M}^{\bar{\alpha}}\in\\{\mathcal{M}^{\alpha}\\}_{\alpha}$ such that
$\mathcal{M}^{\alpha}\subset\mathcal{M}^{\bar{\alpha}}$ for all $\alpha$.

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
$\mathcal{M}^{\alpha}$: it corresponds to the closure of the algebra generated
by the algebra $\cup_{\alpha}\mathcal{M}^{\alpha}$ w.r.t. the distance (5.2).
In particular, being countably generated, it coincides with the measure sub-
algebra generated by countably many elements
$\Omega_{n}\in\cup_{\alpha}\mathcal{M}^{\alpha}$.

Let the sub-algebras $\mathcal{M}^{\alpha_{n}}$ be such that
$\Omega_{n}\in\mathcal{M}^{\alpha_{n}}$, and let $\mathcal{M}^{\bar{\alpha}}$
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
$\\{\mathcal{M}^{\alpha}\\}_{\alpha}$ is closed under countable intersection:
for every sequence $\\{\mathcal{M}^{\alpha_{n}}\\}_{n}$ there exists
$\mathcal{M}^{\bar{\alpha}}$ such that

$\mathcal{M}^{\bar{\alpha}}\subset\bigcap_{n}\mathcal{M}^{\alpha_{n}}.$ (5.4)

######  Proposition 5.2.

There exists
$\mathcal{M}^{\bar{\alpha}}\in\\{\mathcal{M}^{\alpha}\\}_{\alpha}$ such that
$\mathcal{M}^{\bar{\alpha}}\subset\mathcal{M}^{\alpha}$ for all $\alpha$.

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

$\Omega\notin\bigcap_{\alpha}\mathcal{M}^{\alpha}$

there is $\alpha_{\Omega}$ such that
$\Omega\notin\mathcal{M}^{\alpha_{\Omega}}$, and then there exists
$r_{\Omega}>0$ such that the set

$B^{\mathcal{M}}_{R_{\Omega}}(\Omega)=\big{\\{}\Omega^{\prime}:d(\Omega,\Omega^{\prime})<r_{\Omega}\big{\\}}$

has empty intersection with $\mathcal{M}^{\alpha_{\Omega}}$.

By separability, we need only a countable family of balls
$\\{B^{\mathcal{M}}_{R_{\Omega_{n}}}(\Omega_{n})\\}_{n}$ to cover

$\mathcal{M}\setminus\bigcap_{\alpha}\mathcal{M}^{\alpha}.$

Hence we deduce that if

$B^{\mathcal{M}}_{R_{\Omega_{n}}}(\Omega_{n})\cap\mathcal{M}^{\alpha_{n}}=\emptyset,$

then

$\bigcap_{\alpha}\mathcal{M}^{\alpha}=\bigcap_{n}\mathcal{M}^{\alpha_{n}},$

which is the statement by (5.4). ∎

In terms of $\sigma$-algebras, the two statements above can be rewritten as
follows. Consider a family of $\mu$-complete $\sigma$-algebras
$\mathcal{M}^{\alpha}\subset\mathcal{M}$, where $\mathcal{M}$ is the
$\mu$-completion of a separable $\sigma$-algebra.

######  Corollary 5.3.

The following holds:

  1. (1)

if $\\{\mathcal{M}^{\alpha}\\}_{\alpha}$ is closed under countable union, then
there exists an element $\mathcal{M}^{\bar{\alpha}}$ such that

$\forall\alpha,\Omega\in\mathcal{M}^{\alpha}\,\exists\Omega^{\prime}\in\mathcal{M}^{\bar{\alpha}}\big{(}\mu(\Omega\Delta\Omega^{\prime})=0\big{)};$

  2. (2)

if $\\{\mathcal{M}^{\alpha}\\}_{\alpha}$ is closed under countable
intersection, then there exists an element $\mathcal{M}^{\bar{\alpha}}$ such
that

$\forall\alpha,\Omega\in\mathcal{M}^{\bar{\alpha}}\,\exists\Omega^{\prime}\in\mathcal{M}^{\alpha}\big{(}\mu(\Omega\Delta\Omega^{\prime})=0\big{)}.$

The same statement can be rewritten in terms of equivalence relations, by
reversing the intersection and union.

######  Corollary 5.4.

The following holds:

  1. (1)

if $\\{E^{\alpha}\\}_{\alpha}$ is closed under countable intersection, then
there exists an element $E^{\bar{\alpha}}$ such that if $\mathcal{M}^{\alpha}$
is the complete $\sigma$-algebra generated by $E^{\alpha}$ according to (5.3)
then

$\forall\alpha,\Omega\in\mathcal{M}^{\alpha}\,\exists\Omega^{\prime}\in\mathcal{M}^{\bar{\alpha}}\big{(}\mu(\Omega\Delta\Omega^{\prime})=0\big{)};$

  2. (2)

if $\\{E^{\alpha}\\}_{\alpha}$ is closed under countable union, then there
exists an element $E^{\bar{\alpha}}$ such that if $\mathcal{M}^{\alpha}$ is
the complete $\sigma$-algebra generated by $E^{\alpha}$ according to (5.3)
then

$\forall\alpha,\Omega\in\mathcal{M}^{\bar{\alpha}}\,\exists\Omega^{\prime}\in\mathcal{M}^{\alpha}\big{(}\mu(\Omega\Delta\Omega^{\prime})=0\big{)}.$

###  5.1. Application to test plans

The statement of Proposition 5.1 in our setting will be used as follows.

######  Definition 5.5.

Let $\\{\Omega_{\alpha}\\}_{\alpha}$ be a family of subsets of a set $X$. The
equivalence relation generated by $\\{\Omega_{\alpha}\\}_{\alpha}$ is the
smallest equivalence relation $E$ such that

$\bigcup_{\alpha}\Omega_{\alpha}\times\Omega_{\alpha}\subset E.$

######  Lemma 5.6.

The equivalence relation generated by $\\{\Omega_{\alpha}\\}_{\alpha}$ is the
following:

$xEx^{\prime}\quad\Longleftrightarrow\quad\exists
n\in\mathbb{N},\\{\Omega_{\alpha_{i}}\\}_{i=1}^{n},x_{i},x_{i+1}\in\Omega_{\alpha_{i}}\big{(}x_{1}=x\
\wedge\ x_{n+1}=x^{\prime}\big{)}.$

###### Proof.

Immediate from the definition. ∎

We apply the above construction to our case, the sets $\Omega_{\alpha}$ being
the range of the curves $\\{\gamma\in A_{\pi}\\}$.

######  Definition 5.7.

We will say that two points $x,x^{\prime}$ in $X$ are equivalent according to
the set $A_{\pi}\subset\Gamma$ and we will write
$x\lx@glossaries@gls@link{main}{EApi}{\leavevmode{\color[rgb]{0,0,1}E^{A_{\pi}}}}x^{\prime}$
if they are equivalent according to Definition 5.5 where the sets
$\Omega_{\alpha}$ are the family of sets $\\{\gamma([0,1]),\gamma\in
A_{\pi}\\}$. Equivalently, there exists a finite set
$\\{\gamma_{i}\\}_{i}\subset A_{\pi}$ of curves and a path $\gamma\in\Gamma$
such that

$\operatorname{Graph}\gamma\subset\bigcup_{i}\operatorname{Graph}\gamma_{i}\quad\text{and}\quad\gamma(0)=x,\
\gamma(1)=x^{\prime}.$

For every test plan $\pi\in\text{{Adm}}(\mu)$ consider the family of sets
$A_{\pi}$ satisfying Theorem 3.12 (i.e. every closed subloop of $A_{\pi}$ is
of integer cicuitation), and let $E^{A_{\pi}}$, $\mathcal{M}^{E^{A_{\pi}}}$ be
the corresponding equivalence relation and $\sigma$-algebra, respectively (see
(5.3)).

######  Proposition 5.8.

There exists $\bar{A}_{\pi}$ such that $\pi(\bar{A}_{\pi})=1$ and
$\mathcal{M}^{E^{\bar{A}_{\pi}}}$ is maximal.

In particular, for every subset $A_{\pi}\subset\bar{A}_{\pi}$ such that
$\pi(A_{\pi})=1$ it holds

$\mathcal{M}^{E^{\bar{A}_{\pi}}}=\mathcal{M}^{E^{A_{\pi}}}\quad\text{as
measure algebras.}$ (5.5)

We will denote $\mathcal{M}^{E^{\bar{A}_{\pi}}}$ as $\mathcal{M}^{\pi}$, and
the corresponding equivalence relation by
$\lx@glossaries@gls@link{main}{Epi}{\leavevmode{\color[rgb]{0,0,1}E^{\pi}}}=E^{\bar{A}_{\bar{\pi}}}$.
We stress that the only unique object is $\mathcal{M}^{\pi}$ considered as a
complete $\sigma$-subalgebra of $\mathcal{M}$ or as a measure algebra.

###### Proof.

We need only to verify that the family $\\{E^{A_{\pi}}\\}_{A_{\pi}}$ is closed
under countable intersection: this follows from the implications

$\forall
i\big{(}\pi(A_{\pi}^{i})=1\big{)}\quad\Rightarrow\quad\pi\bigg{(}\bigcap_{i}A^{i}_{\pi}\bigg{)}=1,$

and

$E^{\cap_{i}A^{i}_{\pi}}\subset\bigcap_{i}E^{A^{i}_{\pi}}.\qed$

We next apply Proposition 5.2 to sequences of test plans.

######  Proposition 5.9.

There exists
$\lx@glossaries@gls@link{main}{barpi}{\leavevmode{\color[rgb]{0,0,1}\bar{\pi}}}\in\text{{Adm}}(\mu)$
for every $\pi\in\text{{Adm}}(\mu)$ it holds

$\mathcal{M}^{\bar{\pi}}\subset\mathcal{M}^{\pi}\quad\text{as measure
algebras.}$

###### Proof.

Consider a sequence of minimal equivalence relations $\\{E^{\pi_{i}}\\}_{i}$,
and define the test plan

$\hat{\pi}=\frac{1}{\sum_{i}\frac{2^{-i}}{C_{i}}}\sum_{i}\frac{2^{-i}}{C_{i}}\pi_{i},$

where $C_{i}$ is the compressibility constant (3.1) (as before we normalized
the sum of measures in order to be a probability). Let
$\bar{A}_{\hat{\pi}},\bar{A}_{\pi_{i}}$ be a set of curves satisfying
Proposition 5.8 above for $\hat{\pi},\pi_{i}$ respectively.

If we consider the sets

$\hat{A}_{\pi_{i}}=\bar{A}_{\hat{\pi}}\cap\bar{A}_{\pi_{i}},$

then $\pi_{i}(\hat{A}_{\pi_{i}})=1$ because $\pi_{i}\ll\hat{\pi}$, and then by
maximality of $\mathcal{M}^{E^{\bar{A}_{\pi_{i}}}}$ we conclude that by (5.5)
that

$\mathcal{M}^{E^{\bar{A}_{\pi_{i}}}}=\mathcal{M}^{E^{\hat{A}_{\pi_{i}}}}=\mathcal{M}^{\pi_{i}}\quad\text{as
measure algebras.}$

We thus deduce from $E^{\hat{A}_{\pi_{i}}}\subset E^{\bar{A}_{\hat{\pi}}}$
that $\mathcal{M}^{\hat{\pi}}\subset\mathcal{M}^{\pi_{i}}$ for all $i$, which
shows that we are in the situation of Point (2) of Corollary 5.3: the minimal
measure algebra $\mathcal{M}^{\bar{\pi}}$ satisfies the statement. ∎

Let $\bar{A}_{\bar{\pi}}$ be a $\sigma$-compact set where $\bar{\pi}$ is
concentrated such that $E^{\bar{A}_{\bar{\pi}}}$ generates the maximal measure
algebra $\mathcal{M}^{\bar{\pi}}$, which by the previous proposition is the
smallest among all maximal measure algebras $\mathcal{M}^{\pi}$.

Let $\\{\Omega_{n}\\}_{n}$ be a countable family of
$E^{\bar{A}_{\bar{\pi}}}$-saturated sets generating $\mathcal{M}^{\bar{\pi}}$
up to negligible sets, and define the equivalence relation

$\lx@glossaries@gls@link{main}{barE}{\leavevmode{\color[rgb]{0,0,1}\bar{E}}}=\bigcap_{n}\big{(}\Omega_{n}\times\Omega_{n}\big{)}\cup\big{(}(X\setminus\Omega_{n})\times(X\setminus\Omega_{n})\big{)}.$
(5.6)

By construction

$E^{\bar{A}_{\bar{\pi}}}\subset\bar{E},$

and in general the inclusion is strict. The equivalence relation $\bar{E}$ is
made of the atoms of the $\sigma$-algebra generated by the
$\\{\Omega_{n}\\}_{n}$, whose completion coincides with
$\mathcal{M}^{\bar{\pi}}$.

######  Remark 5.10.

Consider another set $\bar{A}^{\prime}_{\bar{\pi}}$ generating
$\mathcal{M}^{\bar{\pi}}$, and let $\\{\Omega^{\prime}_{n}\\}_{n}$ be
$E^{\bar{A}^{\prime}_{\bar{\pi}}}$-saturated sets generating
$\mathcal{M}^{\bar{\pi}}$. Since $\mathcal{M}^{\bar{\pi}}$ is unique and it is
the completion of the $\sigma$-algebra generated by $\bar{E}$, then there are
$\bar{E}$-saturated sets $\\{\Omega^{\prime\prime}_{n}\\}_{n}$ such that

$\mu(\Omega^{\prime\prime}_{n}\Delta\Omega^{\prime}_{n})=0.$

Defining

$N=\bigcup_{n}\Omega^{\prime\prime}_{n}\Delta\Omega^{\prime}_{n},\quad\mu(N)=0,$

we deduce that the equivalence relation obtained by (5.6) using the family
$\\{\Omega_{n}^{\prime}\\}_{n}$ is equal to the one constructed through the
sets $\\{\Omega^{\prime\prime}_{n}\\}_{n}$ up to a $\mu$-negligible set.
Indeed the equivalence relation generated by
$\\{\Omega^{\prime\prime}_{n}\setminus N\\}_{n}$ contains $\bar{E}$ since the
sets $\Omega^{\prime\prime}_{n}$ are $\bar{E}$-saturated. Hence

$E^{\bar{A}^{\prime}_{\bar{\pi}}}\setminus N\times N\subset\bar{E}.$

Reversing the argument we obtain that there exists a $\mu$-negligible set
$N^{\prime}$ such that

$\bar{E}\setminus N^{\prime}\times N^{\prime}\subset
E^{\bar{A}^{\prime}_{\bar{\pi}}},$

and then $E^{\bar{A}^{\prime}_{\bar{\pi}}}$ is equal to $\bar{E}$ up to a
$\mu$-negligible set on $X$.

Let $\pi\in\text{{Adm}}(\mu)$ and $\bar{A}_{\pi}$ generating the maximal
$\mathcal{M}^{\pi}$ as in Proposition 5.8. By minimality of
$\mathcal{M}^{\bar{\pi}}$, for every $\Omega_{n}$ there is a set
$\Omega^{\pi}_{n}\in\mathcal{M}^{\pi}$ such that

$\mu\big{(}\Omega_{n}\Delta\Omega_{n}^{\pi}\big{)}=0.$ (5.7)

In particular, following the reasoning of the previous remark, there exists a
$\mu$-negligible set $N\subset X$ such that if $\bar{E}$ is given by (5.6)
then

$E^{\bar{A}_{\pi}}\cap(X\setminus N\times X\setminus
N)\subset\bar{E}\cap(X\setminus N\times X\setminus N).$ (5.8)

Since $\gamma\in\bar{A}_{\pi}$ takes values in a single class of
$E^{\bar{A}_{\pi}}$, we conclude that

######  Corollary 5.11.

Every $\pi\in\text{{Adm}}(\mu)$ is concentrated on a family of trajectories
$A_{\pi}$ such that

$\forall\gamma\in A_{\pi}\ \big{(}\gamma(s)\bar{E}\gamma(t)\ \text{up to a
$\mathcal{L}^{1}$-negligible subset of $[0,L(\gamma)]$}\big{)}.$

###### Proof.

Since we know that $\gamma(t)E^{\bar{A}_{\pi}}\gamma(s)$ for
$s,t\in[0,L(\gamma)]$ by construction, then we deduce from (5.8) that

$\gamma(s)\bar{E}\gamma(t)\quad\forall
s,t\in[0,L(\gamma)]\setminus\gamma^{-1}(N).$

Being $N$ $\mu$-negligible,

$\int\bigg{[}\int_{0}^{L(\gamma)}1\\!\\!\mathrm{I}_{N}(\gamma(t))dt\bigg{]}\pi(d\gamma)=0,$

so that we need only to remove the negligible set of curves $\gamma$ such that
$\mathcal{L}^{1}(\gamma^{-1}(N))>0$. ∎

######  Remark 5.12.

The presence of the negligible set $N$ is natural: take for example the space
$X=\mathbb{S}^{1}$, $\mu=\mathcal{L}^{1}$, and the equivalence relation

$E=\\{(0,0)\\}\cup(\mathbb{S}^{1}\setminus\\{0\\})^{2}.$

The measure algebra is the trivial one, but any curve passing through $0$ sees
$N=\\{0\\}$.

##  6\. Construction of the function $w$

In the previous section we have constructed an equivalence relation
$E^{\bar{A}_{\bar{\pi}}}$ for the test plan $\bar{\pi}$ such that the measure
algebra $\mathcal{M}^{\bar{\pi}}$ generated by the saturated sets
$\mathcal{M}^{\bar{\pi}}$ satisfies the following properties:

  1. (1)

$\bar{A}_{\bar{\pi}}$ satisfies Theorem 3.12,

  2. (2)

it is the maximal measure algebra among all measure algebra generated by sets
where $\bar{\pi}$ is concentrated,

  3. (3)

all others test plans $\pi$ are concentrated on a set of curves
$\bar{A}_{\pi}$ generating a measure algebra which contains
$\mathcal{M}^{\bar{\pi}}$.

Nothing is said about the strong consistency of the disintegration w.r.t.
$E^{\bar{A}_{\bar{\pi}}}$: if

$\mu=\int\mu_{\alpha}\zeta(d\alpha),\quad\zeta=\mu\llcorner_{\mathcal{M}^{\bar{\pi}}},$
(6.1)

then in general the conditional probability $\mu_{\alpha}$ is not concentrated
on the corresponding equivalence class $E^{\bar{A}_{\bar{\pi}}}_{\alpha}$:
this implies that $\bar{E}\supseteq E^{\bar{A}_{\bar{\pi}}}$.

######  Assumption 6.1.

We assume that the minimal disintegration w.r.t. $E^{\bar{A}_{\bar{\pi}}}$ is
strongly consistent for every $\bar{A}_{\bar{\pi}}$ generating
$\mathcal{M}^{\bar{\pi}}$, i.e.

$\mu_{\alpha}(E^{\bar{A}_{\bar{\pi}}})=1\quad\text{for $\zeta$-a.e.
$\alpha$}.$

We recall that this is equivalent to each of the following conditions ([BC09]
or [Fre02b, Section 452]):

  1. (1)

the equivalence relation $E^{\bar{A}_{\bar{\pi}}}$ is essentially countably
generated, and in particular we can assume that
$\bar{E}=E^{\bar{A}_{\bar{\pi}}}$ up to a $\mu$-negligible set, where
$\bar{E}$ is given by (5.6);

  2. (2)

there exists a $\mu$-measurable map $f:X\to[0,1]$ whole level sets are the
equivalence classes:

$\big{\\{}(x,x^{\prime}):f(x)=f(x^{\prime})\big{\\}}=E^{\pi};$ (6.2)

Moreover, strong consistency implies that, up to a $\mu$-negligible set of
equivalence classes of $E^{\bar{A}_{\bar{\pi}}}$, 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 $E^{\bar{A}_{\bar{\pi}}}$.

We collect the previous observations in the following statement.

######  Lemma 6.2.

Under Assumption 6.1, there exists a test plan $\bar{\pi}$ concentrated on a
set of trajectories $\bar{A}_{\bar{\pi}}$ satisfying Theorem 3.12 such that:

  1. (1)

the equivalence relation $\bar{E}=\bar{E}^{\bar{A}_{\bar{\pi}}}$ generated by
$\bar{A}_{\bar{\pi}}$ admits a Borel section $\bar{S}$ and has the following
equivalent properties (up to a $\mu$-negligible saturated set):

    1. (a)

the disintegration (6.1) is consistent,

    2. (b)

the atoms of the separable $\sigma$-algebra $\mathcal{M}^{\bar{\pi}}$ are the
equivalence classes of $\bar{E}$;

  2. (2)

every two points of an equivalence class are connected by finitely many curves
in $\bar{A}_{\bar{\pi}}$.

Assumption 6.1 is necessary to prove that the integer circuitation condition
of $\Lambda$ implies the existence of the function $w$: indeed if the
disintegration is not consistent, then there are counterexamples as the
following one.

######  Example 6.3.

Consider the Cantor $1/3$-set

$K=[0,1]\setminus\bigcup_{n\in\mathbb{N}}\bigcup_{s\in\\{0,2\\}^{n-1}}\sum_{i=1}^{n-1}s_{i}3^{-i}+3^{-n}(1,2),$

and let $V:[0,1]\to[0,1]$ be the Vitali function

$V\in C^{0}([0,1]),\qquad
V\bigg{(}\sum_{i=1}^{n-1}s_{i}3^{-i}+3^{-n}(1,2)\bigg{)}=\sum_{i=1}^{n-1}s_{i}2^{-1-i}+\\{2^{-n}\\},\quad
s_{i}\in\\{0,2\\}.$

Define for $\alpha\in[0,1]\setminus\mathbb{Q}$

$f:K\to K,\quad f(x)=V^{-1}\circ(\mathrm{id}+\alpha\mod 1)\circ V(x),$

which is defined up to the countably many points belonging to the boundary $Z$
of the intervals $\sum_{i=1}^{n-1}s_{i}3^{-i}+3^{-n}(1,2)$,
$n\in\mathbb{N},s_{i}\in\\{0,2\\}$. For definiteness we can assume that those
points are removed from $K$, in which case the set $K\setminus Z$ is a
$G_{\delta}$-set, i.e. still a Polish space.

It is fairly easy to construct a family of totally disconnected curves
$\gamma:[0,1]\times K\to\mathbb{R}^{3}$ of length $1$ such that

$\gamma(0,x)=(0,0,x),\quad\gamma(1,x)=(0,0,f(x)).$

The MMS $(X,d,\mu)$ is the the union of all these trajectories with the
Euclidean distance in $\mathbb{R}^{3}$, and with the measure
$\mathcal{L}^{1}\times\mathcal{H}^{1/3}$ in the coordinates
$(t,x)\in[0,1)\times K$ see Figure 1.

$x$
---
$f(x)$
---
$f^{2}(x)$
---
$K$
---
$o(t,x)$
---
$f^{3}(x)$
---
$(y,\theta/2\pi)$
---
$(y+\alpha,\theta/2\pi+\beta)$
---
Figure 1. The Cantor set $K$ together with a piece of the orbit $o$, and the
evolution of the point $(y,\theta)$ in the angular coordinates.

Let $o(\cdot,x):\mathbb{R}\to\mathbb{R}^{3}$ be the orbit starting from $x\in
K$ and defined by

$o(t,x)=\gamma(t\mod 1,f^{\lfloor t\rfloor}(x)),\quad\lfloor t\rfloor\
\text{integer part of $t$}.$

The fact that the segments $\gamma$ are totally disconnected implies that the
only admissible plans $\pi$ are concentrated on curves which are subsets of
$o(\cdot,x)$, and in particular any measurable function $v:[0,1]\times
K\to\mathbb{R}$ gives an integer (or free) circuitation cotangent vector
field, being the tangent space one dimensional in each point.

Setting $v=2\pi\beta$ constant with $z\cdot(\alpha,\beta)\notin\mathbb{Z}$ for
all $z\in\mathbb{Z}^{2}\setminus 0$, we obtain that any candidate $w$ such
that $\dot{w}=iwv$ should satisfy

$w(0,0,f(x))=w(0,0,x)e^{2\pi i\beta},$

which, calling

$w^{\prime}(y)=w(0,0,V^{-1}(y)),$

by definition of $f$ gives

$w^{\prime}(y+\alpha\mod 1)=e^{i\theta(y+\alpha\mod 1)}=w^{\prime}(y)e^{2\pi
i\beta}=e^{i\theta(y)+2\pi i\beta},$

where the function $\theta:[0,1]\to\mathbb{S}^{1}$ is the angle.

If we consider the torus $\mathbb{T}^{2}$, then the values of the angles
$\theta(y)$ on a single orbit corresponds to the trajectory of the points

$n\mapsto\bigg{(}y+n\alpha\mod 1,\frac{\theta(y)}{2\pi}+n\beta\mod 1\bigg{)}.$

We are looking for the invariant graph $\theta=\theta(y)$. Consider the
measure $\varpi=\theta_{\sharp}\mathcal{L}^{1}$ and its Fourier transform

$\hat{\varpi}=\sum_{z\in\mathbb{Z}^{2}}c_{z}e^{2\pi iz\cdot(y,\theta)}.$

The invariance implies

$\sum_{z\in\mathbb{Z}^{2}}c_{z}e^{2\pi
iz\cdot(y,\theta)}=\sum_{z\in\mathbb{Z}^{2}}c_{z}e^{2\pi
iz\cdot(y,\theta)+2\pi iz\cdot(\alpha,\beta)},$

i.e.

$\forall c_{z}\not=0\ \big{(}z\cdot(\alpha,\beta)\in\mathbb{Z}^{2}\big{)}.$

This is impossible for $z\not=0$ by the assumptions on $\alpha,\beta$, hence
there are no integrals $w$ to $Dw=iwv$.

We can slightly change the space $X$ and the test plan $\bar{\pi}$ in order to
see that the assumption that the disintegration is strongly consistent for all
$\bar{A}_{\bar{\pi}}$ is essential. Set

$\tilde{X}=X\cup[0,1]\times\\{0\\},$

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

$\bar{A}_{\bar{\pi}}=\big{\\{}(t,0,0),t\in[0,1]\big{\\}}\cup\bigcup_{x\in
K,n\in\mathbb{N}}\big{\\{}o(t,x),t\in[0,1/2-2^{-n}]\big{\\}}\cup\big{\\{}o(-t,x),t\in[0,1/2-2^{-n}]\big{\\}}.$

It is fairly easy to see that $\bar{A}_{\bar{\pi}}$ satisfies Theorem 3.12,
and the equivalence class is $\tilde{X}\setminus\cup_{x}\\{o(1/2,x)\\}$.
However removing the negligible curve $(t,0,0)$ the measure algebra
$\mathcal{M}$ is not the minimal one $\mathcal{M}^{\bar{\pi}}$.

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
$\bar{A}_{\bar{\pi}}$.

######  Proposition 6.4.

There exists a Borel function $\check{w}$ such that for all sub-curves
$\gamma$ of $\bar{A}_{\bar{\pi}}$ it holds

$\check{w}(\gamma(1))=\check{w}(\gamma(0))e^{i\int_{0}^{1}\Lambda_{\gamma}(t)\dot{|\gamma|}(t)dt}.$

###### Proof.

Let $\bar{S}$ be a Borel section of $\bar{E}$, and let
$\Gamma_{\bar{A}_{\bar{\pi}},\bar{S}}$ be the family of subcurves of
$\bar{A}_{\bar{\pi}}$ passing through $\bar{S}$. We will assume that $\bar{S}$
has a point in each equivalence class, by removing a $\mu$-negligible set of
equivalence classes.

For every curve $\gamma\in\Gamma_{\bar{A}_{\bar{\pi}},\bar{S}}$ let

$s(\gamma)=\max\big{\\{}t:\gamma(t)\in\bar{S}\big{\\}}.$

Notice that by the assumption that $\bar{A}_{\bar{\pi}}$ generates $\bar{E}$,
every point $x\in X$ is connected to the unique point $\bar{S}\cap Ex$ by a
subcurve $\gamma\in\Gamma_{\bar{A}_{\bar{\pi}},\bar{S}}$. Being
$\\{(t,\gamma),\gamma(t)\in\bar{S}\\}$ a Borel set, it follows that $s$ is
Borel.

Define the function

$\check{w}:[0,1]\times\Gamma_{\bar{A}_{\bar{\pi}},\bar{S}}\to
L^{\infty}((0,1)),\quad\check{w}(t,\gamma)=e^{i\int_{s(\gamma)}^{t}\Lambda(\gamma)(\tau)\dot{|\gamma|}(\tau)d\tau}.$
(6.3)

It is easy to see that this function is Borel, being $s(\gamma)$ and $\Lambda$
Borel. Moreover, from the integer circuitation assumption on
$\bar{A}_{\bar{\pi}}$, if $\gamma(t)=\gamma^{\prime}(t^{\prime})$,
$\gamma,\gamma^{\prime}\in\Gamma_{\bar{A}_{\bar{\pi}},\bar{S}}$, then

$\frac{\check{w}(\gamma^{\prime}(t^{\prime}))}{\check{w}(\gamma(t))}=e^{i\big{(}\int_{s(\gamma^{\prime})}^{t^{\prime}}\Lambda(\gamma^{\prime})(\tau)\dot{|\gamma|}(\tau)d\tau+\int_{t}^{s(\gamma)}\Lambda(\gamma)(\tau)\dot{|\gamma|}(\tau)d\tau\big{)}}=1,$

because the curve

$\gamma(\tau)=\begin{cases}R_{s(\gamma^{\prime}),t^{\prime}}(\gamma^{\prime})(2\tau)&\tau\in[0,1/2],\\\
R_{t,s(\gamma)}(\gamma)(2(1-\tau))&\tau\in(1/2,1],\end{cases}$

is a sub-loop of $\bar{A}_{\bar{\pi}}$. This shows that $\check{w}(t,\gamma)$
is the graph of a Borel function $\check{w}$ defined on $X$.

Finally, let $\gamma\in\Gamma_{\bar{A}_{\bar{\pi}}}$ be a subcurve of
$\bar{A}_{\bar{\pi}}$. Let
$\gamma_{0},\gamma_{1}\in\Gamma_{\bar{A}_{\bar{\pi}},\bar{S}}$ be subcurves
connecting $\gamma(0),\gamma(1)$ to a point $\bar{S}$, respectively: such
curves exists because $\gamma$ takes values in a single equivalence class, and
each point in an equivalence class is connected to $\bar{S}$ by a subcurve.
The subloop made by these 3 curves has integer circuitation, and then

$\begin{split}\check{w}(\gamma(1))&=\check{w}(\gamma(0))e^{-i\int_{1}^{-s(\gamma_{0})}\Lambda(\gamma_{0})|\dot{\gamma}_{0}|\mathcal{L}^{1}+\int_{s(\gamma_{1})}^{1}\Lambda(\gamma_{1})|\dot{\gamma}_{1}|\mathcal{L}^{1}}\\\
&=\check{w}(\gamma(0))e^{i\int_{0}^{1}\Lambda(\gamma)\dot{|\gamma|}\mathcal{L}^{1}}e^{-i\int_{0}^{1}\Lambda(\gamma)\dot{|\gamma|}\mathcal{L}^{1}-i\int_{1}^{-s(\gamma_{0})}\Lambda(\gamma_{0})|\dot{\gamma}_{0}|\mathcal{L}^{1}+\int_{s(\gamma_{1})}^{1}\Lambda(\gamma_{1})|\dot{\gamma}_{1}|\mathcal{L}^{1}}\\\
&=\check{w}(\gamma(0))e^{i\int_{0}^{1}\Lambda(\gamma)\dot{|\gamma|}\mathcal{L}^{1}}.\end{split}$
(6.4)

where we have used that the curve obtained by joining
$\gamma,\gamma_{0},\gamma_{1}$ is a subloop of $\bar{A}_{\bar{\pi}}$. ∎

The next proposition shows the importance that $\mathcal{M}^{\bar{\pi}}$ is
the maximal measure algebra generated by $\bar{\pi}$.

######  Proposition 6.5.

If $w$ satisfies

$\frac{d}{dt}w(\gamma(t))=0,\quad\pi\text{-a.e.}\ \gamma,$ (6.5)

for every test plan $\pi$, then up to a $\mu$-negligible set $w$ is constant
in each equivalence class.

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 $\mu$-measurable.

###### Proof.

Consider the test plan $\bar{\pi}$ generating the minimal equivalence relation
$\bar{E}$.

Assumption (6.5) implies that $t\mapsto w(\gamma(t))$ is equal to a constant
$h(\gamma)$ for $\mathcal{L}^{1}$-a.e. $t\in[0,1]$ for $\bar{\pi}$-a.e.
$\gamma$. Define the set

$W=\Big{\\{}(\gamma,\gamma^{\prime}):\gamma([0,1])\cap\gamma^{\prime}([0,1])\not=\emptyset,h(\gamma)\not=h(\gamma^{\prime})\Big{\\}}\subset\Gamma\times\Gamma.$

The same analysis of Step 1 of Theorem 3.12 implies that there is a
$\bar{\pi}$-negligible set $N\subset\Gamma$ such that

$\forall\gamma,\gamma^{\prime}\in\Gamma\setminus
N\Big{(}\gamma([0,1])\cap\gamma^{\prime}([0,1])\ \Rightarrow\
h(\gamma)=h(\gamma^{\prime})\Big{)}.$

Indeed, for every measure $0<\xi\in\Pi^{\leq}(\bar{\pi},\bar{\pi})$
concentrated on $W$ the test plan obtained by (3.11) would not satisfy the
assumption of the proposition.

Hence, $t\mapsto w(\gamma(t))$ is constant in each subcurve $\gamma$ of
$\Gamma\setminus N$ for $\mathcal{L}^{1}$-a.e. $t$. In particular, it is
constant in each equivalence class of the equivalence relation
$E^{\bar{A}_{\bar{\pi}}\setminus N}$ generated by
$\bar{A}_{\bar{\pi}}\setminus N$.

This equivalence relation $E^{\bar{A}_{\bar{\pi}}\setminus N}$ is smaller than
$\bar{E}$, but by the maximality of $\mathcal{M}^{\bar{E}}$ it generates the
same measure algebra $\mathcal{M}^{\bar{E}}=\mathcal{M}^{\bar{\pi}}$. If $w$
is not constant in each equivalence class of $\bar{E}$ up to a
$\mu$-negligible set $N^{\prime}\subset X$, then it follows that taking $f$ as
in (6.2) for the equivalence relation $\bar{E}$ and defining

$f^{\prime}(x)=\big{(}f(x),w(x)),$

we obtain a Borel map such that its equivalence relation
$\bar{E}^{\prime}=(f^{\prime},f^{\prime})^{-1}(\mathrm{id})$ is smaller than
$\bar{E}$ and the measure algebra $\mathcal{M}^{\bar{E}^{\prime}}$ strictly
contains $\mathcal{M}^{\bar{\pi}}$, 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

$\frac{d}{dt}w\circ\gamma=iw(\gamma)\Lambda_{\gamma}(t)\dot{|\gamma|}(t),$

rewritten in (6.6) below.

######  Theorem 6.6.

Given a function
$\lx@glossaries@gls@link{main}{omega}{\leavevmode{\color[rgb]{0,0,1}\omega}}\in
L^{\infty}(\zeta,\mathbb{S}^{1})$ ($\zeta$ given by (6.1)), there exists a
function $w\in L^{\infty}(\mu,\mathbb{S}^{1})$ solving the equation for every
test plan $\pi$

$w(\gamma(t))=w(\gamma(s))e^{i\int_{s}^{t}\Lambda_{\gamma}\dot{|\gamma|}\mathcal{L}^{1}},\quad\pi\text{-a.e.}\
\gamma,\ s,t\in[0,1]\setminus N_{\gamma},\mathcal{L}^{1}(N_{\gamma})=0,$ (6.6)

and such that

$w(x)=\omega(y)\check{w}(x)\quad\text{for $\zeta$-a.e.
$y\in\bar{S}\cap\bar{E}x$},$

where $\check{w}$ is the function constructed in Proposition 6.4.

Viceversa, if $w$ is a solution to the equation (6.6) above, then the ratio
$w/\check{w}$ is constant on each equivalence class $\bar{E}_{y}$ for
$\zeta$-a.e. $y$ and defines a function $\omega\in
L^{\infty}(\zeta,\mathbb{S}^{1})$ on the quotient space $X/\bar{E}$.

In particular the map

$L^{\infty}(\zeta,\mathbb{S}^{1})\ni\omega(y)\longleftrightarrow\omega(\bar{E}x)\check{w}(x)\in
L^{\infty}(\mu,\mathbb{S}^{1})$

is bijective on the set of solutions.

###### Proof.

Given a function $\omega\in L^{\infty}(\zeta,\mathbb{S}^{1})$, the following
extension of formula (6.3) constructs the candidate function for the first
part of the statement:

$w(x;\omega)=\omega(\bar{E}x)e^{i\int_{s(\gamma)}^{1}\Lambda(\gamma)\dot{|\gamma|}\mathcal{L}^{1}}=\omega(Ex)\check{w},$
(6.7)

where $\gamma\in\Gamma_{\bar{A}_{\bar{\pi}},\bar{S}}$ is a subcurve of
$\bar{A}_{\bar{\pi}}$ connecting $y$ to $x$.

We need to prove that the integral formula (6.6) holds for all $\pi$. Let
$\pi$ be a test plan: by removing a negligible set of curves, we can assume
that $(\pi+\bar{\pi})/2$ is concentrated on a set of integer circuitation
curves, together with its subloops (Corollary 3.13). Being the equivalence
relation $\bar{E}$ maximal in the sense of Proposition 5.9, we can apply
Corollary 5.11: the curves $\gamma$ on which $\pi$ is concentrated are such
that

  1. (1)

$\gamma(t)$ belongs to the same equivalence class of $\bar{E}$ up to a
$\mathcal{L}^{1}$-negligible subset of $[0,1]$;

  2. (2)

the function $w(\gamma(t);\omega)$ given by formula (6.7) is defined
$\mathcal{L}^{1}$-a.e.

Since the disintegration is strongly consistent, for every $\gamma$ as above
there is a subcurve curve $\gamma^{\prime}$ of the set where $\bar{\pi}$ is
concentrated connecting $\gamma(s)$ to $\gamma(t)$ (by Point (1) above and
Assumption 6.1). Being $\pi+\bar{\pi}$ 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 $\bar{\pi}$-a.e. $\gamma$

$\frac{d}{dt}\frac{w(\gamma(t))}{\check{w}(\gamma(t))}=\frac{w(\gamma(t))}{\check{w}(\gamma(t))}\big{(}\Lambda(\gamma)(t)\dot{|\gamma|}(t)-\Lambda(\gamma)(t)\dot{|\gamma|}(t)\big{)}=0,$

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
$\mu=|f|^{2}m/\|f\|_{2}^{2}$, where $m$ is now the ambient measure on $X$ and
$f\in W^{1,2}(X,m)$. 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 $x\in X$ there exist two sequence of numbers $r_{n},R_{n}$,
$r_{n}\leq R_{n}$, converging to $0$ and a test plan
$\pi_{n}\in\mathcal{P}(\Gamma)$ such that for some constant $C_{x}$

$\int_{0}^{1}e(t)_{\sharp}(\dot{|\gamma|}\pi_{n})dt\leq\frac{C_{x}}{m(B_{r_{n}}(x))}m\llcorner_{B_{R_{n}}(x)},\quad
L(\gamma)\leq 2R_{n},\ m(B_{R_{n}}(x))\leq C_{x}m(B_{r_{n}}(x)),$

and

$(e(0),e(1))_{\sharp}\pi_{n}=\frac{1}{m(B_{r_{n}}(x))^{2}}m\times
m\llcorner_{B_{r_{n}}(x)\times B_{r_{n}}(x)}.$

Up to the restriction that the curves $\gamma$ used by $\pi_{n}$ are localized
inside the ball $B_{R_{n}}(x)$, the test plans $\pi_{n}$ are the Democratic
plans of [LV07].

######  Example 7.2.

The above condition holds for example for $[0,1]^{d}$, $d\in\mathbb{N}$, with
$m=\prod_{i}\mathcal{L}^{1}\llcorner_{[0,1]}$: indeed one considers the image
measure

$((1-t)x+ty)_{\sharp}(m\times m)$

which is equivalent to $m$ by coarea formula, being

$\sqrt{\sum_{i}(\det M_{i})^{2}}=\sqrt{((1-t)^{2}+t^{2})^{d}}\simeq 1,$

where $M_{i}$ are the $d\times d$-minors of the matrix $[(1-t)\mathrm{id}\
t\mathrm{id}]$. 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 $MCP(K,N)$ [Oht07, Definition 2.1], just by
adapting the proof of [LV07, Theorem 3.1].

In the case of branching spaces, the same condition holds if (locally, being
all computations done in a small ball)

$\frac{1}{C}\leq\frac{m(B_{r}(x))}{r^{N}}\leq C$ (7.1)

and for every $x\in X$, $0<r<1$, there is a transport plan
$\pi_{x,r}\in\mathrm{Geo}(X)$ concentrated on geodesics such that

$e(0)_{\sharp}\pi_{x,r}=\delta_{x},\
e(1)_{\sharp}\pi_{x,r}=\frac{m\llcorner_{B_{r}(x)}}{m(B_{r}(x))},\quad
e(t)_{\sharp}\pi_{x,r}\leq\frac{C}{t^{N}}\frac{m\llcorner_{B_{rt}(x)}}{m(B_{r}(x))}.$
(7.2)

In particular, it holds for branching $MCP(K,N)$-spaces with the Hausdorff
measure $\mathcal{H}^{N}$ as ambient measure $m$ [Raj].

To prove this fact, consider any ball $B_{\bar{r}}(\bar{x})$, $0<\bar{r}<1/2$,
and define

$\bar{\pi}=\fint_{B_{\bar{r}}(\bar{x})}\tilde{\pi}_{y,\bar{x},\bar{r}}m(dy),\quad\tilde{\pi}_{y,\bar{x},\bar{r}}=\frac{m(B_{2\bar{r}}(y))}{m(B_{\bar{r}}(\bar{x}))}\pi_{y,2\bar{r}}\llcorner_{\\{\gamma(1)\in
B_{\bar{r}}(\bar{x})\\}}.$

Being $\pi_{x,\bar{r}}\in\mathcal{P}(\mathrm{Geo}(X))$, i.e. it is
concentrated on the set of geodesic, it follows that $L(\gamma)\leq 2\bar{r}$,
and thus $e(t)_{\sharp}\pi_{y,\bar{x},\bar{r}}$ is concentrated on
$B_{2\bar{r}}(\bar{x})$ for $y\in B_{\bar{r}}(\bar{x})$. The measure
contraction property (the last estimate of (7.2)) gives that
$m(B_{2\bar{r}}(\bar{x}))\leq C2^{N}m(B_{\bar{r}}(\bar{x}))$. Moreover by
construction

$(e(0),e(1))_{\sharp}m=\frac{1}{m(B_{\bar{r}}(\bar{x}))^{2}}m\times
m\llcorner_{B_{\bar{r}}(\bar{x})\times B_{\bar{r}}(\bar{x})}.$

Finally, consider a ball $B_{r}(x)$ and compute

$\begin{split}e(t)_{\sharp}\bar{\pi}(B_{r}(x))&=\fint
e(t)_{\sharp}\pi_{y,\bar{x},\bar{r}}(B_{r}(x))m(dy)\leq\min\bigg{\\{}1,\frac{Cm(B_{r}(x))}{t^{N}m(B_{\bar{r}}(\bar{x}))}\bigg{\\}}\frac{m(B_{r+2t\bar{r}}(x))}{m(B_{\bar{r}}(\bar{x}))},\end{split}$

where we have used the last estimate in (7.2) and the simple observation

$e(t)_{\sharp}\bar{\pi}_{y,\bar{x},\bar{r}}(B_{r}(x))\leq\begin{cases}1&y\in
B_{r+2t\bar{r}}(x),\\\ 0&\text{otherwise}.\end{cases}$

Using again the measure contraction property one obtains

$m(B_{r+t\bar{r}}(x))\leq
C\bigg{(}1+\frac{t\bar{r}}{r}\bigg{)}^{N}m(B_{r}(x)),$

and then for $t\geq r/\bar{r}$ and using (7.1)

$\begin{split}\frac{Cm(B_{r}(x))}{t^{N}m(B_{\bar{r}}(\bar{x}))}\frac{m(B_{r+2t\bar{r}}(x))}{m(B_{\bar{r}}(\bar{x}))}&\leq
C^{2}\bigg{(}\frac{1}{t}+\frac{2\bar{r}}{r}\bigg{)}^{N}\frac{m(B_{r}(x))^{2}}{m(B_{\bar{r}}(\bar{x}))^{2}}\leq\frac{C^{2}(3\bar{r})^{N}}{r^{N}}\frac{m(B_{r}(x))^{2}}{m(B_{\bar{r}}(\bar{x}))^{2}}\\\
&\leq C^{4}\frac{m(B_{r}(x))}{m(B_{\bar{r}}(\bar{x}))}.\end{split}$

For $t\leq r/\bar{r}$ we have $m(B_{r+2t\bar{r}}(x))\leq C4^{N}m(B_{r}(x))$,
and thus we obtain

$e(t)_{\sharp}\bar{\pi}(B_{r}(x))\leq\frac{C^{\prime\prime}}{m(B_{\bar{r}}(x))}m(B_{r}(x))$

for some constant $C^{\prime\prime}$. This shows that
$e(t)_{\sharp}\bar{\pi}\leq C^{\prime\prime}m/m(B_{\bar{r}}(\bar{x}))$.

######  Lemma 7.3.

There exists a test plan $\pi$ satisfying Assumption 7.1 and such that the
equivalence relation $\bar{E}$ given by (5.6) is made of countably many
equivalence classes of positive measure.

###### Proof.

Take a point $x$ as in Assumption 7.1, fix $n$ and let $A_{\pi_{n}}$ be a set
of curves where $\pi_{n}$ is concentrated: by assumption $m$-a.e. point
$x_{1}\in B_{r_{n}}(x)$ is connected by a curve to $m$-a.e. point $x_{2}\in
B_{r_{n}}(x)$. By the transitivity of the equivalence relation, all these
points are in the same equivalence class, i.e. $B_{r_{n}}(x)$ 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 $\\{\Omega_{n}\\}_{n}=\\{B_{r_{n}}(x_{n})\\}_{n}$ be the family of balls
as above covering $m$-a.e. $x\in X$. Let $\pi_{nn^{\prime}}$ be a test plan
(if it exists) such that

$e(0)_{\sharp}\pi_{nn^{\prime}}\ll m\llcorner_{B_{r_{n}}(x_{n})},\quad
e(1)_{\sharp}\pi_{nn^{\prime}}\ll
m\llcorner_{B_{r_{n^{\prime}}}(x_{n^{\prime}})}.$

The test plan of the statement is

$\bar{\pi}=\sum_{n}c_{n}\pi_{n}+\sum_{nn^{\prime}}c_{nn^{\prime}}\pi_{nn^{\prime}},$

with $c_{n},c_{nn^{\prime}}$ chosen as in (3.12) to have a probability
measure. ∎

We next consider a function $f\in W^{1,2}(X,m)$ and define probability measure

$\mu=\frac{1}{\|f\|_{2}^{2}}|f|^{2}m.$

We assume that we are given the cotangent vector field $v\in L^{2}(T^{*}X)$ on
the space $(X,\mu)$, and we denote with $\Lambda_{\gamma}(t)$ the
corresponding function as in (3.6).

######  Proposition 7.4.

There exists a test plan $\bar{\pi}$ (w.r.t. the measure $\mu$) 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
$\bar{\pi}$ generating an equivalence class of positive measure essentially
invariant w.r.t. the choice of the sets $A_{\bar{\pi}}$.

Let $\bar{x}\in X$ be a Lebesgue point for $|f|$ and $|Df|$ (being $|Df|$ the
upper gradient), i.e. for definiteness

$\fint_{B_{\bar{r}}(\bar{x})}|f-1|^{2}m,\fint_{B_{\bar{r}}(\bar{x})}||Df|-1|^{2}m<\epsilon^{2}.$
(7.3)

Consider the ball $B_{r_{n}}(\bar{x})$, with $R_{n}\leq\bar{r}$, and the test
plan $\pi_{n}$ defined in Assumption 7.1: we have

$\begin{split}\int\bigg{[}\int_{0}^{1}|Df|(\gamma(t))\dot{|\gamma|}(t)dt\bigg{]}\pi_{n}(d\gamma)&=\int
L(\gamma)\pi_{n}(d\gamma)+\int\bigg{[}\int_{0}^{1}\big{[}|Df|(\gamma(t))-1\big{]}\dot{|\gamma|}(t)dt\bigg{]}\pi_{n}(d\gamma)\\\
\big{[}L(\gamma)\leq 2R_{n},\ e(t)_{\sharp}\pi_{n}\leq
C_{\bar{x}}m/m(B_{r_{n}}(\bar{x}))\big{]}\quad&\leq
2R_{n}+C_{\bar{x}}\int_{B_{R_{n}}(\bar{x})}||Df|-1|\frac{m}{m(B_{r_{n}}(\bar{x}))}\\\
&\leq
2R_{n}+C_{\bar{x}}\epsilon\frac{m(B_{R_{n}}(\bar{x}))}{m(B_{r_{n}}(\bar{x}))}\leq
2R_{n}+C_{\bar{x}}^{2}\epsilon.\end{split}$

In particular, using Chebyshev inequality, the $\pi$-measure of the set
$\mathcal{W}$ of trajectories for which

$\int_{0}^{1}|Df|(\gamma(t))\dot{|\gamma|}(t)dt>\frac{1}{2}$

is bounded by

$\pi(\mathcal{W})=\pi\bigg{(}\bigg{\\{}\gamma:\int_{0}^{1}|Df|(\gamma(t))\dot{|\gamma|}(t)dt>\frac{1}{2}\bigg{\\}}\bigg{)}<4R_{n}+2C_{\bar{x}}^{2}\epsilon<\frac{1}{64}$

for $n\gg 1$.

By the disintegration w.r.t. the map $e(0):\gamma\mapsto\gamma(0)$,

$\pi_{n}=\fint_{B_{r_{n}}(\bar{x})}\pi_{n,x}m(dx),$

it follows that

$\frac{1}{m(B_{r_{n}}(\bar{x}))}m\bigg{(}\bigg{\\{}x:\pi_{n,x}(\mathcal{W})\geq\frac{1}{8}\bigg{\\}}\bigg{)}<\frac{1}{8}.$

For all other points, there exists a family of geodesics of
$\pi_{n,x}$-measure greater than $7/8$ such that

$\int_{0}^{1}||Df(\gamma(t))|-1|^{2}\dot{|\gamma|}dt\leq\frac{1}{2}.$ (7.4)

By Assumption 7.1,

$e(1)_{\sharp}\check{\pi}_{n,x}=\frac{m}{m(B_{r_{n}}(\bar{x}))},$

so that the end points of the previous trajectories cover a subset of
$B_{\delta}(\bar{x})$ with $m$-measure larger than $7m(B_{r_{n}}(\bar{x}))/8$.

The set of $x$ such that $|f(x)-1|>1/4$ has measure smaller than
$16\epsilon^{2}m(B_{r_{n}}(\bar{x}))$, 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
$>7m(B_{r_{n}}(\bar{x}))/8$. In particular the trajectories starting from each
of these points must intersect, and since

$|f(\gamma(t))|\geq|f(\gamma(0))|-\int_{0}^{1}|Df(\gamma(s))|\dot{|\gamma|}(s)ds\geq\frac{1}{4},$

then they live in an equivalence class of positive measure. ∎

######  Corollary 7.5.

The disintegration w.r.t. the equivalence classes of $E^{\bar{A}_{\bar{\pi}}}$
for the measure $\mu$ is strongly consistent for every $\bar{A}_{\pi}$, 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. ∎

Finally, we consider the case $f\in W^{1,2}(X,m)$ and $v\in L^{2}(T^{*}X)$
cotangent vector field in $(X,\mu)$. The above computations can be applied to
the equivalence classes of $\mu=f^{2}m$, and then by Theorem 6.6 we can
construct a function $w\in L^{\infty}(\mu,\mathbb{S}^{1})$ such that

$w(\gamma(1))-w(\gamma(0))=\int_{0}^{1}\Lambda(\gamma)\dot{|\gamma|}\mathcal{L}^{1},\quad\pi\text{-a.e.
}\gamma,\ \pi\ \text{test plan of $(X,\mu)$}.$

Let now $\pi^{\prime}$ be a test plan for $(X,m)$: we can assume that $\pi$ is
concentrated on a family of curves $A_{\pi}$ such that $f\circ\gamma\in
H^{1}([0,L(\gamma)])$ for all $\gamma\in A_{\pi}$. In particular,
$f\circ\gamma$ is continuous and for every $n\in\mathbb{N}$ define

$\big{\\{}|f\circ\gamma|>2^{-n}\big{\\}}=\bigcup_{k}\big{(}t^{-}_{k,n}(\gamma),t^{+}_{k,n}(\gamma)\big{)}.$
(7.5)

The next lemma assures that the dependence w.r.t. $k,\gamma$ of
$t^{\pm}_{k,n}$ is Borel.

######  Lemma 7.6.

There exists a family of countable maps of Borel regularity

$\gamma\mapsto t^{-}_{k,n}(\gamma),t^{+}_{k,n}(\gamma)$

such that (7.5) holds.

###### Proof.

Being $(t,\gamma)\mapsto e(t,\gamma)=\gamma(t)$ continuous, by selecting a
Borel representative of $f$ we obtain that

$\tilde{f}(t,\gamma)=\big{[}|f(\gamma(t))|-2^{-n}\big{]}^{+}$

is Borel, and by the choice of $A_{\pi}$ it is continuous w.r.t. $t$ for every
$\gamma\in\Gamma$ by setting $\tilde{f}(t,\gamma)=0$ if $\gamma\notin
A_{\pi}$. We will prove the statement for the function $\tilde{f}$.

Let $s_{\ell}\in[0,1]\cap\mathbb{Q}$ be a countable dense sequence of times,
set

$A_{\pi,k}=\\{\gamma\in A_{\pi}:\tilde{f}(s_{k},\gamma)>0\\}$

and define the function $\tilde{t}^{+}_{k}:A_{\pi,k}\to[0,1]$ as the function
whose graph is

$\big{\\{}\gamma:\tilde{t}^{+}_{k}(\gamma)\geq t\big{\\}}=\bigcap_{s_{k}\leq
s_{\ell}\leq t}A_{\pi,\ell}.$

In the same way the function $\tilde{t}^{-}_{k}:A_{\pi,k}\to[0,1]$ is given by

$\big{\\{}\gamma:\tilde{t}^{-}_{k}(\gamma)\leq t\big{\\}}=\bigcap_{t\leq
s_{\ell}\leq s_{k}}A_{\pi,\ell}.$

Begin $A_{\pi,k}$ Borel, the functions $\tilde{t}^{\pm}_{k}$ are Borel with
domain $A_{\pi,k}$. It is clear that with this procedure we have

$\\{\tilde{f}>0\\}=\bigcup_{\gamma}\bigcup_{k}\big{(}t^{-}_{k}(\gamma),t^{+}_{k}(\gamma)\big{)}.$

The only step left is to have that every interval of
$\\{\tilde{f}(\cdot,\gamma)>0\\}$ is counted once. Set
$t_{1,n}^{\pm}(\gamma)=\tilde{t}_{1}^{\pm}(\gamma)$, and in general

$t^{\pm}_{k,n}(\gamma)=\tilde{t}^{\pm}_{k}(\gamma)\quad\text{if}\
k=\min\big{\\{}j:s_{j}\in(\tilde{t}^{-}_{k}(\gamma),\tilde{t}^{+}_{k}(\gamma))\big{\\}}.$

Being the last set a Borel set, the functions $t^{\pm}_{k}$ are again Borel. ∎

By defining

$\gamma_{k,n}=\gamma\llcorner_{(t^{-}_{k,n}(\gamma),t^{+}_{k,n}(\gamma))},\quad\pi_{k,n}=(\gamma_{k,n})_{\sharp}\pi,$

we have that $\pi_{k,n}$ is a test plan for the measure
$\mu=f^{2}m/\|f\|_{2}^{2}$, with compressibility constant
$2^{2n}C/\|f\|_{2}^{2}$, $C$ being the compressibility constant of $\pi$.
Along $\pi_{k,n}$-a.e. curve, by elementary 1d computations we have that

$\frac{d}{dt}(fw)(\gamma(t))=\frac{d}{dt}f(\gamma(t))w(\gamma(t))+f(\gamma(t))\frac{d}{dt}w(\gamma(t)),$
(7.6)

so that, recalling $w\in\mathbb{S}^{1}$, for $\pi_{n,k}$-a.e. $\gamma$ we
deduce

$\bigg{|}\frac{d}{dt}(fw)\circ\gamma(t)\bigg{|}\leq|Df|(\gamma(t))+|f(\gamma(t))||Dw(\gamma(t))|=|Df|(\gamma(t))+|f(\gamma(t))||v(\gamma(t))|.$
(7.7)

The last function belongs to $L^{2}(X,m)$, because $f\in W^{1,2}(X,m)$ and
$|Dw|\in L^{2}(X,\mu)$, i.e.

$\int|Dw|^{2}|f|^{2}\mu<\infty.$

By letting $n\to\infty$ and extending $t\mapsto(fw)(\gamma(t))$ as a
continuous function, we deduce that (7.6), (7.7) holds for $\pi$-a.e.
$\gamma$, and since trivially $|fw|=|f|\in L^{2}(X,m)$ we conclude with the
following proposition.

######  Proposition 7.7.

Under the assumptions $f\in W^{1,2}(X,m)$, $v\in L^{2}(X,|f|^{2}m)$, there
exists a solution $w$ such that $fw\in W^{1,2}(X,m)$ and

$D(fw)=wDf+fDw.$ (7.8)

Moreover any other solution $w^{\prime}$ satisfies

$\frac{w(x)}{w^{\prime}(x)}=\omega(Ex),$

where $E$ is the equivalence relation constructed in Section 5 in the space
$(X,\mu)$ with $\mu=f^{2}m/\|f\|_{2}^{2}$.

In particular there are at most $(\mathbb{S}^{1})^{\mathbb{N}}$-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). ∎

## Glossary

$\mathrm{Adm(\mu)}$

     $\mu$-admissible transference plans $\pi\in\mathcal{P}(\Gamma)$
$A(n)$

    set of curves made of up to pieces of $n$-different curves
$\Sigma^{1}_{1}$

    analytic sets, i.e. projection of closed sets
$A_{\pi}$

    set of curves where the test plan $\pi$ is concentrated
$B^{X}_{r}(x)$

    ball of radius $r$ centered in $x\in X$
$B^{X}_{r}(A)$

     $r$-neighborhood of the set $A\subset X$
$\bar{A}_{\pi}$

    set of curves where the test plan $\pi$ is concentrated generating the minimal equivalence relation
$\bar{E}$

    equivalence relation generated by countably many saturated sets
$\bar{\pi}$

    test plan generating a maximal equivalence relation
$\bar{S}$

    Borel section of an equivalence relation
$\mathcal{B}(X)$

    Borel $\sigma$-algebra of the topological space $X$
$\mathcal{C}(n)$

    subset of $A(n)$ made of connected curves
$\mathcal{M}$

    (complete) $\sigma$-algebra, often identificated with the corresponding measure algebra
$\mathcal{M}^{E}$

     $\sigma$-algebra made of $E$-saturated sets, of the corresponding measure algebra
$\mathcal{M}^{\pi}$

    maximal measure algebra generated by $\pi$
$\sharp A$

    cardinality of the set $A$
$C$

    compressibility constant of a test plan $\pi$
$\mathrm{const.}$

    generic constant
$d$

    distance on the metric space $X$
$Df$

    cotangent vector field corresponding to the function $f\in S^{1}$
$\operatorname{Dom}(f)$

    domain of the function $f$
$\dot{|\gamma|}$

    metric derivative of $\gamma$
$E^{A_{\pi}}$

    equivalence relation generated by the curves in $A_{\pi}$
$E^{\pi}$

    minimal equivalence relation generated by $\pi$
$Ex$

    equivalence class containing $x$
$E$

    equivalence relation
$e(t)$

    evaluation map $\Gamma\ni\gamma\mapsto\gamma(t)\in X$
$F_{\sigma}$

    countable union of closed sets
$\Gamma$

    space of Lipschitz curves on $X$
$\gamma$

    Lipschitz curves on $X$
$\Gamma_{A}$

    set of all subcurves of a given family of curves
$\Gamma_{\bar{A}_{\bar{\pi}},\bar{S}}$

    subcurves of $\bar{A}_{\bar{\pi}}$ passing trhough $\bar{S}$
$\Gamma_{c}$

    space of closed Lipschitz curves on $X$
$G_{\delta}$

    countable intersection of open sets
$f$

    generic function on $X$
$\overrightarrow{[x,y]}$

    geodesic connecting $x$ to $y$
$\operatorname{Graph}(f)$

    Graph of a function $f$
$\mathcal{H}^{d}$

     $d$-dimensional Hausdorff measure
$\hat{\pi}$

    test plan on constant curves with $e(t)_{\sharp}\hat{\pi}=1$
$d_{H}(K_{1},K_{2})$

    Hausdorff distance on $\mathcal{K}(X)$
$\mathcal{I}_{c}$

    set of closed path with integer circuitation
$\mathbb{Z}$

    integer numbers
$\mathcal{K}(X)$

    compact subsets of $X$, with the Haudorff topology
$L^{1}(T^{*}X)$

    cotangent bundle of $L^{1}$ covectors
$\Lambda_{\gamma}(t,v)$

    evaluation of the cotangent vector field $v$ on the curve $\gamma$
$\mathcal{L}^{d}$

    Lebesgue measure in $\mathbb{R}^{d}$
$L(\gamma)$

    length of the Lipschitz curve $\gamma$
$L^{\infty}(TX)$

    tangent bundle of $L^{\infty}$ vector fields
$\mathrm{Lip}([0,1];X)$

    Lipschitz functions from $[0,1]$ to $X$
$L^{p}(\mu)$

    standard integral space of functions
$m$

    probability on $X$ used for defining $W^{1,2}$
$\mu$

    an element of $\mathcal{P}(X)$
$\mu_{y}$

    conditional probabilities of the disintegration of $\mu$
$\mathbb{N}$

    natural numbers
$\omega$

     $L^{\infty}(m)$-function defined on $\bar{S}$ with values in $\mathbb{S}^{1}$
$\pi$

    test plan
$\mathcal{P}(X)$

    probability measure over $X$
$\mathtt{p}_{i}$

    projection on the $i$-th component of the space $\prod_{i}X_{i}$
$\mathtt{p}_{X}$

    projection on the space $X$
$f_{\sharp}\mu$

    push forward of $\mu$ by $f$
$\mathtt{e}$

    quotient map for the equivalence relation $E$
$f(X)$

    range of the function $f$
$\mathbb{Q}$

    rational numbers
$\mathbb{R}$

    real numbers
$R_{s,t}(\gamma)$

    restriction of the curve $\gamma$ to the interval $[s,t]$
$R(\gamma)(t)=\gamma(1-t)$

    time reversal map
$\mathbb{S}^{1}$

    unitary circle in $\mathbb{R}^{2}$
$S^{1}(X)$

    space of functions with upper gradient $|Df|$ in $L^{1}(X)$
$S$

    section of the equivalence relation $E$
$\sigma(\Sigma^{1}_{1})$

     $\sigma$-algebra generated by analytic sets
$|Df|$

    upper gradient of a function $f$
$\varpi$

    image measure of the disintegration of $\mu$
$v$

    element of the cotangent bundle $L^{1}(T^{*}X)$
$W^{1,2}(X)$

    space of functions in $L^{2}(X)$ with upper gradient in $L^{2}(X)$
$X$

    separable metric space
$x$

    generic point in the metrix space $X$

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