Michael Goldman
email: goldman@math.univ-paris-diderot.fr

Martin Huesmann
email: huesmann@iam.uni-bonn.de

Felix Otto
email: felix.otto@mis.mpg.de

A large-scale regularity theory for the Monge-Ampère equation with rough data and application to the optimal matching problem

The aim of this paper is to obtain quantitative bounds for solutions to the optimal matching problem in dimension two. These bounds show that up to a logarithmically divergent shift, the optimal transport maps are close to be the identity at every scale. These bounds allow us to pass to the limit as the system size goes to infinity and construct a locally optimal coupling between the Lebesgue measure and the Poisson point process which retains the stationarity properties of the Poisson point process only at the level of second-order differences. Our quantitative bounds are obtained through a Campanato iteration scheme based on a deterministic and a stochastic ingredient. The deterministic part, which can be seen as our main contribution, is a regularity result for Monge-Ampère equations with rough right-hand side. Since we believe that it could be useful in other contexts, we prove it for general space dimensions. The stochastic part is a concentration result for the optimal matching problem which builds on previous work by Ambrosio, Stra and Trevisan.

Introduction

We are interested in the optimal matching problem between the Lebesgue measure and the Poisson point process µ on the torus

Q L := -L 2 , L 2
d (i.e. Q L with the periodized induced metric | • | per from R d ): W 2 2,per µ(Q L ) |Q L | χ Q L , µ , (1.1)
where W 2 2,per denotes the squared L 2 -Wasserstein distance on Q L with respect to | • | per . This problem and some of its variants such as generalizations to L p -costs or more general reference measures, have been the subject of intensive work in the past thirty years (see for instance [START_REF] Talagrand | Upper and lower bounds for stochastic processes: modern methods and classical problems[END_REF][START_REF] Barthe | Combinatorial optimization over two random point sets[END_REF][START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF][START_REF] Ledoux | On optimal matching of Gaussian samples[END_REF][START_REF]On optimal matching of Gaussian samples II[END_REF]). As far as we know, essentially all the previous papers investigating (1.1) were focusing on estimating the mean of (1.1), e.g. [START_REF] Ajtai | On optimal matchings[END_REF][START_REF] Dobrić | Asymptotics for transportation cost in high dimensions[END_REF][START_REF] Barthe | Combinatorial optimization over two random point sets[END_REF][START_REF] Talagrand | The transportation cost from the uniform measure to the empirical measure in dimension ≥ 3[END_REF][START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF], or deviation from the mean, e.g. [START_REF] Dereich | Constructive quantization: approximation by empirical measures[END_REF][START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF], by constructing on the one hand sophisticated couplings whose costs are asymptotically optimal and proving on the other hand ansatz free lower bounds. The only exception is [START_REF] Huesmann | Optimal transport from Lebesgue to Poisson[END_REF] where for d ≥ 3, stationary couplings between the Lebesgue measure and a Poison point process on R d minimizing the cost per unit volume are constructed. From these works, in fact since [START_REF] Ajtai | On optimal matchings[END_REF], it is understood that d = 2 is the critical dimension for (1.1). Indeed, while

for d ≥ 3, E L 1 L d W 2 2,per µ(Q L ) |Q L | χ Q L ,
µ is of order 1, it is logarithmically diverging for d = 2 (see Section 1.1.1). We focus here on the critical dimension d = 2 and aim at a better description of the optimal transport maps, that is, of minimizers of (1.1). Building on a large-scale regularity theory for convex maps solving the Monge-Ampère equation

∇ψ#dx = µ, (1.2)
which we develop along the way, we prove that from the macroscopic scale L down to the microscopic scale the solution of (1.1) is close to the identity plus a shift. Here closeness is measured with respect to a scale-invariant L 2 norm. Our main result is the following: < ∞ and such that if i r * ,L L there exists

x L = x L (µ) ∈ Q L with ii |x L | 2 r 2 * ,L log 3 L r * ,L (1.3)
such that if T = T µ,L is the minimizer of (1.1), then for every 2r * ,L ≤ ≤ L, there holds

1 4 B (x L ) |T -(x -x L )| 2 log 3 r * ,L r * ,L 2
.

(1.4)

Note that the bounds in (1.3) and (1.4) are probably not optimal. Indeed, in both estimates one would rather expect a linear dependence on the logarithms. Similarly, using a similar proof in dimension d ≥ 3 we would get bounds of the order of log 2 (L) for the shift x L even though it is expected to be of order one. In order to improve our bounds, one would need to better capture some cancellation effects. Notice however that the proof of (1.4) lead to the optimal estimate (cf. Let us point out that Theorem 1.1 would also hold for the Euclidean transport problem on Q L . The motivation for considering instead the transport problem on the torus comes from the good stationarity properties of the optimal transport maps in this setting. Indeed, since the bound (1.4) is uniform in L one can try to construct a covariant and locally optimal coupling between the Lebesgue measure and the Poisson point process on R 2 by taking the limit L → ∞ in T µ,L . A similar strategy has been implemented in [START_REF] Huesmann | Optimal transport from Lebesgue to Poisson[END_REF] for d ≥ 3 using ergodic-type arguments iii . One of the key ingredients used in that paper, namely the fact that the minimal cost per unit volume is finite, is missing for d = 2. The presence of the logarithmically divergent shift x L in (1.4) can be seen as a manifestation of the logarithmic divergence of the minimal cost per unit volume. In order to pass to the limit we thus need to renormalize the transport map by subtracting this shift. Because of this renormalization the limiting map will loose its stationarity properties which will roughly speaking only survive at the level of the gradient. Also since we do not have any uniqueness statement for the limit objects, we will have to pass to the limit in the sense of Young measures.

In order to state our second main result, we need more notation. It is easier to pass to the limit at the level of the Kantorovich potentials rather than for the corresponding transport maps. Also, since the Lebesgue measure on R 2 is invariant under arbitrary shifts while the Poisson point process on Q L (extended by periodicity to R 2 ) is not, it is more natural to make the shift in the domain and keep the image unchanged. This could also serve as motivation for centering the estimate (1.4) around x L (which is approximately equal to T -1 (0)) rather than around 0.

To be more precise, denote by ψ = ψ µ,L the Kantorovich potential defined on R 2 and associated to T µ,L , i.e. T µ,L = ∇ ψ µ,L , satisfying ψ µ,L (0) = 0 (see Section 2.3). Define, ψ µ,L (x) := ψ µ,L (x + x L )

with corresponding Legendre dual ψ * µ,L (y) = ψ * µ,L (y) -x L • y. With this renormalization we still have

∇ψ µ,L # µ(Q L ) |Q L | = µ.
Denote the space of all real-valued convex functions ψ : R 2 → R by K and the space of all locally finite point configurations by Γ. We equip K with the topology of locally uniform convergence and Γ with the topology obtained by testing against continuous and compactly supported functions. There is a natural action on Γ by R 2 denoted by θ z and given by θ z µ := µ(• + z) for z ∈ R 2 . Define the map Ψ L : Γ → K by µ → ψ µ,L and denote by P L the Poisson point process on Q L . For each dyadic L ≥ 1 we define the Young measure associated to Ψ L by q L := (id, Ψ L )#P L = P L ⊗ δ Ψ L .

Then, we have the following result:

Theorem 1.2. The sequence (q L ) L of probability measures is tight. Moreover, any accumulation point q satisfies the following properties:

(i) The first marginal of q is the Poisson point process P L ;

(ii) q almost surely ∇ψ#dx = µ;

(iii) for any h, z ∈ R 2 and f ∈ C b (Γ × C 0 (R 2 )) there holds

Γ×K f (µ, D 2 h ψ * )dq = Γ×K f (θ -z µ, D 2 h ψ * (• -z))dq,
where D 2 h ψ * (y) := ψ * (y + h) + ψ * (y -h) -2ψ * (y).

Part (iii) of Theorem 1.2 says that for any h, under the measure q the random variable (µ, ψ) → (µ, D 2 h ψ * ) is stationary with respect to the action induced by the natural shifts on Γ and K. Observe that as already pointed out, while the second-order increments of the potentials ψ are stationary, the induced couplings (id, ∇ψ)#dx are not. This is due to the necessary renormalization by x L . It is an interesting open problem to understand whether one can prove that the sequence (Ψ L ) L actually converges and get rid of the Young measures. A slightly weaker open problem is to show non/uniqueness of the accumulation points of (q L ) L .

1.1 Main ideas for the proof of Theorem 1.1

The proof of Theorem 1.1 is inspired by (quantitative) stochastic homogenization, in the sense that it is based on a Campanato iteration scheme which allows to transfer the information that (1.4) holds at the "thermodynamic" scale (here, the scale L ↑ ∞ of the torus) by [START_REF] Ajtai | On optimal matchings[END_REF][START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF] to scales of order one (here, the scale r * ,L ). This is reminiscent of the approach of Avellaneda and Lin [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF] to a regularity theory for (linear) elliptic equations with periodic coefficients: The good regularity theory of the homogenized operator, i.e. the regularity theory on the thermodynamic scale, is passed down to the scale of the periodicity. This approach has been adapted by Armstrong and Smart [START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF] to the case of random coefficients; the approach has been further refined by Gloria, Neukamm and the last author [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF] (see also [START_REF] Armstrong | Quantitative stochastic homogenization and large-scale regularity[END_REF]) where the random analogue of the scale of periodicity, and an analogue to r * ,L in this paper, has been introduced and optimally estimated (incidentally also by concentration-of-measure arguments as in this paper).

The Campanato scheme is obtained by a combination of a deterministic and a stochastic argument. The deterministic one is similar in spirit to [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF]. It asserts that if at some scale R > 0 the excess energy is small and if the Wasserstein distance of µ B R to µ(B R )

|B R | χ B R is also small then up to an affine change of variables the excess energy is well controlled by these two quantities at scale θR for some θ 1. The aim of the stochastic part is to prove that with overwhelming probability, the Euclidean

L 2 -Wasserstein distance 1 R 4 W 2 2 µ B R , µ(B R ) |B R |
is small for every (dyadic) scale R between L and 1 so that the Campanato scheme can indeed be iterated down to the microscopic scale. Since our proof of the stochastic estimate is based on the results of [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF] which are stated for cubes, we will actually prove the stochastic estimate on cubes instead of balls. We now describe these two parts separately in some more detail. We start with the stochastic aspect since it is simpler.

The stochastic part

For a measure µ on Q L and ≤ L we denote its restriction to Q ⊂ Q L by µ := µ Q . Then, the main stochastic ingredient for the proof of Theorem 1.1 is the following result: < ∞ and such that for every dyadic with 2r * ,L ≤ ≤ L,

1 4 W 2 2 µ , µ(Q ) 2 ≤ log r * ,L r * ,L 2 .
The proof of this result relies on an adaptation of the concentration argument put forward in [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF]Remark 4.7]. One of the differences between this article and [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF] is that in [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF] the more classical version of (1.1), namely the matching of the empirical measure of n iid uniformly distributed points X 1 , . . . , X n on the cube

Q = [-1 2 , 1
2 ) d to their reference measure is considered:

C n,d := E W 2 2 χ Q , 1 n n i=1 δ X i = 1 n 2 d E W 2 2 1 n χ Q n 1 d , 1 n n i=1 δ n 1 d X i . (1.5)
Since the typical distance between nearby points X i and X j is of order n -

1 d it is expected that C n,d ∼ n -2
d . However, it turns out that this is only true in d ≥ 3. Since the seminal work [START_REF] Ajtai | On optimal matchings[END_REF], it is known that in dimension two an extra logarithmic factor appears. In dimension one the correct scaling is of order 1 n so that we can summarize [START_REF] Barthe | Combinatorial optimization over two random point sets[END_REF][START_REF] Ledoux | On optimal matching of Gaussian samples[END_REF].

C n,d ∼        1 n , d = 1, cf. [11] log n n , d = 2, cf. [1] 1 n 2 d , d ≥ 3, cf.
Based on a linearization ansatz of the Monge-Ampère equation suggested by [START_REF] Caracciolo | Scaling hypothesis for the euclidean bipartite matching problem[END_REF] in the physics literature, [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF] significantly strengthened the two-dimensional case to

lim n→∞ n log n C n,2 = 1 4π .
Additionally, it is remarked in [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF]Remark 4.7] that the mass concentrates around the mean. Combining this concentration argument with conditioning on the number of points of µ in Q L and a Borel-Cantelli argument we show Theorem 1.3.

The deterministic part

As already alluded to, the deterministic ingredient is one step of a Campanato scheme for solutions of the Monge-Ampère equation with arbitrary right-hand side (1.2). Since we believe that this far-reaching generalization of [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF]Proposition 4.7] (see also [START_REF]An ε-regularity result for optimal transportation maps between continuous densities[END_REF]) could have a large range of applications we prove it for arbitrary dimension d ≥ 2. Let us point out that while [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF] gives an alternative proof of the partial regularity result for the Monge-Ampère equation with "regular" data previously obtained in [START_REF] Figalli | Partial regularity of Brenier solutions of the Monge-Ampère equation[END_REF][START_REF]Partial regularity for optimal transport maps[END_REF] (see [START_REF] Philippis | Partial regularity results in optimal transportation[END_REF] for a nice informal presentation of this approach), it is unclear if that other approach based on maximum principles could also be used in our context. Given for some R > 0 a bounded set Ω ⊃ B 2R and an arbitrary measure µ, denote by T the optimal transport map between χ Ω and µ and let O ⊃ B 2R be an open set.

We have the following result:

Theorem 1.4. For every 0 < τ 1, there exist positive constants ε(τ ), C(τ ), and

0 < θ < 1 such that if 1 R d+2 B 2R |T -x| 2 + 1 R d+2 W 2 2 µ O, µ(O) |O| ≤ ε(τ ), (1.6)
then there exists a symmetric matrix B and a vector b ∈ R d such that

|B -Id| 2 + 1 R 2 |b| 2 1 R d+2 B 2R |T -x| 2 ,
and letting x := B -1 x, Ω := B -1 Ω and then

T (x) := B(T (x) -b)
and μ := T #χ Ωdx, so that T is the optimal transport map between χ Ω and μ, we have

1 (θR) d+2 B 2θR | T -x| 2 ≤ τ R d+2 B 2R |T -x| 2 + C(τ ) R d+2 W 2 2 µ O, µ(O) |O| . (1.7)
The only reason for not taking O = B 2R is that in our application, the control on the data term Since the estimates in Theorem 1.4 are scale-invariant, it is enough to prove it for R = 1. We then let for notational simplicity

E := B 2 |T -x| 2 and D := W 2 2 µ O, µ(O) |O| .
The main ingredient for the proof of Theorem 1.4 is the following result, which is the counterpart of [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF]Proposition 4.6] and which states that if E + D 1, that is, if the energy is small and if the data is close to a constant in the natural W 2 2 -topology, then T -x is quantitatively close to the gradient ∇ϕ of a solution to a Poisson equation. This quantifies the well-known fact that the Monge-Ampère equation linearizes to the Poisson equation around the constant density. Proposition 1.5. For every 0 < τ 1, there exist positive constants ε(τ ) and C(τ ) such that if E + D ≤ ε(τ ), then there exists a function ϕ with harmonic gradient in B 1 4 and such that

B 1 4 |T -(x + ∇ϕ)| 2 τ E + C(τ )D and B 1 4 |∇ϕ| 2 E.
As in [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF], Proposition 1.5 is actually proven at the Eulerian (or Benamou-Brenier) level. That is, if we let for t ∈ [0, 1], T t := (1 -t)Id + tT , ρ := T t #χ Ω , and

j := T t #(T -x)χ Ω , the couple (ρ, j) solves min (ρ,j) Ω 1 0 1 ρ |j| 2 : ∂ t ρ + ∇ • j = 0, ρ 0 = χ Ω , ρ 1 = µ (1.8)
and we show Proposition 1.6. For every 0 < τ 1, there exist positive constants ε(τ ) and C(τ ) such that if E + D ≤ ε(τ ), then there exists a function ϕ with harmonic gradient in B 1 such that

B 1 2 1 0 1 ρ |j -ρ∇ϕ| 2 τ E + C(τ )D (1.9)
and

B 1 |∇ϕ| 2 E.
As in [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF], this is proven by first choosing a good radius where the flux of j is well controlled in order to define ϕ, then obtaining the almost-orthogonality estimate

B 1 2 1 0 1 ρ |j -ρ∇ϕ| 2 B 1 1 0 1 ρ |j| 2 - B 1 |∇ϕ| 2 + τ E + C(τ )D (1.10)
and finally constructing a competitor and using the minimality of (ρ, j) for (1.8) in order to estimate the term inside the brackets in (1.10). However, each of these steps is considerably harder than in [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF]. This becomes quite clear considering that by analogy with [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF], letting for R > 0, f := 1 0 j • ν where ν is the outward normal to B R , one would like to define ϕ as a solution of

∆ϕ = 1 -µ in B R ∂ϕ ∂ν = f on ∂B R for some well chosen R ∈ ( 1 2 , 3
2 ). However if µ is a singular measure ∇ϕ will typically not be in L 2 since this would require L 2 bounds (actually H -1 2 would be enough) on f which cannot be obtained from the energy through a Fubini type argument since the L ∞ norm of ρ t typically blows up as t → 1. Similar issues were tackled in [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF][START_REF] Ledoux | On optimal matching of Gaussian samples[END_REF] by mollification of µ with smooth kernels (the heat and the Mehler kernel respectively). Here instead we introduce a small time-like parameter τ and work separately in (0, 1 -τ ) and in the terminal layer (1 -τ, 1). In (0, 1 -τ ), we take care of the flux going through ∂B R in that time interval. We first modify the definition of f and let f := 1-τ 0 j • ν, and then change ϕ so that it connects in B R the constant density equal to 1 to the constant density equal to 1

-1 |B R | ∂B R f , i.e. ϕ solves ∆ϕ = 1 |B R | ∂B R f in B R . (1.11)
Regarding the Neumann boundary conditions, we face here the problem that even though ρ ∈ L ∞ (B 2 × (0, 1 -τ )), its L ∞ bound blows up as τ → 0. This leads to L 2 bounds on f which are not uniform in τ . To overcome this difficulty, we need to replace f by a better behaved density on ∂B R . This is the role of Lemma 3.4.

Treating separately the incoming and outgoing fluxes f ± , we construct densities

ρ ± on ∂B R with ∂B R ρ 2 ± E and W 2 2 (ρ ± , f ± ) E d+3 d+2 .
(1.12)

The densities ρ ± can be seen as rearrangements of f ± through projections on ∂B R .

Considering the time-dependent version of the Lagrangian problem, since E 1 the particles hitting ∂B R in (0, 1 -τ ) must come from a small neighborhood of ∂B R at time 0. A key point in deriving (1.12) is that at time 0 the density is well-behaved (since it is constant) and thus the number of particles coming from such a small neighborhood of ∂B R is under control. The estimate in (1.12) on the Wasserstein distance between ρ ± and f ± is important in view of the construction of a competitor. Indeed, this indicates that if we know how to construct a competitor having ρ ± as boundary fluxes, we will then be able to modify it into a competitor with the correct f ± boundary conditions (see in particular Lemma 2.6). We then complement (1.11) with the boundary conditions

∂ϕ ∂ν = ρ + -ρ - on ∂B R .
The almost-orthogonality estimate (1.10) is proven in Proposition 3.6. It is readily seen that assuming for simplicity that R = 1 is the good radius,

B 1 2 1 0 1 ρ |j -ρ∇ϕ| 2 B 1 1 0 1 ρ |j| 2 - B 1 |∇ϕ| 2 + B 1 1-τ 0 ρ -1 |∇ϕ| 2 + B 1 ϕρ 1-τ + ∂B 1 ϕ f -(ρ + -ρ -) + τ E. (1.13)
While in [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF] the first term in the second line was easily estimated since in that case, up to a small error we had ρ ≤ 1, we need here a more delicate argument. In order to estimate the second term, we use that, again up to choosing a good radius, we may assume that

W 2 2 ρ 1-τ B 1 , ρ 1-τ (B 1 ) |B 1 | χ B 1 τ 2 E + D, (1.14)
see Lemma 3.5. Ignoring issues coming from the flux through ∂B 1 , and thus assuming that ρ 1-τ (B 1 )

|B 1 | = µ(B 1 ) |B 1 | = µ(O) |O| (where O ⊃ B 2 is the open set in the definition of D), (1.14) follows from W 2 2 (ρ 1-τ B 1 , µ B 1 ) τ 2 E by displacement interpolation, W 2 2 (µ B 1 , µ(B 1 ) |B 1 |
) D by definition and triangle inequality. The last term in (1.13) is estimated thanks to the W 2 2 estimate given in (1.12). Let us finally describe the construction of the competitor given in Proposition 3.7. As explained in the beginning of this discussion, we employ a different strategy for the time intervals (0, 1 -τ ) and (1 -τ, 1). In (0, 1 -τ ), forgetting the issue of connecting ρ ± to f ± , we mostly take as competitor

( ρ, j) = 1 - t 1 -τ 1 |B 1 | ∂B 1 f , ∇ϕ + (s, q),
where (s, q) are supported in the annulus B 1 \B 1-r × (0, 1 -τ ) for some boundary layer size r 1 and satisfy the continuity equation,

s 0 = s 1-τ = 0 and q • ν = j • ν -f on ∂B 1 × (0, 1 -τ ).
The existence of such a couple (s, q) satisfying the appropriate energy estimates is given by [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF]Lemma 3.4] which in turn is inspired by a similar construction from [START_REF] Alberti | Uniform energy distribution for an isoperimetric problem with long-range interactions[END_REF]. In the terminal time layer (1 -τ, 1), we connect the constant 1 -1 |B 1 | ∂B 1 f to the measure µ. The outgoing flux is easily treated by pre-placing on ∂B 1 the particles which should leave the domain in (1 -τ, 1). For the incoming flux, we use the corresponding part of (ρ, j) as competitor. Finally, the remaining part of the measure µ ≤ µ which was not coming from particles entering

∂B 1 in (1 -τ, 1) is connected to 1 -1 |B 1 | ∂B 1 f thanks to the estimate W 2 2 µ , µ (B 1 ) |B 1 | τ 2 E + D,
which is obtained as (1.14) in Lemma 3.5.

Outline

The plan of the paper is the following. In Section 2 we first set up some notation and then prove a few more or less standard elliptic estimates. We then write in Section 2.3 a quick reminder on optimal transportation. In Section 2.4, we give the definition and first properties of the Poisson point process and then prove our main concentration estimates (see Theorem 2.8 and Theorem 2.10). Section 3 is the central part of the paper and contains the proof of Theorem 1.4. We first explain in Section 3.1 how to choose a good radius before proving Proposition 1.6. This is a consequence of Proposition 3.6 and Proposition 3.7 which contain the proof of the almost-orthogonality property (1.10) and the construction of a competitor respectively. In Section 4, we combine the stochastic and deterministic ingredients to perform the Campanato iteration and prove both the quantitative bounds of Theorem 1.1 and perform the construction of the locally optimal coupling between Lebesgue and Poisson given in Theorem 1.2.

Preliminaries

Notation

In this paper we will use the following notation. The symbols ∼, , indicate estimates that hold up to a global constant C, which typically only depends on the dimension d. For instance, f g means that there exists such a constant with f ≤ Cg, f ∼ g means f g and g f . An assumption of the form f 1 means that there exists ε > 0, typically only depending on dimension, such that if f ≤ ε, then the conclusion holds. We write log for the natural logarithm. We denote by H k the k-dimensional Hausdorff measure. For a set E, ν E will always denote the external normal to E. When clear from the context we will drop the explicit dependence on the set. We write |E| for the Lebesgue measure of a set E and χ E for the indicator function of E. When no confusion is possible, we will drop the integration measures in the integrals. Similarly, we will often identify, if possible, measures with their densities with respect to the Lebesgue measure. For R > 0 and x 0 ∈ R d , B R (x 0 ) denotes the ball of radius R centered in x 0 . When x 0 = 0, we will simply write B R for B R (0). We denote the gradient (resp. the Laplace-Beltrami operator) on ∂B R by ∇ bdr (resp. ∆ bdr ). For L > 0, we denote by

Q L := -L 2 , L 2
d the cube of side length L.

Elliptic estimates

We start by collecting a few more or less standard elliptic estimates which we will need later on.

Lemma 2.1. For f ∈ L 2 (∂B 1 ) let ϕ be the (unique) solution of

   ∆ϕ = 1 |B 1 | ∂B 1 f in B 1 ∂ϕ ∂ν = f on ∂B 1 , with ∂B 1 ϕ = 0, then letting p := 2d d-1 , B 1 |∇ϕ| p 1 p ∂B 1 f 2 1 2 . (2.1)
Moreover, for every 0 < r ≤ 1,

B 1 \B 1-r |∇ϕ| 2 r ∂B 1 f 2 . (2.2)
Proof. Replacing ϕ by ϕ -|x| 2 2d|B 1 | ∂B 1 f , we may assume that ∂B 1 f = 0. By Pohozaev's identity (see [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF]) and Poincaré's inequality, we have that ϕ ∈ H 1 (∂B 1 ) with

∂B 1 |∇ϕ| 2 ∂B 1 f 2 .
(2.3) Estimate (2.2) follows from (2.3) together with the sub-harmonicity of |∇ϕ| 2 in the form

∂Br |∇ϕ| 2 ≤ ∂B 1 |∇ϕ| 2 for r ≤ 1.
We are just left to prove (2.1). Since by (2.3) we have ∂B 1 |∇ϕ| 2 ∂B 1 f 2 , and since ∇ϕ is harmonic, it suffices to show that for every harmonic function ϕ with ∂B 1 ϕ = 0 and thus B 1 ϕ = 0,

B 1 |ϕ| p 1 p ∂B 1 ϕ 2 1 2
.

(2.4)

The argument for (2.4) roughly goes as follows: By L 2 -based regularity theory, the H 1 2 (B 1 )-norm of ϕ is estimated by the L 2 (∂B 1 )-norm of its Dirichlet data, so that (2.4) reduces to a fractional Sobolev inequality on B 1 . If one wants to avoid fractional Sobolev norms, in view of their various definitions on bounded domains, one needs to construct an extension φ of ϕ to the (semi-infinite) cylinder B 1 × (0, ∞), preserving

B 1 φ = 0 and with the H 1 (B 1 × (0, ∞))-norm of φ estimated by the H 1 2 (B 1 )-norm of ϕ, which combines to B 1 ∞ 0 |∇ φ| 2 + (∂ t φ) 2 1 2 ∂B 1 ϕ 2 1 2
.

(

It then remains to appeal to the (Sobolev-type) trace estimate

B 1 |ϕ| p 1 p B 1 ∞ 0 |∇ φ| 2 + (∂ t φ) 2 1 2
.

(2.6)

For the sake of completeness, we now give the arguments for (2.5) and (2.6). Starting with (2.5), we first argue that it suffices to consider the case where ϕ |∂B 1 is an eigenfunction of the Laplace-Beltrami operator -∆ bdr , say for eigenvalue λ ≥ 0. Then we have

ϕ(x) = r α ϕ(x) with r := |x|, x := x r , α(d -2 + α) = λ, (2.7
) With φ being another function of this form we have with ∇ bdr ϕ denoting the tangential part of the gradient of ϕ

which follows from ∆ = 1 r d-1 ∂ r r d-1 ∂ r + 1 r 2 ∆ bdr
(∇ φ • ∇ φ + ∂ t φ∂ t φ )(x, t) = exp(-(α + α )t)r α+α -2 ∇ bdr ϕ • ∇ bdr ϕ + αα (1 + r 2 )ϕϕ (x),
and thus by integration by parts on ∂B 1 we have for every r, t

∂Br×{t} ∇ φ • ∇ φ + ∂ t φ∂ t φ = exp(-(α + α )t)r α+α -2 (λ + αα (1 + r 2 )) ∂B 1
ϕϕ .

From this we learn that the L 2 (∂B 1 )-orthogonality of the eigenspaces transmits to the extensions; hence we may indeed restrict to an eigenfunction. Integrating in (r, t) the last identity for ϕ = ϕ we obtain

B 1 ∞ 0 |∇ φ| 2 + (∂ t φ) 2 = 4α + λ + λ 2α 4α 2 -1 ∂B 1 ϕ 2 .
Since (2.7) implies that λ α 2 + 1, we get (2.5).

We now turn to (2.6), which we deduce from the (mean-value zero) Sobolev estimate

B 1 ∞ 0 | φ| q 1 q B 1 ∞ 0 |∇ φ| 2 + (∂ t φ) 2 1 2
where q = 2(d + 1)

d -1 , (2.8)
which holds because the analogue estimate holds on every

B 1 × (n -1, n), n ∈ N, since B 1 ×(n-1,n) φ = 0. From | d dt B 1 | φ| p | ≤ p B 1 | φ| p-1
|∂ t φ| and Cauchy-Schwarz's inequality (using 2(p -1) = q) we have

∞ 0 d dt B 1 | φ| p ≤ p B 1 ∞ 0 | φ| q 1 2 B 1 ∞ 0 (∂ t φ) 2 1 2
.

Therefore, using that B 1 | φ| p → 0 as t → ∞, we get

B 1 |ϕ| p 1 p B 1 ∞ 0 | φ| q 1 2p B 1 ∞ 0 (∂ t φ) 2 1 2p
(2.8)

B 1 ∞ 0 |∇ φ| 2 + (∂ t φ) 2 q 4p + 1 2p = B 1 ∞ 0 |∇ φ| 2 + (∂ t φ) 2 1 2
, that is (2.6).

For the choice of a good radius (see Lemma 3.5 below) we will need the following not totally standard elliptic estimate.

Lemma 2.2. For every c > 0 and every (z, f ), with 0 ≤ z ≤ c and Spt z ⊂ B 1 \B 1 2 , the unique solution ϕ (up to additive constants) of

   ∆ϕ = z -1 |B 1 | B 1 z -∂B 1 f in B 1 ∂ϕ ∂ν = f on ∂B 1 satisfies B 1 |∇ϕ| 2 ∂B 1 f 2 + c B 1 (1 -|x|)z.
Proof. Without loss of generality, we may assume that B 1 ϕ = 0 and that ∂B 1 f 2 + B 1 (1 -|x|)z < ∞, otherwise there is nothing to prove. Moreover, by scaling we may assume that c = 1. Using integration by parts, the trace inequality for Sobolev functions together with the Poincaré inequality for functions of mean zero,

B 1 |∇ϕ| 2 = ∂B 1 ϕf - B 1 ϕ∆ϕ ≤ ∂B 1 ϕ 2 1 2 ∂B 1 f 2 1 2 + B 1 zϕ B 1 |∇ϕ| 2 1 2 ∂B 1 f 2 1 2 + B 1 zϕ .
Using Young's inequality, it is now enough to prove that

B 1 zϕ B 1 |∇ϕ| 2 1 2 B 1 (1 -|x|)z 1 2
.

(2.9)

For ω ∈ ∂B 1 , letting

z(ω) := 1 0 z(rω)r d-1 dr, we claim that ∂B 1 z 2 B 1 (1 -|x|)z. (2.10)
Indeed, momentarily fixing ω ∈ ∂B 1 and setting ψ(r) := r d-1 z(rω) for r ∈ [0, 1], we have 0 ≤ ψ ≤ 1 and

1 0 ψ = z(ω),
so that for almost every ω ∈ ∂B 1 ,

1 0 (1 -r)r d-1 z(rω) ≥ min 0≤ ψ≤1 ψ=z(ω) 1 0 (1 -r) ψ(r) z 2 (ω),
where the last inequality follows since the minimizer of min

0≤ ψ≤1 ψ=z(ω) 1 0 (1 -r) ψ(r)
is given by the characteristic function of (1 -z(ω), 1). Using that by hypothesis

Spt z ⊂ B 1 \B 1 2
, we can thus write

B 1 zϕ ≤ ∂B 1 ϕz + ∂B 1 1 1 2 (ϕ(rω) -ϕ(ω))z(rω)r d-1 drdω ∂B 1 ϕ 2 1 2 ∂B 1 z 2 1 2 + ∂B 1 1 1 2 |ϕ(rω) -ϕ(ω)|z(rω)r d-1 drdω (2.10) B 1 |∇ϕ| 2 1 2 B 1 (1 -|x|)z 1 2 + ∂B 1 1 1 2 |ϕ(rω) -ϕ(ω)|z(rω)r d-1 drdω,
where in the last line we used once more that

∂B 1 ϕ 2 B 1 |∇ϕ| 2 . Since for r ∈ ( 1 2 , 1) |ϕ(rω) -ϕ(ω)| ≤ 1 1 2 |∂ r ϕ(sω)| 2 ds 1 2 1 0 |∂ r ϕ(sω)| 2 s d-1 ds 1 2
, estimate (2.9) follows from

∂B 1 1 1 2 |ϕ(rω) -ϕ(ω)|z(rω)r d-1 drdω ∂B 1 1 1 2 1 0 |∂ r ϕ(sω)| 2 s d-1 ds 1 2 z(rω)r d-1 drdω = ∂B 1 1 0 |∂ r ϕ(sω)| 2 s d-1 ds 1 2 z(ω)dω ≤ ∂B 1 1 0 |∂ r ϕ(sω)| 2 s d-1 ds 1 2 ∂B 1 z 2 1 2
(2.10)

B 1 |∇ϕ| 2 1 2 B 1 (1 -|x|)z 1 2
.

The optimal transport problem

In order to set up notation, let us quickly recall some well known facts about optimal transportation. Much more can be found for instance in the books [START_REF] Villani | Topics in optimal transportation[END_REF][START_REF]Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF] to name just a few. We will always work here with transportation between (multiples) of characteristic functions and arbitrary measures so that we restrict our presentation to this setting. For a measure Π on R d × R d we denote its marginals by Π 1 and Π 2 , i.e. Π 1

(A) = Π(A × R d ), Π 2 (A) = Π(R d × A).
For a given bounded set Ω, a positive constant Λ and a measure µ with compact support and such that µ(R d ) = Λ|Ω| any measure Π on R d × R d with marginals Π 1 = Λχ Ω and Π 2 = µ is called a transport plan or coupling between Λχ Ω and µ. We define the Wasserstein distance between Λχ Ω and µ as

W 2 2 (Λχ Ω , µ) := min Π 1 =Λχ Ω , Π 2 =µ R d ×R d |x -y| 2 dΠ = min T #Λχ Ω =µ Ω |T -x| 2 Λdx. (2.11)
By Brenier's Theorem [31, Theorem 2.12], the minimizer of the right-hand side of (2.11) exists, is called optimal transport map, and is uniquely defined a.e. on Ω as the gradient of a convex map ψ. Conversely, for every convex map ψ, every Λ > 0 and every bounded set Ω, ∇ψ is the solution of (2.11) for µ := ∇ψ#Λχ Ω (see [START_REF] Villani | Topics in optimal transportation[END_REF]Theorem 2.12] again).

By [START_REF] Villani | Topics in optimal transportation[END_REF]Theorem 5.5], we have the time-dependent representation of optimal transport

W 2 2 (Λχ Ω , µ) = min X Ω 1 0 | Ẋ(x, t)| 2 Λdx : X(x, 0) = x, X(•, 1)#Λχ Ω = µ
(2.12) and if T is the solution of (2.11), the minimizer of (2.12) is given by X(x, t) = T t (x) := (1 -t)x + tT (x) which are straight lines. We will often drop the argument x and write X(t) = X(x, t).

As in [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF], a central point for our analysis is the Eulerian version of optimal transportation, also known as the Benamou-Brenier formulation (see for instance [START_REF] Villani | Topics in optimal transportation[END_REF]Theorem 8.1] or [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]Chapter 8]). It states that

W 2 2 (Λχ Ω , µ) = min (ρ,j) R d 1 0 1 ρ |j| 2 : ∂ t ρ + ∇ • j = 0, ρ(0) = Λχ Ω and ρ(1) = µ ,
(2.13) where the continuity equation and the boundary data are understood in the distributional sense, i.e. for every

ζ ∈ C 1 c (R d × [0, 1]), R d 1 0 ∂ t ζρ + ∇ζ • j = R d ζ(x, 1)dµ - R d ζ(x, 0)Λχ Ω (x)dx, (2.14)
and where

R d 1 0 1 ρ |j| 2 = R d 1 0 |v| 2 dρ
if j ρ with dj dρ = v and infinity otherwise (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]Theorem 2.34]). Let us point out that in particular, the admissible measures for (2.13) are allowed to contain singular parts with respect to the Lebesgue measure. We also note that if K ⊂ R d is a compact set and if (ρ, j) are measures on K × [0, 1], then we have the equality (see [START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF]Proposition 5.18])

K 1 0 1 2ρ |j| 2 = sup ξ∈C 0 (K×[0,1],R d ) K 1 0 ξ • j - |ξ| 2 2 ρ. (2.15)
If we let for t ∈ [0, 1],

ρ t := T t #Λχ Ω and j t := T t #[(T -Id)Λχ Ω ], (2.16)
then j is absolutely continuous with respect to ρ and (ρ, j) is the minimizer of (2.13) (see [START_REF] Villani | Topics in optimal transportation[END_REF]Theorem 8.1] or [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]Chapter 8] and [START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF]Proposition 5.32] for the uniqueness).

Notice that by [31, Proposition 5.9], for t ∈ [0, 1), ρ t and j t are absolutely continuous with respect to the Lebesgue measure and 1 ρ |j| 2 agrees with its pointwise definition. By Alexandrov's Theorem [START_REF]Grundlehren der Mathematischen Wissenschaften[END_REF]Theorem 14.25], T is differentiable a.e. and by [START_REF] Villani | Topics in optimal transportation[END_REF]Theorem 4.8] for t ∈ [0, 1), the Jacobian equation

ρ t (T t (x)) det ∇T t (x) = Λχ Ω (x)
(2.17) holds a.e. We say that a map T is monotone if for a.e. (x, y), (T (x)-T (y))•(x-y) ≥ 0.

In particular, since the optimal transport map for (2.11) is the gradient of a convex function, it is a monotone map. Let us recall the following L ∞ bound for monotone maps proven in iv [23, Lemma 4.1].

Lemma 2.3. Let T be a monotone map. Let R > 0 be such that 1 R d+2 B 2R |T -x| 2 1. Then sup B 7R 4 |T -x| + sup B 3R 2 dist(y, T -1 (y)) R 1 R d+2 B 2R |T -x| 2 1 d+2
.

(2.18)

Moreover, letting for t ∈ [0, 1], T t = (1 -t)Id + tT , T -1 t (B 3R 2 ) ⊂ B 2R . (2.19)
Let us show how together with displacement convexity this implies an L ∞ bound for (ρ t , j t ).

Lemma 2.4. Let Λ = 1 and assume that B 2 ⊂ Ω and

E := B 2 |T -x| 2
1, where T is the optimal transport map for (2.11). Then, for a.e.

0 < t < 1, if (ρ t , j t ) is given by (2.16), sup B 3 2 ρ t ≤ 1 (1 -t) d
and sup

B 3 2 |j t | E 1 d+2 1 (1 -t) d .
(2.20)

Moreover, B 3 2 1 ρ t |j t | 2 ≤ E and B 3 2 ρ t 1. (2.21)
Proof. We start by proving (2.20). The estimate on ρ is a direct consequence of displacement convexity: By concavity of det

1 d on positive symmetric matrices, det 1 d (∇T t ) ≥ (1 -t)det 1 d Id + tdet 1 d ∇T ≥ (1 -t) (2.

22)

iv This bound was proven there for optimal transport maps but a quick inspection of the proof shows that only monotony is used. Similarly, it is stated there for R = 1 but a simple rescaling gives the present version of the estimate. and thus by (2.17),

ρ t (x) = 1 det ∇T t (T -1 t (x)) ≤ 1 (1 -t) d .
(2.23)

We turn to the estimate on j. For ξ ∈ C c (B 3 2 , R d ), and t ∈ (0, 1)

B 3 2 ξ • j t (2.16) = T -1 t (B 3 2 ) ξ(T t ) • (T -x) ≤ sup T -1 t (B 3
2

)
|T -x|

T -1 t (B 3 2 ) |ξ(T t )| (2.19)&(2.18) E 1 d+2 B 3 2 |ξ|ρ t (2.23) E 1 d+2 1 (1 -t) d B 3 2 |ξ|.

Estimate (2.21) then follows from (2.19):

B 3 2 1 ρ t |j t | 2 (2.16) = T -1 t (B 3 2 ) |T -x| 2 ≤ B 2 |T -x| 2 = E and B 3 2 ρ t = |T -1 t (B 3
2

)| 1.
Before closing this section, let us spend a few words about optimal transportation on the torus and on the sphere since both problems will appear later on. Let us start with the periodic setting. For L > 0, we let

Q L := [-L 2 , L 2
) d be the centered cube of side length L. We say that a measure µ on R d is Q L -periodic v if for every z ∈ (LZ) d , and every measurable set A, µ(A + z) = µ(A) so that we may identify measures on the flat torus of size

L > 0 with Q L -periodic measures on R d . If µ is a Q L -periodic measure and Λ := µ(Q L )
L d , then we can define

W 2 2,per (Λ, µ) := min T #Λ=µ Q L |T -x| 2 per Λdx, (2.24)
where | • | per denotes the distance on T L i.e. |x -y| per = min z∈(LZ) d |x -y + z|. By [31, Theorem 2.47] (see also [START_REF] Cordero-Erausquin | Sur le transport de mesures périodiques[END_REF][START_REF] Ambrosio | Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case[END_REF] for a simpler proof), there exists a unique (up to additive constants) convex function ψ on R d such that if T is the unique solution of (2.24), then for

x ∈ Q L , T (x) = ∇ψ(x) and for (x, z) ∈ R d × (LZ) d , ∇ψ(x + z) = ∇ψ(x) + z. (2.

25)

v when it is clear from the context we will simply call them periodic measures Remark 2.5. We will often identify T and ∇ψ. Let us point out that although T is defined only Lebesgue a.e., it will be sometimes useful to consider a pointwise defined map which we then take to be an arbitrary but fixed measurable selection of the subgradient ∂ψ of ψ.

Notice that of course for every

Q L -periodic measure µ, W 2 2,per (Λ, µ) ≤ W 2 2 (Λχ Q L , µ Q L ) . Finally for f 1 and f 2 two non-negative densities on ∂B 1 with ∂B 1 f 1 = ∂B 1 f 2 , we define W 2 ∂B 1 (f 1 , f 2 ) := min T #f 1 =f 2 ∂B 1 d 2 ∂B 1 (T (x), x)df 1 , (2.26)
where

d
W 2 ∂B 1 (f 1 , f 2 ) is comparable to the Wasserstein distance in R d between f 1 H d-1 ∂B 1 and f 2 H d-1 ∂B 1 , that is W 2 2 (f 1 , f 2 ) ≤ W 2 ∂B 1 (f 1 , f 2 ) W 2 2 (f 1 , f 2 ). (2.27)
As for (2.12), we have the time-dependent formulation [31, Theorem 5.6]

W 2 ∂B 1 (f 1 , f 2 ) = min X ∂B 1 1 0 | Ẋ(x, t)| 2 df 1 : X(x, 0) = x, X(•, 1)#f 1 = f 2 ,
(2.28) and the Benamou-Brenier formulation [START_REF]Grundlehren der Mathematischen Wissenschaften[END_REF]Theorem 13.8]

W 2 ∂B 1 (f 1 , f 2 ) = min (ρ,j) ∂B 1 1 0 1 ρ |j| 2 : ∂ t ρ + ∇ bdr • j = 0, ρ(0) = f 1 and ρ(1) = f 2 ,
(2.29) where we stress that j is tangent to ∂B 1 . Even though it is more delicate than in the Euclidean case, the analog of (2.17) also holds in this case. Indeed, by [START_REF]Grundlehren der Mathematischen Wissenschaften[END_REF]Theorem 13.8] and [START_REF]Grundlehren der Mathematischen Wissenschaften[END_REF]Theorem 11.1] (see also [START_REF] Cordero-Erausquin | A Riemannian interpolation inequality à la Borell, Brascamp and Lieb[END_REF]Theorem 4.2] and [15, Lemma 6.1] for more details) if vi T (x) = exp x (∇ψ(x)) is the minimizer of (2.26), letting for t ∈ [0, 1], T t (x) := exp x (t∇ψ(x)) and then ρ t := T t #f 1 , we have that ρ t is a minimizer of (2.29) and for every t ∈ [0, 1] and a.e. x ∈ ∂B 1 , the Jacobian equation

ρ t (T t (x))J t (x) = f 1 (x) (2.30)
holds, with J t the Jacobian determinant (see for instance [START_REF] Cordero-Erausquin | A Riemannian interpolation inequality à la Borell, Brascamp and Lieb[END_REF]Lemma 6.1] for its definition). The main point for us is that similarly to (2.22), it satisfies by [START_REF]Grundlehren der Mathematischen Wissenschaften[END_REF]Theorem 14.20] (see also [START_REF] Cordero-Erausquin | A Riemannian interpolation inequality à la Borell, Brascamp and Lieb[END_REF]Lemma 6.1] for a statement closer to ours)

J 1 d-1 t ≥ (1 -t) + tJ 1 d-1 1 , (2.31)
where we used the fact that since the sphere has positive Ricci curvature, its volume distortion coefficients are larger than one. We finally prove a simple lemma which will be useful in the construction of competitors in Proposition 3.7 below.

vi here exp x denotes the exponential map on ∂B 1 Lemma 2.6. Let f 1 and f 2 be two non-negative densities on ∂B 1 of equal mass. For every 0 ≤ a < b ≤ 1 and 0 ≤ c ≤ d ≤ 1 with a < c and b < d, there exists (ρ, j)

supported on ∂B 1 × [a, d] such that for every ζ ∈ C 1 (∂B 1 × [0, 1]) ∂B 1 1 0 ∂ t ζρ + ∇ bdr ζ • j = 1 d -c ∂B 1 d c ζdf 2 - 1 b -a ∂B 1 b a ζdf 1 (2.32)
and

∂B 1 1 0 1 ρ |j| 2 W 2 ∂B 1 (f 1 , f 2 ) (d -b) -(c -a) log d -b c -a , (2.33)
with the understanding that for c = d,

1 d -c ∂B 1 d c ζdf 2 = ∂B 1 ζ(•, c)df 2 and for d -b = c -a, 1 (d -b) -(c -a) log d -b c -a = 1 c -a . (2.34) Proof. For x ∈ ∂B 1 and t ∈ [0, 1], let X(x, t) ∈ ∂B 1 be the minimizer of (2.28), i.e. for ζ ∈ C 0 (∂B 1 ), ∂B 1 ζ(X(x, 1))df 1 = ∂B 1 ζdf 2 and W 2 ∂B 1 (f 1 , f 2 ) = ∂B 1 1 0 | Ẋ| 2 df 1 . ( 2
t ∈ [s, ψ(s)] X s (x, t) := X x, t -s ψ(s) -s , so that X s (x, s) = x and X s (x, ψ(s)) = X(x, 1). ( 2

.36)

We then let ρ be the non-negative measure on

∂B 1 × [0, 1] defined for ζ ∈ C 0 (∂B 1 × [0, 1]) by ∂B 1 1 0 ζdρ := 1 b -a ∂B 1 b a ψ(s) s ζ(X s (x, t), t)dtdsdf 1 and j the R d -valued measure defined for ξ ∈ C 0 (∂B 1 × [0, 1], R d ) by ∂B 1 1 0 ξ • dj := 1 b -a ∂B 1 b a ψ(s) s ξ(X s (x, t), t) • Ẋs (x, t)dtdsdf 1 .
It is readily seen that with this definition j ρ. Let us establish (2.32). For

ζ ∈ C 1 (∂B 1 × [0, 1]) a test function, ∂B 1 1 0 ∂ t ζρ + ∇ bdr ζ • j = 1 b -a ∂B 1 b a ψ(s) s ∂ t ζ(X s , t) + ∇ζ(X s , t) • Ẋs dtdsdf 1 = 1 b -a ∂B 1 b a ζ(X s (x, ψ(s)), ψ(s)) -ζ(X s (x, s), s)dsdf 1 (2.36) = 1 b -a ∂B 1 b a ζ(X(x, 1), ψ(s)) -ζ(x, s)dsdf 1 (2.35) = ∂B 1 b a 1 b -a ζ(x, ψ(s))df 2 ds - ∂B 1 b a 1 b -a ζ(x, s)dsdf 1 = ∂B 1 d c 1 d -c ζdf 2 dŝ - ∂B 1 b a 1 b -a ζdsdf 1 ,
where we made the change of variables ŝ = ψ(s) in the last equality. We now turn to (2.33). Using (2.15) and the definition of (ρ, j),

∂B 1 1 0 1 2ρ |j| 2 = sup ξ∈C 0 (∂B 1 ×[0,1],R d ) ∂B 1 1 0 ξ • j - |ξ| 2 2 ρ = sup ξ∈C 0 (∂B 1 ×[0,1],R d ) 1 b -a ∂B 1 b a ψ(s) s ξ(X s (x, t), t) • Ẋs (x, t) - |ξ(X s (x, t), t)| 2 2 dtdsdf 1 ≤ 1 b -a ∂B 1 b a ψ(s) s sup ξ∈R d ξ • Ẋs (x, t) - 1 2 |ξ| 2 dtdsdf 1 = 1 b -a ∂B 1 b a ψ(s) s 1 2 | Ẋs (x, t)| 2 dtdsdf 1 (2.36) = 1 b -a ∂B 1 b a ψ(s) s 1 (ψ(s) -s) 2 Ẋ x, t -s ψ(s) -s 2 dtdsdf 1 t= t-s ψ(s)-s = 1 b -a ∂B 1 b a 1 0 1 ψ(s) -s | Ẋ(x, t)| 2 d tdsdf 1 = W 2 ∂B 1 (f 1 , f 2 ) b -a b a ds ψ(s) -s . Let us finally estimate 1 b-a b a 1 ψ(s)-s . By definition of ψ, 1 b -a b a 1 ψ(s) -s = 1 b -a b a 1 c -s + d-c b-a (s -a) s=(1-t)a+tb = 1 0 1 (1 -t)(c -a) + t(d -b) = 1 (d -b) -(c -a) log d -b c -a ,
where we take as convention (2.34) if d -b = c -a.

2.4 Poisson process, optimal matching, and concentration

The Euclidean problem

Let Γ be the set of all locally finite counting measures on

R d Γ = µ : µ = i δ y i , y i ∈ R d , µ(K) < ∞, ∀ K compact ,
where Γ is equipped with the σ-field F generated by the mappings µ → µ(A) for Borel sets A ⊂ R d . We say that (µ, P) We let θ : R d × Γ → Γ be the shift operator, that is for (z, µ) ∈ R d × Γ and a Borel set

(
A ⊂ R d , θ z µ(A) := µ(A + z), (2.38)
which we write shortly as θ z µ(•) = µ(• + z). Moreover, we note that P is stationary in the sense that it is invariant under the action of θ, i.e. P • θ = P.

For > 0, we recall that Q = [-2 , 2 ) d .
The optimal matching problem consists in understanding the behavior as

→ ∞ of vii W 2 2 µ Q , µ(Q ) d χ Q (2.39)
together with the properties of the corresponding optimal transport maps. We will use as shorthand notation µ := µ Q and when it is clear from the context we will identify a constant Λ > 0 with the measure Λχ Q . As explained in the introduction,

it is known that E W 2 2 µ , µ(Q ) d ∼ d for d ≥ 3 whereas in dimension
two there is an extra logarithmic factor in the scaling of (2.39) resulting from larger shifts of the mass. In particular, the transport cost per unit volume diverges logarithmically. Our aim is to investigate the behavior of the associated transport maps. Although our techniques also allow us to say something for the case of d ≥ 3 we focus on the case d = 2 where an additional renormalization of the maps is needed in order to be able to pass to the limit. Hence, we assume from now on d = 2. The main stochastic ingredient is a control at every scale 1 < ∞ of (2.39). This estimate is a quite direct consequence of a result of Ambrosio, Stra and Trevisan [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF] which we now recall. Since the results of [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF] are not stated for the Poisson point process but rather for a deterministic number n → ∞ of uniform iid random variables X i on a given domain Q , we need to introduce some more notation. For a given n ∈ N, we let the probability P n on Γ be defined as

P n [F ] := P [F ∩ {µ (Q ) = n}] P [µ (Q ) = n] ,
and let E n be the associated expectation. Note that by (2.37), we have

p n := P [µ (Q ) = n] = exp(-2 ) 2n n! . (2.40)
Equipped with this probability measure, µ can be identified with n uniformly iid random variables X i on Q . A simple rescaling shows that

1 2 log n E n W 2 2 µ , µ(Q ) 2 is independent of and [6, Theorem 1.1] states that, lim n→∞ 1 2 log n E n W 2 2 µ , µ(Q ) 2 = 1 4π . (2.41)
Arguing as in [6, Remark 4.7] and using the fact that the uniform measure on [0, 1] satisfies a log-Sobolev inequality to replace exponential bounds by Gaussian bounds, this gives that there exists c 0 > 0 such that for every M ≥ 1 (since we pass from a deviation to a tail estimate) and n large enough uniformly viii in ,

P n 1 2 log n W 2 2 µ , µ(Q ) 2 ≥ M ≤ exp(-c 0 M log n). ( 2
lim →∞ 1 2 log E W 2 2 µ , µ(Q ) 2 = 1 2π (2.43)
viii Notice that the left-hand side of (2.42) actually does not depend on .

and there exists a universal constant c independent of and M such that for ≥ 2, M ≥ 1,

P 1 2 log W 2 2 µ , µ(Q ) 2 ≥ M ≤ exp(-cM log ) . (2.44)
Proof. We first start by noting that by Cramér-Chernoff's bounds for the Poisson distribution with intensity 2 (see [START_REF] Boucheron | Concentration inequalities[END_REF]), there exists a constant c such that

P µ(Q ) 2 / ∈ 1 2 , 2 ≤ exp(-c 2 ), (2.45)
and for M 1

P µ(Q ) 2 ≥ M ≤ exp(-c 2 M ). ( 2

.46)

Let us now prove (2.43). Recall p n from (2.40). By definition of E n we have

1 2 log E W 2 2 µ , µ(Q ) 2 = 1 2 log n / ∈[ 2 /2,2 2 ] p n E n W 2 2 µ , µ(Q ) 2 + 1 2 log n∈[ 2 /2,2 2 ] p n E n W 2 2 µ , µ(Q ) 2 . (2.47) Using the crude transport estimate W 2 2 µ , µ(Q ) 2 ≤ 2 µ(Q ) together with (2.40
) we can estimate the first term as

1 2 log n / ∈[ 2 /2,2 2 ] p n E n W 2 2 µ , µ(Q ) 2 ≤ 1 log n / ∈[ 2 /2,2 2 ] np n = 1 log n / ∈[ 2 /2,2 2 ] exp(-2 ) 2n (n -1)! 2 log P µ(Q ) 2 / ∈ [ 1 2 , 2]
(2.45) 2 exp(-c 2 ) log , which goes to zero as → ∞. The second term in (2.47) can be rewritten as

1 2 log n∈[ 2 /2,2 2 ] p n E n W 2 2 µ , µ(Q ) 2 = n∈[ 2 /2,2 2 ] p n log n log 1 2 log n E n W 2 2 µ , µ(Q ) 2 .
Since by (2.41),

1 2 log n E n W 2 2 µ , µ(Q ) 2 converges uniformly in to 1 4π , it is enough to show that lim →∞ n∈[ 2 /2,2 2 ] p n log n log = 2.
This is a simple consequence of (2.45) and the fact that

2 log -log 2 log n∈[ 2 /2,2 2 ] p n ≤ n∈[ 2 /2,2 2 ] p n log n log ≤ 2 log + log 2 log n∈[ 2 /2,2 2 ]
p n .

We now turn to (2.44). For 1 ≤ M 2 log , by definition of

P n P 1 2 log W 2 2 µ , µ(Q ) 2 ≥ M = n / ∈[ 2 /2,2 2 ] p n P n 1 2 log W 2 2 µ , µ(Q ) 2 ≥ M + n∈[ 2 /2,2 2 ] p n P n 1 2 log W 2 2 µ , µ(Q ) 2 ≥ M ≤ P µ(Q ) 2 / ∈ 1 2 , 2 + n∈[ 2 /2,2 2 ] p n P n 1 2 log W 2 2 µ , µ(Q ) 2 ≥ M (2.45) ≤ exp(-c 2 ) + n∈[ 2 /2,2 2 ] p n P n 1 2 log n W 2 2 µ , µ(Q ) 2 ≥ log 2 log + 2 M (2.42) ≤ exp(-c 2 ) + n∈[ 2 /2,2 2 ] p n exp(- c 0 4 M log n) ≤ exp(-c 2 ) + exp(- c 0 4 M log ) ≤ exp(-c 1 M log ),
while for M 2 log , using once more the estimate < ∞ such that for every dyadic with 2r * ≤ ,

W 2 2 µ , µ(Q ) 2 ≤ 2 µ(Q ) together with (2.46), we obtain P 1 2 log W 2 2 µ , µ(Q ) 2 ≥ M ≤ P µ(Q ) 2 ≥ M log 2 ≤ exp(-cM log ).
1 4 W 2 2 µ , µ(Q ) 2 ≤ log r * r * 2 .
(2.48)

ix notice that we keep implicit here the dependence on µ

Proof. We first prove that there exist a constant c > 0 and a random variable Θ with E [exp(cΘ)] < ∞ such that for all dyadic with 1,

1 2 log W 2 2 µ , µ(Q ) 2 ≤ Θ. (2.49) For k ≥ 1, let k := 2 k and put Θ k := 1 2 k log k W 2 2 µ k , µ(Q k ) 2 k χ Q k , Θ := sup k≥1 Θ k . (2.50)
We claim that the exponential moments of Θ k given by Proposition 2.7 translate into exponential moments for Θ. Indeed fix 1 c > 0. Then, we estimate

E[exp(cΘ)] ≤ exp(c) + M ∈N P[Θ ≥ M ] exp(c(M + 1)) ≤ exp(c) + M ∈N exp(c(M + 1)) k≥1 P[Θ k ≥ M ] (2.44) ≤ exp(c) + exp(c) k≥1 M ∈N exp(-M (c log k -c)) k =2 k & c 1 exp(c) + exp(c) k≥1 exp(-ck) < ∞.
Hence, Θ has exponential moments and (2.49) is satisfied for every large enough dyadic .

Define r * ≥ 1 via the equation We remark that r * inherits all stationarity properties of the Poisson process as a measurable function of the Poisson process (similarly for r * ,L in Theorem 2.10). We will not explicitly mention this in the sequel.

r 2 * log (2r * ) = Θ log 2 , ( 2

The periodic problem

Since our aim is to construct a coupling on R 2 between Lebesgue and Poisson which keeps some of the shift-invariance properties of the Poisson point process, it is more convenient to work for finite-size cubes also with a shift-invariant point process. For L > 1 let us introduce the Q L -periodic Poisson point process (which can be identified with the Poisson point process on the flat torus of size L). For µ ∈ Γ, we let µ per L be the Q L -periodic extension of µ Q L and then

P L := µ per L #P. (2.52)
We denote by E L the expectation with respect to P L . We then call (µ, P L ) (or simply µ when it is clear from the context) a Q L -periodic Poisson point process (or Poisson point process on the torus). Since P is invariant under θ, so is P L . Notice that for ≤ L the restriction of a Q L -periodic Poisson point process to Q is a Poisson point process on Q in the sense of Section 2.4.1. For µ ∈ Γ and L > 1, our main focus will be to understand at every scale 1 ≤ L the structure of the optimal transport map T µ,L on the torus between µ(Q L ) L 2 and µ, i.e. T µ,L is the unique minimizer of (2.24). We will often identify T µ,L with ∇ψ µ,L where ψ µ,L is the convex potential given in (2.25). When it is clear from the context, we drop the dependence of T µ,L and ψ µ,L on either µ, L or both. Uniqueness of the optimal transport map solving (2.24) implies that T µ is covariant in the sense that

T µ (x + z) = T θzµ (x) + z x, z ∈ R 2 .
(2.53)

For future reference, let us prove a corresponding stationarity property of the potentials.

Lemma 2.9. Let µ be a Q L -periodic Poisson point process and let T µ = ∇ψ µ be the optimal transport map between µ(Q L )

L 2
and µ on the torus. Then, for z ∈ R 2 ,

ψ θzµ (x) = ψ θzµ (0) -ψ µ (z) + ψ µ (x + z) -z • x ∀x ∈ R 2 (2.54)
and if ψ * is the convex conjugate of ψ,

ψ * θzµ (y) = ψ * θzµ (0) -ψ * µ (z) + ψ * µ (y + z) -z • y ∀y ∈ R 2 . (2.55) As a consequence, if we let for h ∈ R 2 , D 2 h ψ(x) := ψ(x + h) + ψ(x -h) -2ψ(x), D 2 h ψ θzµ (x) = D 2 h ψ µ (x + z) and D 2 h ψ * θzµ (y) = D 2 h ψ * µ (y + z).
(2.56)

Proof. Equation (2.54) is a direct consequence of (2.53) so that we just need to prove that it implies (2.55). By definition,

ψ * θzµ (y) = sup x [x • y -ψ θzµ (x)] (2.54) = sup x [x • y -ψ θzµ (0) + ψ µ (z) -ψ µ (x + z) + z • x] = -ψ θzµ (0) + ψ µ (z) -y • z -|z| 2 + sup x [(x + z) • (y + z) -ψ µ (x + z)] = -ψ θzµ (0) + ψ µ (z) -y • z -|z| 2 + ψ * µ (y + z).
Applying this to y = 0, we obtain

ψ * θzµ (0) -ψ * µ (z) = -ψ θzµ (0) + ψ µ (z) -|z| 2 ,
so that (2.55) follows.

Let us finally translate the result of Theorem 2.8 into the periodic setting. In particular, the following result contains Theorem 1.3.

Theorem 2.10. There exists a universal constant c such that for L = 2 k , k ∈ N, dyadic and µ a Q L -periodic Poisson point process, there exists a family of random variables r * ,L ≥ 1 with sup L E L exp

cr 2 * ,L log(2r * ,L ) < ∞ such that if 2r * ,L ≤ L we have µ(Q L ) L 2 ∈ 1 2 , 2 , Spt µ ∩ B r * ,L = ∅ (2.57)
and for every dyadic with 2r * ,L ≤ ≤ L,

1 4 W 2 2,per µ , µ(Q ) 2 ≤ 1 4 W 2 2 µ , µ(Q ) 2 ≤ log r * ,L r * ,L 2 .
(2.58)

Proof. Let first r * ,L = r * ,L (µ) be defined by r * ,L := inf r :

1 4 W 2 2 µ , µ(Q ) 2 ≤ log r r 2
, for all dyadic with 2r ≤ ≤ L ,

where we take the convention that r * ,L = L/2 if the set on the right-hand side is empty. We then define r * ,L = r * ,L (µ) by

r * ,L := L 2 if µ(Q L ) L 2 / ∈ 1 2 , 2 max( r * ,L , min Spt µ |y|) otherwise. Since P L µ(Q L ) L 2 / ∈ 1 2 , 2 ≤ exp(-c 1 L 2 ) for some universal c 1 > 0 (cf. (2.45)) and for r ≤ L/2, P L (Spt µ ∩ B r = ∅) = exp(-r 2 ), it is enough to prove that r * ,L satisfies sup L E L exp c 2 r 2 * ,L log(2 r * ,L )
< ∞ for some c 2 > 0. Now this follows directly from (2.48) and the fact that for general potentially non-periodic µ ∈ Γ, r * ,L (µ per L ) ≤ r * (µ). The first inequality in (2.58) follows from the fact that W 2 2,per ≤ W 2 2 .

Remark 2.11. To avoid confusion between periodic and Euclidean objects, we would like to stress a few things which will be important in Section 4. While the map T µ is defined through the periodic optimal transport problem, estimate (2.58) gives a bound on the Euclidean Wasserstein distance between the restrictions µ and the corresponding multiples of the Lebesgue measure on Q . The reason for the two different transport problems is that on the one hand we want to work with a map which has good stationarity properties and on the other hand, for the iteration argument below, it is more natural to have bounds on the Euclidean Wasserstein distances between µ and µ(Q ) 2 χ Q . Presently, conditions (2.57) come out of the blue but they will be very useful in Section 4. Similarly, the first inequality in (2.58) will allow us to initialize the iteration argument in Theorem 1.1.

The main regularity result

In this section we prove our main regularity result which states that for every dimension d ≥ 2, given an optimal transport map T between a bounded set Ω and a measure µ, if at some scale R > 0 both the excess energy

E(µ, T, R) := 1 R d+2 B 2R |T -x| 2 (3.1)
and the local squared Wasserstein distance of µ in O ⊃ B 2R to a constant

D(µ, O, R) := 1 R d+2 W 2 2 µ O, µ(O) |O| χ O (3.2)
are small, then on B R , T is quantitatively close (in terms of E and D) to an harmonic gradient field. This is similar to [23, Proposition 4.6] with the major difference that here the measure µ is arbitrary and can be in particular singular. Let us point out that we allow for O = B 2R only because of the application we have in mind to the optimal matching problem where we have good control on cubes instead of balls (see Theorem 2.8 and the proof of Theorem 1.1). By scaling we will mostly work here with R = 1 and will use the notation E for E(µ, T, 1) and similarly for D. The global strategy is similar to the one used in [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF] and goes through an estimate at the Eulerian level (2.13). However, as opposed to [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF], if (ρ, j) is the minimizer of (2.13) it does not satisfy a global L ∞ bound (see Lemma 2.4). We will thus need to introduce a terminal layer. In [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF], regularization by the heat flow is used as an alternative approach to tackle this issue. For τ > 0, let

ρ := 1-τ 0 ρ t dt.
By (2.20), we have

ρ τ -(d-1) in B 3 2 . (3.3)
As in [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF], we would like to use the flux of j as boundary data for the solution of the Poisson equation we will consider. This requires choosing a good radius R for which j satisfies good estimates on ∂B R . In our setting, this is much more complex than in [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF] and is the purpose of the next section. Let us point out that since the estimates we want to use are on the L 2 scale, we would need that the flux of j through ∂B R is well controlled in L 2 . Since this is in general not the case, we will also need to replace this flux by a more regular one (see Lemma 3.4 below).

f R - f R + X(t R -) X(t R + ) ρ t = 1 t = 1 -τ t = 0 ∂B R 0 ∂B R ρ 1 = µ ρ 0 = 1 Figure 1: The definition of f R ± .

Choice of a good radius

Let X(x, t) = T t (x) be the solution of the time-dependent version of optimal transport (2.12). Let us recall that the corresponding trajectories t → T t (x) are straight segments and that we often drop the dependence in x when it is not necessary to specify it. For R > 0, and a given trajectory X passing through B R we let (see Figure 1)

t R -:= min{t ∈ [0, 1] : X(t) ∈ B R } and t R + := max{t ∈ [0, 1] : X(t) ∈ B R }
be the entrance and exit times. If X(t) does not intersect B R , we set t R -= 1 and t R + = 0. Notice that for t R -< 1 we have X(t R -) ∈ ∂B R and likewise t R + > 0 implies X(t R + ) ∈ ∂B R . We now define the flux f R of j through ∂B R (formally f R = j •ν B R , where ν B R denotes the outward normal to B R ) by its action on functions

ζ ∈ C 0 c (R d × [0, 1]) as R d 1 0 ζdf R := Ω χ 0≤t R -<t R + <1 (X)ζ(X(t R + ), t R + ) - Ω χ 0<t R -<t R + ≤1 (X)ζ(X(t R -), t R -). (3.4) Note that the measure f R is supported on ∂B R × [0, 1]
. The integration in (3.4) is with respect to the Lebesgue measure dx (the integrand depends on x since X and t R ± depend on x). Our first lemma states that f R really acts like boundary values for (ρ, j). Lemma 3.1. Let (ρ, j) be a minimizer of (2.13) and let f R be defined by (3.4).

Then, for every ζ ∈ C 1 c (R d × [0, 1]), B R 1 0 ∂ t ζρ + ∇ζ • j = B R ζ 1 dµ - B R ζ 0 + R d 1 0 ζdf R . (3.5)
As a consequence, (ρ, j) is a local minimizer of (2.13) in the sense that for every ( ρ, j) with Spt ( ρ, j) ⊂ B R × [0, 1] and satisfying (3.5),

B R 1 0 1 ρ |j| 2 ≤ B R 1 0 1 ρ | j| 2 . (3.6)
Proof. Once (3.5) is established, local minimality of (ρ, j) follows from the fact that (ρ, ĵ) defined as ( ρ, j) in B R × [0, 1] and (ρ, j) outside, is admissible for (2.13). We thus only need to prove that (3.5) holds. By (3.4), for every

ζ ∈ C 1 c (R d × [0, 1]), R d 1 0 ζdf R = Ω χ 0≤t R -<t R + <1 (X)ζ(X(t R + ), t R + ) - Ω χ 0<t R -<t R + ≤1 (X)ζ(X(t R -), t R -) = Ω t R + t R - d dt ζ(X, t) + χ t R -=0 (X)ζ(X(t R -), t R -) -χ t R + =1 (X)ζ(X(t R + ), t R + ) = Ω 1 0 χ B R (X) ∂ t ζ(X, t) + ∇ζ(X, t) • Ẋ + Ω χ B R (X(0))ζ(X(0), 0) -χ B R (X(1))ζ(X(1), 1). Since ρ t = T t #χ Ω and j t = T t #(T -x)χ Ω , and since X(x, t) = T t (x) = (1 -t)x + tT (x), for every ζ ∈ C 1 c (R d × [0, 1]), B R 1 0 ∂ t ζdρ = Ω 1 0 χ B R (X)∂ t ζ(X, t) and B R 1 0 ∇ζ • dj = Ω 1 0 χ B R (X)∇ζ(X, t) • Ẋ.
This together with X(0) = Id and X(1) = T concludes the proof.

We now prove that f R satisfies a bound analog to (2.20).

Lemma 3.2. The measure f R is absolutely continuous with respect to the measure

H d-1 ∂B R ⊗ dt and for t ∈ (0, 1), sup ∂B R |f R t | E 1 d+2 1 (1 -t) d . (3.7) Proof. Let ζ ∈ C 1 c (R d × (0, 1
)) be fixed and for 0 < r R, let η r be a smooth radial function such that η r

(x) = 0 if |x| ≤ R-r, η r (x) = 1 for |x| ≥ R and sup |∇η r | r -1 . Testing (3.5) with ζη r , we obtain since f R is supported on ∂B R × [0, 1], R d 1 0 ζdf R B R \B R-r 1 0 |∂ t ζ|ρ + |∇ζ||j| + r -1 |ζ||j| (2.20) B R \B R-r 1 0 (|∂ t ζ| + E 1 d+2 |∇ζ|) 1 (1 -t) d + r -1 E 1 d+2 |ζ| (1 -t) d .
Letting r → 0, we obtain

R d 1 0 ζdf R ∂B R 1 0 E 1 d+2 |ζ| (1 -t) d ,
from which the claim follows.

We define the outgoing and incoming fluxes as (see Figure 1)

R d 1 0 ζdf R + := Ω χ 0≤t R -<t R + <1 (X)ζ(X(t R + ), t R + )
and

R d 1 0 ζdf R -:= Ω χ 0<t R -<t R + ≤1 (X)ζ(X(t R -), t R -), (3.8) so that f R = f R + -f R -.
Now for a given layer size 0 < τ < 1, we define the cumulated fluxes

R d ζdf R + := Ω χ 0≤t R -<t R + <1-τ (X) ζ(X(t R + ))
and

R d ζdf R -:= Ω χ t R -<t R + (X)χ 0<t R -<1-τ (X) ζ(X(t R -)), (3.9)
and then let

f R (x) := 1-τ 0 f R (x, t)dt so that f R = f R + -f R -. (3.10) Lemma 3.3. There holds f R + ⊥ f R -and sup ∂B R f R ± E 1 d+2 τ -(d-1) . (3.11)
Proof. Let us prove that f R + ⊥ f R -. Estimate (3.11) will then follow from (3.7). To this end we consider the space-time points on the cylinder through which particles exit

A := {(y, t) ∈ ∂B R × (0, 1) : ∃x ∈ Ω such that t R -< t R + and (X(t R + ), t R + ) = (y, t)}
and the original positions of particles that enter at the same space-time where another particle exits

B := x ∈ Ω : 0 < t R -< t R + , ∃ x ∈ Ω with t R -< t R + = t R -and X(x, t R -) = X( x, t R + ) .
We claim that B = ∅. Recall (cf. Remark 2.5) that T is given as a measurable selection of the subgradient of a convex function. In particular, we have (T (x) -T ( x)) • (x -x) ≥ 0 for all x, x.

Hence, if x is such that 0 < t R -< t R + and x such that t R -< t R + = t R -we cannot have X(x, t R -) = T t R - (x) = T t R - ( x) = X( x, t R + ) and thus B = ∅. Now since f R + (A c ) (3.8) = Ω χ 0≤t R -<t R + <1 (X)χ A c (X(t R + ), t R + ), we have f R + (A c ) = 0 by definition of A. Similarly, f R -(A) (3.8) = Ω χ 0<t R -<t R + ≤1 (X)χ A (X(t R -), t R -) = |B| = 0, which proves that f R + ⊥ f R -.
We will also need the analog of f R + for the outgoing flux in (1 -τ, 1). Let f R,lay + be the measures defined for

ζ ∈ C 0 c (R d ), by R d ζdf R,lay + := Ω χ t R -<1-τ <t R + <1 (X) ζ(X(t R + )). (3.12)
Notice that we consider only the particles which leave B R after t = 1 -τ but were already inside B R at time 1 -τ . For later use, let also

R d 1 0 ζdf R,lay + := Ω χ t R -<1-τ <t R + <1 (X) ζ(X(t R + ), t R + ). (3.13)
We can now show that there exists a good radius R.

Lemma 3.4. Assume that E + D 1, then there exists R ∈ ( 1 2 , 3 2 ) such that ∂B R 1-τ 0 (f R ) 2 τ -d E (3.14)
and there exist densities ρ R ± and ρ R,lay

+ on ∂B R such that ∂B R (ρ R ± ) 2 E and W 2 ∂B 1 (ρ R ± , f R ± ) E d+3 d+2 , (3.15)
and

∂B R (ρ R,lay + ) 2 τ 2 E + D and W 2 ∂B 1 (ρ R,lay + , f R,lay + ) τ 3 E d+3 d+2 + τ E 1 d+2 D. (3.16) Moreover, sup ∂B R ρ R ± E 1 d+2 . (3.17)
Proof. Let us start by (3.14). For this, given

ζ ∈ C 1 c (R d × (0, 1 -τ )), integrating (3.5) in R ∈ ( 1 2 , 3
2 ), we obtain Letting ω(x) := χ B R (x)dR and using Fubini, we obtain

3 2 1 2 R d 1-τ 0 ζdf R = 3 2 1 2 B R 1-τ 0 ∂ t ζρ + ∇ζ • j. ∂B R ρ R + Π X(t R + ) t = 0 t = 1 -τ t = 1
3 2 1 2 R d 1-τ 0 ζdf R = R d 1-τ 0 ω (∂ t ζρ + ∇ζ • j) = R d 1-τ 0 ζ∇ω • j,
where in the second line we used the fact that (ρ, j) satisfies the continuity equation on R d × (0, 1). By the Cauchy-Schwarz inequality together with the estimate on ρ given by (2.20) and by (2.21), we thus obtain

3 2 1 2 R d 1-τ 0 ζdf R   B 3 2 1-τ 0 ρζ 2   1 2   B 3 2 1-τ 0 1 ρ |j| 2   1 2 τ -d 2 E 1 2   B 3 2 1-τ 0 ζ 2   1 2
, from which we obtain by duality

3 2 1 2 ∂B R 1-τ 0 (f R ) 2 τ -d E. (3.18)
We now turn to (3.15) and (3.17). Notice that by (2.27), it is enough to prove (3.15) for W 2 2 instead of W 2 ∂B 1 . Let ρ R ± be the measures supported on ∂B R such that for ζ ∈ C 0 c (R d ) (see Figure 2)

∂B R ζdρ R + := Ω χ t R -<t R + <1-τ (X)ζ R X(0) |X(0)| and ∂B R ζdρ R -:= Ω χ t R -<t R + (X)χ 0<t R -<1-τ (X)ζ R X(0) |X(0)| . (3.19)
Since the proofs are almost identical for ρ R -, we focus for brevity on ρ R + . We start with the L 2 bound. We introduce the measure ρ R + of all original particles that spend time in B R but exit before 1 -τ , that is

R d ζdρ R + := Ω χ t R -<t R + <1-τ (X)ζ (X(0)) . (3.20)
Let us point out that ρ R + ≤ χ Ω and note that ρ R + is nothing else than the push-forward of ρ R + under the map x → R x |x| so that on the level of densities, we have for

x ∈ ∂B R ρ R + (x) = +∞ 0 r R d-1 ρ R + r x |x| dr.
Notice that because of the L ∞ bound (2.18) on T -x, the integral above can be restricted to (R -CE

1 d+2 , R + CE 1 d+2
) ⊂ (R/2, 4R/3). This directly implies (3.17). Arguing as for (2.10), we obtain

∂B R (ρ R + ) 2 R d |R -|x||dρ R + ,
so that we are just left to prove that

3 2 0 R d |R -|x||dρ R + E. (3.21)
Note that since X are straight lines, ρ R + a.s. we have |R -|X(0)|| ≤ |X(1) -X(0)| so that

3 2 0 R d |R -|x||dρ R + (3.20) = 3 2 0 Ω χ t R -<t R + <1-τ (X)|R -|X(0)|| ≤ 3 2 0 Ω χ t R -<t R + <1-τ (X)|X(1) -X(0)| = Ω |X(1) -X(0)| 3 2 0 χ t R -<t R + <1-τ (X) ≤ Ω χ |X(0)|<2 |X(1) -X(0)| 2 (3.1)

E,

where we used again the L ∞ bound (2.18) for T -x and the fact that for every x such that X(0) = X(1) and every t 1 , t 2 ∈ [0, 1],

H 1 (R : ∃t ∈ [t 1 , t 2 ] with X(t) ∈ ∂B R ) ≤ |X(t 1 ) -X(t 2 )|, (3.22)
to obtain that for given x such that X(1) = X(0),

3 2 0 χ t R -<t R + <1-τ (X) ≤ χ |X(0)|<2 H 1 (R : ∃t ∈ (0, 1 -τ ) with X(t) ∈ ∂B R ) ≤ χ |X(0)|<2 |X(0) -X(1)|
This shows (3.21) and thus

3 2 1 2 ∂B R (ρ R + ) 2 E. (3.23)
Let us now turn to the W 2 2 estimate in (3.15). We consider the coupling Π between f R + (recall (3.9)) and ρ R + defined for ζ ∈ C 0 c (R d × R d ) by (see Figure 2)

R d ×R d ζdΠ := Ω χ t R -<t R + <1-τ (X)ζ X(t R + ), R X(0) |X(0)| .
Using that for |X| R (which Π a.e. is the case by the L ∞ bound on T -x) the radial projection on ∂B R is Lipschitz continuous and thus

|X(t R + ) -R X(0) |X(0)| | |X(t R + ) -X(0)|, we get W 2 2 (ρ R + , f R + ) ≤ Ω χ t + R <1-τ (X) X(t R + ) -R X(0) |X(0)| 2 Ω χ t + R <1-τ (X)|X(t R + ) -X(0)| 2 ≤ Ω χ t + R <1-τ (X)|X(1) -X(0)| 2 ,
where in the last step we used once again that X is a straight line. Integrating in R and arguing as above, we obtain

3 2 0 W 2 2 (ρ R + , f R + ) Ω χ |X(0)|< 7 4 |X(1) -X(0)| 3 sup B 7 4 |T -x| Ω χ |X(0)|<2 |X(1) -X(0)| 2 E d+3 d+2 , (3.24)
where in the last step we have used once more Lemma 2.3 and definition (3.1).

We finally turn to (3.16), the proof of which is similar to the one of (3.15). As above, thanks to (2.27), it is enough to prove (3.16) for W 2 2 instead of W 2 ∂B 1 . The definition of ρ R,lay + is a little bit more complex than the one of ρ R + , since we need to couple the trajectories X to the ones given by the optimal coupling for D

= W 2 2 (µ O, µ(O) |O| χ O ). Consider first the coupling Π 12 (x, y) := χ O (y)(Id × T )#χ Ω (x, y) so that for ζ ∈ C 0 c (Ω × O) Ω×O ζdΠ 12 = Ω χ O (T (x))ζ(x, T (x)).
In particular, the second marginal of Π 3)

R d ζdρ R,lay + := Ω×O×O χ t R -<1-τ <t R + <1 (X)ζ R z |z| dΠ(x, y, z).
We also define the unprojected density

R d ζdρ R,lay + := Ω×O×O χ t R -<1-τ <t R + <1 (X)ζ (z) dΠ(x, y, z).
By the argument used for (2.10), we have

3 2 0 ∂B R ρ R,lay + 2 3 2 0 R d |R -|x||dρ R,lay + = 3 2 0 Ω×O×O χ t R -<1-τ <t R + <1 (X)|R -|z||dΠ ≤ 3 2 0 Ω×O×O χ t R -<1-τ <t R + <1 (X) |X(t R + ) -y| + |y -z| dΠ.
By definition of Π and since the trajectories of X are straight lines, we have

Ω×O×O χ t R -<1-τ <t R + <1 (X)|X(t R + )-y|dΠ = Ω×O×O χ t R -<1-τ <t R + <1 (X)|X(t R + )-X(1)|dΠ ≤ Ω χ t R -<1-τ <t R + <1 (X)|X(1 -τ ) -X(1)|,
which by (3.22) and (2.18) leads to

3 2 0 Ω×O×O χ t R -<1-τ <t R + <1 (X)|X(t R + ) -y|dΠ ≤ Ω χ |X(0)|<2 |X(1 -τ ) -X(1)| 2 .
Since (3.22) and (2.18) also yield we have using that |X(t) -X(1

3 2 0 Ω×O×O χ t R -<1-τ <t R + <1 (X)|y -z|dΠ = Ω×O×O |y -z| 3 2 0 χ t R -<1-τ <t R + <1 (X) dΠ ≤ Ω×O×O |y -z|χ |X(0)|<2 |X(1 -τ ) -X(1)|dΠ Young Ω χ |X(0)|<2 |X(1 -τ ) -X(1)| 2 + Ω×O×O |y -z| 2 dΠ = Ω χ |X(0)|<2 |X(1 -τ ) -X(1)| 2 + O×O |y -z| 2 dΠ 23 , ∂B R O µ O t = 0 t = 1 X(t R + ) Π ρ R,lay + t = 1 -τ µ(O) |O|
)| = (1 -t)|X(0) -X(1)|, 3 2 0 ∂B R ρ R,lay + 2 τ 2 Ω χ |X(0)|<2 |X(0) -X(1)| 2 + O×O |y -z| 2 dΠ 23 ≤ τ 2 E + D. (3.25)
In order to obtain the second estimate in (3.16), we consider the coupling Π between f R,lay + (recall (3.12)) and ρ R,lay

+

given by

R d ×R d ζd Π := Ω×O×O χ t R -<1-τ <t R + <1 (X)ζ X(t R + ), R z |z| dΠ.
This is indeed a coupling between f R,lay + and ρ R,lay

+ since for x ∈ Ω such that t R -< 1 -τ < t R + < 1, we have X(1) ∈ B 2 ⊂ O and therefore, R d ×R d ζ(x)d Π = Ω×O×O χ t R -<1-τ <t R + <1 (X)ζ X(t R + ) dΠ = Ω χ t R -<1-τ <t R + <1 (X)χ O (X(1))ζ X(t R + ) = R d ζdf R,lay + .
Arguing as for (3.24) we have

3 2 0 W 2 2 (f R,lay + , ρ R,lay + ) ≤ 3 2 0 Ω×O×O χ t R -<1-τ <t R + <1 (X) X(t R + ) -R z |z| 2 dΠ ≤ 3 2 0 Ω×O×O χ t R -<1-τ <t R + <1 (X) X(t R + ) -z 2 dΠ 3 2 0 Ω χ t R -<1-τ <t R + <1 (X) X(t R + ) -X(1) 2 + 3 2 0 Ω×O×O χ t R -<1-τ <t R + <1 (X) |y -z| 2 dΠ 3 2 0 Ω χ t R -<1-τ <t R + <1 (X) |X(1 -τ ) -X(1)| 2 + 3 2 0 Ω×O×O χ t R -<1-τ <t R + <1 (X) |y -z| 2 dΠ Ω χ |X(0)|< 7 4 |X(1 -τ ) -X(1)| 3 + Ω×O×O χ |X(0)|< 7 4 |y -z| 2 |X(1 -τ ) -X(1)|dΠ τ 3 sup B 7 4 |T -x| Ω χ |X(0)|<2 |X(1) -X(0)| 2 + τ sup B 7 4 |T -x| O×O |y -z| 2 dΠ 23 τ 3 E d+3 d+2 + τ E 1 d+2 D.
Putting this together with (3.18) Let µ R be the part of µ B R coming from trajectories which were inside B R before the time 1 -τ . That is, for

ζ ∈ C 0 c (R d ) R d ζdµ R := Ω χ t -≤1-τ (X)χ t + =1 (X)ζ(X(1)). (3.26)
We then have

Lemma 3.5. Assume that E + D 1, then there exists R ∈ ( 1 2 , 3
2 ) such that the conclusions of Lemma 3.4 hold and

W 2 2 ρ 1-τ B R , ρ 1-τ (B R ) R | χ B R + W 2 2 µ R , µ R (B R ) |B R | χ B R τ 2 E + D. (3.27)
Proof. We are only going to show that

3 2 1 2 W 2 2 ρ 1-τ B R , ρ 1-τ (B R ) |B R | χ B R τ 2 E + D, (3.28)
since the estimate for

W 2 2 µ R , µ R (B R ) |B R | χ B R is similarly obtained.
For notational simplicity, in this proof we will often drop the R dependence in our notation. Put

Λ := ρ 1-τ (B R ) |B R | and Γ := µ(O)
|O| . We will not distinguish between Λ and the function Λχ B R and similarly for Γ. Since E 1, (2.18) and (2.19) imply that Λ ∼ Γ ∼ 1. Let X be the optimal trajectories for W 2 2 (χ Ω , µ) and Π 23 be the optimal coupling for D = W and let µ ≤ µ be the measure defined by (see Figure 4)

R d ζd µ := Ω χ |X(1-τ )|≤R (X)ζ(X(1)) = Ω×O χ |X(1-τ )|≤R (X)ζ(y)dΠ 12 ,
or in words, µ is the part of µ which originates from

ρ 1-τ B R along X. Notice that µ B R = µ R (recall (3.26)).
Recall that if Π is the coupling obtained by the Gluing Lemma applied to Π 12 and Π 23 , we can then define g ≤ Γχ O by

R d ζdg := Ω×O×O χ |X(1-τ )|≤R (X)ζ(z)dΠ(x, y, z),
which is the part of Γχ O which originates from µ through Π 23 . We then project the parts of µ and g outside B R onto ∂B R :

R d ζdf µ := R d χ B c R (y)ζ R y |y| d µ = Ω×O χ |X(1-τ )|≤R (X)χ B c R (y) ζ R ∂B R O Γ ρ 1-τ t = 0 t = 1 -τ t = 1 g f g f µ µ
Figure 4: The definition of µ, g, f µ and f g .

and

R d ζdf g := O χ B c R (z) ζ R z |z| dg = Ω×O×O χ |X(1-τ )|≤R (X)χ B c R (z)ζ R z |z| dΠ.
Since g ≤ Γ 1 we can argue as for (2.10) to obtain

x 3 2 0 ∂B R f 2 g 3 2 0 Ω×O×O |R -|z||χ |X(1-τ )|≤R (X)χ B c R (z)dΠ (3.22) ≤ Ω×O×O |X(1 -τ ) -z| 2 χ |X(0)|<2 (X)dΠ Ω×O×O |X(1 -τ ) -y| 2 χ |X(0)|<2 (X)dΠ + Ω×O×O |z -y| 2 dΠ ≤ τ 2 E + D. (3.29)
We then let μ := µ B R + f µ and ĝ := g B R + f g .

Since projecting from outside B R reduces the distances

W 2 2 (ρ 1-τ B R , μ) ≤ W 2 2 (ρ 1-τ B R , µ) ≤ Ω χ |X(0)|<2 |X(1 -τ ) -X(1)| 2 = τ 2 B 2 |T -x| 2 = τ 2 E.
For the same reason, we also have

W 2 2 (μ, ĝ) ≤ W 2 2 ( µ, g) ≤ D.
Therefore by triangle inequality

W 2 2 (ρ 1-τ B R , Λ) W 2 2 (ρ 1-τ B R , μ)+W 2 2 (μ, ĝ)+W 2 2 (ĝ, Λ) τ 2 E +D+W 2 2 (ĝ, Λ).
(3.30) We are thus left with the estimate of W 2 2 (ĝ, Λ). For this we first claim that

W 2 (ĝ, Λ) W 2 1 2 (ĝ + Λ), Λ .
Indeed, by triangle inequality and monotonicity of the transport cost

W 2 (Λ, s) ≤ W 2 Λ, 1 2 (Λ + s) + W 2 1 2 (Λ + s), s ≤ W 2 Λ, 1 2 (Λ + s) + W 2 1 2 Λ, 1 2 s = W 2 Λ, 1 2 (Λ + s) + 1 √ 2 W 2 (Λ, s).
x notice that we cannot assert the same thing for f µ Now let ϕ g be the solution of

   ∆ϕ g = Λ -g in B R ∂ϕ g ∂ν = f g on ∂B R , (3.31)
with B R ϕ g = 0. Notice that since by definition of Λ and g,

Λ = 1 |B R | g(R d ) so that by definition of f g , B R (Λ -g) = g(B c R ) = ∂B R f g ,
so that this equation is indeed solvable. Let

ρ := (1 -t)Λ + t 1 2 (Λ + ĝ) and j := 1 2 ∇ϕ g .
The pair ( ρ, j) is admissible for the Benamou-Brenier formulation (2.13) of W 2 2 (Λ, 1 2 (Λ+ ĝ)) since (3.31) implies in a distributional sense

∇ • j = 1 2 (Λ -g -f g ) = 1 2 (Λ -ĝ) in R d ,
where we think of j as being extended by zero from B R to R d . Hence, as desired,

∂ t ρ + ∇ • j = 0 in R d × (0, 1) in a distributional sense. Noticing that ρ ≥ 1 2 Λ, we thus have W 2 2 (Λ, 1 2 (Λ + ĝ)) 1 Λ B R |∇ϕ g | 2 .
(3.32)

Let g -:= (Γ -g)χ B R so that by definition of g,

B R ζdg -= Ω×O×O χ |X(1-τ )|>R (x)χ B R (z)ζ(z)dΠ.
Thanks to the L ∞ bound (2.18) on the transport, we have that Spt

g -⊂ B R \B R/2 . We can rewrite ∆ϕ g = Λ -g = g --(Γ -Λ) so that by Lemma 2.2, B R |∇ϕ g | 2 ∂B R f 2 g + B R (R -|x|)dg -.
Arguing as for (3.21), we get

3 2 1 2 B R (R -|x|)dg -D,
so that using (3.29), (3.30) and (3.32) we obtain (3.28). From this we see that we may find R ∈ ( 1 2 , 3 2 ) such that both the conclusions of Lemma 3.4 and (3.27) hold.

The main estimate

To ease notation, we shall now assume that R = 1 and we will drop the index R.

The main goal of this section is to prove Proposition 1.6 which states that for every fixed τ 1, there exists a constant C(τ ) > 0 such that if E and D are small enough, then there exists an harmonic gradient field ∇ϕ in B 1 such that

B 1 2 1 0 1 ρ |j -ρ∇ϕ| 2 τ E + C(τ )D. (3.33)
From this Eulerian estimate, Proposition 1.5 which is the Lagrangian counterpart, is readily obtained. This in turn leads to the proof of Theorem 1.4, which is one step in a Campanato iteration scheme.

We now proceed with the definition of ϕ. Recall ρ ± from Lemma 3.4 and let ϕ be the (unique) solution of

   ∆ϕ = 1 |B 1 | ∂B 1 (ρ + -ρ -) in B 1 ∂ϕ ∂ν = ρ + -ρ - on ∂B 1 , (3.34)
such that B 1 ϕ = 0. Notice that by (2.1), the Hölder inequality and (3.15)

B 1 |∇ϕ| 2 ∂B 1 ρ 2 + + ρ 2 - E. (3.35)
Moreover, by Pohozaev, we also have

∂B 1 |∇ϕ| 2 E. (3.36)
The proof of (3.33) is divided into two parts. The first is an almost orthogonality property (see (3.37)) and the second is a construction of a competitor to estimate

B 1 1 0 1 ρ |j| 2 - B 1 |∇ϕ| 2 ,
see (3.52). We start with the almost orthogonality property.

Proposition 3.6. For every 0 < τ 1, there exist constants ε(τ ) > 0 and C(τ ) > 0 such that if E + D ≤ ε(τ ), then letting ϕ be defined via (3.34), we have the function ∇ϕ is smooth, the measure ρ∇ϕ is well defined. Furthermore, since clearly j -ρ∇ϕ ρ also the left-hand side of (3.37) is well defined. We start by noting that

B 1 2 1 0 1 ρ |j -ρ∇ϕ| 2 B 1 1 0 1 ρ |j| 2 - B 1 |∇ϕ| 2 + τ E + C(τ )D. ( 3
B 1 2 1 0 1 ρ |j -ρ∇ϕ| 2 = B 1 2 1-τ 0 1 ρ |j -ρ∇ϕ| 2 + B 1 2 1 1-τ 1 ρ |j -ρ∇ϕ| 2
and that since by harmonicity of ∇ϕ, sup B 1 2

|∇ϕ| 2 B 1 |∇ϕ| 2 , B 1 2 1 1-τ 1 ρ |j -ρ∇ϕ| 2 B 1 2 1 1-τ 1 ρ |j| 2 + sup B 1 2 |∇ϕ| 2 B 1 2 1 1-τ ρ (2.21)&(3.35) τ E.
Therefore,

B 1 2 1 0 1 ρ |j -ρ∇ϕ| 2 B 1 1-τ 0 1 ρ |j -ρ∇ϕ| 2 + τ E (3.38)
and we are left with bounding the first term on the right-hand side. Notice that since in (0, 1 -τ ), ρ and j are bounded functions by (2.20), the right-hand side of (3.38) is well defined in a pointwise sense. Recalling that ρ = 1-τ 0 ρ, we may now compute

B 1 1-τ 0 1 ρ |j -ρ∇ϕ| 2 = B 1 1-τ 0 1 ρ |j| 2 - B 1 |∇ϕ| 2 (3.39) -2 B 1 1-τ 0 j - 1 1 -τ ∇ϕ • ∇ϕ + B 1 (ρ -1)|∇ϕ| 2 .
Step 2. In this step we show that Let the boundary layer size r 1 to be fixed later and let η be a smooth cut-off function with χ B 1-2r ≤ η ≤ χ B 1-r and |∇η| r -1 . We split the integral:

B 1 (ρ -1)|∇ϕ| 2 τ -d(d-1) γ d (τ ) 1 d+2 E d+3 d+2 + τ E, ( 3
B 1 (ρ -1)|∇ϕ| 2 = B 1 (ρ -1)(1 -η)|∇ϕ| 2 + B 1 (ρ -1)η|∇ϕ| 2 .
(3.42)

The first term may be estimated as follows

B 1 (ρ -1)(1 -η)|∇ϕ| 2 ≤ B 1 \B 1-2r |ρ -1||∇ϕ| 2 (3.3) τ -(d-1) B 1 \B 1-2r |∇ϕ| 2 (2.2)
rτ - (d-1)

∂B 1 ρ 2 + + ρ 2 - (3.15)
rτ -(d-1) E.

(3.43)

We now turn to the second term. By

B 1 (ρ -1)η|∇ϕ| 2 ≤ B 1 (ρ -(1 -τ ))η|∇ϕ| 2 + τ B 1 |∇ϕ| 2 (3.35) B 1 (ρ -(1 -τ ))η|∇ϕ| 2 + τ E, it is enough to estimate B 1 (ρ -(1 -τ ))η|∇ϕ| 2 .
To this purpose we give an alter-

native representation: since -(1 -τ -t)η|∇ϕ| 2 ∈ C ∞ c (B 1 × [0, 1]
) we can extend it by zero for t ∈ [1 -τ, 1] and test (3.5) with it to obtain

B 1 ρη|∇ϕ| 2 = B 1 1-τ 0 ρ∂ t (-(1 -τ -t)η|∇ϕ| 2 ) = B 1 1-τ 0 (1 -τ -t)j • ∇(η|∇ϕ| 2 ) + B 1 (1 -τ )η|∇ϕ| 2 .
Therefore,

B 1 (ρ -(1 -τ ))η|∇ϕ| 2 = B 1 1-τ 0 (1 -τ -t)j • ∇(η|∇ϕ| 2 ) ≤ B 1 1-τ 0 1 ρ |j| 2 1 2 B 1 1-τ 0 (1 -τ -t) 2 ρ|∇(η|∇ϕ| 2 )| 2 1 2
(2.20)

E 1 2 1-τ 0 1 (1 -t) d-2 B 1 |∇(η|∇ϕ| 2 )| 2 1 2 (3.44) γ 1 2 d (τ )E 1 2 B 1 |∇(η|∇ϕ| 2 )| 2 1 2
, where we recall that γ d is defined in (3.41). By Leibniz rule and Cauchy-Schwarz we have

B 1 |∇(η|∇ϕ| 2 )| 2 1 r 2 B 1-r |∇ϕ| 4 + B 1-r |∇ϕ| 2 |∇ 2 ϕ| 2 1 r 2 B 1-r |∇ϕ| 4 + r 2 B 1-r |∇ 2 ϕ| 4 .
By the mean value formula for ∇ϕ, for every x ∈ B 1-r ,

|∇ 2 ϕ|(x) 1 r 1 |B r 2 | B r 2 (x)
|∇ϕ| so that integrating, using Jensen inequality and Fubini,

r 2 B 1-r |∇ 2 ϕ| 4 1 r 2 B 1-r 2 |∇ϕ| 4
from which the above estimate simplifies to

B 1 |∇(η|∇ϕ| 2 )| 2 (3.15) 1 r 2 B 1-r 2 |∇ϕ| 4 .
Let p = 2d d-1 . By the mean value formula for ∇ϕ and Jensen's inequality, sup

B 1-r 2 |∇ϕ| 1 r d B 1 |∇ϕ| p 1 p (2.1) r -d p ∂B 1 ρ 2 + + ρ 2 - 1 2 r -d p E 1 2 .
We then have

1 r 2 B 1-r 2 |∇ϕ| 4 ≤ 1 r 2 sup B 1-r 2 |∇ϕ| 4-p B 1 |∇ϕ| p r -2 r -d p E 1 2 4-p E p 2 = r -d E 2 .
Collecting all the previous estimates we obtain

B 1 (ρ -1)η|∇ϕ| 2 r -d 2 γ 1 2 d (τ )E 3 2 + τ E
and thus plugging this and (3.43) into (3.42), we get

B 1 (ρ -1)|∇ϕ| 2 rτ -(d-1) E + r -d 2 γ 1 2 d (τ )E 3 2 + τ E.
Optimizing in r through r = τ 2(d-1) γ d (τ )E 1 d+2 and using γ

1 2 d (τ )E 1 2
τ -(d-1) to ensure that r 1, we obtain the aimed estimate (3.40).

Step 3. We now estimate

B 1 1-τ 0 j - 1 1 -τ ∇ϕ • ∇ϕ.
For this we want to use (3.5) for ζ = χ (0,1-τ ) ϕ. Notice first that since ρ, j and f ± (recall the definition (3.9)) are bounded densities in (0, 1 -τ 2 ) (see Lemma 2.4 and Lemma 3.3), by density we can apply (3.5)

to ζ ∈ H 1 (B 1 × (0, 1)) with Spt ζ ⊂ B 1 × [0, 1 -τ /2]. Let ϕ δ ∈ C 0 (B 1 ) be a mollification of ϕ so that by continuity of t → ρ t in W 2 , 1 ε 1-τ +ε 1-τ B 1 ϕ δ ρ t → B 1 ϕ δ ρ 1-τ . (3.45)
Then, apply (3.5) to η ε (t)ϕ δ (x) where for ε > 0

η ε (t) =      1 for t ∈ (0, 1 -τ ] 1 -ε -1 (t -(1 -τ )) for t ∈ (1 -τ, 1 -τ + ε) 0 for t ≥ 1 -τ + ε
to obtain for ε → 0 using (3.45)

B 1 1-τ 0 ∇ϕ δ • j = B 1 ϕ δ ρ 1-τ - B 1 ϕ δ + R d 1-τ 0 ϕ δ df.
Letting δ → 0 using that ∆ϕ = constant and B 1 ϕ = 0 and recalling the definition of ϕ in (3.34) and (3.10), we thus obtain

B 1 1-τ 0 j - 1 1 -τ ∇ϕ •∇ϕ = B 1 ϕρ 1-τ + ∂B 1 ϕ[(f + -ρ + )-(f --ρ -)]. (3.46)
Let us estimate the first term. Let ( ρ, j) be given by the Benamou-Brenier theorem and such that

B 1 1 0 1 ρ | j| 2 = W 2 2 ρ 1-τ (B 1 ) |B 1 | χ B 1 , ρ 1-τ B 1 (3.27) τ 2 E + D. (3.47)
If T is the optimal transport map between ρ 1-τ (B 1 )

|B 1 | χ B 1 and ρ 1-τ B 1 , ρ -1 d t = ρ 1-τ (B 1 ) |B 1 | -1 d det 1 d ∇ T t ( T -1 t ) (2.22)
≥ ρ 1-τ (B 1 ) |B 1 | -1 d (1 -t) + tdet 1 d ∇ T ( T -1 t ) = (1 -t) ρ 1-τ (B 1 ) |B 1 | -1 d + tρ -1 d 1-τ ( T -1 t ) (2.20) ≥ (1 -t) ρ 1-τ (B 1 ) |B 1 | -1 d + tτ
and thus since ρ 1-τ (B 1 )

|B 1 | ∼ 1 thanks to (2.18), ρ ((1 -t) + tτ ) -d . (3.48)
We then have because of

B 1 ϕ = 0, B 1 ϕρ 1-τ = B 1 1 0 ∇ϕ • j ≤ B 1 1 0 ρ|∇ϕ| 2 1 2 B 1 1 0 1 ρ | j| 2 1 2
(3.47)

B 1 1 0 ρ - ρ 1-τ (B 1 ) |B 1 | |∇ϕ| 2 + ρ 1-τ (B 1 ) |B 1 | B 1 |∇ϕ| 2 1 2 τ 2 E + D 1 2 (3.35) B 1 1 0 ρ - ρ 1-τ (B 1 ) |B 1 | |∇ϕ| 2 + E 1 2 τ 2 E + D 1 2 τ B 1 1 0 ρ - ρ 1-τ (B 1 ) |B 1 | |∇ϕ| 2 + τ E + τ -1 D, (3.49)
where in the last line we used Young's inequality. The term

B 1 1 0 ρ -ρ 1-τ (B 1 ) |B 1 |
|∇ϕ| 2 is estimated as in Step 2. Indeed, choosing for r 1 a smooth cut-off function η with χ B 1-2r ≤ η ≤ χ B 1-r , we obtain as in (3.43) that

B 1 1 0 ρ - ρ 1-τ (B 1 ) |B 1 | (1 -η)|∇ϕ| 2 rτ -(d-1) E.
Using that

ρ - ρ 1-τ (B 1 ) |B 1 | = 1 0 (1 -t)∂ t ρ,
we obtain as in (3.44)

B 1 1 0 ρ - ρ 1-τ (B 1 ) |B 1 | η|∇ϕ| 2 B 1 1 0 1 ρ | j| 2 1 2 B 1 1 0 (1 -t) 2 ((1 -t) + tτ ) d |∇(η|∇ϕ| 2 )| 2 1 2 (3.47) γ 1 2 d (τ )(τ 2 E + D) 1 2 B 1 |∇(η|∇ϕ| 2 )| 2 1 2 r -d 2 γ 1 2 d (τ )(τ 2 E + D) 1 2 E.
Optimizing in r, we get

B 1 1 0 ρ - ρ 1-τ (B 1 ) |B 1 | |∇ϕ| 2 τ -d(d-1) γ d (τ ) 1 d+2 (τ 2 E + D) 1 d+2 E.
Plugging this into (3.49) we obtain for some C(τ ) 1,

B 1 ϕρ 1-τ C(τ )(τ 2 E + D) 1 d+2 + τ E + τ -1 D. (3.50)
We now turn to the second term in (3.46). It is enough to bound

∂B 1 ϕ(f + -ρ + )
since the other term is treated analogously. Let (ρ, ĵ) be the minimizer of (2.29), i.e.

∂B 1 1 0 1 ρ | ĵ| 2 = W 2 ∂B 1 (ρ + , f + ) (3.15) E d+3 d+2 .
Arguing as for (3.48) but using (2.30) and (2.31) together with (3.17) and (3.11), we obtain ρ-

1 d-1 ((1 -t) + tτ ) E - 1 (d-1)(d+2) so that 1 0 ρ τ -(d-2) E 1 d+2 , with the convention that for d = 2, τ -(d-2) = | log τ |.
By integration by parts we then have

∂B 1 ϕ(f + -ρ + ) = ∂B 1 1 0 ∇ bdr ϕ • ĵ ≤ ∂B 1 1 0 ρ|∇ϕ| 2 1 2 ∂B 1 1 0 1 ρ | ĵ| 2 1 2 (3.36) τ -(d-2) E 1 d+2 1 2 E 1 2 E d+3 d+2 1 2 τ -d-2 2 E d+3 d+2 , with the convention that for d = 2, τ -d-2 2 = | log τ | 1 2
. This estimate together with (3.50) yields

B 1 1-τ 0 j - 1 1 -τ ∇ϕ • ∇ϕ C(τ ) (τ 2 E + D) 1 d+2 + E 1 d+2 + τ E + τ -1 D.
(3.51) Putting together (3.39), (3.40) and (3.51), we conclude that

B 1 2 1 0 1 ρ |j -ρ∇ϕ| 2 - B 1 1 0 1 ρ |j| 2 - B 1 |∇ϕ| 2 C(τ ) (τ 2 E + D) 1 d+2 + E 1 d+2
+ τ E + τ -1 D, so that (3.37) follows if E + D ≤ ε(τ ) for some ε(τ ) small enough.

We may now use the minimality of (ρ, j) to estimate

B 1 1 0 1 ρ |j| 2 -B 1 |∇ϕ| 2 .
Proposition 3.7. For every 0 < τ 1, there exist ε(τ ) and C(τ ) such that if E + D ≤ ε(τ ), then letting ϕ be defined via (3.34), we have

B 1 1 0 1 ρ |j| 2 - B 1 |∇ϕ| 2 τ E + C(τ )D. (3.52)
Proof. Recall the measure f from (3.4). We are going to construct a competitor ( ρ, j) supported in B 1 × [0, 1] of the form

ρ t = ρ bdr t H d-1 ∂B 1 + ρ in t dx B 1 and j t = j bdr t H d-1 ∂B 1 + j in t dx B 1 ,
where j bdr t is tangent to ∂B 1 , and such that for every

ζ ∈ C 1 (R d × [0, 1]) ∂B 1 1 0 ∂ t ζρ bdr + ∇ bdr ζ • j bdr + B 1 1 0 ∂ t ζρ in + ∇ζ • j in = B 1 ζ 1 dµ - B 1 ζ 0 + ∂B 1 1 0 ζf. (3.53)
By (3.6), we then have

B 1 1 0 1 ρ |j| 2 - B 1 |∇ϕ| 2 ≤ R d 1 0 1 ρ | j| 2 - B 1 |∇ϕ| 2 . (3.54)
For the construction we will decompose ( ρ, j) into (see Figures 5678)

ρ in := ρ bulk + ρ lay j in := j bulk + j lay ρ bdr := ρ bdr,bulk + ρ bdr,lay j bdr := j bdr,bulk + j bdr,lay .

The bulk terms will live in the time interval (0, 1 -τ ) while the layer terms will allow to treat the boundary layer (in time) but will be defined for all t ∈ (0, 1). One of the crucial points for the estimate is that

1 4 ≤ ρ in ≤ 2 for t ∈ (0, 1 -τ ). (3.55)
Indeed, we will then have (recall that for t ∈ (1 -τ, 1), ρ in = ρ lay )

B 1 1 0 1 ρ | j| 2 - B 1 |∇ϕ| 2 = B 1 1-τ 0 1 ρ in |j in | 2 - B 1 1 0 |∇ϕ| 2 + ∂B 1 1 0 1 ρ bdr |j bdr | 2 + B 1 1 1-τ 1 ρ lay |j lay | 2 ≤ B 1 1-τ 0 1 ρ in |j in | 2 -ρ in |∇ϕ| 2 + ∂B 1 1 0 1 ρ bdr |j bdr | 2 + B 1 1 1-τ 1 ρ lay |j lay | 2 (3.55) B 1 1-τ 0 |j bulk | 2 -ρ in |∇ϕ| 2 + B 1 1-τ 0 |j bulk | 2 1 2 B 1 1-τ 0 |j lay | 2 1 2 + B 1 1-τ 0 |j lay | 2 + ∂B 1 1 0 1 ρ bdr |j bdr | 2 + B 1 1 1-τ 1 ρ lay |j lay | 2 .
Since for (ρ, j) compactly supported (recall (2.15))

R d 1 0 1 2ρ |j| 2 = sup ξ∈C 0 (R d ×[0,1],R d ) R d 1 0 ξ • j - |ξ| 2 2 ρ
is subadditive, we have

∂B 1 1 0 1 ρ bdr |j bdr | 2 ≤ ∂B 1 1 0 1 ρ bdr,bulk |j bdr,bulk | 2 + ∂B 1 1 0 1 ρ bdr,lay |j bdr,lay | 2 so that B 1 1 0 1 ρ | j| 2 - B 1 |∇ϕ| 2 1-τ 0 B 1 |j bulk | 2 -ρ in |∇ϕ| 2 + B 1 1-τ 0 |j bulk | 2 1 2 B 1 1-τ 0 |j lay | 2 1 2 + ∂B 1 1 0 1 ρ bdr,bulk |j bdr,bulk | 2 + ∂B 1 1 0 1 ρ bdr,lay |j bdr,lay | 2 + B 1 1-τ 0 |j lay | 2 + B 1 1 1-τ 1 ρ lay |j lay | 2 .
(3.56)

We now define and estimate the various contributions to the energy.

Step 1. We start by constructing and estimating (ρ bulk , j bulk ). The main estimate of this step is

B 1 1-τ 0 |j bulk | 2 -ρ in |∇ϕ| 2 τ E + τ -d E d+2 d+1 + D. (3.57)
Note that the first right-hand side term of (3.56) involves ρ lay through ρ in = ρ bulk + ρ lay . However, for its estimate in this substep we only need that for t ∈ (0, 1 -τ ),

ρ lay τ 2 E + D 1 2 1. (3.58)
Recalling the definition (3.12) of f lay + and that f ± are defined in (3.9) similarly, we let

m lay + := ∂B 1 f lay + and f :=            0 for t ∈ [0, τ ) -2 1-2τ f -for t ∈ [τ, 1 2 ) 2 1
-2τ f + for t ∈ [ 1 2 , 1 -τ ). (3.59)
Notice that since ∂B 1 f lay + = ∂B 1 ρ lay + , by Cauchy-Schwarz and (3.16),

m lay + τ 2 E + D 1 2 . (3.60) Moreover, since f + ⊥ f -(recall Lemma 3.3), ∂B 1 1-τ 0 f 2 ∂B 1 f 2 + + f 2 - (3.10) = ∂B 1 f 2 ∂B 1 1-τ 0 f 2 ,
where in the last inequality we used Jensen's inequality. This yields

∂B 1 1-τ 0 (f -f ) 2 ∂B 1 1-τ 0 f 2 (3.14) τ -d E. (3.61) τ 1 - m lay + |B1| f - 2τ ρ + -ρ - 1 -2τ 1 - m lay + |B1| -1 |B1| ∂B1 ρ + -ρ - 1 2 -f + 1 -τ ρ bulk ∂B 1 ∂B 1
Figure 5: The definition of ρ bulk .

For a boundary layer size r τ

-d E 1 d+1 ∂B 1 1-τ 0 (f -f ) 2 1 d+1 let A r := B 1 \B 1-
r , and let (s, q) be given by [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF]Lemma 3.4] applied to f -f and to the time interval (0, 1 -τ ) instead of (0, 1). We recall that (s, q) is such that it has support in

A r × [0, 1 -τ ], |s| ≤ 1 2 and for ζ ∈ C 1 (R d × [0, 1]), B 1 1 0 ∂ t ζs + ∇ζ • q = Ar 1-τ 0 ∂ t ζs + ∇ζ • q = ∂B 1 1-τ 0 ζ(f -f ).
In addition it satisfies the estimate

Ar 1-τ 0 |q| 2 r ∂B 1 1-τ 0 (f -f ) 2 (3.61) rτ -d E. (3.62)
We then let (ρ bulk , j bulk ), supported in B 1 × [0, 1 -τ ], be defined through (see Figure 5)

ρ bulk := 1 - m lay + |B 1 | + s - 1 |B 1 | ∂B 1 (ρ + -ρ -) ×      0 for t ∈ [0, 2τ ] t-2τ 1-4τ for t ∈ [2τ, 1 -2τ ] 1 for t ∈ [1 -2τ, 1 -τ ]
and

j bulk := q + ∇ϕ ×      0 for t ∈ [0, 2τ ]
so that by definition (3.34) of ϕ

B 1 1 0 ∂ t ζρ bulk + ∇ζ • j bulk = B 1 ζ 1-τ 1 - m lay + |B 1 | - 1 |B 1 | ∂B 1 (ρ + -ρ -) -ζ 0 1 - m lay + |B 1 | + ∂B 1 1-τ 0 ζ χ (2τ,1-2τ ) 1 1 -4τ (ρ + -ρ -) + f -f . (3.63)
Notice that thanks to (3.58) also (3.55) is satisfied. Now, we can start estimating

B 1-r 1-τ 0 |j bulk | 2 -ρ in |∇ϕ| 2 .
By definition of ρ bulk , since s vanishes in

B 1-r × (0, 1 -τ ), |1 -ρ in | ρ lay + m lay + + ∂B 1 (ρ + + ρ -) (3.58)&(3.60)
τ 2 E + D 1 2 + ∂B 1 ρ 2 + + ρ 2 - 1 2 (3.15) D 1 2 + E 1 2 1 in B 1-r × (0, 1 -τ ).
Therefore, since q also vanishes on B 1-r × (0, 1 -τ ),

B 1-r 1-τ 0 |j bulk | 2 -ρ in |∇ϕ| 2 (D 1 2 + E 1 2 + τ ) B 1 |∇ϕ| 2 (3.35) E 3 2 + D 1 2 E + τ E E 3 2 + τ E + D,
where in the last line we used Young's inequality together with the fact that since Combining these two estimates and taking into account that since τ 1 and E 1,

E 1, E 2 E
E 3 2 τ -d E d+2 d+1
, we find (3.57). Notice also for further reference that using the same argument, we obtain

B 1 1-τ 0 |j bulk | 2 E + τ -d E d+2 d+1 .
(3.64)

1 2 1 2 f - τ ρ - 1 -2τ 1 -τ f + 1 -2τ ρ + 2τ ∂B 1 ρ bdr 1 2τ ∂B 1 ρ bdr 2 ∂B 1 ∂B 1
Figure 6: The definition of ρ bdr,bulk .

Step 2. We now define (ρ bdr,bulk , j bdr,bulk ), supported in ∂B 1 × [0, 1 -τ ] so that

∂B 1 1 0 ∂ t ζρ bdr,bulk +∇ bdr ζ•j bdr,bulk = ∂B 1 1-τ 0 ζ f -χ (2τ,1-2τ ) 1 1 -4τ (ρ + -ρ -) (3
∂ t ζρ bulk + ∇ζ • j bulk + ∂B 1 1 0 ∂ t ζρ bdr,bulk + ∇ bdr ζ • j bdr,bulk = B 1 ζ 1-τ 1 - m lay + |B 1 | - 1 |B 1 | ∂B 1 (ρ + -ρ -) -ζ 0 1 - m lay + |B 1 | + ∂B 1 1-τ 0 ζf. (3.67)
We make the ansatz (ρ bdr,bulk , j bdr,bulk ) := (ρ bdr 1 + ρ bdr 2 , j bdr 1 + j bdr 2 ) (see Figure 6) requiring that

∂B 1 1 0 ∂ t ζρ bdr 1 +∇ bdr ζ•j bdr 1 = ∂B 1 1-τ τ ζ χ ( 1 2 ,1-τ ) 2 1 -2τ f + -χ (2τ,1-2τ ) 1 1 -4τ ρ + (3.68) and ∂B 1 1 0 ∂ t ζρ bdr 2 +∇ bdr ζ•j bdr 2 = ∂B 1 1-τ τ ζ χ (2τ,1-2τ ) 1 1 -4τ ρ --χ (τ, 1 2 ) 2 1 -2τ f -, (3.69
) so that by definition (3.59) of f , (3.65) holds. Let (ρ bdr 1 , j bdr 1 ) be given by Lemma 2.6 for

f 1 = ρ + , f 2 = f + , a = 2τ, b = 1 -2τ, c = 1 2 , and d = 1 -τ so that 1 (d -b) -(c -a) log d -b c -a = - 2 1 -6τ log 2τ 1 -4τ | log τ |.
Thanks to (2.32), we have (3.68) and by (2.33) combined with (3.15), we have

∂B 1 1 0 1 ρ bdr 1 |j bdr 1 | 2 | log τ |E d+3 d+2 . (3.70)
Similarly, using Lemma 2.6 with

f 1 = f -, f 2 = ρ -, a = τ, b = 1 2
, c = 2τ, and d = 1 -2τ to define (ρ bdr 2 , j bdr 2 ), we obtain that (3.69) holds and that (3.70) is also satisfies by (ρ bdr 2 , j bdr 2 ). By subadditivity this proves (3.66)

Step 3. We now define and estimate the quantities related to the terminal layer (in time). In this step we deal with the construction in the time interval [0, 1 -τ ] (see Figure 7) and define (ρ bdr,lay , j bdr,lay ) supported ∂B 1 × [0, 1 -τ ] and (ρ lay , j lay ) supported in B 1 × [0, 1 -τ ] such that (recall the definition (3.12) of f (3.72) Let ϕ lay be the solution of

     ∆ϕ lay = 1 |B 1 | ∂B 1 ρ lay + = m lay + |B 1 | in B 1 ∂ϕ lay ∂ν = ρ lay + on ∂B 1 ,
with ∂B 1 ϕ lay = 0 (recall that ρ lay + was defined in Lemma 3.4). By (2.1) and Hölder's inequality combined with (3.16), and extend them by zero for t ∈ ( 1 2 , 1 -τ ). Note that (3.60) automatically implies the smallness hypothesis (3.58). In view of the boundary value problem defining ϕ lay we have Let (ρ bdr,lay , j bdr,lay ) be defined by Lemma 2.6 with

B 1 1-τ 0 ∂ t ζρ lay + ∇ζ • j lay = ∂B 1 1 2 0 2ζρ lay + - B 1 ζ 0 m lay + |B 1 | , (3.74
f 1 = ρ lay + , f 2 = f lay + , a = 0, b = 1 2 , and c = d = 1 -τ so that 1 (d -b) -(c -a) log d -b c -a = 1 2 log 1 -2τ 2(1 -τ ) 1.
For these choices, (2.32) turns into Step 4. We are left with the construction in [1 -τ, 1]. We define (ρ bdr,lay , j bdr,lay ) supported ∂B 1 × [1 -τ, 1] and (ρ lay , j lay ) supported in

B 1 × [1 -τ, 1] such that B 1 1 1-τ ∂ t ζρ lay + ∇ζ • j lay + ∂B 1 1 1-τ ∂ t ζρ bdr,lay + ∇ bdr ζ • j bdr,lay = B 1 ζ 1 dµ- B 1 ζ 1-τ 1 - m lay + |B 1 | - 1 |B 1 | ∂B 1 (ρ + -ρ -) - ∂B 1 ζ 1-τ f lay + + ∂B 1 1
∂ t ζρ lay + ∇ζ • j lay + ∂B 1 1 0 ∂ t ζρ bdr,lay + ∇ bdr ζ • j bdr,lay = ∂B 1 1 1-τ ζf + B 1 ζ 1 dµ - B 1 ζ 1-τ 1 - m lay + |B 1 | - 1 |B 1 | ∂B 1 (ρ + -ρ -) -ζ 0 m lay + |B 1 | . (3.78)
The construction of (ρ bdr,lay , j bdr,lay ) takes care of the outgoing flux f lay + (recall (3.13)) in [1 -τ, 1] by defining ρ bdr,lay := and thus at no cost. The construction of (ρ lay , j lay ) for t ∈ (1 -τ, 1) is done in several steps (see Figure 8). We first take care of the incoming flow f -(recall (3.8)) in [1 -τ, 1] and to this purpose take the corresponding bulk density from X itself,

B 1 1 1-τ ζdρ lay -:= Ω χ 1-τ <t -<t + ≤1 (X) t + t - ζ(X, t)
and bulk flux

B 1 1 1-τ ξ • dj lay -:= Ω χ 1-τ <t -<t + ≤1 (X) t + t - ξ(X, t) • Ẋ.
With these definitions, it is readily seen that j lay ρ lay -. Let µ -:= ρ lay -(•, 1) and denote by f thr + the flux coming from particles that enter and leave B 1 during (1-τ, 1), i.e. the particles passing through B 1 ,

∂B 1 1 1-τ ζdf thr + := Ω χ 1-τ ≤t -<t + ≤1 (X)ζ(X(t + ), t + ), so that f + = f lay + + f thr + on ∂B 1 × [1 -τ, 1]
(recall the definitions (3.8) and (3.13)). By the same argument that led to (3.5) we have

B 1 1 1-τ ∂ t ζρ lay -+ ∇ζ • j lay -= B 1 ζ 1 dµ -+ ∂B 1 1 1-τ ζ(f thr + -f -), so that B 1 1 1-τ ∂ t ζρ lay -+ ∇ζ • j lay -+ ∂B 1 1 1-τ ∂ t ζρ bdr,lay + ∇ bdr ζ • j bdr,lay = B 1 ζ 1 dµ -- ∂B 1 ζ 1-τ f lay + + ∂B 1 1 1-τ ζf. (3.79)
Furthermore, by (2.15)

B 1 1 1-τ 1 2ρ lay - |j lay -| 2 = sup ξ∈C 0 (B 1 ×[1-τ,1],R d ) B 1 1 1-τ ξ • j lay -- |ξ| 2 2 ρ lay - = sup ξ∈C 0 (B 1 ×[1-τ,1],R d ) Ω χ 1-τ <t -<t + ≤1 (X) t + t - ξ(X, t) • Ẋ - |ξ| 2 2 ≤ Ω χ 1-τ <t -<t + ≤1 (X) t + t - 1 2 | Ẋ| 2 .
Let us point out that using an approximation argument one could actually show that equality holds in the previous inequality. Using that the trajectories X are straight lines, we have for 1 -τ < t -< t + ≤ 1, (3.63) to what remains from the target measure µ after subtracting ρ lay -(•, 1) = µ -i.e. we need to connect Λ to µ := µ -µ -. Since µ coincides with the measure defined in (3.26), by (3.27), W 2 2 (Λ, µ ) τ 2 E + D. We can thus use the Benamou-Brenier formulation of optimal transport (2.13) to find (ρ lay in , j lay in ) rescaled from [0, 1] to [1 -τ, 1] and such that,

t + t - | Ẋ| 2 = (t + -t -)|X(1) -X(0)| 2 ≤ τ |X(1) -X(0)| 2 so that B 1 1 1-τ 1 ρ lay - |j lay -| 2 ≤ τ Ω χ 1-τ <t -<t + ≤1 (X)|X(1) -X(0)| 2 (2.18) ≤ τ Ω χ |X(0)|≤2 (X)|X(1) -X(0)| 2 = τ E. (3.80) It remains to connect ρ bulk 1-τ = 1 - m lay + |B 1 | -1 |B 1 | ∂B 1 (ρ + -ρ -) =: Λ, cf.
B 1 1 1-τ ∂ t ζρ lay in + ∇ζ • j lay in = B 1 ζ 1 dµ - B 1 ζ 1-τ 1 - m lay + |B 1 | - 1 |B 1 | ∂B 1 (ρ + -ρ -)
(3.81) and

B 1 1 1-τ 1 ρ lay in |j lay in | 2 τ E + τ -1 D. (3.82)
We thus let for t ∈ [1 -τ, 1], ρ lay := ρ lay -+ ρ lay in and j lay := j lay -+ j lay in . Combining (3.79) and (3.81) we obtain (3.76). Moreover, using the subadditivity of 1 ρ |j| 2 , (3.80), (3.82) and the fact that in [1 -τ, 1], j bdr,lay = 0 we conclude the proof of (3.77).

Step 5. Combining (3.67) and (3.78), we see that (3.53) holds. Plugging (3.57), (3.64), (3.66), (3.72) and (3.77), into (3.56), we find

R d 1 0 1 ρ | j| 2 - B 1 |∇ϕ| 2 τ E + (τ -d E) d+2 d+1 + D + E + (τ -d E) d+2 d+1 1 2 τ 2 E + D 1 2 + τ 2 E + D + | log τ |E d+3 d+2 + τ E + τ -1 D τ E + (τ -d E) d+2 d+1 + | log τ |E d+3 d+2 + τ -1 D,
where we used Young's inequality together with the fact that τ

1 and E + D 1. Since (τ -d E) d+2 d+1 + | log τ |E d+3 d+2 is super-linear in E, there exists 0 < ε(τ ) 1 such that if E ≤ ε(τ ), (τ -d E) d+2 d+1 + | log τ |E d+3 d+2 τ E. Therefore, if E + D ≤ ε(τ ), R d 1 0 1 ρ | j| 2 - B 1 |∇ϕ| 2 τ E + τ -1 D,
which together with (3.54) proves (3.52).

Combining (3.37) and (3.52), we obtain our main estimate, Proposition 1.6 which we now recall for the reader's convenience.

Proposition. For every 0 < τ 1, there exist positive constants ε(τ ) and C(τ ) such that if E + D ε(τ ), then letting ϕ be defined via (3.34) we have Theorem. For every 0 < τ 1, there exist positive constants ε(τ ), C(τ ) and θ > 0 such that if E(µ, T, R) + D(µ, O, R) ≤ ε(τ ), then there exists a symmetric matrix B and a vector b ∈ R d such that

B 1 2 1 0 1 ρ |j -ρ∇ϕ| 2 τ E + C(τ )D. ( 3
|B -Id| 2 + 1 R 2 |b| 2 E(µ, T, R), ( 3
we have

E(μ, T , θR) ≤ τ E(µ, T, R) + C(τ )D(µ, O, R). ( 3

.88)

Proof. The proof is analogous to the one of [START_REF] Goldman | A variational proof of partial regularity for optimal transportation maps[END_REF]Proposition 4.7] with minor modifications. Still, we give the proof for the reader's convenience. By rescaling, we may assume that R = 1 and we then recall that E = E(µ, T, 1) and D = D(µ, O, 1). Let τ to be fixed later on and let then ϕ be given by Proposition 1.5 for τ . We define b := ∇ϕ(0), A := ∇ 2 ϕ(0) and then B := e -A 2 so that B is symmetric. Since ∇ϕ is harmonic, we obtain from (3.85) and the mean value formula that (3.86) holds.

1 4 W 2 2 μ , μ(Q ) 2 log r * ,L r * ,L 2 (4.2)
so that by (2.58) also

1 L 4 W 2 2,per (μ, 1) log L r * ,L L r * ,L 2 . ( 4.3)
Let us recall (see Section 2.3 and in particular Remark 2.5 for more details) that by [START_REF] Cordero-Erausquin | Sur le transport de mesures périodiques[END_REF], there exists a convex function ψ on R 2 such that the map T can be identified on R 2 with a measurable selection of the subgradient ∂ψ of ψ.

Let y L = y L (µ) := argmin Spt µ |y| (which is uniquely defined P L -a.e.) and x L the barycenter of its pre-image under T , i.e.

x L = x L (µ) := 1 |T -1 (y L )| T -1 (y L ) xdx, so that the map µ → x L is P L -measurable. Let us show that T (x L ) = y L . By convexity of ψ the set (∂ψ) -1 (y L ) = ∂ψ * (y L ) is convex. Since ψ is differentiable a.e., we have |(∂ψ) -1 (y L )\T -1 (y L )| = 0 and |(∂ψ) -1 (y L )| = µ(y L ) > 0 so that x L
lies in the interior of (∂ψ) -1 (y L ). Since ∇ψ = y L a.e. on this set, ψ is affine inside (∂ψ) -1 (y L ) and thus ψ is differentiable at Finally, we can prove our main result, Theorem 1.1 which we recall for the convenience of the reader.

Theorem. Let L 1 be dyadic and µ be a

Q L -periodic Poisson point process. Then, if µ is such that r * ,L L, |x L | 2 r 2 * ,L log 3 L r * ,L (4.5)
and for every 2r * ,L ≤ ≤ L,

1 4 B (x L ) |T -(x -x L )| 2 log 3 r * ,L r * ,L 2 . ( 4.6)
Proof.

Step 1.[The setup] By periodicity we have

E := 1 L 4 B L |T -x| 2 1 L 4 Q L |T -x| 2 = 1 L 4 W 2 2,per (μ, 1) (4.3) log L r * ,L L r * ,L 2 . (4.7)
Let µ := ∇ψ#(χ B L dx) so that T is the Euclidean optimal transport map between χ B L dx and µ. By the L ∞ bound (2.18), we have that µ = μ on Q L . Let finally

T 0 (x) := T (x L + x) -x L and µ 0 := µ(• + x L ) (4.
E 0 := E(µ 0 , T 0 , 0 ) = 1 (2 0 ) 4 B 2 0 |T 0 -x| 2 and D 0 := D(µ 0 , Q L -x L , 0 ) = 1 4 0 W 2 2 µ Q L , µ(Q L ) |Q L | satisfy for L large enough E 0 + D 0 ≤ ε(τ ). (4.10)
We also let

B 0 := Id, b 0 := 0, Ω 0 := B L -x L and O 0 := Q L -x L .
Let θ > 0 be given by Theorem 1.4. Without loss of generality we may assume that θ is dyadic i.e. θ = 2 -j for some j ∈ N. For k ≥ 1, let k := θ k 0 and notice that k is also dyadic since 0 is. It is of course enough to show that (4.6) holds for = k . We now prove by induction that there exist C * , C 0 , C 1 , C 2 > 0 sufficiently large but universal such that for every k ≥ 0 such that k ≥ C * r * ,L , we can find a symmetric matrix B k and a vector b k such that

|B k -Id| 2 + 1 2 k |b k | 2 ≤ C 0 log k r * ,L k r * ,L 2 , (4.11)
and letting

T k (x) := B k (T k-1 (B k x) -b k ), Ω k := B -1 k Ω k-1 and µ k := T k #χ Ω k we have for E k := E(µ k , T k , k ), E k ≤ C 1 log k r * ,L k r * ,L 2 , (4.12)
and T k is the optimal transport map between χ Ω k and µ k . Moreover, we may find a target set O k such that letting

D k := D(µ k , O k , k ), we have B 2 k ⊂ O k and D k ≤ C 2 log k r * ,L k r * ,L 2 . ( 4

.13)

As we shall argue, letting A 0 = Id, a 0 = 0 and then for k ≥ 1

A k := B k A k-1 and a k := B k a k-1 + B k b k , (4.14)
that is

A k = B k B k-1 • • • B 0 and a k = k i=0 B k B k-1 • • • B i b i , this entails T k (x) = A k T 0 (A * k x) -a k (4.8) = A k T (A * k x + x L ) -(A k x L + a k ), (4.15)
where A * denotes the transpose of A,

|A k -Id| 2 log k r * ,L k r * ,L 2 and |a k | 2 k 2 r 2 * ,L log L r * ,L . (4.16)
Notice that (4.12) and (4.13) in particular imply that if k ≥ C * r * ,L , then

E k + D k ≤ ε(τ ). (4.17)
Step 2.[The iteration argument] By (4.10) the induction hypothesis is satisfied for k = 0. Let us assume that it holds for k -1.

Step 2.1.[Proof of (4.11) and (4.12)] Thanks to (4.17) we may apply Theorem 1.4 with R = k-1 and O = O k-1 (recall that we fixed τ 1) to find a symmetric matrix B k and a vector b k ∈ R 2 such that

|B k -Id| 2 + 1 2 k |b k | 2 ≤ CE k-1 (4.12) ≤ CC 1 log k-1 r * ,L k-1 r * ,L 2 ≤ C 0 log k r * ,L k r * ,L 2 ,
if C 0 is taken large enough (depending only on C 1 ). From this we see that (4.11) is satisfied. Moreover, by (3.88)

E k ≤ τ E k-1 + C(τ )D k-1 (4.12) ≤ C 1 τ + C(τ )C 2 C 1 log k-1 r * ,L k-1 r * ,L 2 . Now if C 1 ≥ 1 1-τ C(τ )C 2 , since the function f (t) := log t t 2 is decreasing for t large enough, if k = θ k-1 ≥ C * r * ,L for some universal constant C * large enough then f ( k /r * ,L ) ≥ f ( k-1 /r * ,L ) and thus E k ≤ C 1 f k-1 r * ,L ≤ C 1 f k r * ,L = C 1 log k r * ,L k r * ,L 2 ,
which proves (4.12).

Step 2.2.[Optimality of T k ] Since T k-1 is the optimal transport map between χ Ω k-1 and µ k-1 , by Brenier's Theorem [31, Theorem 2.12], there exists a convex map

ψ k-1 such that T k-1 = ∇ψ k-1 . Then T k = ∇ψ k for the convex function ψ k (x) := ψ k-1 (B k x) -b k • B k
x so that T k is the optimal transport map between χ Ω k and µ k (see [START_REF] Villani | Topics in optimal transportation[END_REF]Theorem 2.12]).

Step 2.3.[Derivation of (4.16)] For k > 1 and i ≤ k we first prove that for k /r * ,L 1

k j=i    log j r * ,L j r * ,L 2    1 2 log 1 2 k r * ,L k r * ,L . (4.18)
Since the function log

1 2 t t 2
is decreasing for t large enough, we obtain

k j=i    log j r * ,L j r * ,L 2    1 2 k j=i j r * ,L j+1 r * ,L log 1 2 t t 2 = i r * ,L k+1 r * ,L log 1 2 t t 2 ≤ ∞ k+1 r * ,L log 1 2 t t 2 1 + log 1 2 k+1 r * ,L k+1 r * ,L log 1 2 k r * ,L k r * ,L
, which proves (4.18).

We can now make a downward induction on i to show that (4.11) implies that for k > 1 and i ≤ k

|B k B k-1 • • • B i -Id| ≤ 2 C 0 k j=i    log j r * ,L j r * ,L 2    1 2 (4.19)
which combined with (4.18) implies

|B k B k-1 • • • B i -Id| 2 log k r * ,L k r * ,L 2 . ( 4
|B k B k-1 • • • B i b i | (4.20) ≤ 2|b i | (4.11) ≤ 2 C 0 r 2 * ,L log L r * ,L 1
B k-1 • • • B i | ≤ 2 for k r * ,L
large enough so that we can estimate

|B k B k-1 • • • B i-1 -Id| ≤ |B k B k-1 • • • B i -Id| + |B k B k-1 • • • B i (B i-1 -Id)| ≤ 2 C 0 k j=i    log j r * ,L j r * ,L 2    1 2 + |B k • • • B i ||B i-1 -Id| (4.11) ≤ 2 C 0 k j=i    log j r * ,L j r * ,L 2    1 2 + 2 C 0    log i-1 r * ,L i-1 r * ,L 2    1 2 ≤ 2 C 0 k j=i-1    log j r * ,L j r * ,L 2    1 2 .
Step 2.4.[Proof of (4.13)] We first notice that since θ 1, k k-1 and we recall that k-1 is dyadic. We set

O k := A k Q k-1 -(A k x L + a k ) (4.21)
and notice that by the L ∞ bound (2.18) applied to T k we have

|A k y L -(A k x L + a k )| (4.15)&(4.9) = |T k (0)| k E 1 4
k so that from (4.12) in the form of E k 1, the first part of (4.16) in the form of |A k -Id| 1 and (4.9),

|A k x L + a k | ≤ |A k y L -(A k x L + a k )| + |A k y L | k E 1 4 k + r * ,L k . (4.22)
Therefore, using again that |A k -Id| 1 together with the fact that

B 2 k ⊂ Q k-1 2 imply that B 2 k ⊂ A k Q k-1 -(A k x L + a k ) = O k
so that the first part of (4.13) holds.

Let us prove that the second part of (4.13) also holds. Let T k be the optimal transport map between the constant measure on Q k-1 and the restriction of the measure µ to this set i.e.

W 2 2 µ Q k-1 , µ(Q k-1 ) |Q k-1 | χ Q k-1 = Q k-1 | T k -y| 2 µ(Q k-1 ) |Q k-1 | .
We then let for z ∈ O k , T k (z) :

= A k T k (A -1 k (z + A k x L + a k )) -(A k x L + a k ). We first show that T k # µ k (O k ) |O k | χ O k = µ k O k .
For this we notice that by definition of µ k , if

µ Q k-1 = α 0 i δ y i then µ k O k = α 0 | det A k | i δ A k y i -(A k x L +a k ) , (4.23
O k ζ T k # µ k (O k ) |O k | = O k ζ(A k T k (A -1 k (z + A k x L + a k )) -(A k x L + a k )) µ k (O k ) |O k | = Q k-1 ζ(A k T k -(A k x L + a k )) µ k (O k ) |O k | | det A k | = Q k-1 ζ(A k y -(A k x L + a k )) µ k (O k ) |O k | |Q k-1 | µ(Q k-1 ) | det A k |d µ(y) = Q k-1 ζ(A k y -(A k x L + a k ))| det A k | -1 d µ(y)
D k ≤ 1 4 k O k | T k -z| 2 µ k (O k ) |O k | = 1 4 k O k |A k T k (A -1 k (z + A k x L + a k )) -(A k x L + a k ) -z| 2 µ k (O k ) |O k | (4.16)&(4.21) 1 4 k Q k-1 | T k -y| 2 µ(Q k-1 ) |Q k-1 | (4.2) log k r * ,L k r * ,L 2 ,
where we used that µ = μ in Q k-1 . This concludes the proof of the second part of (4.13).

Step

3.[Conclusion]

We can thus iterate this procedure up to K = Using the L ∞ bound (2.18) for T K , we obtain |T - Since by definition (recall (4.14)) a i = B i a i-1 + B i b i , we have A -1 i a i -A -1 i-1 a i-1 = A -1 i-1 b i and thus for 1 ≤ k < K, we have |T -

|x L + A -1 K a K | 2
(x + A -1 k a k )| 2 ≤ 1 4 k+1 A * k B k +x L |T -(x + A -1 k a k )| 2 (4.16) 1 4 k B k |A k T (A * k y + x L ) -A k (A * k y + x L + A -1 k a k )| 2 1 4 k B k |A k T (A * k y + x L ) -(A k x L + a k ) -y| 2 + 1 4 k B k |(Id -A k A * k )y| 2
|A -1 K a K -A -1 k a k | 2 ≤ K i=k+1 |A -1 i a i -A -1 i-1 a i-1 | 2 = K i=k+1 |A -1 i-1 b i |
(x + A -1 k a k )| 2 + 1 2 k |A -1 K a K -A -1 k a k | 2 log 3 k r * ,L k r * ,L 2 ,
and obtain (4.6).

Remark 4.1. We would like to highlight that, although our estimate (4.6) is not optimal with respect to the power on the logarithmic term, estimate (4.26) leads to the optimal estimate inf ξ∈R 2 The suboptimal rate in (4.6) comes from the bound (4.16) which does not take cancellations into account.

Locally optimal couplings between Lebesgue and Poisson

In this section we show how Theorem 1.1 can be used to derive locally optimal couplings between the Lebesgue measure and the Poisson measure on R 2 . For this we will use the optimal transport maps T L = T µ,L constructed above and pass to the limit as L → ∞. Since the transport cost per unit volume diverges logarithmically, see [START_REF] Ambrosio | A PDE approach to a 2-dimensional matching problem[END_REF] or Section 2.4, we will need to use a renormalization procedure. Therefore, while the approximating couplings enjoy strong stationarity properties, cf. (2.53), the limiting couplings themselves will not. However, the shift stationarity property will be shown to survive in the second-order increments of the corresponding Kantorovich potentials. In order to set up the limit procedure, we need to equip the configuration space Γ (see Section 2.4) and the set of potentials with a topology. We equip Γ with the topology obtained by testing against continuous and compactly supported functions. Denote the space of all real-valued convex functions ψ : R 2 → R by K. We equip K (and C 0 (R 2 )) with the topology of uniform convergence on compact sets. Let us point out that with these topologies, both Γ and K are metrizable, which makes them Polish spaces. On P(Γ × K), we will consider the weak topology given by testing against functions in C b (Γ × K).

Denote by ψ L the convex function on R 2 such that T L = ∇ ψ L on Q L and (2.25) holds, i.e. ∇ ψ L (x + z) = ∇ ψ L (x) + z for all (x, z) ∈ R 2 × (LZ) 2 and ∇ ψ L #dx = L 2 µ(Q L ) µ.

Since x L (defined above Theorem 1.1) is logarithmically diverging in L, we will need to translate either the Lebesgue measure or the Poisson measure by a logarithmically diverging factor in order to pass to the limit. Since the Lebesgue measure (on R 2 ) is invariant under such translations while the Poisson point process is not, it is better to make this shift in the domain rather than in the image and set By adding a constant to ψ L we may assume that ψ L (0) = 0. Notice that by (2.56) of Lemma 2.9 and recalling that D Let us point out that because of the shift introduced in (4.28), the same invariance does not hold for ψ L .

For a given µ ∈ Γ, the bound (4.6) directly translates into locally uniformly L 2 -bounds for ∇ψ µ,L which by convexity of ψ µ,L yields compactness of (ψ µ,L ) L in K (see (4.32) below). Therefore, up to subsequence, ψ µ,L converges locally uniformly to a convex function ψ µ satisfying ∇ψ µ #dx = µ. However since we do not have any uniqueness property of this limit, the subsequence depends a priori on µ and we need to pass to the limit in the sense of Young measures. For this purpose, we first define the map

Ψ L : Γ → K, µ → ψ L ,
which is measurable and depends only on µ Q L , and then define the probability measure q L ∈ P(Γ × K) by

q L := (id, Ψ L )#P L = P L ⊗ δ Ψ L . (4.31)
We will show that the sequence (q L ) L is tight and that any limit point q gives full mass to pairs (µ, ψ) such that ∇ψ#dx = µ and such that the second-order increments of ψ * are shift covariant. The crucial ingredient is the following lemma that gives us a uniform control on the potentials ψ L .

Lemma 4.2. There exists a constant C > 1 such that for every dyadic L and every µ ∈ Γ such that r * ,L L, there holds for x ∈ R 2

Q L |∇ψ L -x| 2 ≤ 1 L 4 Q L |∇ψ L -x| 2 (4.2) 1
and put G := ∩ n∈N ∪ k∈N G k,n . The claim would follow provided we can prove that q(G) = 1. We first show that for fixed k, n ∈ N the set G k,n is closed. Let (µ m , ψ m ) m∈N ∈ G k,n be a sequence converging to some (µ, ψ) ∈ Γ × K. Since F k is closed, we have ψ ∈ F k and we only need to prove that 1 -

1 n µ ≤ ∇ψ#dx ≤ 1 + 1 n µ, (4.36)
which by weak convergence of µ m to µ and the fact that (µ m , ψ m ) satisfies (4.36), would be proven provided we show that ∇ψ m #dx weakly converges up to subsequence to ∇ψ#dx.

Let f ∈ C c (R 2 ) be fixed and let us prove that up to subsequence, This gives the claim and shows that G k,n is indeed a closed set. Since G k,n is measurable, G = ∩ n ∪ k G k,n is also measurable. Let q be an accumulation point of (q L ) L so that up to subsequence q L → q. For a given ε > 0, let us prove that for every n and for k large enough (depending only on ε) and every L large enough (depending only on k, n and ε),

q L (G k,n ) ≥ 1 -ε. (4.38) Since q L ((G k,n ) c ) ≤ q L (Γ×(F k ) c )+q L (µ, ψ) : 1 - 1 n µ ≤ ∇ψ#dx ≤ 1 + 1 n µ c ,
Since for fixed z ∈ R 2 , θ -z is continuous on Γ, for every such z and λ > 0 the map (µ, ψ) → f (θ -z µ, D 2 h ψ * ) ∈ C b (Γ × F λ ) so that by weak convergence q L → q combined again with (4.35) we have

1

 1 Main ideas for the proof of Theorem 1.1 . . . . . . . . . . . . . . . . . 5 1.1.1 The stochastic part . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 The deterministic part . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Theorem 1 . 1 .

 11 Assume that d = 2. There exists c > 0 such that for each dyadic L ≥ 1 there exists a random variable r * ,L = r * ,L (µ) ≥ 1 satisfying the exponential bound sup L E L exp cr 2 * ,L log 2r * ,L

2 ∀

 2 2r * ,L ≤ ≤ L.

Theorem 1 . 3 .

 13 For dyadic L ≥ 1 and µ a Poisson point process on Q L there exists a universal constant c > 0 and a family of random variables r * ,L ≥ 1 satisfying sup L E L exp cr 2 * ,L log(2r * ,L )

  , and define the extension φ(x, t) = exp(-αt)ϕ(x).

  .35) Let ψ be the affine function defined on [a, b] through ψ(a) = c and ψ(b) = d and let then for

Figure 2 :

 2 Figure 2: The definition of ρ R + .

Figure 3 :

 3 Figure 3: The definition of ρ R,lay + .

2 2 (

 2 µ O, Γχ O ). Let us recall that Π 12 is the measure defined on Ω × O by Ω×O ζdΠ 12 := Ω χ O (T (x))ζ(x, T (x))

2 |

 2 log τ | for d = 3 τ -(d-3) otherwise. (3.41)

3 2 . 1 d+1 , we have Ar 1 -τ 0 |j bulk | 2 -ρ in |∇ϕ| 2 ( 3

 2110223 Choosing r to be a large multiple of τ -d E

  bulk |j bdr,bulk | 2 | log τ |E

∂ 1 1-τ 0 ∂ 1 ζ 1 1 ρ

 10111 t ζρ lay + ∇ζ • j lay + ∂B t ζρ bdr,lay + ∇ bdr ζ • j bdr,lay = ∂B bdr,lay |j bdr,lay | 2 τ 2 E + D.

B 1 |∇ϕ 1 Figure 7 :

 117 Figure 7: The definition of ρ lay and ρ bdr,lay .

|j lay | 2 τ 2 E

 2 + D. (3.75)

∂ 1 ζ 1 1 1-τ 0 1 ρ 1 d+2 D τ 2 E

 111112 t ζρ bdr,lay + ∇ζ • j bdr,lay = ∂B with (3.74) we find (3.79). By (2.33) and (3.16) we also obtain ∂B bdr,lay |j bdr,lay | 2 τ 3 E d+3 d+2 + τ E 1 d+2 D, which combined with (3.75) and the fact that τ 3 E d+3 d+2 + τ E + D gives (3.72).

1 Figure 8 :

 18 Figure 8: The definition of ρ lay -and ρ lay in .

1 ρ

 1 bdr,lay |j bdr,lay | 2 τ E + τ -1 D. (3.77) Note that combining (3.71) and (3.76), we get B 1 1 0

  .86) and letting x := B -1 x, Ω := B -1 Ω and then T (x) := B(T (x) -b) and μ := T #χ Ωdx,

  x L with T (x L ) = ∇ψ(x L ) = y L . Therefore, by the definition of y L and (4.1) we have for µ such that 2r * ,L ≤ L, T (x L ) = y L and |y L | ≤ r * ,L . (4.4)

8 )

 8 so that (4.4) becomesT 0 (0) = y L -x L with |y L | ≤ r * ,L .(4.9)Denote by 0 the largest dyadic such thatB 2 (x L ) ⊂ Q L . Fix 0 < τ 1 forwhich Theorem 1.4 applies. By (4.7) and the L ∞ bound (2.18), |x L | E 1 4 L L so that 0 ∼ L. By (4.7) we thus have that (recall (3.1) and (3.2))

  )so that µ k (O k ) = µ(Q k-1 ) | det A k | and |O k | = | det A k ||Q k-1 |. For ζ ∈ C 0 (O k ) we thus have

  proving that indeed T k # µ k (O k ) |O k | χ O k = µ k O k .If we now use T k as competitor for the optimal transport problem betweenµ k (O k ) |O k | χ O k and µ k O k , we obtain

  L . By (4.16) we have|A -1 K a K | 2 r 2 * ,L log 3 L r * ,L .(4.24)

(4. 16 ) 2 ( 4 . 15 ) 2 ( 4

 16241524 |A K T (x L ) -(A K x L + a K )| 2 + |T (x L )| |T K (0)| 2 + |y L | (4.24) gives (4.5). We now prove (4.6). Since B k+1 (x L ) ⊂ A * k B k + x L and recalling that T k (x) = A k T (A * k x + x L ) -(A k x L + a k ) (see (4.15)), we can first estimate 14 k+1 B k+1 (x L )

|T k -y| 2 + 2 ( 4

 224 |Id -A k A * k |

27 ) 1 4 1 4

 2711 Noticing that by(4.25), it is enough to prove (4.6) with A -1 K a K instead of -x L , we conclude by (4.26) and(4.27) that k+1 B k+1 (x L ) |T -(x + A -1 K a K )| 2 k+1 B k+1 (x L )

1 4B 2 ∀

 12 (x L ) |T -(x -ξ)| 2 log r * ,L r * ,L 2r * ,L ≤ ≤ L.

ψ

  L (x) := ψ L (x + x L ).∇ψ L (0) = 0 and the Legendre conjugate ψ * L of ψ L satisfies ψ * L (y) = ψ * L (y) -x L • y.

  2 h ψ * (y) = ψ * (y + h) + ψ * (y -h) -2ψ * (y), we still have D 2 h ψ * θzµ,L (y) = D 2 h ψ * µ,L (y + z).(4.30)

1 4 |x| 2 - 2 *For 4 Q

 224 Cr 2 * ,L ≤ ψ L (x) ≤ |x| 2 + Cr 2 * ,L .(4.32)Therefore, letting for λ ∈ R,F λ := ψ ∈ K : 1 4 |x| 2 -λ ≤ ψ ≤ |x| 2 + λ (4.33) this means that if r * ,L L, ψ L ∈ F λ for every λ ≥ Cr ≤ L,this directly follows from (4.6) and the definition of ψ L . If now = kL for some k ∈ N, Q L -periodicity of the function (∇ψ L -x) yields 1 |∇ψ L -x| 2 = 1 L 2 2

R 2 f 2 f 4 |x| 2 - 4 |x| 2 -

 224242 (∇ψ m ) → R (∇ψ). (4.37) By local uniform convergence of the convex functions ψ m to ψ, if p m ∈ ∂ψ m (x) with p m → p, then p ∈ ∂ψ(x). Therefore, ∇ψ m converges a.e. to ∇ψ. Let r > 0 be such that Spt f ⊂ B r . In order to apply the dominated convergence theorem and conclude the proof of (4.37), we need to prove that there exists R depending only on k and r such that if |x| ≥ B R , then |∇ψ m (x)| ≥ r. This is a simple consequence of the fact that ψ m ∈ F k and the monotonicity of ∇ψ m . Indeed, since ψ m ∈ F k , 1 k ≤ ψ m ≤ |x| 2 + k so that at every point x of differentiability of ψ m , since 1 2k ≤ ψ m (x) -ψ m (0) ≤ ∇ψ m (x) • x ≤ |∇ψ m (x)||x|, we have |∇ψ m (x)| ≥ 1 4 |x| -2k |x| .

Γ×Kf

  (θ -z µ, D 2 h ψ * (• -z))dq L → Γ×K f (θ -z µ, D 2 h ψ * (• -z))dqwhich implies the thesis.

  ∂B 1 is the geodesic distance on ∂B 1 . Let us point out that a minimizer exists by McCann's extension of Brenier's Theorem [31, Theorem 2.47]. Notice that

  Let µ be a Poisson point process on R 2 . Then,

	.42)
	Let us now show how (2.41) and (2.42) translate into our setting.
	Proposition 2.7.

  12 is equal to µ O. Let then Π 23 be the optimal coupling between µ O and µ(O) |O| χ O . By the Gluing Lemma (see [31, Lemma 7.6]), there exists a measure Π on Ω × O × O with marginals Π 12 on Ω × O and Π 23 on O × O. We now define ρ R,lay

	+	in analogy to (3.19) by (see Figure

  , (3.23), (3.24) and (3.25), we see that we may choose R ∈ ( 1 2 , 3 2 ) such that (3.14), (3.15) and (3.16) hold.

2 .

 2 It thus remains to prove (4.19) which clearly holds for i = k by (4.11). Assume (4.19) holds for i. Then as already pointed out, (4.20) implies |B k

vii in order to have shift-invariance properties, we will actually consider periodic variants of (2.39), see below.

y |y| dΠ 12

1-4τfor t ∈ [2τ, 1 -2τ ] 0 for t ∈ [1 -2τ, 1 -τ ],

Acknowledgements

MH gratefully acknowledges partial support by the DFG through the CRC 1060 "The Mathematics of Emerging Effects" and by the Hausdorff Center for Mathematics. MG and MH thank the Max Planck Institute MIS for its warm hospitality.

Defining T and μ as in (3.87), we get

Recalling that B = e -A 2 , b = ∇ϕ(0) and A = ∇ 2 ϕ(0), we conclude using again the mean value formula for ∇ϕ

Choosing first θ small enough so that C(E + θ 2 ) ≤ τ 2 and then τ small enough so that Cθ -(d+2) τ ≤ τ 2 , we see that we can guarantee that (3.88) is satisfied.

4 Application to the optimal matching problem

Quantitative bounds on T

We now turn back to the optimal matching problem and combine Theorem 2.10 and Theorem 1.4 to obtain the desired quantitative estimate on the transport map. Let us recall that we work here in dimension d = 2. Let us recall that for for every dyadic L, we consider µ a realization of the Q L -periodic Poisson point process (see Section 2.4) and let T = T µ,L be the optimal transport map between µ(Q L )

and µ for the periodic transport problem (2.24), i.e. a minimizer of W 2 2,per ( µ(Q L ) L 2 , µ). By Theorem 2.10, there exist a constant c > 0 and a random variable r * ,L such that sup L E L exp

For µ such that 2r * ,L ≤ L, we let μ := L 2 µ(Q L ) µ so that T is also the optimal transport map (for the periodic transport problem (2.24)) between the Lebesgue measure and μ. By (2.58) of Theorem 2.10, we have that for all dyadic with 2r * ,L ≤ ≤ L, so that (4.34) can be also obtained for ≥ L. Letting f (x) := ψ L (x) -|x| 2 2 , this implies together with the L ∞ -bound given by Lemma 2. . Going back to the definition of f , this concludes the proof of (4.32).

This lemma endows us with the necessary compactness to prove the main result of this subsection, the convergence of (Ψ L ) L in terms of Young measures. This is precisely Theorem 1.2 which we recall for the convenience of the reader.

Theorem. The sequence of probability measure (q L ) L (cf (4.31)) is tight in P(Γ×K). Moreover, any accumulation point q satisfies the following properties:

(i) The first marginal of q is the Poisson point process;

(ii) q almost surely ∇ψ#dx = µ;

Proof.

Step 1. We start with tightness. Since trivially µ Q L → µ in Γ we have that P L → P weakly in P(Γ). In particular, the sequence (P L ) L is tight and for any ε > 0 there is a compact set Γ ε ⊂ Γ such that for all L we have P L (Γ ε ) ≥ 1 -ε. Since by Theorem 2.10 we have sup L E L exp

< ∞, there is a constant λ such that for each L large enough

Because of convexity, local boundedness yields local compactness in uniform topology. Thus setting K ε := Γ ε × F λ we have that K ε is compact and

which proves tightness. Moreover, since P L → P weakly in P(Γ) item (i) is shown.

Step 2. To show (ii) we define for k, n ∈ N the set G k,n ⊂ Γ × K by (recall the definition of F k given in (4.33))

it is enough to prove that each of the terms on the right-hand side are smaller than ε 2 for k, n and L large enough. The first term is estimated in (4.35) and we just need to consider the second term. For every L > 0 and µ ∈ Γ, we have by (4.29)

which by Cramér-Chernoff's bounds for the Poisson distribution with intensity L 2 (see [START_REF] Boucheron | Concentration inequalities[END_REF]) gives

concluding the proof of (4.38). Now for fixed k, n ∈ N large enough, since G k,n is closed, we have by (4.38)

Using that for every k, n ∈ N, G k,n+1 ⊂ G k,n and that G = ∩ n ∪ k G k,n , we obtain that for every ε > 0, q(G) ≥ 1 -ε, which concludes the proof.

Step 3. To show (iii) fix an accumulation point q and a subsequence, still denoted by (q L ) L converging weakly to q. Since for fixed λ > 0, the Legendre transform ψ → ψ * is continuous from F λ to K, for every h ∈ R 2 and λ > 0 the map ψ → D 2 h ψ * is continuous on F λ (recall (4.33)) with values in C 0 (R 2 ). Hence, the convergence q L → q together with (4.35) readily implies for all f ∈ C b (Γ × C 0 (R 2 )) that also