L^{p,q} estimates on the transport density

In this paper, we show a new regularity result on the transport density {\sigma} in the classical Monge-Kantorovich optimal mass transport problem between two measures, {\mu} and {\nu}, having some summable densities, f^+ and f^-. More precisely, we prove that the transport density {\sigma} belongs to L^{p,q}({\Omega}) as soon as f^+, f^- \in L^{p,q}({\Omega}).


Introduction
Let µ and ν be two given non-negative Borel measures on a compact convex domain Ω ⊂ R d , satisfying the mass balance condition µ(Ω) = ν(Ω). Let | · | stand for the Euclidean norm in R d . The classical Monge problem (MP) [9] consists of finding a transport map T * : Ω → Ω minimizing the functional This problem may have no solutions: this happens, for instance, when µ is a Dirac mass and ν is not. In [8], Kantorovich proposed a notion of weak solution to this transport problem. He suggested to look for transport plans instead of transport maps, i.e. non-negative measures γ on Ω × Ω whose marginals are µ and ν. Formally, this means that (Π x ) # γ = µ and (Π y ) # γ = ν, where Π x and Π y : Ω × Ω → Ω are the canonical projections. Denoting by Π(µ, ν) the class of transport plans, he wrote the following minimization problem (KP) min Ω×Ω |x − y| dγ : γ ∈ Π(µ, ν) .
Due to the convexity of the new constraint γ ∈ Π(µ, ν) and the linearity in γ of the functional, it turns out that weak topologies can be effectively used to provide existence of solutions to (KP). In fact, if (γ n ) n is a minimizing sequence, then, using Prokhorov's theorem, we have, up to a subsequence, γ n ⇀ γ with γ ∈ Π(µ, ν). Moreover, one has Ω×Ω |x − y| dγ n → Ω×Ω |x − y| dγ, which implies directly that γ is optimal for (KP).
The connection between the Kantorovich formulation of the transport problem and Monge's original one can be seen noticing that any transport map T induces a transport plan γ, defined by (Id, T ) # µ, which means that this plan is concentrated on the graph of T in Ω × Ω. We also see that the converse holds, i.e. whenever γ is concentrated on a graph, then γ is induced by a transport map. Since any transport map induces a transport plan with the same cost, it turns out that inf (MP) ≥ min (KP).
We also note that the equality inf (MP) = min (KP) holds as soon as there is an optimal transport plan γ which is concentrated on a graph y = T (x). By the way, this map T will be optimal for (MP). Yet, it has been really hard to give some answer about the existence of such an optimal transport plan which is induced by a map.
On the other hand, it is well known that the dual setting (DP) for the Monge-Kantorovich problem consists of finding a function u (called Kantorovich potential) which maximizes the functional over all v ∈ Lip 1 (Ω), where Lip 1 (Ω) stands for the set of Lipschitz continuous functions on Ω with Lipschitz constant one. This duality min (KP) = sup (DP) implies that optimal γ and u satisfy u(x) − u(y) = |x − y| on spt(γ).
We call transport ray any non-trivial (i.e., different from a singleton) segment [x, y] such that u(x) − u(y) = |x − y| that is maximal for the inclusion among segments of this form. Following this definition, we see that an optimal transport plan has to move the mass along the transport rays. And, it is well known that two different transport rays cannot intersect at an interior point of one of them (see, for instance, [11]).
Coming back to the problem of existence of optimal transport maps, Evans and Gangbo [6] have made a remarkable progress showing by differential methods the existence of such a map, under the assumption that the two measures µ and ν are absolutely continuous with respect to L d , that their densities f + and f − are Lipschitz with compact supports and that spt(f + ) ∩ spt(f − ) = ∅ (we note that after the work of Evans and Gangbo, Ambrosio in [1] has proved that there exists an optimal transport map for the Monge problem provided that µ ≪ L d ). A solution to the classical Monge-Kantorovich problem can be constructed by studying the p − Laplacian equation in the limit as p → +∞. They show that u p → u uniformly, where u is a Kantorovich potential between f + and f − , and at the same time, they prove the existence of a special non-negative function a such that The diffusion coefficient a in the PDE above plays a special role in the theory. Indeed, one can show that the measure σ := a · L d (the so-called transport density) can be represented in several different ways, and in particular as for some optimal transport plan γ, where H 1 stands for the 1-dimensional Hausdorff measure. Its physical meaning is the work for transporting the mass through the set A. It has been proven in [7,10] that if either µ or ν is absolutely continuous with respect to L d , then σ is unique (i.e. does not depend on the choice of the optimal plan γ) and it is also absolutely continuous with respect to L d . Moreover, the authors of [4,5,10] proved the following L p result on the transport density σ: On the other hand, the transport density σ (see (1.1)) is the total variation of a vector measure v solving the following problem (which is the continuous transportation problem proposed by Beckmann in [2]) where M(Ω, R d ) denotes the space of vector measures on Ω. In fact, for a given optimal transport plan γ, let us define a vector measure v γ and a scalar one σ γ as follows where w x,y is a curve parameterizing the straight line segment connecting x to y. Recalling (1.1), we observe easily that σ γ is nothing but the transport density between µ and ν. It is not difficult to see that v γ = −σ γ ∇u, where u is a Kantorovich potential in the transportation of µ onto ν. In addition, one can show that the vector measure v γ is, in fact, a minimizer for (BP) and, one has Moreover, every minimizer v for (BP) is of the form v = v γ (1.2), for some optimal transport plan γ (see [11]). This implies that if the source measure µ or the target one ν is absolutely continuous with respect to L d , then (BP) has a unique minimizer which is, by the way, in L 1 (Ω, R d ). So, this provides existence and uniqueness of the minimizer for the following minimal flow problem We recall that the unique minimizer v of (1.4) belongs to L p (Ω, R d ) as soon as f ∈ L p (Ω). The goal of this paper is to generalize this result by showing the following novel one about the L p,q regularity of the transport density σ:

New estimates on the transport density
In [4,5,10], the authors have already showed, using different techniques, the following L p summability on the transport density σ: Here, our aim is to extend this result to the Lorentz space (see Appendix 3). This means that we will prove the following implication In this way, (2.1) becomes a particular case of (2.2), when q = p. The strategy of the proof (which is already used in [10]) is based on a displacement interpolation and an approximation by discrete measures. In all that follows at least one between µ and ν will be absolutely continuous with respect to L d . Then, there will exist an optimal transport map T for (MP) from µ to ν (or, from ν to µ) and one unique transport density σ associated to those measures (independent of the optimal transport plan γ). First, let us suppose that the target measure ν is finitely atomic and let us denote by (x i ) i=1,...,n its atoms, that is , then the unique transport density σ associated with the transport of µ onto ν belongs to L p,q (Ω).
Proof. Let γ be an optimal transport plan from µ to ν and let σ be the unique transport density between them. Let µ t be the standard interpolation between the two measures µ and ν, that is where Π t (x, y) = (1 − t)x + ty. We see that µ 0 = µ and µ 1 = ν. Since the domain Ω is bounded, it is evident, recalling (1.3), that we have Yet, then it is easy to see that . Now, using Minkowski's inequality in the Lorentz space L p,q (Ω), we get the following estimate ||σ|| L p,q (Ω) ≤ C p 1 0 ||µ t || L p,q (Ω) dt.
Since µ ≪ L d , then there exists an optimal transport map T from µ to ν. But ν is finitely atomic, then Ω is the disjoint union of a finite number of sets Ω It is not difficult to see that Ω i (t) are essentially disjoint. Yet, µ t is absolutely continuous and its density f t coincides on each set Ω i (t) with the density of a homothetic image of f + , the homothetic ratio being (1 − t). This means that f t is concentrated on the union of Ω i (t) and, for any i ∈ {1, .., n}, one has For a fixed s > 0, we have Then, one has Hence, σ ∈ L p,q (Ω).
Lemma 2.2. Suppose that ν n ⇀ ν. If γ n is an optimal transport plan between µ and ν n , then there exists a subsequence (γ n k ) n k such that γ n k ⇀ γ and γ ∈ Π(µ, ν) is an optimal transport plan between µ and ν.
Proof. For each n, let u n be a Kantorovich potential between µ and ν n such that min u n = 0. Then, we see easily that there is a subsequence (u n k ) n k such that u n k → u uniformly in Ω. Yet, we have Then, passing to the limit, we get This implies that γ is an optimal transport plan between µ and ν, and u is the corresponding Kantorovich potential. Proposition 2.3. If µ = f + · L d with f + ∈ L p,q (Ω) and ν is any non-negative measure on Ω, then, if p < d ′ := d/(d − 1), the unique transport density σ associated with the transport of µ onto ν belongs to L p,q (Ω).
Proof. Let us consider a regular grid G n ⊂ Ω composed of approximately Cn d points (take G n = 1 n Z d ∩ Ω) and let P n be the projection map from Ω to G n . Set ν n := (P n ) # ν.
Then, ν n is atomic with at most Cn d atoms and ν n ⇀ ν. Let σ n be the transport density associated with the transport of µ onto ν n . By Proposition 2.1, we have that σ n ∈ L p,q (Ω) and This inequality, which is true in the discrete case, stays true at the limit as well, indeed, by Lemma 2.2, σ n ⇀ σ, where σ is the unique transport density associated with the transport of µ onto ν. But (σ n ) n is bounded in L p,q (Ω), then and σ ∈ L p,q (Ω).
Remark 2.1. If we denote by µ n,t the interpolation between the two measures µ and ν n , then µ n,t ⇀ µ t .
Moreover, if f n,t denotes the density of µ n,t , then (f n,t ) n is bounded in L p,q (Ω) and so, we By Remark 2.1, we see that the measures µ t inherit some regularity (L p,q summability) from µ exactly as it happens for homotheties of ratio 1 − t. This regularity degenerates as t → 1, but we saw that this degeneracy produced no problem for L p,q estimates on the transport density σ, provided p < d/(d − 1). Yet, for p ≥ d/(d − 1), we need to exploit another strategy: suppose both f + and f − share some regularity assumption (belong to L p,q ). Then we can give estimate on f t for t → 0 starting from f + and for t → 1 starting from f − . This will avoid the degeneracy.
In fact, this strategy works but we must pay attention to one thing: in the previous estimates, f t is obtained as a limit from discrete approximations and so, it doesn't share a priori the same behavior of piecewise homotheties of f + . And, when we pass to the limit, we do not know which optimal transport plan γ will be selected as a limit of the optimal transport plans γ n . This was not an issue in Proposition 2.3, thanks to the uniqueness of the transport density σ (since any optimal transport plan γ induces the same transport density σ). But here we want to glue together estimates on f t for t → 0 which have been obtained by approximating f − and estimates on f t for t → 1 which come from the approximation of f + . Should the two approximations converge to two different transport plans, we could not put together the two estimates and deduce anything on σ. So, the lack of uniqueness of optimal transport plans may create a problem. Hence, the idea is to consider a strictly convex cost as |x − y| 1+ε , where ε > 0, instead of |x − y| since, in this case, the corresponding optimal transport plan γ ε will be unique (see, for instance, [11,12]). We note that this strategy is different from the one given in [10] where the author shows that the "monotone optimal transport plan" can be approximated in both directions.
Finally, we get the following: Proposition 2.4. Suppose that µ = f + · L d and ν = f − · L d with f ± ∈ L p,q (Ω) and let σ be the unique transport density associated with the transport of µ onto ν. Then, σ belongs to L p,q (Ω) as well.
Proof. Let γ ε be the unique optimal transport plan in the transportation of µ onto ν with transport cost |x − y| 1+ε . Then, we see easily that γ ε ⇀ γ, where γ is an optimal transport plan in the transportation of µ onto ν with transport cost |x−y|. This implies that µ ε,t ⇀ µ t .
This means that the quasinorms ||.|| L p,q and ||.|| p,q are equivalent. Moreover, L p,q is a Banach space and, the dual of L p,q is isomorphic to L p ′ ,q ′ , where 1 p + 1 p ′ = 1 and 1 q + 1 q ′ = 1 (L p,q is also reflexive for 1 < p , q < ∞).