Factorization Polar and Monotone Rearrangement of Vector-Valued Functions
YANN BRENIER
Universite` de Paris VI
Abstract Given a probability space ( X , p ) and a bounded domain R in R d equipped with the Lebesgue measure 1 . I (normalized so that 10I = I ), it is shown (under additional technical assumptions on X and Q) that for every vector-valued function u E L p ( X ,p; R d ) there is a unique "polar factorization" u = V$.s, where $ is a convex function defined on R and s is a measure-preserving mapping from ( X , p ) into ( Q , I . I), provided that u is nondegenerate, in the sense that p ( u - ' ( E ) ) = 0 for each Lebesgue negligible subset E of Rd. Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real-valued functions are unified. The Monge-Amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate "Monge-Kantorovich" problem.
1. Introduction 1.1. Review of Some Well-Known Results
In this paper, several apparently unrelated classical results are unified through the concept of polar factorization of vector-valued functions. Let us review them.
Polar Coordinates in the Complex Plane
Any complex number can be written as z
Polar Factorization of Real Matrices
=
re", r 2 0, 0 E R/27rZ.
Any real matrix A can be written as the product RU of a symmetric non-negative matrix R by a real unitary matrix U . IfA is regular (det A # 0), then the factorization is unique.
Helmholtz Decomposition of Vector Fields
Let fl be a smooth bounded connected open set in Rd. Then, any smooth vector field z on fl can be written, in a unique way, as z = w V p , where p is a smooth real function, defined on D up to an additive constant, and w is a smooth divergence free vector field, parallel to the boundary of 9.This decomposition theorem is the simplest application of the Hodge theorem on differential forms; see [ 11.
+
Communications on Pure and Applied Mathematics, Vol. XLIV, 375-417 (1991) CCC 0010-3640/9 1/040375-43$04.00 0 1991 John Wiley & Sons, Inc.
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Y. BRENIER
Nondecreasing Rearrangements of Real Functions
Before stating the last classical result considered in this introduction, let us first review some well-known definitions on probability spaces. A "measure-preserving mapping" from a probability space (X, p ) into another probability space (Y, v ) is a mapping s : X + Y such that: for every v-measurable subset A of Y,
(1)
s - ' ( A ) is p-measurable and p ( s - ' ( A ) )= v ( A ) .
Or, equivalently,
for every v-integrable function f, (2)
f o sis p-integrable and
s,
f o s dp
=
Lfdv.
Such a mapping is not necessarily one-to-one. Consider, for instance, the case: p ) = ( Y , v ) = ([0, I], I . I), where I I denotes the Lebesgue measure, and s(x) = min(2x, 2 - 2x). If there is a one-to-one measure-preserving mapping from (X, p ) into (Y, v), then (X, p ) and ( Y , v ) are said to be isomorphic. In this paper, only probability spaces that are isomorphic to ([0, 11, I * I ) will be considered. This restriction is not severe, since, for instance, any separable complete metric space X , equipped with a Bore1 measure p such that p ( X ) = 1 and p( { x } ) = 0, forallxEX,isisomorphicto([O, I], I . ( ) ; s e e [21]. I f p E [ 1, +co [ and d = I , 2 , 3 . ,we call rearrangement of u E L p ( X ,p ; R d ) on (Y, v), any v E L p ( Y ,v; R d ) such that:
(X,
for each
f E C ( R d )such that If(<)
I 5 cst( I
+ II<IIp).
The last classical result deals with nondecreasing rearrangements, on the unit interval, of real-valued functions u E L p ( X ,p; R) (which means d = 1 and ( Y , v ) = ([O, 11, I * 1)). Let(X,p)beaprobabilityspaceisomorphic to([O, 11, I - I ) , a n d p E [ I , +m[. Then, for each u E L p ( X , p ) , there is a unique nondecreasing rearrangement u E Lp(O, ). Moreover, the mapping u + u is continuous from L p ( X ,p) into ` 1 ' LP( 0, 1 ). This result is well known; see [ 141, [ 51, and [ 181. A recent, detailed review of these topics can be found in [2]. Let us briefly indicate some complementary properties:
(4)
1 - u:IILP(o.I) 1 4
5
llUl
- u2IILp(X.d.
VECTOR-VALUED FUNCTIONS
377
For each u E L p ( X ,p),
(6)
there is a measure-preserving mapping s : (X, p) --* ([ 0, 11, I . 1 ), such that u
=
u#.s.
This last statement, proved by Ryff in [ 22 1, is not very well known.
1.2. The Main Results
Our results can be essentially seen as the extension of the classical result on nondecreasing rearrangements of real functions to the case of vector-valued functions u E L p ( X ,p ; R d ) , for d > 1. The unit interval 10, 1 [ is now replaced by a bounded connected open set 3 in Wd. The Lebesgue measure is replaced by a probability measure P on such that P ( d 3 ) = 0. It is assumed that d p ( z ) = p ( z ) dz, for some Lebesgue integrable nonnegative function p, bounded away from 0 on any compact subset of 3. It follows that P has the same negligible sets as the Lebesgue measure, which means that P is absolutely continuous with respect to the Lebesgue measure and conversely. Notice P ) is isomorphic to (10, 11, 1 I ) ; see [21]. that The weighted Sobolev space
(a,
will play an important role and is assumed to be compactly embedded in Lp(3, P ) (which is automatically enforced when 3 is smooth and is bounded away from 0 and +co on G). Under these assumptions, our main results are: Rearrangements of Vector-Valued Functions THEOREM 1.1. u# in the class
For each u E L p ( X ,p; R d ) , there is a unique rearrangement
3, P ) ; # convex} c L p ( 3, p; R d )
(8)
K = {V#; # E
W , p (
and the mapping u + u# is continuous.
Polar Factorization of Vector-Valued Functions
THEOREM Let N be the set of all u E L p ( X ,p; Rd)for which thefollowing 1.2. "nondegeneracy" condition fails:
(9)
p ( u - ` ( E ) )= 0
for each Lebesgue negligible subset E of R d .
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Y . BRENIER
Then, for each u E L p ( X ,p; Rd)\N, there is a unique pair ( d ,) , such that u# s belongs to K , s is a measure-preserving mappingfrom ( X , p) into p ) , and u = u# s . Moreover, ( a ) u# is the unique rearrangement of u in K ( a s in Theorem 1.1 ); ( b ) s is the unique measure-preserving mapping that maximizes u(x). s ( x )dCL(x); ( c ) the mapping u + (u', s ) is continuousfrom L p ( X ,p; Rd)\N into Lp(fl,/3; R ~ x) L ~ ( 1.1; Rd),forall q E [ I , +a[. x,
(a,
sx
Remark. We believe that the result is still true when fl is unbounded, provided t h a t p > l a n d S , I l z l l q f l ( z ) d z < + ~ , w h e r e l / q + 1 / p = 1.
Comments on the "Rearrangement Theorem"
Obviously, K is, with respect to Lp(9,0; d ) ,the generalization of the class of all R nondecreasing functions in Lp(0, 1 ). Thus Theorem 1.1 is a generalization of the classical result on nondecreasing rearrangements of real functions, which correspondstotheparticularcased= l , ( f i , f l ) = ( [ 0 , I], 1.1). As in the case of real arrangements, the mapping u --* u#is continuous. However, properties ( 4 ) and ( 5 ) are not true in general when d > 1. For instance, in the particular case ( X , p) = fl)= I . 1 ), where I . I denotes the Lebesgue measure 9 (normalized so that 1 I = l ) , it will be shown in Section 2.4 that:
(a,
(a,
(10)
uy(x)-u:(x) dx h
is not true in general. 1.3. Recovery of the Classical Results Our "rearrangement theorem" generalizes the classical results on nondecreasing rearrangements of real functions. Let us now check that the polar factorization of real matrices and the Helmholtz decomposition of vector fields are particular cases of the situation considered in our "factorization theorem".
Helmholtz Decomposition
The Helmholtz decomposition turns out to be the linearization of the polar p) factorization of vector-valued functions about the identity map, when (X, =
(% 6 )= (%
1 . I).
u ( x ) = x + Z ( X ) , G 1,
x E 3i.
Let us consider u as a smooth perturbation of the identity map:
For E small enough, u satisfies the nondegeneracy condition (9) (because thejacobian
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379
determinant is bounded away from 0) and u can be written: u = V + # . s . It is natural to seek V$" and s as perturbations of the identity map: + # ( x )= $ IIxI12 + t p ( x ) + 0 ( 2 ) ,
s(x) = x
+ EW(X) + O(t2).
Then, z = V p w immediately follows. To keep + # convex, p needs no special property (provided that p is smooth enough and t is small enough). However, to keep s measure preserving, it is necessary to enforce
+
for each smooth function fdefined on
a. This leads to
which precisely means (in the weak sense) that w is divergence free on R and parallel to dR. So, the Helmholtz decomposition is the linearization of the polar factorization of vector-valued functions. At this point, a very intriguing question can be raised: Is it a general fact that a Hodge decomposition of differential forms (see [ 11) can be seen as the linearization of some "generalized" polar factorization?
Polar Factorization of Matrices
Let us now show that the polar factorization of matrices is a particular case of the polar factorization of vector-valued functions. Here, ( X , p) = P) = I I) and R is a ball centered at the origin, u is a linear mapping: u ( x ) = A x , for some real d X d matrix A . The nondegeneracy condition (9) exactly means that A is nonsingular (det A # 0). The polar factorization A = RU corresponds to the polar factorization u = V + # . s , where: + # ( x ) = f R x . x is convex and s ( x ) = U . x is a measure-preserving mapping from (Q, 1 . I ) into itself. Notice that, due to the special geometry of the ball, for a linear mapping u ( x ) = A . x, each factor of the polar factorization s ( x ) = U . x, u # ( x )= R - is a linear mapping. This cannot be true x for a general geometry. The case of complex numbers z = rei8can be treated in the same way. So our polar factorization theorem unifies all these classical results. Let us now show how our rearrangement theorem is strongly linked to the Monge-AmpZre equation; see [9].
(n,
(a,
1.4. The Rearrangement Theorem and the Monge-AmpGre Equation Theorem 1.1 can be seen as an existence and uniqueness theorem of a "generalized" solution for the following Monge-Am$re problem.
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Y. BRENIER
Monge-Amp6re Problem
Given a E L ( R d ) , a I 0, such that continuous) convex function Q : + R ' be precised ) :
'
(1
+ [Iy 11 ")a( ) dy < 03 , find a (Lipschitz y R that satisfies (in a generalized sense to
+
(12)
V 4 maps thesupport of a into
a.
Usually, the Monge-Amp3-eequation is set on a bounded (often convex) domain with Dirichlet or Neumann boundary conditions; see [ 9 3 . Here, the "range condition" ( 12) replaces the usual boundary conditions. To see the link with this Monge-Amp6re problem, we need a different version of our rearrangement theorem that relies on the following observation: What really matters in Theorem 1.1 is the probability measure a associated with u E LP(X, p; R d ) and defined by
for each compactly supported f C(Rd). Notice that ( 1 positive measure, which means
+
11 y 11 ") da( y ) is a tight
IIyIIp). In and ( 13) still holds for each f C ( R d ) such that I f ( y ) I 4 cst(1 addition, if a sequence (u,) converges to u in L p ( X ,p; Rd), then the corresponding probability measures a,?satisfy
+
and, therefore, converge to (Y in the following sense: J- f da, --* fEC( Rd) s uc ht h at If(y)I $ cst(l IIyIIp). So, it is not hard to see that Theorem 1.1 is a corollary of:
+
f da for every
THEOREM 1.3. For each probability measure a on R d satisfiing j( 1 + 11 y II 2, da( y ) < +a, there is a unique u# = V## in K such that
for eachfE C(R d ) such that
(fI
y ) I S cst( 1
+ 11 y
IIP).
Moreover, i f s f dan + J
VECTOR-VALUED FUNCTIONS
38 I
f d a ,for a sequence o such probability measures a, and any f E C(R d ) such that f If(v)I 5 cst(l + IIYIIP), then
A link can be established with the Monge-AmpZre problem in the case when a is absolutely continuous with respect to the Lebesgue measure
dcu(y) = a ( y ) d y , a E L ' ( R d ) , a 1 0 , which exactly means that u is nondegenerate in the sense of (9) (by LebesgueNikodym theorem). In Theorem 1.3, Gffis convex and V$' maps !? into the support of a, as follows from ( 14). Let us consider the Legendre-Fenchel transform 4' of +# (cf. [13]), defined by
It will be shown (see Section 3 ) that 4' is Lipschitz continuous on Rd, V4' is well defined almost everywhere on Rd, and
V4'(V+"(z))
=
z , for a.e. z E
for
(Y
Q7
v+#(v##(~)) , = y
- a.e. y E
Thus, V$# and V4' are reciprocal and, formally, one gets for every compactly supported fE C( R d , :
by using the change of variable y has
=
V+'( z), z
=
V $ # ( y ) .By definition of qff one
and, thus, 4' satisfies (in a generalized sense) the Monge-AmpZre equation ( 1 1 ) together with the range condition defined by ( 12).
1.5. Origin of the Results
The polar factorization theorem was motivated by the study of the following "projection problem" (introduced in [ 61). Here (X, p ) = (!?, p) = I * I ) and p = 2.
(a,
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Let us denote by H the Hilbert space L 2 (Q; R d ) and by S the set of all measurepreserving mapping from (Q, I * I ) into itself, which means
This set S is a closed (nonconvex, noncompact) bounded subset of H (cf. Section 2). Notice that S is contained in a sphere centered at the origin. Indeed, from ( 17), one gets: llsll = J Ils(x) 11 dx = Jn llxll dx = cst. Let us now consider: n The Projection Problem
( Find s E S that minimizes 11 u - sII = JQ11 u( x) - s x) 11 dx, or, equivalently, that maximizes ( ( u ,s)) = Jn u ( x ) . s ( x ) dx. As mentioned in Theorem 1.2, when the nondegeneracy condition (9) is satisfied, the factor s in the polar factorization of u = u#.s, for u E L 2 (Q ; R d ) , is exactly the u(x) - s ( x )dx and, therefore, is the unique Hilbert unique maximizer in S of projection of u on S . The projection problem is a key to understanding the concept of polar factorization. For instance, the set K , here defined by
K
=
{v+;+ E w ' , ~ ( + convex} c L ~ ( Q ;~ ) , Q); R
is closely linked to the "projection problem". Indeed, it can be shown (cf. Section 2 ) that K is exactly the set of all u E H for which the identity map e is a Hilbert projection of u on S:
or, equivalently, K = (uEH;((u,e-s))>=O,VsES}
Notice that, in terms of convex analysis, K is the "polar cone" of S (or, equivalently, of its convex hull). This "geometrical" description of the polar factorization can be made even more precise, by noticing that S is a semigroup with respect to the composition rule
s,.=s2 S , whenever sI,s2 E S , E
and the identity map e is the neutral element. If S were a group (which is definitely not true since, for instance, in the case I . I) = ([0, 11, I I), s ( x ) = min(2x,
(n,
VECTOR-VALUED FUNCTIONS
383
2 - 2x) is not invertible in S ),then the "polar factorization" theorem would follow from the study of the "projection problem". Indeed, if u E H has a Hilbert projection s on S , where s is invertible in S , then u admits a polar factorization u = k . s, where k is defined by k = u.s-' and clearly belongs to K , since, for any cr E S , because of property ( 17), one has
There is a general answer for the projection problem, given by Edelstein's theorem (see [4]),which asserts that "almost every" u E H , in the sense of Baire, has a unique projection s = r( u ) on S . This is due to the fact that S is a closed, bounded subset of a Hilbert space. Unfortunately, in our case, since S is not a group, there is no reason for s to be in general invertible and, therefore, there is no direct way to recover the factor k of the polar decomposition of u. We have not been able to overcome this difficulty to obtain our main results directly and a more involved proof has been used, through the study of the "Monge-Kantorovich" problem (cf, Section 1.6). Note that it is not a good idea to substitute for S the group G of all invertible measure-preserving mappings in S , since G is not closed in H , which makes impossible the use of Edelstein's theorem. Even if this ``geometrical'' approach does not yield a proof of our main results, it seems to us that the discussion of the polar factorization in terms of group and Hilbert projection is of interest (especially in view of possible generalizations). This is why the second section of this paper is mainly devoted to this approach, in a rather abstract framework: we are given a Hilbert space H , a closed bounded subset S , and a composition rule * on S X H such that ( S , *) is a group and IIs*uII = IIuIJ, for all u E H , s E S . Then it is proved, under additional assumptions, that for almost every u E H there is a unique polar factorization u = s * k , s E S , k E K . Moreover, it is shown that u -+ s is the gradient of the Lipschitz continuous convex function J( u ) = supsEs((u, s))(which is the Legendre-Fenchel transform of the indicator function of S ) . Section 2 can be read independently of the remainder of the paper and can be ignored by those who are interested only in the proofs of our main results. 1.6. A Proof Using the Monge-Kantorovich Problem Our proof is based on the observation that the "projection problem" is a variant of the "mass transference problem" introduced by Monge, in the eighteenth century (see [ 17 1, cf. [ 3 ] ) ,and generalized by Kantorovich in the 1940s; see [ 15 1. In modern terms, the Monge problem (MP) and the Monge-Kantorovich problem (MKP) can be described as follows.
MP
Given two compact metric probability spaces (X, ( Y , v ) and a continuous p), "cost" function c : X X Y OX+, find a one-to-one measure-preserving map--+
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Y. BRENIER
ping from ( X , p) into ( Y , V ) that minimizes the "transportation cost"
s(x))
sx c(x,
44x).
MKP
Given two compact metric probability spaces ( X , p), ( Y , V ) and a continuous "cost" function c : X X Y + R+, find a probability measure y on X X Y with marginals p and V, which means, for each f C ( X )and each g E C( Y ) :
,
that minimizes the "generalized transport cost" c ( x , y ) d y ( x , y ) . The MKP is a "relaxed" version of the MP, in the sense that any admissible solution s to the MP yields an admissible solution y to the MKP, defined by
s
The MKP has many applications and a very complete review can be found in a recent paper by Rachev; see [ 191. The MKP is an infinite dimensional linear program and a key point of the analysis developed by Kantorovich is the study of the dual linear program:
such that
f ( x)
+ g ( y ) 4 c(x, y ) ] .
It can be proved (see [19]) that the MKP has a solution y that satisfies the "strong duality" relation I = c ( x , y ) d y ( x , y ) . A more refined result, due to Sudakov (see [ 19]), asserts that under more specific conditions (that are satisfied when the cost function is a power of some distance function), the solution y is actually of the form d y ( x , y ) = 6 ( y - s( x)) dp( x ) , where s is a one-to-one measurepreserving mapping, which means that the Monge problem has a solution. Our projection problem corresponds to the case when ( X , g ) = ( Y , v ) = I I) and c ( x , y ) = Ilu(x) - y ( J We look for a measure-preserving mapping, not c(x, s ( x ) ) dx. In the correnecessarily one-to-one, s that minimizes the cost sponding MKP problem, we look for a "doubly stochastic" probability measure on a X 0, which means that the marginals of y are 1 1, that minimizes the cost I = Ilu(x) - y ( ( d y ( x , y ) . Notice that the cost function is not necessarily continuous, since u is given in L2(0; Rd). However, the MKP still makes sense because
'.
(a,
s
VECTOR-VALUED FUNCTIONS
385
c is automatically integrable for any doubly stochastic probability measure (this can be easily seen). In the earlier version of this paper (see [6]), we proved, with the help of the strong duality relation and some well-known properties of convex functions, that, when u is nondegenerate in the sense of ( 9 ) , this MKP has a unique solution of the form
for some measure-preserving mapping s, that turns out to be the unique solution of our projection problem. Moreover, the solution of the dual program is linked to a pair of convex conjugate functions ($#, 4`) and V$# is precisely the rearrangement of u in K . In this paper, a slightly different and more efficient approach is used, by considering a different MKP that generalizes the projection problem in a different way. We look for a probability measure p on X with the following marginals:
a a,
for each f E C( that minimizes I = 11 x - y 11 d p ( x , y ) . Clearly, any admissible solution s of the projection problem yields an admissible solution p of this new MKP, defined by
a),
'
for each f C( X function c(x, y ) = 11 x
a a). The main advantage of the new MKP is that the cost
-
y 1) is now simpler and smoother.
1.7. References to Related Works Our polar factorization theorem was introduced, in a weaker form (for instance, u was supposed to be bounded and Riemann integrable), in [ 61 and the proof (using the Monge-Kantorovich problem (MISF')) was sketched. Our motivation was the numerical study of the motion of perfect incompressible fluids. In [ 71, a "Lagrangian" scheme was introduced, where the "projection problem" plays an essential role. Our proof relies on the study of the MKP and Rachev's paper, [ 191, was very useful. We would like to thank G. Strang for communicating it to us. Since our first paper, we have heard (thanks to P.-L. Lions, J. Mossino, J. Norbury, and S . Rachev) of two papers that anticipated our "rearrangement theo-
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Y. BRENIER
rem". Both are due to British scientists, one in statistics and the other in atmospheric sciences. In the second one, by Cullen and Purser (see [lo], followed by [ I I ] ) , a numerical scheme is designed, using a quasi-geostrophic model, for weather forecasting. The solution is described by a vector field that evolves in time. At each time step, this field is rearranged as the gradient of some convex potential, for some stability reasons linked to the physical model. A numerical procedure to compute this rearrangement is also described. This procedure is reminiscent of the method that Pogorelov introduced to prove the existence of generalized solutions for the Monge-Ampire equation; see [ 91. The first paper, recently pointed out to us by S . Rachev, is due to Knott and Smith (see [ 161) and deals with the following problem in statistics. Given two random variables X and Y , with prescribed laws, find a correspondence Y = Z( X )that maximizes the expectation of X . Y . It is shown that the optimality condition for Z is to be the subdifferential of some convex function. Behind this problem, it is not hard to recognize the situation considered in our rearrangement theorem. The solution of this problem, in the one-dimensional case ( d = I ), goes back to Frkchet and Hoeffding (see [ 201 ). So, in these earlier works, the concept of rearrangement for vector-valued functions is more or less explicitly present, although the notion of polar factorization is missing.
1.8. Organization of the Paper
Section 2 is devoted to the abstract concept of "polar factorization". A definition and some examples are considered in Section 2.1. In particular, the polar factorization of vector-valued functions (in the case p = 2, ( X , p) = (h, 1 . 1 )) is discussed. In Section 2.2, an abstract polar factorization theorem is proved and relies on a group property that is not satisfied in the case of vector-valued functions. Sections 2.3 and 2.4 give complementary results in the case of vector-valued functions. Section 2 is completely independent and can be ignored by readers who are only interested in the proof of our main results. These proofs are given in Section 3 and essentially rely on the study of the Monge-Kantorovich problem (MKP). For simplicity the proof is given in the case p = 1, which is less restrictive (indeed ( X , p) is a probability space and, therefore, all spaces L p ( X ,p; R d ) are contained in L ' ( X , p; Rd)). In Section 3. I , the MKP is presented and two essential Q priori results are stated (Propositions 3.1 and 3.2). Their proofs are given in Section 3.2. Then, our rearrangement and polar factorization theorems are deduced in Section 3.3. Finally, the existence proof for the MKP is given in Section 3.4.
2. The Abstract Concept of Polar Factorization
2.1. Definition and Examples
In this section, the notion of "polar factorization" is defined in a general framework that involves: ( a ) a real Hilbert space H , with Hilbert product (( -,.)) and norm 11 * 11;
VECTOR-VALUED FUNCTIONS
387
( b ) a closed bounded subset S of H ; ( c ) a composition rule * : S X H H such that ( S , * ) is a semigroup, with neutral element e, which means:
--f
sI*s2 S E
whenever
sI,s2E S , s*e
=
e*s
=
s, for all s E S .
It is assumed that
Then, the "polar cone"
(19)
K
=
{ u E H ; ( ( u , e - s)) 2 0, Vs E S }
is introduced and the following definition is stated. Abstract Polar Factorization It is said that there is a polar factorization of the Hilbert space H by the semigroup S and the polar cone K if there is a negligible subset N of H (in the sense of Baire: H\N contains a dense countable intersection of open sets), such that ( 1 ) for every u E H\N, there is a unique factorization u = s * k , (s, k ) E S X K, (ii) u (s, k ) is continuous on H\N. Let us review the examples of polar factorization already considered in the introduction.
--f
Example 1. Polar factorization of complex numbers. Here H = 43 = R 2 (43 is considered as a real Hilbert space), llzll = I z / , S = {e", 0 E R/2aZ},and the composition rule is the usual multiplication of complex numbers. Then, one gets K = R N = (0). , Example 2. Polar factorization of real d X d matrices.
Here
S={UEH
suchthat
UTU=UUT=I),
and the composition rule is the usual multiplication of real matrices. Then, the
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Y. BRENIER
"exceptional" set N is the set of all degenerate matrices and K is the convex cone of all symmetric non-negative matrices.
Example 3. Polar factorization of vector-valued functions. Let us compare our factorization theorem, as stated in the introduction, and our abstract definition of polar factorization. For simplicity, we assume ( X , EL) = (fi, P ) = (0, I . I ) and p = 2. Our Hilbert space H now is the space L2(9;R d ) and I(.11 is the L2-norm. The set S is the set of all measure-preserving mappings s from (0, I . I ) into itself, which means
The composition law * is the usual composition law for mappings * = ,. and is well defined on S X H . Indeed, since each s E S preserves the Lebesgue negligible sets and each u E H i s a Borel mapping up to a possible change on a Lebesgue negligible set, uosis a well defined Borel mapping. It is a fairly elementary exercise to check that property ( 18) is satisfied (because of property ( 2 0 ) ) ,(S, .) is a semigroup, and S is a closed bounded subset of H . So, our "factorization theorem" 1.2 would exactly fit into the framework of the abstract definition, if one could prove that ( a ) the set N of all u E L2(Q; R d ) for which the nondegeneracy condition (9) fails is negligible in the sense of Baire (i.e., contained in a countable union of closed subsets with empty interior); ( b) the set {O$; E W ' , 2Q ) , $ convex} is the polar cone defined by ( 19), $ ( namely: (21)
K
=
{ u E L z ( QR d ) ; ;
s,
u ( x ) * ( - s ( x ) ) d~ 2 0, VS E S}. x
We did not succeed in proving the first statement (although it is easy to check that H \ N is dense). However we were able to prove: PROPOSITION (Characterization of the cone K ) . The cone K defined by 2.1 (21)exactlyi.s {V$;$E w'.*(n), $convex}. The proof is at the end of Section 2.3. Let us now get back to the abstract framework of the polar factorization concept.
2.2. An Abstract Polar Factorization Theorem
We do not know exactly which conditions H , S , and * must satisfy to enforce the polar factorization of H by S and K . However, we can prove:
THEOREM (Abstract polar factorization). U ( S ,* ) is a group, then the 2.1 polar factorization o H by S and K holds. Moreover, for each u E H \ N : f ( a ) there is a unique Hilberl projection ?r( u ) on S;
VECTOR-VALUED FUNCTIONS
389
( b ) thefactor s E S in the polarfactorization u = s* k , (s, k ) E S X Kprecisely is T ( u ) ; } ( c ) { ~ ( u ) = d J ( u ) , where J is the Lipschitz continuous convex function J ( u ) = SUP,,S((U, s)). Notice that J is the Legendre-Fenchel transform of the indicator function of S,
IS( u ) = 0 if u E S , = fco otherwise.
The proof is essentially based on the group property (the semigroup property is not sufficient) and Edelstein's theorem [4] that asserts the following: THEOREM (Edelstein's theorem). Let S be a closed-bounded subset ofa 2.2 real Hilbert space H . Then, the set o all u E H for which there is a unique Hilbert f projection T(u) on S contains a dense countable intersection of open sets H\N, defined by H\N (22)
=
{ u E H ; Yt > 0,36 > 0 such that
t,
/Isl- s2)(5
Vsl,
s2 E S , whenever
llsi - u)I S dist(u, S) + 6, i = 1, 2 ) . Moreover K is continuous from H\N into H . The proof given in [4] follows from the Ekeland-Lebourg theorem on local ``cdifferentiability". Let us sketch a more direct proof. Proof of Edelstein's theorem: defined by Let us show that, for a fixed t > 0, the set T,
T, = { u E H ; 36 > 0 such that
(Is, u / ( 5 dist(u, S) -
+ 6, i = 1 , 2 )
is a dense open subset of the Hilbert space H. The rest of the theorem then easily follows. The fact that T, is open is almost obvious. Indeed, assume that u belongs to T, and take any u E H such that IIu - uII 5 6 / 3 . Now, consider the set A = ( s E S ; 1s - 1 1 5 dist(u, S) 6 / 3 } . If s,, i = 1,2, belong to A , then 1
+
which implies llsl - sz(l S t, since u belongs to T,. follows that u also belongs to It T,, which shows that T, is open. To prove that T, is dense in H , let us consider u E H and set d = dist( u , S). If d = 0 (i.e., u belongs to S), then u obviously
390
Y . BRENIER
belongs to T,. Indeed, the diameter of { s E S ; 1s - u 11 5 dist ( u , S ) + t/2 ] is less 1 than t. If d > 0, it is not restrictive to assume u = 0, d = 1. Then, there is so E S such that llsoll = R for some R L 1 arbitrarily close to 1. Let us introduce u, = r s o / R , where r 10, 1 [ is arbitrarily small, and show that u, belongs to T,, if R is appropriately chosen. Let us consider the set A = {s E S ; (Is- u,(( I dist(u,, S ) 6} and find 6 > 0 so that the diameter of A is smaller than t. We have dist (u,, S ) d IIu, - soil = IIrso/R - soil = R - r. Thus, A is contained in the ball B(u,, R - r + 6 ) . Since d = dist(0, S ) = 1, S is contained in { v H ; llvll h 1 ) and, therefore, A is also contained in { u E H ; l\ull 2 1). For any v E A , let us estimate 11 = llvl12 - 2 ( ( v , so)) R2. Because 1 1 - u,II I - r + 6 and u, = R S and 1 1 r s o / R , we get llul12 - 2 r / R ( ( v , so)) r2 5 ( R - r Ilv112( 1 - R / r ) R 2 R / r [ ( R - r 6 ) 2 - r 2 ] . Since R > r a n d llvll 2 1, we deduce 112) - s0ll2 5 1 - R / r R 2 R / r [ ( R - r - r 2 ] = ( R / r - 1)( 1 R 2 26R) a 2 R / r . By choosing first R , close enough to 1, and, then, 6 > 0 sufficiently small, the right-hand side becomes smaller than c2/4,which proves that the diameter of A is smaller than t and shows that u, belong to T,. Since (Iu u, 11 = r is arbitrarily small, this proves that T, is dense in H and achieves the main part of the proof of Edelstein's theorem. Let us now get back to the polar factorization framework. In the case considered in Theorem 2.1, S is a bounded closed subset of H and, therefore, Edelstein's theorem can be applied. Moreover, S is contained in a sphere centered at the origin. Indeed,
+
+
+
+
+
+
+
+
+ +
+
+
immediately follows from ( 18). This allows us to use the following characterization of the projection operator K.
f PROPOSITION (Characterization of K), Let S be a closed subset o a sphere 2.2 centered at the origin in a real Hilbert space H . Then, the projection operator K H + S can be characterized as the gradient of the Lipschitz continuous funclion
More precisely, one has aJ( u ) = { K( u ) } , for all u E H\N, where H\N is defined b (22). Y
Before proving this proposition, let us first deduce Theorem 2.1.
Step 1 (existence of a polar factorization). By Edelstein's theorem and P r o p osition 2.2, we know that, for every u E H\N (where H\N is defined by ( 2 2 ) ) , there is a unique projection s = K( u ) E S. Since S is a group, k = s-' * u is well defined in H and u = s* k . Thus, to prove the existence of a polar factorization of
VECTOR-VALUED FUNCTIONS
39 1
u , it is enough to show that k belongs to the polar cone K . This can be easily deduced from the following calculations: for every a E S ,
( ( k ,e - a))
=
( ( s * k ,s - s * a ) ) (by property (18))
(by definition of k )
= (( u , s - s* a))
=
f { J J u s*cIJ -
- IJu- sIJ
'> (because S is contained in a sphere)
2 0 (since s = x ( u ) ) .
Step 2 (uniqueness of the polar factorization). Let u E H\N. Assume that there is s E S and k E K such that u = s* k . Since k = s-' * u (because of the group property), it is sufficient to prove that s is unique and, more precisely, that s is the unique projection x I() of u on S , that is (( u, s)) 2 (( u , a)), for all a E S, or, ( equivalently, ( ( s * k ,s)) 2 ( ( s * k , a)). By property (18); this exactly means ( ( k , e ) ) L (( k , s-' * a)) and is always true, since k is assumed to belong to K . Step 3 (continuity of the polar factorization). Let u, = s,*k, E H\N that , converges to u = s*k E H\N. By Edelstein's theorem, s = x ( u , ) converges to s = x u ) . Let us prove that k, converges to k . By ( 18) we have: (
Since s a n d u are fixed, s,*s-'*u converges to s * s - ' * u = u (by (18)). Since u, converges to u (by assumption), it follows that k, converges to k . This achieves the proof of Theorem 2.1, provided that we prove Proposition 2.2. Proof of Proposition 2.2: Since S is contained in a sphere centered at the origin, for each u E H , s E S is the Hilbert projection of u on S if and only if s maximizes (( u , s)) (indeed, J J u sJI = )Iu 1) * cst - 2 ( ( u , s))). It follows from the definition of J (24) that
+
To prove that x is the gradient of J , we shall use two elementary lemmas of convex analysis. LEMMA 2.1. For any u E H , dJ( u ) is contained in the closure of the convex hull of S in H . LEMMA 2.2.
For any u E H and any p E dJ( u ) , ( ( p , u ) ) = J ( u )
Before proving these lemmas, let us deduce Proposition 2.2 from them by showing dJ( u ) = { ?r( u ) } for every u H\N.
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Y . BRENIER
Step 1. It is easy to check that T(u ) belongs to dJ( u ) , which means J ( v ) 2 J( u ) + (( T( u ), v - u ) ) , for all 2, E H . By (25), this is equivalent to J ( v ) 2 (( T( u ) , v ) ) , which immediately follows from the definition of J (24).
Step 2. Let us prove that p = T ( u ) for any p E d J ( u ) . By Lemmas 2.1 and 2.2, we know that ( ( a ,u ) ) = J ( u ) and p belongs to the closure of the convex hull of S. Since u belongs to H\N, by definition (22) of H\N in Edelstein's theorem, for each fixed t > 0, there is 6 > 0 such that every s E S
Let us choose y > 0 so that y 11 1.4 11 5 $6'. Because p belongs to the closure of the convex hull of S , there is a convex combination
such that
which implies
Let us introduce
a; = 2 J ( u ) - 2((s;, u ) ) = 2((7r(u), u ) ) - 2((Si, u ) ) =
IIu - s;11*
- IIu
- 7r(u)112.
We have ai 2 0 and C I E Ierar5 6*. Thus, by Chebyshev's inequality, there is a subset I` of Z such that a, 5 6 for each i E I' and CiElt Oi I1 - 6. By (26) we deduce that [IT( u ) - s,11 I 2 for each i E Z'. It follows that t/
(since x i , I \ ~ ` 0; 5 6 and IIsiII = llell). Because y and 6 can be chosen so that y 6llell I where c is arbitrarily small, we conclude that p = T ( u ) , which achieves c/2 the proof of Proposition 2.2 Let us now prove Lemmas 2.1 and 2.2. Lemma 2.1 is a direct consequence of the definition (24) of J (use the Hahn-Banach theorem, for instance).
+
VECTOR-VALUED FUNCTIONS
393
Proof of Lemma 2.2
Ifp E dJ( u ) , we have J( v ) 1 J ( u ) ( ( p , v - u ) ) ,for all v E Ha n d , in particular for v = 0, 0 = J ( 0 ) 2 J ( u ) - ( ( p , u ) ) . Thus J ( u ) S ( ( p , u ) ) . By Lemma 2.1, belongs p to the closure of the convex hull of S . Thus ( ( p , u ) ) 5 sup{ (( s, 2 1 ) ) ; s E S } . Since this supremum precisely is J ( u ) , it follows that ( ( p , u ) ) = J( u ) , which achieves the proof.
+
Finally, the proof of Theorem 2.1 is completed.
2.3. Proof of Proposition 2.1
Let us split the proof into three steps.
Step 1 (each u E K is the gradient of a function in W'.2(R)). An easy way to build a family of Lebesgue measure-preserving mappings from !? into itself is to integrate a smooth, compactly supported in Q , divergence free vector field w ;see [ 11. The corresponding flowmap t E R g( t ) = exp( t w ) gives, for each fixed t , a smooth Lebesgue measure-preserving mapping g( t ) E S (indeed, g( t) is a diffeomorphism from !? into itself and the jacobian determinant det (D,g( t , x)) is identically equal to 1). Moreover, g ( t ) = e tw O ( t 2 ) t + 0, where e denotes the , identity map. By definition ( 2 I ) of K , for each u E K and each s E S , u ( x ) * ( X - s(x)) dx 2 0. Thus
-f
+ +
sn
follows, by setting s = g( t ) and, therefore, u( x ) . w(x ) dx must vanish for any smooth compactly supported divergence free vector field w defined on R. This implies (see [12], Chapter 9A) that there is a distribution # such that u = V# in the sense of distributions and, since u E L 2 (R; R d ) , $ belongs to the Sobolev space w',*(R).
Step 2 ( u is monotone and I.is convex). I convex, it is enough to show that
(27)
To prove that the potential
I )
is
( u ( x I - u ( x Z ) ) . ( x I x 2 ) L 0, for a.e. x I ,x2 E 0. ) -
Because u is Lebesgue integrable, almost every point x E R is a Lebesgue point, which means
where B denotes the unit ball in Rd.
394
Y. BRENIER
Let us consider a pair ( x I, x2)of such points. When t is small enough, then the following mapping s, is Lebesgue measure-preserving from 9 into itself
(29)
s,(x)
+ x2 = x - x2+ x I
x -xI
x,
if x E B ( x l ,t ) if x E B ( x 2 ,t )
otherwise.
Since u belongs to K , we have
u ( x ).(x - s , ( x ) )dx 2 0, that is
which immediately leads to (27) and shows that u = V$ is monotone and, therefore, J/ is convex. Notice that this result could have been obtained in just one step by using the following characterization of the subdifferential u of a convex function $, due to Rockefaeller [ 8 ] ,
c
I =
u ( x ; ). ( X i
-
xi-
1)
10
I.,
for (almost) all finite sequences xl , . . . , x, = xo of points in 9. This inequality would have been directly obtained by using the following volume-preserving mapping
s,(x)
=
(30)
x
- x i +I
+ xi
if x E B ( x i , t )
x , otherwise,
*
for t small enough and for any sequences of Lebesgue points x I, *
,x,
=
xo.
Step 3. To achieve the proof, let us show that V$ belongs to K for each convex function $ in W ' . 2 ( Q )This immediately follows from the convexity property, . since
$(s(x)) 2 $(x)
+ V$(x).(s(x)
-
x ) , a.e. in
D
holds for every Lebesgue measure-preserving mapping s and, after integrating over 9,
which means that V$ belongs to K .
VECTOR-VALUED FUNCTIONS
395
2.4. Proof of a Comment on the Rearrangement Theorem
In the introduction, we asserted that property
s,
U : ( x ) * U : ( x ) dx
2
s,
UI (x) U 2 ( X )
dx
is not true for all uI , u2 in L2( R d ) , where U : = Vgi, i = 1,2, are the rearrangements 9; of u I , u2, when the dimension d is larger than 1 (for d = 1 and Q = 10, 1 [, this property always holds). Let us now justify this statement. If property ( 3 1 ) were true, it would follow that
holds for every pair ($, , $2) of convex functions in 9 ) and any measurepreserving mapping s. By using exactly the same argument as in the previous subsection (where we proved that any u K is monotone), we would deduce
It is easy to fine a counter example. Take, for instance,
where A A2 are two symmetric non-negative real d X d matrices. Then, property ( 3 3 ) would exactly mean that A l A 2 + A 2 A I is symmetric non-negative, which cannot be true in general. Indeed, if
A1=(0 6) ' O
A2=(:
;),
then A I A2
+ A 2 A is not non-negative for 6 > 0 sufficiently small.
3. Proof of the Main Results
3.1. The Monge-Kantorovich Problem
Our proof is based on the study of a particular Monge-Kantorovich problem ( M U ) (see [ 191 for a general review). We are given a probability measure a on R d such that ( 1 11 y 11 ) da( y ) < co and the three following problems are considered.
+
+
396
Y. BRENIER
The Primal MKP
Find E C(Rd)n L l ( R d , a ) and )I E C ( Q ) L ' ( 8 ,P ) that minimize fl and satisfy the following conditions.
(b
da
$(y)
+ $(z) 2 y - z ,
V ( y , 2 ) E R d X 9.
The Dual MKP
Find a probability measure p on R d X fi that maximizes y * zd p ( y , z ) under the a following conditions: 11 y 11 d p ( y , z ) < +a, and P are the marginals ofp on R d and 0. which means
s
s
for each f E C ( R d )such that
If(
y ) I 5 cst( 1 + 11 y 11 ), and
The Mixed MKP
Find I$ E C(Rd)fl L1(Wd, ) , J. E C ( Q ) L 1 ( Qp ) and a probability measurep a fl , on R d X 0 such that
on R d and fi, $) I dB = 0, and $ 4 doc 5 y - z d p ( y , z ) . Notice that, in the dual and the mixed MKP, p necessarily is a tight probability andp(Rd X dQ) = P(dQ) (since measure on R d X 0 (since IIyII d p ( y , z ) < +a) /3 is the marginal of p) = 0 (by assumption, cf. Introduction). Our study of these three problems involves in an essential way the subset KOof the Sobolev space W 'J( Q, p ) , defined by:
$ 11 y 11 d p ( y , z ) < +a,,a and /3 are the marginals of p
s
34 : Rd --* 08 U {+a convex, l.s.c., such that },
tc/
=
$ on Q } .
VECTOR-VALUED FUNCTIONS
397
An existence proof will be given in Section 3.4 for the mixed MKP. From this existence proof, our main results (as listed in the Introduction) will be obtained in Section 3.3, with the help of the following "a priori" results.
3.1. Assume that the mixed MKP has a solution (I), 4,p ) . PROPOSITION Then : ( i ) I belongs to KO ) = by (34)) and IlVI)ll~1(n.~) S IIY /I d 4 ~ ) ; (ii) 4 = I)*, a-almost everywhere on Rd, where I)* is the Legendre transform of I) defined by
(iii) d p ( y , z) = S ( j 1 - VI)(z))P(z) dz; (iv) J 4 d a = S y * z d p ( y , z ) a n d J I ) d P - O ; (v) ( ) 4) is the unique solution to the primal MKP andp is the unique solution I, of the dual MKP. Moreover, if a is absolutely continuous with respect to the Lebesgue measure, then ( i ) z = VI)*( y ) , y = VI)( z ) , p-almost everywhere on R d X (ii) VI)*(VI)( z ) ) = z, @-almosteverywhere on and VI)(VI)*( y ) ) = y , a-almost everywhere on R d ; (iii) dP(Y, Z ) = S ( Y - V I ) * ( Y )d 4 Y ) . )
a;
PROPOSITION Let (a,) be a sequence of probability measures on R d such 3.2. that J f d a , + J fdcY,for a n y f E C ( R d )such that If(y)I 5 cst(1 + IIylI). r f (#,, &, a,,) is a solution to the mixed MKP corresponding to a,, then the mixed p MKP corresponding to a has a unique solution (I), 4, ) and 4, + I) uniformly on P any compact subset ofRd, I), + I) in W ' , ` ( Q ,) , f dpn + f dp, for each f E C(Rdx St) such that If( y , z ) I 5 cst( 1 + II y 11).
s
s
From the existence result (proved in Section 3.4) and Propositions 3.1 and 3.2, we finally get:
THEOREM (Solution of the MKP). The mixed MKP has a unique solution 3.1 (I), 4, p ) , (I), 4)is the unique solution of the primal MKP, and p is the unique solution of the dual MKP. All the properties listed in Propositions 3. I and 3.2 are satisfied.
3.2. Proof of Propositions 3.1 and 3.2
Notice that, since C2 is bounded, fi is contained in the ball B( 0, r ) for some r > 0. To prove Propositions 3 . I and 3.2, we shall use a preliminary result on the set KO defined by ( 34 ) .
398
Y . BRENIER
PROPOSITION (Main properties of the set K O ) . 3.3 ( a ) For each in KO,thefollowing properties hold: ( i ) - 2 r M 5 + ( z ) , o r a l l z E Q, whereM= J IVl dP, f I+l (ii-a) +( z ) 5 2r 11 V+( z ) 11, almost everywhere on Q, (ii-b) +( z ) 5 C( F)rM,for all z E Q, where F = dist( z , an), (iii) J [ + I dp 5 4rM. (iv) The Legendre-Fenchel transform +* of +, defined by
+
is Lipschitz continuous on R d and satisjes - r y 11 5 +*( y ) i II y II + 2rM, for all r r y E Rd, Lip(+*) i and +( z ) = +* *( z ) = sup,, R d { y ' z - +*( y ) } ,for all z E 0. ( b ) Let (+,,) be a sequence in KOsuch that 5 I V ,11 d p 5 M . Then, there is a I+, subsequence, still labelled by n , and a pair (+, 4) such that
+,, +
+?
+ in L ' (0, 0)and uniformly on any compact subset of 52,
4 uniformly on any compact subset of R d
+
Proof of Proposition 3.3. Let us first recall that Q is supposed to be contained in the ball B( 0, r ) for some r > 0 and that W I.'( Q, p ) is supposed to be compactly embedded into L ' ( R, p ) .
Step 1. Let E KO. Since is continuous on Q and J d o = 0, there is zo f Q such that +(zo) = 0. Since belongs to W ' . ' ( Q p ) , V+ is defined almost ,
+
+
+
+
everywhere on Q (with respect to both ,f3 and the Lebesgue measure, since these measures have the same negligible sets, by assumption). By convexity, one gets +(zo) 2 + ( z ) + V+(z)*(zo - z , for almost every z E 0, and, thus, + ( z ) S ) -V+( z).( zo - z) 5 2r IV( z)11 (since z,zoE R C B( 0, r ) ) . We also have for I+ almost every z , 2 E Q, +( z ) 2 +( 2 + V ( 2 (z- .?), and, hence, after integrating ) + ) this inequality over Q with respect to 5,
+ ( z ) s d p > s + ( i ) / 3 ( i ) d 2 + V+(F)-(z-i)p(Z)d2.
Since p is a probability measure and follows that
s
J + dp = 0 (because + belongs to K O ) ,it
VECTOR-VALUED FUNCTIONS
399
+ ( z )2
J O$(Z) . ( z - z)P(z")di L
-
2r
J 11vrc.11 dp
holds for almost every z E R. So far, we have proven -2rM 5 $(z ) I IIV$( z ) 11, 2r almost everywhere on R. By integrating this inequality over R, we get
Let us now fix z E R and set 6 = dist(z, dR) > 0. If B denotes the unit ball in Rd, then Q contains both z 6 / 2 B and 0 = { ZE R; dist( 2, do) 2 6 / 2 } . By assumption (cf. Introduction), p is essentially bounded away from 0 on 0 by some constant p( 6) > 0. Thus, $ is Lebesgue integrable on W and
+
By convexity, $(z) 5 Thus (since W contains z
I z +'-I.;
J
zidl2B
$(i) di
+ 6/2B),
which means that there is a constant C(6) such that $(z) 5 C(G)rM,for all z E 9, where 6 = dist(z, d o ) .
Step 2. By definition (35), $* is well defined and satisfies $ * ( y ) 2 z o . y $(zo) = z o . y (by definition of zo), L -rllyll, and
-
Moreover $* is Lipschitz continuous on R d ( indeed in definition (35) the supremum is taken over R C B(0, r ) ) and Lip($*) I r. Since $ = on R, where $ is a convex 1.s.c function R d + R U { +oo } ,we also have $* * = $ on R (this is a classical result in convex analysis; see [ 131).
4
Step 3. Let ($,,) be a sequence in KOsuch that former estimates, it follows that
11 V& 11
dp 5 M . From the
400
Y . BRENIER
(i) ( $ n ) is a bounded sequence in W'.'(O,p ) and, therefore, by assumption (cf. Introduction); ( qn)has a convergent subsequence, still labelled by n , in L ' ( Q, p ) . Moreover, since -2rM 5 (Cln(z)I C(G)rM,where 6 is dist(z, an), for any z E a,the $H are uniformly bounded and, thus, uniformly Lipschitz continuous (cf. [ 131) on any compact subset of R; (ii) ($$) is uniformly Lipschitz continuous on R d and
l$:(Y)l
r(2M+
Ilvll),
VYERd.
Thus, there is a subsequence, still labelled by n , and a pair 4 E C(Rd),$ E c(n) n L'(s2, p ) , such that
$$ -+ 4, uniformly on any compact subset of Ed,
tc,,
+
tc, in L ' ( 52, P ) and uniformly on any compact subset of Q.
Moreover 4 satisfies I 4( y ) I 5 r( 2 M
+ 11 y 11 ). Notice that
v z E n, y E Rd
4 ( y )+ $ ( z ) L y . 2 ,
immediately follows from the definition of $ and the convergence properties. This : achieves the proof of Proposition 3.3. Proof of Proposition 3.1: The proof of Proposition 3.1 which, in our opinion, is the most important of our intermediary results, relies on the following wellknown property of convex conjugate functions $, $*, namely $( z ) + $*( y) = y * z , if and only if z E a$*( y) and y E a$( 2). The uniqueness and the precise characterization of the solution to the mixed MKP (whenever it exists) follow from this elementary property.
Step 1. Let (4, $, p ) be any solution to the mixed MKP. Let us first show that $ E KOand (4, $, p) are linked together by the following relations
4
=
$*, a - a.e.,
s s
4 da
=
y . z d p ( y , z ) and
Let us first introduce
Since (4, $, p) is a solution to the mixed MKP, we have
4( y) + $( z) 2 y .z ,
vz E n, y E Rd.
VECTOR-VALUED FUNCTIONS
40 1
Thus b(y ) 5 4( y ) , for all y E Rd. For some fixed zo E 52, one has b(y ) 2 y * zo l( zo), for all y E Rd. So is finite everywhere and, because R is contained in B(0, j r ) , it follows from definition (36) that 4 is convex and Lipschitz continuous on R d with Lip($) 5 r. Let us now introduce
Here & is a well defined convex 1.s.c. function from R d into R U { +a} and 3, are convex conjugates (see [ 131): & = i*, Moreover, one gets from defi= $*. nitions [ 361 and [ 37 ]
4
4
& ( z )5 l j ( Z ) ,
$ ( z )2
-4(O)
> -a,
vz E R.
Since we know that J- ( I + Ilyll) d a ( y ) < f a , ( 1 + IjzlPz dz < +a, l())() it follows from the bounds on & and 4 that they respectively belong to L ( Q , P ) and L ' ( R d ,a ) . In addition,
'
and, by definition of &, & y ) & ( z )2 y * z ,for all y , z E Rd. Thus, since a and P are the marginals of p , one deduces
+
The right-hand side of the last inequality is not larger than 0, since (4, p ) is a lj, solution to the mixed MKP. It follows that these inequalities actually are equalities and (since S 6,4 S lj)
4
i 4, a - a.e., =
138)
=
3. = lj, P - a.e.,
p - a.e. on I W x ~
=
$(y)
=
+ $ ( z >= y - z ,
2
0,
J ida J 6 da J y .
dp( y , z ) ,
J 4 dp J lj dp = 0.
402
Y. BRENIER
4 is equal to the continuous function $, ,&almost everywhere on Q and /3 has the same negligible sets as the Lebesgue measure, it follows that 5/ is locally Lipschitz
continuous on f (cf. [ 131) and, therefore, equal to $ everywhere on Q. Moreover 2 V$ is well defined (up to a Lebesgue negligible set) as a Borelian mapping from Q into R d and a$( z) = { V$( z ) } holds almost everywhere on Q (here a denotes the subdifferential of a convex function; see [ 1 3 ] ). According to a well-known result in convex analysis (see [ 13]), it follows from properties ( 3 8 ) that
z E & ( y ) , y E a$(z)p
-
By definition of the mixed MKP, $ belongs to C( Q ) . Since the convex function
a.e. ( y , z) E
x s2.
Since (Y and /3 are the marginals of p on fi and Rd, and because /3 has the same negligible sets as the Lebesgue measure, we deduce
P({ ( y , z ) R d x
a;
M z )
=
+ {V$(ZH 1) = 0
(after noticing that p ( R d X dQ) = P ( d Q ) By combining these properties, we get
0, by assumption (cf. Introduction).
(39)
y
=
v$(z), p - a.e. ( y , z ) E R~ x
SZ.
Let us now consider an arbitrarily chosen function f E C(R d X
n) such that
If(
Y , z) 5 cst( 1 + II y II 1,
v (v,2) E R x 32.
Since (4, $, p ) is a solution to the mixed MKP, ( 1 + 11 y 11 ) d p ( y , z ) must be finite, and, therefore f is p-integrable. Because of property (39), we have
and the right-hand side of this equality is Jf(V$(z), z)p(z) d z , because p is the marginal o f p on 0. This shows that d p ( y , z) = 6 ( y - V$(z))p(z) dz. Moreover, in the particular case whenf( y , z) = 11 yll, one gets
which shows that $ belongs to the Sobolev space W1.l 8 , p ) (we already know that ( $ belongs to C( Q ) U L 1 ( p ) ) . Finally, since $ = li/ on Q, we see that $ belongs Q, to the set KO, IlV$IIL~(n:@,= J llyll d 4 Y ) . So, the proof of the first four statements of Proposition 3.1 is completed.
Step 2. Let us now prove the fifth statement of Proposition 3.1 by showing that, for any solution (4, $, p) to the mixed MKP, (4, $) is the unique solution to the primal MKP and p is the unique solution to the dual MKP.
VECTOR-VALUED FUNCTIONS
403
Let us consider a solution (4, p ) to the mixed MKP, any solution $, ) to the primal MKP, and any solution pI to the dual MKP. It is straightforward to check that (4, pI) and ( d I , p ) solve the mixed MKP. Indeed, $,
s
y . z dp(y, z ) 5
s
y.2
&I(Y, z )
(because pl solves the dual MKP)
(because 4 1 ( y )+ $ , ( z ) 2 y . z for every ( y , z) E R d X fl and p r ( R dX dfl) D(dfl) = 0 )
=
(since a and , are the marginals of p~) 8
(because ( 4 1 , ) solves the primal MKP)
(as just shown). Thus, all these inequalities become equalities. It follows that $ I ,p ) and (4, pI) also solve the mixed MKP. According to the first step of the $, proof, it follows that
$1
E KO,
4I = $ 7 ,
a
-
a.e.,
y
y
=
=
V $ , ( z ) , p - a.e. on
R~ x R* x
0,
v() +z,
p - a.e.
on
a.
Thus pI = p and V$l = V$, almost everywhere on fi (since p is the marginal of p on fi and p has the same negligible sets as the Lebesgue measure). Since $ belong to KO and $2 is supposed to be connected, we deduce = $ and dl = $7 = $* = 4,a-almost everywhere. This shows that (4, is the unique solution to the primal MKP and p is the $) unique solution to the dual MKP (notice, however, that 4 is uniquely defined up to some a-negligible subset of R d ) .
404
Y. BRENIER
Step 3. Let us now consider the particular case when a is absolutely continuous with respect to the Lebesgue measure. We know that $* is a Lipschitz continuous convex function defined on Rd, and, therefore, V$* is a well defined (up to some Lebesgue negligible set) Bore1 mapping from R d into itself. Since a is absolutely continuous with respect to the Lebesgue measure, the set { y E Rd; d $ * (y) # { O $ * ( y ) } } is a-negligible. It follows that a$*( y ) = { V$*( y ) } is true for p-almost everywhere ( y , z ) E Rd X since a is the marginal ofp on R ~ . We already know that z E a$*( y ) and y = O$( z ) for p-almost everywhere ( y , z ) . Thus, z = V$*( y ) and y = V$(z) holds for p-almost everywhere ( y , z ) , and, since a and /3 are the marginals of p , we get z = O$*(V( $( z ) ) for /3-almost every z E and y = V$(V$*( y ) ) for a-almost every y E Rd. Thus V$ and O$* are reciprocal. Moreover d p ( y , z ) = 6 ( z - O # * ( y ) ) d a ( y ) immediately follows, which achieves the proof of Proposition 3.1.
a,
Proof of Proposition 3.2: Let (a,) be a sequence of probability measures on R d such that
$ , From Let (6n, p,) be a solution to the mixed MKP corresponding to ( a n ) . Proposition 3.1, we know that
From Proposition 3.3, we deduce that, for a subsequence labelled by m ,there is a pair (4, such that $)
@ ( y ) $ ( z )2 y . z ,
$m +
+
V ( y , z ) E Rd x Q.
$ in L I ( Q, 0) and uniformly on any compact subset of 3,
6m+ 6 uniformly on any compact subset of Rd,
VECTOR-VALUED FUNCTIONS
405
Because (&, #, p,) is a solution to the mixed MKP, we have ,
Thus # dP = 0 and 4 da d lim 4, dam.(This follows from ( i ) the uniform convergence of (&) on every compact subset of Rd, (ii) the uniform bounds
s
s
s
Let us now consider the sequence (p,). The marginals ofp, are a,,, and P. Moreover, by assumption,
Thus, for a subsequence, still labelled by m , there is a positive measure p on R d X Q , such that
for any compactly supported continuous function f o n R d X
G, and
This implies that (i) p is a (tight) probability measure on R d X fi with marginals a,P and that (ii) equation ( 4 1 ) holds for any continuous function on R d X i7 such that
In particular,
So, J y z dp( y , z ) 2 4 da, J # dp = 0 and, therefore, (4, #, p ) satisfies all the conditions required to solve the mixed MKP. By Proposition 3.1, ( 4 = $*, $) is the unique solution to the primal MKP, p is the unique solution to the dual MKP, and
s
406
Y . BRENIER
Now, it is easy to deduce that (qm) strongly converges to $ in J+".'( Q, p ) (instead of L ( 9, p ) , as already obtained! ) . Indeed, we get from equations ( 4 1 ), (40), and (43)
'
for any continuous function satisfying (42), which is possible only if $rnstrongly converges to $ in W','(Q, 0 ) . Because of the uniqueness of the solution (4, $, p ) to the MKP, the whole sequence ($,) converges to ($) in W I - ' ( 9, p ) . In the same way, 4, + 4 uniformly on any compact subset of R d and p , converges t o p (in the sense of ( 4 1 ) and (42)). This achieves the proof of Proposition 3.2.
3.3. Proof of the Main Theorems
Let us recall that the Rearrangement Theorem 1.1 is a corollary of Theorem 1.3 that we are going to prove with the help of Theorem 3.1 and the following characterization result. PROPOSITION (Characterization of O$) 3.4
Let $ E KOsuch that
(44)
for e a c h f E C ( R d ) such that IJ(y)I 5 cst( 1 + IIyII). Then (4 unique solution to the MKP corresponding to a,where
=
$*, $, p ) is the
Before proving this result, let us first deduce Theorem 1.3 from Theorem 3.1 and Proposition 3.4. The first part of Theorem 1.3 directly follows from these results. Let us briefly prove the second part. Let a, be a sequence of probability measures such that J f da, + J f da, for every f C ( R d ) such that I f ( y ) I S cst( 1 + Ilyll). From the results on the MKP (Proposition 3 . 2 ) , we know that the MKP corresponding to a, has a unique solution (4, = $ X , $ p,) that converges , to the unique solution (4= $*, $, p ) of the MKP corresponding to a. In particular, $ converges to $ in W ' , I ( 0, p), which achieves the proof. ,
VECTOR-VALUED FUNCTIONS
407
Proof of Proposition 3.4: By assumption, $ belongs to KO,thus V$ belongs to L ' ( 9,P; R d ) and can be considered as a Bore1 mapping (up to a possible modification on a Lebesgue negligible set) from fi into Rd (recall that P has the same negligible sets as the Lebesgue measure and @(an) 0). So p is well defined as a = (tight) probability measure on R d X fi by equation (45) and
holds for all f C(Rd X fi) such that I f ( y , z ) I 5 cst( 1 11 y 11 ). Indeed, since $ belongs to KO,J ( 1 + llvll) &(Y, 2) = J ( I + IIV$(z) I )P(z> < +a. asdz BY sumption (44) and definition (45), the marginals of p precisely are (Y and p on R d and fi. Since $ E KO,$ is convex, locally Lipschitz continuous on 9 ...