Some extensions of the Prékopa Leindler inequality using Borell s stochastic approach

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1 Some extensions of the Prékopa Leindler inequality using Borell s stochastic approach Dario Cordero-Erausquin and Bernard Maurey Institut de Mathématiques de Jussieu-PRG, UPMC Paris 6 December, 215 Abstract We present an abstract form of the Prékopa Leindler inequality that includes several known and a few new related functional inequalities on Euclidean spaces. The method of proof and also the formulation of the new inequalities are based on Christer Borell s stochastic approach to Brunn Minkowski type inequalities. 1 Introduction and main statement The Brunn Minkowski inequality asserts that for A, B Borel subsets of and t [, 1], the volume of the Minkowski combination 1 ta + tb = {1 ta + tb : a A, b B} verifies the inequality 1 ta + tb A 1 t B t, where E denotes the volume of a Lebesgue-measurable subset E of. There is a long story of functional generalizations of this inequality, that we do not recall here; let us just mention that Borell s 1975 paper [5] remains a milestone in the subject. A somewhat definitive form is given by the following theorem. Theorem 1 Prékopa Leindler inequality. Let t [, 1] and let f, f 1, g : R {+ } be Borel functions such that, for every x, x 1, g 1 tx + tx 1 1 tf x + tf 1 x 1. Then 1 t t e gx dx e fx dx e f1x dx. 1

2 Accepting the value + enables us to reach directly indicator functions 1 E = e f E, by letting f E be on E and + outside. However, we can restrict ourselves to Borel functions that are not Lebesgue-almost everywhere equal to +. Such Borel functions f will be said to be L-proper, which is equivalent to saying that e fx dx >. One can reasonably argue that the interest of the Prékopa Leindler inequality resides not only in its consequences, which are numerous some are recalled for instance in [8, 11], but also in the emphasis it has put on log-concavity, and in the related techniques of proof it has originated, such as mass transportation or semi-group techniques, and more recently L 2 -methods as in [7]. Here, we will concentrate on Borell s stochastic approach [6] to the inequality above. It somehow reduces the inequalities under study to the convexity of 2, the square of the Euclidean norm on. It will allow us to obtain some unexpected inequalities, for instance that of the following proposition. Proposition 2. Let f, f 1, g, g 1 be four Borel functions from to R {+ } such that, for every x, x 1, Then g 2x /3 + x 1 /3 + g 1 x /3 + 2x 1 /3 f x + f 1 x 1. 1 e gx dx e g1x dx e fx dx e f1x dx. We will see that it is rather natural to arrive to this type of inequality using Borell s stochastic approach, whereas it seems not to be the case with other methods, for instance those based on transportation methods. The point here is to split the values of the functions f i s at some points into the values of two functions g j s at some related points. This is not interesting in the case the functions f i s take only the values + and, and the functional inequality above does not give anything new when applied to the case where the functions e f i s are indicators of sets, as we will explain in the Section 4 below. The previous proposition and its proof suggest actually more general inequalities. Writing the conclusion as j log e g j i log e f i, we may think about an extension of the results, where the finite families of functions f i, g j are replaced by families f s, g t depending upon continuous parameters s, t as for instance in [2]. In the basic assumption 1, the two values x, x 1 at the right of the inequality will be replaced, for example, by a selection x = {xs} s [,1] of points of, and the values y = {yt} t [,1] at the left will be obtained from a linear transformation A acting on this data x, i.e. y = Ax. So, roughly speaking, under appropriate assumptions on the operator A, we may expect that if for all x = {xs} we have 1 g t Axt dt 1 f s xs ds 2

3 then it will follow that 1 log e gt dt 1 log e fs ds. Actually, there was no reason, when replacing the sums by integrals, to use the uniform distributions dt and ds rather than probability measures µdt and νds on [, 1], which include the discrete case when these measures are convex combinations of Dirac measures. Anyway, the main question is to understand what are the appropriate conditions to impose on the operator A. Several points must be set before proceeding. We say that a real function F on a measure space Ω, Σ, µ is µ-semi-integrable if at least one of F + = maxf, or F = max F, is µ-integrable. The integral of F takes then a definite value in [, + ]. This assumption is needed for F s = log e fs in order to make sense of the preceding integrals. We introduce the following abstract setting. Setting 3. We are given Two measure spaces X 1 = Ω 1, Σ 1, µ 1 and X 2 = Ω 2, Σ 2, µ 2, where Σ i is a σ-algebra of subsets of Ω i, i = 1, 2. We assume that Ω 1 is a Polish topological space and Σ 1 its Borel σ-algebra. An integer n 1 and a continuous linear operator A : L 2 X 1, L 2 X 2,, where the L 2 -norms of the -valued functions are computed with respect to the Euclidean norm on and the measures µ 1 and µ 2, respectively. We assume that i the operator A satisfies the norm condition A 1, ii the operator A acts as the identity on the constant vector valued functions, i.e., for any v, the constant function Ω 1 s v is sent by A to the constant function Ω 2 t v. Two families {f s } s Ω1 and {g t } t Ω2 of Borel functions from to R {+ } satisfying iii the functions s, x f s x and t, x g t x are measurable with respect to the σ-algebras Σ i B, i = 1, 2, respectively, the notation B being for the Borel σ-algebra of, iv the functions s Ω 1 log e fsx dx and are semi-integrable with respect to µ 1, µ 2, respectively. 3 t Ω 2 log e gtx dx

4 Unfortunately, there is a tradeoff between the generality of the statement we can reach and the technical assumptions on the functions required in the proof. We will start with a fairly general situation. In most applications, the technical assumptions on the functions in the statement below are either easy to impose or to discard, as explained for instance in the Remark 5 and as shown in Corollary 7 below. So, our most abstract version of the Prékopa Leindler inequality reads as follows. Theorem 4. Under the Setting 3 with µ 1 and µ 2 having the same finite mass, µ 1 Ω 1 = µ 2 Ω 2 < +, we make the additional assumptions on the functions: for every s Ω 1, the function f s is non-negative, and for every t Ω 2, the function g t is non-negative and lower semicontinuous, for some ε >, we have that log exp f s x ε x 2 dx dµ 1 s < +. 2 Ω 1 Then, if for every α L 2 X 1, we have g t Aαt dµ 2 t Ω 2 f s αs dµ 1 s, Ω 1 3 it follows that log Ω 2 e gtx dx dµ 2 t log Ω 1 e fsx dx dµ 1 s. 4 Remark 5. We can easily relax the restrictions f s x, g t x. Suppose indeed that f s x as, g t x bt, where as, bt are non-negative functions on Ω 1, Ω 2 that are µ 1, µ 2 -integrable respectively. Assuming, as we may, that Ω 1 a dµ 1 = Ω 2 b dµ 2, we see that the basic assumption 3 and the conclusion 4 are unchanged when passing from f s, g t to the non-negative functions f s + as, g t + bt. So, in the Theorem above, we are free to assume only that the functions are bounded from below in a reasonable way with respect to the parameters. Remark 6. By classical arguments of measure theory, we may replace f s Borel by f s l.s.c. such that f s f s and e f s e fs, but we do not know how to do it on the g t side in a way that 3 remains true. However, when the family {g t } consists of a single function g, it is easy to replace g by a l.s.c. function. The real issue is with the values of the f s functions that are equal to +. If f s, g t are merely non-negative Borel functions, with f s locally bounded, it is possible to reduce the problem to continuous functions {f s } and {g t } by convolution with non-negative compactly supported continuous kernels, hence reducing this case to the preceding theorem. On the other hand, it is hard to believe that our l.s.c. assumption is necessary for the validity of Theorem 4. We rather tend to think that the result is true for general Borel functions as we can prove it in the discrete case of Corollary 7; see also Remark 18. 4

5 Suppose that the conditions of Setting 3 and Theorem 4 are satisfied for {f s } and {g t }. Using the norm condition i, we see that the basic assumption 3 remains true if we add a same multiple ε x 2 of x 2, ε >, to all the functions f s and g t. We can see that after this addition, the other conditions of Setting 3 and Theorem 4 remain obviously true, with two exceptions that are either less obvious or not always true: the semi-integrability condition iv remains true because f s and g t are also assumed to be non-negative, thus e fsx ε x 2 dx Cε and the same for g t, and if we assume < ε < ε, then the condition 2 remains true, with ε replaced by ε ε. It follows that the conclusion 4 holds if we replace the inside integration with respect to the Lebesgue measure on by integration with respect to the isotropic Gaussian probability measure γ n,τ on defined by dγ n,τ x = e x 2 /2τ dx/2πτ n/2, provided 2τ > ε 1. Actually, the proof of Theorem 4 will start with the Gaussian case and obtain the Lebesgue measure case from it, the Lebesgue case being the flat extremal case when τ +. Note that, indeed, the log of the normalization constant 2πτ n/2 on the two sides of 4 then cancels since µ 1 and µ 2 have the same finite mass. In many applications, X 1 and X 2 are finite probability spaces, and we are then working with finite families of objects parameterized by Ω 1 and Ω 2, or rather by the supports suppµ 1 and suppµ 2 ; in particular, the mappings α : Ω 1 are families of Ω 1 vectors of and the linear operator A : Ω 1 Ω 2 is norm-one for the operator norm associated to the l 2 -norms weighted by the µ i s, with the property that the vector x,..., x Ω 1 is sent to x,..., x Ω 2, for every x. The proof of the next corollary will be given in Section 6. Corollary 7. Under the Setting 3, assume in addition that µ 1 and µ 2 are measures with finite support and with µ 1 Ω 1 = µ 2 Ω 2 < +. Then the assumption 3 implies 4 with no further restriction on the functions {f s } and {g t }. Let us describe a particular instance of Theorem 4. Assume that Ω is a Polish space, Σ its Borel σ-algebra and µ a probability measure on Ω, Σ. Take Σ 1 = Σ, let Σ 2 Σ be a sub-σ-algebra of Σ, and let µ 1 = µ 2 = µ. Then the conditional expectation A = E [ Σ 2 ] : L 2 Σ, L 2 Σ 2, L 2 Σ, verifies the norm condition i and the constant functions condition ii from Setting 3. A simple and already interesting case is when we take Σ 2 to be trivial, Σ 2 = {, Ω}, in which case A is simply the µ-mean, Aα = αs dµs, Ω and the space Ω 2 can be then considered as being a one-point space, say Ω 2 = {}. In this case, the family {g t } consists of a single function g and Theorem 4 reads as: 5

6 Corollary 8. Let Ω be a Polish space, Σ its Borel σ-algebra, µ a probability measure on Ω, Σ. Suppose that we are in Setting 3 with Ω 1, Σ 1, µ 1 = Ω, Σ, µ and Ω 2 = {}. Let {f s } s Ω satisfy the assumptions of Theorem 4, and let g be a bounded below l.s.c. function on. If for every α L 2 Ω, Σ, µ,, we have g αs dµs f s αs dµs, Ω Ω then log e g log e fs dµs. Ω The Prékopa Leindler inequality follows by taking Ω to be a two point probability space, for instance Ω = {, 1}, and taking µ of the form µ = 1 tδ + tδ 1 for some t [, 1]. Actually, it is easily seen and certainly known that the previous corollary can be deduced from the Prékopa Leindler inequality, by discretizing µ and by a simple induction procedure. Therefore, it only serves as motivation for the abstract setting above, but does not bring new information. Let us explain how to deduce the more surprising Proposition 2 above. We apply Corollary 7 to Ω 1 = Ω 2 = {, 1}, to the finite measures µ 1 = µ 2 = δ + δ 1 and to the linear mapping A : defined by Ax, x 1 = 2x /3 + x 1 /3, x /3 + 2x 1 /3. We are in Setting 3 since Ax, x = x, x and 2x /3+x 1 /3 2 + x /3+2x 1 /3 2 x 2 + x 1 2. The result therefore follows from Corollary 7. In the next section, we present Borell s approach and establish in Proposition 12 a Gaussian version of Theorem 4; the proof of this Proposition can be considered as the heart of the present paper. In Section 3, we present the proof of Theorem 4. Then, in the following section, we discuss some consequences of it. In Section 5 we present a generalization of Theorem 4, when the functions f s and g t live on Euclidean spaces of different dimensions. The result will include as particular cases the Brascamp Lieb inequality in geometric form and its reverse form devised by Franck Barthe [1]. Several technical proofs that have little geometric interest, and involve mostly measure theoretic arguments, are gathered in Section 6. 2 Borell stochastic approach and Gaussian inequality Borell s stochastic proof of the Prékopa Leindler inequality relies on the representation formula given in the next lemma. Let B r r be a standard Brownian motion with values in, starting at, with filtration F = F r r. Let P r = e r /2, r, be the heat semigroup on associated with this Brownian motion, P r fx = E fx + B r = fx + ye y 2 /2r dy R 2πr, x n n/2 Rn, 6

7 for f bounded and continuous on and r >. An -valued drift u = {u r } r T will be called of class D 2 on [, T ] if it is F-progressively measurable on [, T ] and T E u r 2 dr < +. Lemma 9. Let T > be fixed. For every bounded continuous real function f : R, we have log P [ T T e f = inf f B T + u r dr + 1 T ] u r 2 dr, u 2 5 where the infimum is taken over -valued drifts {u r } r T of class D 2. Moreover, the infimum is attained. Proof see Borell [6]. We begin by assuming that f is bounded with bounded derivatives of order 2. For a drift u = {u r } r T of class D 2, we define Xr u := B r + r u ρ dρ, which satisfies the stochastic differential equation X u =, dx u r = db r + u r dr on the interval [, T ]. For r T, define f r = f T r by e fr = P T r e f, that is to say f r x = logp T r e f x, x. 6 The function r, x f r x on, T satisfies the partial differential equation r f r = 1 2 f r f r 2. By a direct application of the Itō formula, we see that the process M r := f r X u r r is a submartingale for any drift u in D 2, since u ρ 2 dρ, r [, T ], dm r = f r Xr u db r + 1 fr Xr u + u r 2 dr. 2 This implies the inequality case of 5, namely, an upper bound of logp T e f for every drift u in D 2. Indeed, note that M = f X = log P T e f and f T = f, and so the inequality in 5 immediately follows by considering E M r at r = and r = T. Moreover, if u r = f r Xr u, i.e., if Xr u is the process solving the stochastic differential equation X =, dx r = db r f r X r dr, 7 then M r becomes a martingale, thus giving an equality case in 5. Assume now that f is bounded and continuous on. Adding a constant to f, we may suppose that f. Define f r as in 6, and note that f r is bounded above and. 7

8 Since f is bounded, P r e f is bounded away from, so the function f T ε = log P ε e f has bounded derivatives of all orders, for every ε, T ]. Writing T ε = T ε and e fr = P Tε rp ε e f = P Tε re f Tε, r Tε, we are back to the good setting on [, T ε ]. Hence the optimal representation by the martingale M r = f r B r + r u ρ dρ r u ρ 2 dρ = M + r f ρ X ρ db ρ, with M = f, u r = f r X r and X r = B r + r u ρ dρ is valid for r T ε. Since f r, we see that r u ρ 2 dρ 2M r and we obtain that r r E M r M 2 = E f ρ X ρ 2 dρ = E u ρ 2 dρ 2 E M r = 2f. We have T 1 E T u ρ dρ 2 T E u ρ 2 dρ 2f, hence u D 2, X r converges a.s. and in L 2 to X T = B T + T u ρ dρ. We see that M r r<t is an L 2 -bounded martingale, thus M r converges in L 2 -norm, as r T, to a limit M T such that E M T = M. On the other hand, M r converges almost surely to T f B T + u ρ dρ + 1 T u ρ 2 dρ 8 2 when r T since f r converges locally uniformly to the bounded continuous function f. This implies that M T is equal to the expression in 8, hence the expectation of 8 is equal to M = f, that is, it implies formula 5 with equality. In the inequality case of 5, the drift u is in D 2 by assumption, hence X u r r<t defined as above converges a.s. and in L 2 to X u T, as r T. The submartingale M r r<t is L 2 - bounded, hence converges a.s. and in L 2 to the expression in 8, and the result follows. Remark 1. If f is bounded and upper semicontinuous on, then for every x and ε >, there is r < T and a neighborhood V of x such that P T r e f y > e fx ε for y V, r < r < T, that is to say, a : f r y < fx + ε. When f is lower semicontinuous, the inequality is reversed, b : f r y > fx ε. When f is u.s.c. it follows from a that in the inequality case of 5, the limit of M r r<t as r T is less than or equal to the expression in 8, so the inequality case remains true. For every bounded Borel function f, we can find f u.s.c. such that f f and e f dγ e f dγ. This implies that the inequality case in 5 is true for every bounded Borel function f. Using the reverse inequality b for l.s.c. functions, we see that the equality in 5 remains true for bounded lower semicontinuous functions but the infimum need not be achieved by an optimal martingale. We shall need more than the case of bounded functions. The proof of the next lemma will be given in Section 6; it requires a closer look at the argument given above. 8

9 Lemma 11. The formula 5 remains valid when f is continuous, bounded below and satisfies an exponential upper bound of the form fx ae b x, a, b, x. 9 The inequality case in 5 is valid for any bounded below continuous function f. The next proposition provides a rather direct and simple link between Borell s lemma and our Theorem. Proposition 12. Under the Setting 3 with finite measures µ 1 and µ 2, assume that f s, g t are continuous on or bounded from below as in Remark 5, and that f s satisfies the following exponential bound: there are non-negative measurable functions as, bs on Ω 1 such that f s x ase bs x, x. 1 Then, for every isotropic Gaussian probability measure γ = γ n,τ on, the basic assumption 3 implies that log e gtx dγx dµ 2 t log e fsx dγx dµ 1 s. 11 Ω 2 Ω 1 As seen from the proof of Corollary 7 which indeed reduces the study to the situation of this Proposition, we can remove the additional assumptions on f s and g t when the measures µ 1 and µ 2 have finite support. Proof. The proof will be an application of the formula 5, as extended in Lemma 11, to the functions f s and g t. We may assume that the right-hand side of 11 is not equal to +. The Gaussian measure γ = γ n,τ is the distribution of B T for T = τ >, so that log e fsx dγx = logp T e fs < +, µ 1 -a.e. Let ε > be given. By Lemma 11, for almost every s Ω 1, we can introduce Us D 2, an almost optimal drift for the function f s, s Ω 1, namely, a drift Us = {U r s} r T such that T E [f s B T + U r s dr + 1 T ] U r s 2 dr < logp T e fs + ε We will show later Claim 28 that this process {Us} s Ω1 can be chosen to be Σ 1 - measurable. Note also that 12 and f s ensure that T E Ω 1 U r s 2 dr dµ 1 s 2 logp T e fs dµ 1 s + εµ 1 Ω 1 < Ω 1 9

10 Assume that the Brownian motion B r is defined on a probability space E, A, P. We have a -valued random process U, that will be denoted by one of Us, r, ω = U r sω = U r s, ω = U r,ω s, s Ω 1, r [, T ], ω E. By 13, we know that U L 2 Ω 1 [, T ] E, = L 2 [, T ] E, L 2 Ω 1,. We shall estimate P T e gt using the inequality case of formula 5 given by Lemma 11, with the drift {V r t} r T = {AU r t} r T, namely V r,ω t := AU r,ω t = A s U r,ω s t, t Ω 2, 14 where U r,ω L 2 X 1, for almost every r, ω. We shall use the basic assumption 3 on the families {f s } and {g t } for the random function α ω = B T ω + β ω, where β ω is defined by β ω s := T U r s, ω dr, 15 and where we consider B T ω as a constant function of the s variable. We know by 13 that β ω and α ω are in L 2 X 1, for almost every ω. The constant functions condition ii on A ensures that Aα ω t = AB T ω + β ω t = B T ω + Aβ ω t = B T ω + T V r t, ω dr. 16 As we said, we apply the inequality case of formula 5 for g t with the drift {V r } and then we integrate in t, in order to get that log P T e gt T dµ 2 t [g t B T + V r t dr + 1 T ] V r t 2 dr dµ 2 t. Ω 2 2 Ω 2 E For future reference, we state an obvious consequence of 16, g t Aαω t = g t ABT ω+β ω t = g t BT ω+aβ ω t = g t B T ω+ Ω 2 g t Ω 1 f s T V r t, ω dr. Using 3 and 17, we have on the set E the pointwise inequality T T B T ω + V r t, ω dr dµ 2 t B T ω + U r s, ω dr dµ 1 s, and since A 1, we have another pointwise inequality, for every r [, T ], V r t, ω 2 dµ 2 t Ω 2 U r s, ω 2 dµ 1 s. Ω 1 18 Finally log P T e gt dµ 2 t Ω 2 Ω 1 E 17 T [f s B T + U r s dr + 1 T ] U r s 2 dr dµ 1 s 2 1

11 < log P T e fs dµ 1 s + εµ 1 Ω 1. Ω 1 We conclude by letting ε. As the reader has noticed, the proof is rather short, provided one has put forward the abstract properties contained in the four equations 14, 15, 17 and 18 that allow us to run Borell s argument. 3 Proof of Theorem 4 Suppose that we try to obtain the conclusion 4 of Theorem 4 for {f s } and {g t } as limit case of good cases for which the conclusion is known, say {f s,k } and {g t,k }, k N, such that f s,k f s, g t,k g t when k +. The next elementary lemma will help us to do it. Consider a measure space Ω, Σ, µ and a measure ν on, B, where µ and ν are σ-finite. Let h s,k x, k N and h s x, s Ω, x be Σ B -measurable from Ω to R {+ }. Define H k and H by e Hks = e hs,kx dνx, e Hs = e hsx dνx, 19 that are Σ-measurable functions by the theory behind Fubini s theorem. Note that H + s = log e hsx dνx, H s = log + e hsx dνx. Lemma 13. Let H k s, Hs be defined by 19. µ-semi-integrable. Assume that H k, k N and H are a Suppose that h s,k x h s x pointwise on Ω, and that H s > for µ-almost every s Ω. Then: a1 H k s Hs µ-a.e. and Hs dµs lim sup Ω k H Ω ks dµs. a2 If in addition Ω H s dµs < +, then Hs dµs = lim Ω k H Ω ks dµs. b Suppose that h s,k x h s x pointwise on Ω. Then: b1 H k s Hs pointwise on Ω, and Hs dµs lim inf Ω k H Ω ks dµs. b2 If in addition Ω H+ s dµs < +, then Hs dµs = lim Ω k H Ω ks dµs. Proof. Apply repeatedly one of the classical results of Integration Theory: the Fatou lemma, the monotone convergence theorem and the dominated convergence theorem. In case a, we have e hs,kx e hsx, and H s > allows us to apply dominated convergence to ν and deduce that H k s Hs. Next, H + k s H+ s and H k s H s. In H = Ω Ω H+ Ω H that makes sense by the semi-integrability assumption, apply monotone convergence for H + and Fatou for H. If Ω H dµ < +, replace Fatou by dominated convergence for H. The proof of b is similar and left to the reader. 11

12 Proof of Theorem 4. Since µ 1 and µ 2 have the same finite mass it is enough, as mentioned after the statement of the theorem, to have an inequality involving P T e gt and dx P T e fs = 2πT, 2T > 1/ε. n/2 e fsx x 2 /2T The result will follow by letting T +. Indeed, f s x + ε x 2 decreases to f s x, g t x + ε x 2 decreases to g t x when ε ; for ε [, ε ], define F ε s, G ε t as in 19, with ν being the Lebesgue measure, that is to say, let e Fεs = e fsx ε x 2 dx and use the similar expression for G ε t. By the semi-integrability condition iv, G t is semi-integrable, and we may assume that G t is integrable, otherwise the left side of 4 is, an obvious case. We also know by the assumption 2 that F ε + s = log e fsx ε x 2 dx is integrable. This yields that F ε s and G ε t are semi-integrable for ε [, ε ]. Use Lemma 13, b to see that F ε s F s, G ε t Gt for every s, t. For the F side, we may use b2 because F ε + is integrable. We can therefore pass to the limit as ε and conclude that F ε s dµ 1 s = Ω 1 log Ω 1 e fsx ε x 2 dx dµ 1 s log e fs dµ 1 s. Ω 1 For the analogous expressions with g t, we use b1 of Lemma 13. Now we have to consider the problem with γ = γ n,τ replacing the Lebesgue measure. We know by 2 that s log e fsx dγx is µ 1 -integrable when 2τ > 1/ε. We shall need the existence of a selection α L 2 X 1, such that Ω 1 f s α s dµ 1 s < +, which is granted by Lemma 29. Fix k > and define f s,k, g t,k by inf-convolution of f s, g t with h k x = k x 2, f s,k x = inf u fs x + u + k u 2, g t,k x = inf u gt x + u + k u 2. 2 Clearly, f s,k f s, g t,k g t, and f s,k, g t,k are continuous on. By Lemma 29, we may find negligible Borel sets N 1 Σ 1, N 2 Σ 2 such that s, x f s,k x, t, x g t,k x are Borel functions on Ω 1 \ N 1 and Ω 2 \ N 2. We have that f s,k x f s α s + k x α s 2, x, fitting the exponential bound 9 needed to apply Proposition 12. Let α be in L 2 X 1,. For any fixed ε >, we may find a measurable selection us Lemma 29 such that f s αs + us + k us 2 ε < f s,k αs f s α s + k αs α s 2. Since f s, this shows that u L 2 X 1,. We can write [ g t,k Aαt dµ 2 t ] gt Aαt + Aut + k Aut 2 dµ 2 t, Ω 2 Ω 2 12

13 which is bounded, using the basic assumption 3 and the norm condition A 1, by [ ] fs αs + us + k us 2 dµ 1 s f s,k αs dµ 1 s + εµ 1 Ω 1. Ω 1 Ω 1 Hence, f s,k and g t,k also satisfy the basic assumption. By Proposition 12, we conclude that log e gt,kx dγx dµ 2 t log e fs,kx dγx dµ 1 s. Ω 2 Ω 1 When k tends to infinity, f s,k increases to a l.s.c. function f s f s, and g t,k x increases to g t x because x g t x is l.s.c. on. We apply again Lemma 13, this time with ν = γ, defining F k s, F s F s, Gk t, Gt from f s,k, fs, f s, g t,k, g t as in 19. Since the functions are, e fs,kx, e gt,kx are bounded by 1, we have F k s, G k t because γ is a probability measure. Thus F s, G t > and F k s F s, G k t Gt by a. The conclusion follows, by a1 for F, and by a2 for G since G =. Remark 14. Assume that f s, g t are continuous on. Let D be a countable dense subset of. Since f s and g t are continuous on, we may define the inf-convolution by f s,k x = inf fs x + u + k u 2, g t,k x = inf gt x + u + k u 2. u D u D It is now clear that s, x f s,k x, t, x g t,k x are Σ i B -measurable, as countable infima of measurable functions. Similarly, the possibility of selecting us is now evident. 4 Other formulations and consequences Let us first comment on Corollary 8. The classical Prékopa Leindler inequality corresponds to the case of a two point space. If we replace the two point space i.e., Bernoulli variables by the unit circle S 1 = {e iθ ; θ R} i.e., Steinhaus variables, then we obtain: Corollary 15. Under the assumptions of Corollary 8 with Ω = S 1 and µ = dθ/2π, let {f ξ } ξ S 1 be non-negative or properly bounded from below as in Remark 5 Borel functions on and let g be a bounded from below l.s.c. function on. If for every α L 2 S 1,, we have 2π g αe iθ dθ 2π f 2π e iθ αe iθ dθ 2π, then it follows that 2π log e g f dθ log e eiθ R 2π. n A consequence of the previous Corollary is one of the Berndtsson s plurisubharmonic extensions of Prékopa s theorem the relatively easy tube case. 13

14 Corollary 16 Berndtsson[3]. Let ϕ : C C n R be plurisubharmonic on C n+1 and such that, for every z C, w C n, we have that ϕz, w = ϕz, Rw. Then, the function ψz = log e ϕz,x dx is subharmonic on C. Proof. We will check that ψ is subharmonic at z =, say. We can assume that ϕ is bounded below for all z close to. We want to prove that ψ 2π ψre iθ dθ for any 2π fixed r > small enough. We take r = 1 to simplify notations. Take gx = ϕ, x and f e iθx := ϕe iθ, x for x C n. To conclude, it suffices to check that these functions verify the hypothesis in the previous corollary. Let α L 2 S 1, and let α be the harmonic extension of α to the unit disc D. In particular α = 2π αe iθ dθ. We 2π can write this α as the real part RH of an holomorphic function H : D C n such that H = α. We conclude by noticing that z ϕz, Hz is subharmonic on D, and using ϕz, Hz = ϕz, αz we obtain that 2π g αe iθ dθ 2π = ϕ, H ϕe iθ, He iθ dθ 2π 2π 2π = f e iθαe iθ dθ 2π. So far, we have presented consequences of Corollary 8, and therefore more or less straightforward consequences of the Prékopa Leindler inequality itself. The situation is different with other instances of Theorem 4 such as Proposition 2. It seems difficult to guess the existence of such inequalities without having in mind Borell s proof. Theorem 17. Let µ be a Borel probability measure on [, 1] and set m := t dµt [, 1]. Under Setting 3 with Ω 1 = {, 1} and Ω 2 = [, 1], µ 2 = µ, let f, f 1 and {g t } t [,1] be bounded from below as in Remark 5 Borel functions from to R {+ }. Assume g t l.s.c. for every t in Ω 2. If for every x, x 1 we have 1 then it follows that 1 g t 1 tx + tx 1 dµt 1 mf x + mf 1 x 1, log e gt dµt 1 m log e f m log e f 1. The deduction of this result from Theorem 4 is as follows. Take X 1 = Ω 1, µ 1 = {, 1}, 1 mδ + mδ 1, X2 = Ω 2, µ 2 = [, 1], µ and for v, v 1 L 2 X 1, define Av, v 1 L 2 X 2, by Av, v 1 : t 1 tv + tv 1. 14

15 Then the constant functions condition ii is satisfied, as Av, v v, and by the convexity of the square of the Euclidean norm on, letting H i = L 2 X i,, i = 1, 2, we have Av, v 1 2 H 2 = 1 1 tv + tv 1 2 dµt 1 m v 2 + m v 1 2 = v, v 1 2 H 1. Remark 18. It is possible to prove Theorem 17 without assuming g t l.s.c. and f i bounded from below, see Claim 26 for a sketch of proof. As a particular case of the previous proposition, if we take only one function g t g : R, we have that when µ is a probability measure on [, 1] with barycenter m, and x, x 1, 1 g 1 tx + tx 1 dµt 1 mf x + mf 1 x 1, 21 then 1 m m. e g e f e f 1 The conclusion is independent of µ, so given the functions f, f 1, g, one can try to find an optimal µ for which the condition 21 holds. The classical Prékopa Leindler inequality corresponds to the choice µ = δ m, and this is indeed the optimal choice when g is convex, as seen by using Jensen s inequality in 21. Does the above result really improve on the Prékopa Leindler inequality when g is non-convex? Another particular case of the previous proposition in the case of convex combination of two Dirac measures is the following extension of Proposition 2. Proposition 19. Fix s, t, r [, 1] and set m := 1 rs + rt [, 1]. Let f, f 1, g, g 1 be four Borel functions from to R {+ } such that for every x, x 1, 1 rg 1 sx + sx 1 + rg1 1 tx + tx 1 1 mf x + mf 1 x 1. Then 1 r r 1 m m e gx dx e g1x dx e fx dx e f1x dx. Proof. Although the result is a particular case of Theorem 17 with µ = 1 rδ s + rδ t, it is better to go back to Corollary 7 since in the case of finitely many functions there are no technical conditions. We take Ω 1 := {{, 1}, 1 mδ + mδ 1 } and Ω 2 = {{, 1}, 1 rδ + rδ 1 }, and the linear mapping A : H 1 H 2 with H i := L 2 Ω i, defined by Ax, x 1 := 1 sx + sx 1, 1 tx + tx 1, x, x 1. We have Av, v = v, v for every v and note that for x, x 1, 1 r 1 sx + sx r 1 tx + tx rs1 s + rt1 t x x 1 2 = 1 m x 2 + m x

16 and so in particular 1 r 1 sx + sx r 1 tx + tx m x 2 + m x 1 2 which exactly means that A 1. Let us mention that Proposition 2 and also the above results do not give anything more than the Brunn Minkowski inequality when applied to the case of indicator functions, i.e., to e f = 1 A, e f 1 = 1 A1, e g = 1 2A /3+A 1 /3, e g 1 = 1 A /3+2A 1 /3. Indeed, by Brunn Minkowski, one has that 2A /3 + A 1 /3 A 2/3 A 1 1/3, A /3 + 2A 1 /3 A 1/3 A 1 2/3, and in this case, the result of Proposition 2 is obtained by taking the product of these two inequalities. It seems that the extra information comes from applying the results to true functions. This is consistent with the fact that we do not know how to reach our functional inequalities using other classical proofs of the Prékopa Leindler inequalities. For instance, it is not clear that Proposition 2 can be proved using the mass transportation argument of McCann [12]; precisely, this transportation argument uses some form of localization inside the integral amounting to reduce the problem to sets ellipsoids, actually and eventually matrices. Example 2 Gaussian self-improvement and generalized τ-property. Let α, 1. We shall comment on Proposition 19 in the case s = α, t = 1 α = 1 s and r = 1/2 and so m = 1/2. The goal is to get improved Gaussian inequalities by exploiting identity 22 instead of an inequality. Let f, f 1 be real Borel functions on. We start first with the case of the Prékopa-Leindler inequality and consider the following variant g of inf-convolution, defined for every x by gx = inf{1 αf x + αf 1 x 1 + α1 α x x 1 2 /2 : x = 1 αx + αx 1 }. We assume that gx >. Using the following particular case of22, 1 αx + αx α1 α x x 1 2 = 1 α x 2 + α x 1 2, 23 it follows that gx + x 2 /2 1 α f x + x 2 /2 + α f 1 x 1 + x 1 2 /2 whenever x = 1 αx + αx 1, and we obtain by Prékopa Leindler that 1 α α, e g dγ n e f dγ n e f 1 dγ n where γ n is the standard Gaussian measure γ n,1 on. This infimal convolution inequality ensures the τ-property from [1] for the Gaussian measure. For α, β, λ, 1, we may generalize g as g α,β,λ x = inf{1 βf x + βf 1 x 1 + λα1 α x x 1 2 : x = 1 αx + αx 1 }, 16

17 the former g being equal to g α,α,1/2. The Prékopa Leindler inequality does not seem to apply to values of the parameters other than triples of the form α, α, 1/2. Combining g α,β,λ x α 1 βf x + βf 1 x 1 + λα1 α x x 1 2 and the corresponding inequality for g 1 α,1 β,1 λ x 1 α, we get, with g = g α,β,λ and g 1 = g 1 α,1 β,1 λ, that g x α + g 1 x 1 α f x + f 1 x 1 + α1 α x x 1 2, The identity 22 in the case s = α, t = 1 α and r = 1/2 and so m = 1/2 rewrites as 1 αx + αx αx + 1 αx α1 α x x 1 2 = x 2 + x 1 2 and so we find g x α + x α 2 /2 + g 1 x 1 α + x 1 α 2 /2 f x + x 2 /2 + f 1 x 1 + x 1 2 /2. Therefore, by Proposition 19 with s = α, t = 1 α and r = 1/2 = m, we arrive at the following generalized infimal convolution inequality: e g α,β,λ dγ n e g 1 α,1 β,1 λ dγ n e f dγ n e f 1 dγ n. Example 21 Exotic situation. All our examples so far of linear maps A are of convex type, meaning that ϕ Aαt dµ 2 t ϕαs dµ 1 s Ω 2 Ω 1 for every convex function ϕ on, while the norm condition i on A ensures this property for ϕx = k x 2 only, k. Here is a non convex example, with Ω 1 = Ω 2 = Ω = {, 1} and µ 1 = µ 2 = δ + δ 1. The space L 2 Ω, is equal to, the matrix A can be represented by blocks of size n n, A1 B A = 1. C 1 D 1 By the constant functions condition ii, the image of the constant function equal to x is A 1 x + B 1 x, C 1 x + D 1 x = x, x, which is equivalent to A 1 + B 1 = C 1 + D 1 = I n, the identity matrix. One can thus write I B B A =. C I C Since A 1, each block must have norm 1, and this implies that the diagonal coefficients of B, C are in [, 1]. The condition A 1 means that A A I 2n, which translates to B B + C C B B C C B + B B C B B C C B B + C C B C C + C. 17

18 In the simpler case C = B, this amounts to 2B B B + B, or Bx 2 Bx x, for every x. For an elementary explicit example, take n = 2 and b ε B = ε b with b, 1 and b 2 + ε 2 b. We see that B = bid 2 εr, with R the rotation by π/2 in the plane R 2. Then, with b = 1/3 and ε = 2/3, if we know that 2 g 3 x x 1 3 Rx 1 x + g 1 3 x x Rx 1 x f x + f 1 x 1 for every x, x 1, we get the same conclusion as that of Proposition 2. Remark 22. Following the previous construction, it is natural to ask if we can construct an exotic Prékopa-Leindler situation. The answer is no. Let α [, 1]. The only n n real matrix B such that I n Bx + By 2 1 α x 2 + α y 2 for all x, y is B = αi n try x = u + tαv, y = u t1 αv and t. In other words, there is no exotic example in the Prékopa-Leindler case. More generally, if B 1,..., B k are n n matrices such that k j=1 B j = I n and k j=1 B jx j 2 k j=1 α j x j 2, with α j and k j=1 α j = 1, then B j = α j I n, j = 1,..., k. 5 Generalized Brascamp Lieb and reverse Brascamp Lieb inequalities By slightly modifying the Setting 3, it is possible to recover and extend, almost for free, the Brascamp Lieb inequalities and their reverse forms. Let us mention that it has been known for some time that the Brascamp Lieb inequalities can be recovered using Borell s technique see e.g. [9]. If the functions {f s } are defined on R m and {g t } on, with m n, then the linear operator A of Setting 3 should now act from L 2 X 1, R m to L 2 X 2, and the constant functions condition ii has to be revised. We shall do it by using projections from the larger space, R m or, onto the smaller. To be precise, our projections are adjoint to isometries from the smaller space into the larger. For example, if n < m and if T is an isometry from into R m, its adjoint Q = T is a mapping from R m onto such that QQ = Id, and Q Q is the orthogonal projection of R m onto the range of T. Then, for v R m, we can compare the image As v of the constant function Ω 1 s v with the constant function Ω 2 t Qv. Actually, the new setting will be notably more complicated, introducing a family of projections Qt, t Ω 2, and comparing the image As v with the function t Qtv. We can also view the new setting as giving a measurable family of n-dimensional subspaces Xt of R m, parameterized by T t : Xt, and Qt = T t being the composition of the orthogonal projection πt of R m onto Xt with the inverse map of T t. 18

19 Except for what concerns the linear mapping A, the modifications are straightforward, and will be just indicated without rewriting completely the modified assumption. Setting 3. The definition of the measure spaces X 1 = Ω 1, Σ 1, µ 1, X 2 = Ω 2, Σ 2, µ 2 is as in Setting 3. Two integers m, n 1 are given, and the linear operator A acts now as A : L 2 X 1, R m L 2 X 2, with A 1, where the norms are computed with respect to the Euclidean norm on R m, and the measures µ 1 and µ 2, respectively. The constant functions condition is modified as follows: ii 1 ii 2 if m n, there exists a Σ 1 -measurable family Ω 1 s P s of projections P s from onto R m such that, for every vector w, A s P sw t = w for µ 2 -almost every t Ω 2, if m n, there exists a Σ 2 -measurable family Ω 2 t Qt of projections Qt from R m onto such that, for every vector v R m, As v t = Qtv for µ 2 -almost every t Ω 2. Note that formally, ii 2 is the adjoint situation to ii 1. For the conditions on {f s }, {g t }, we need only replace by R m for what concerns f s in the measurability condition iii and in the semi-integrability condition iv. We arrive to the following extension of Theorem 4. Theorem 23. Under Setting 3, we make the additional assumptions that the functions f s, g t are non-negative with g t lower semicontinuous on, that for some ε > we have log exp f s x ε x 2 dx dµ 1 s < +, 24 Ω 1 R m and that the measures µ 1, µ 2 are such that m µ 1 Ω 1 = n µ 2 Ω 2 < +. If for every α L 2 X 1, R m, we have g t Aαt dµ2 t f s αs dµ 1 s, 25 Ω 2 Ω 1 then log Ω 2 e gt dµ 2 t log Ω 1 R m e fs dµ 1 s. Not only the argument for deriving Theorem 23 follows the proof of Theorem 4, but in its Gaussian version, Theorem 23 is already contained in Proposition 12. Indeed, after reducing to the Gaussian case, we further approximate as before the functions f s and g t by inf-convolution, in order to be in a position to apply the proposition below, just as we 19

20 did to prove Thorem 4. The assumption m µ 1 Ω 1 = n µ 2 Ω 2 is needed to pass to the limit from the Gaussian case for P T, T +, in respective dimensions m and n. For the Gaussian version below, we will have not much to add, except maybe for the case ii 2 of the Setting 3. So Proposition 12 almost includes all possible geometric situations around the Prékopa Leindler inequality, including these generalized Brascamp Lieb inequalities. Proposition 24. Under the Setting 3 with finite measures µ 1 and µ 2, assume that f s, g t are continuous on R m and, respectively or bounded from below as in Remark 5, and that the f s satisfy the exponential bound 1 on R m. Then, the basic assumption 25 implies that log e gty dγ n,τ y dµ 2 t log e fsx dγ m,τ x dµ 1 s. 26 Ω 2 Ω 1 R m As before, in the case the measures µ 1 and µ 2, have finite support, we can remove all the additional assumptions on f s and g t. Proof. We want to apply Proposition 12 but we have to deal with the fact that dimensions m and n are now different. We shall do it by reducing to the case when both dimensions are equal to maxm, n. In case ii 1, when m n, we extend f s to by defining f 1,s y = f s P sy, y, so that e f1,sy dγ n,τ y = e fsx dγ m,τ x. 27 R m For every α L 2 X 1,, define α 1 L 2 X 1, R m by α 1 s = P sαs and set A 1 α = Aα 1. Then A 1 is linear from L 2 X 1, to L 2 X 2,, and clearly A 1 A 1. If w is a fixed vector in, then A 1 s w = As P sw = w by ii 1, so A 1 verifies the constant functions condition ii. Next, by the basic assumption of Theorem 23 applied to α 1, we get that g t A1 αt dµ 2 t = g t Aα1 t dµ 2 t f s α1 s dµ 1 s Ω 2 Ω 2 Ω 1 = f s P sαs dµ1 s = f 1,s αs dµ 1 s. Ω 1 Ω 1 We see that A 1, {f 1,s } and {g t } satisfy the assumptions of Proposition 12, including the basic assumption 3 on. The result follows therefore from 27 and from the conclusion of Proposition 12 for {f 1,s } and {g t }. In case ii 2, when m n, let T t be the isometry from into R m adjoint to the projection Qt, so that QtT t = Id. The function g t is extended to R m by setting g 1,t x = g t Qtx, x R m. For α L 2 X 1, R m, we set A 1 αt = T taαt and get that QtA 1 αt = Aαt. We have that A 1 = A 1. If α = v is 2

21 a constant function from Ω 1 to R m, we know by ii 2 that Aα t = Qtv. Then A 1 α t = T taα t = T tqtv, not equal to v in general. The constant functions condition ii is not satisfied by A 1, but still we have g 1,t A1 v + αt = g 1,t v + A 1 αt, t Ω 2, 28 for every α L 2 X 1, R m. This is because g 1,t x 1 = g 1,t x 2 when Qtx 1 = Qtx 2, and because we have by ii 2 that QtA 1 v + αt = Av + αt = Qtv + Aαt = Qt v + A 1 αt. The basic assumption 3 is satisfied for the families {f s }, {g 1,t } of functions on R m and the mapping A 1 : observing that g 1,t A1 αt = g t Aαt, we get g 1,t A1 αt dµ 2 t = g t Aαt dµ2 t f s αs dµ 1 s. Ω 2 Ω 2 Ω 1 We stressed during the proof of Proposition 12 that the equality 28 is enough to run the argument see equation 17 and the lines around it. Finally, as before, observe that e gty dγ n,τ y = e g1,tx dγ m,τ x, R m and use Proposition 12 to get the desired conclusion. Remark 25. The normalization A 1 can be replaced by the following assumptions. Let κ := A > and assume that κ 2 m µ 1 R m = n µ 2. If for every α in L 2 X 1, R m, we have g t Aαt dµ 2 t κ 2 Ω 2 f s αs dµ 1 s, Ω 1 then log Ω 2 e gt dµ 2 t κ Ω 2 log e fs dµ 1 s. 1 R m The reader will just look at inequality 18 and the inequality before it. We shall examine now particular cases of Theorem 23. We will see that it contains both Brascamp Lieb and reverse Brascamp Lieb inequalities in their geometric form. Let us give unit vectors u 1,... u N in the Euclidean space R d and positive reals c 1,..., c N that decompose the identity of R d : N c i u i u i = Id R d. 29 i=1 21

22 This is equivalent to saying that x x = N i=1 c i u i x 2 for every x R d. If we consider the one-point space E 1 = {} equipped with the trivial probability measure ν 1, and the measure space {1, } N E 2, ν 2 =... N, c i δ i, ν 2 E 2 = i=1 N c i = d, i=1 then 29 is equivalent to saying that the mapping U : x R d x u 1,..., x u N 3 is an isometry from L 2 E 1, ν 1, R d R d into L 2 E 2, ν 2, R R N. In order to recover the reverse Brascamp Lieb inequalities, consider X 1 = E 2, ν 2, m = 1, X 2 = E 1, ν 1, n = d, mµ 1 Ω 1 = ν 2 E 2 = d = nµ 2 Ω 2. Define A : R N L 2 X 1, R m R d L 2 X 2, to be the linear operator At 1,..., t N = N c i t i u i, t i R. i=1 Then A is adjoint to the into isometry U of 3 between our L 2 spaces, and A verifies the condition ii 1 of Setting 3, with the family of projections P i v = v u i, i Ω 1, because we have by 29 that N Ai P i v = c i v u i u i = v. The conclusion reads as follows. If for every t 1,..., t N R we have N g t i c i u i i=1 N c i f i t i i=1 i=1 then R d e g N ci e i f. R This is the reverse Brascamp Lieb inequality of Barthe [1] in its geometric form. Actually, in the same way, we can see that the formulation above contains the continuous statement given in [2] that can be easily derived by approximation, anyway. To recover the classical Brascamp Lieb inequalities, apply the theorem with X 1 = E 1, ν 1, m = d, X 2 = E 2, ν 2, n = 1 and the linear operator A : L 2 X 1, R m R d L 2 X 2, R N defined by Ax = x u 1,..., x u N. Then A = U, the into isometry 3 between the corresponding L 2 spaces, and it satisfies the assumption ii 2 with the family of projections Q j x = x u j, j Ω 2 = {1,..., N}. The conclusion reads as follows: given f : R d R and g 1,..., g N : R R such that x R d, N c j g j x u j fx, j=1 22 we have i=1 N j=1 R e g cj j R d e f.

23 This is the geometric Brascamp Lieb inequality, usually stated with the best possible f, which is defined by the equality in place of inequality above, namely exp R d N j=1 c j g j x u j dx N j=1 R e g j cj. It is possible to state and prove in the same way a more general form of Theorem 23 where each function f s resp. g t is defined on a different Euclidean space E s resp. F t, and to deduce from it the multidimensional geometric Brascamp Lieb inequalities. In this situation, we assume that all spaces E s, F t are subspaces of a given R l. We may consider that the space E s is R ms and that F t is t, with 1 ms, nt l and we assume that ms dµ 1 s = nt dµ 2 t. Ω 1 Ω 2 The mapping A is defined on the closed subspace H of L 2 X 1, R l consisting of those α with αs E s for every s Ω 1. For every square integrable choice α H, the image Aα L 2 X 2, R l is such that Aαt belongs to F t for every t Ω 2. We give projections P s, Qt from R l onto E s, F t respectively. The constant functions condition says now that for every w R l, we have that A s P sw t = Qtw. The proof mixes the two cases ii 1 and ii 2 of the proof of Theorem 23. We leave this to the reader. 6 Technicalities Proof of Corollary 7. Under the Setting 3 in the discrete case, assume that f, f 1,..., f p and g, g 1,..., g q are Borel functions from to R {+ }, and that for all x, x 1,..., x p in we have q p p ν j g j a j,i x i µ i f i x i, 31 j= i= where µ i, ν j >, p i= µ i = q j= ν j = 1, where the matrix A = a j,i satisfies the norm condition i and the constant functions condition ii. We want to prove that q j= ν j log e g j p i= i= µ i log e f i. 32 If e f i = for one i, then by the semi-integrability assumption iv, the other e f i are finite, the right-hand side of 32 is + and this case is obvious. We may therefore assume that all f i are L-proper. It follows that e g j >, j =,..., q. Indeed, if for 23

24 example e g =, then g is + almost everywhere. We know that p i= a,i = 1 since A preserves constant functions by ii. Assume a,1 for example. Since all f i are L-proper, we may fix x i with f i x i < + for i 1, and we see from 31 that for almost every x 1, + = ν g a,1 x 1 + a,i x i + i 1 j p ν j g j a j,i x i µ i f i x i µ 1 f 1 x 1, i= i 1 hence f 1 is + almost everywhere, a contradiction. For ε > and i =,..., p, j =,..., q, the functions f i,ε x = f i x + ε x 2, g j,ε x = g j x + ε x 2, still satisfy 31 by the assumption A 1. We may reduce the problem to proving 32 for f i,ε, g j,ε, since we can pass to the limit as ε in e g jx εx 2 dx, e f ix εx 2 dx, obtaining 32 by monotone convergence. In other words, we may keep f i, g j but replace the Lebesgue measure in 32 by a Gaussian measure dγx = e ε x 2 dx. If e g j dγ = + for one j, there is nothing to prove: the other integrals e g j dγ, j j are >, and the left-hand side of 32 is. Otherwise, for N N, define g j,n = ming j, N and f i,n = maxf i, N, that trivially satisfy 31, and observe that it is enough to give the proof for f i,n, g j,n ; indeed, since e mingj, e g j + 1 is integrable with respect to dγ, we may again pass to the decreasing limit in the integrals e g j,n dγ as N +, and use monotone convergence for e f i,n dγ. Now we reduced the question to the case g j N and f i N. Thanks to the discrete situation, we may further assume that f i is bounded above by 2N/µ i for each i, since q p ν j g j a j,i x i µ i f i x i N + 1 µ i N < 2N, i i j= i= and for the same reason, we may also assume g j bounded below by 2N/ν j, while still keeping 31 true. Finally, we restricted the problem to bounded Borel functions f i, g j and a Gaussian measure γ. If all f i, g j are translated by a same vector, then 31 remains true by the constant functions condition ii, therefore 31 is stable by convolution with non-negative kernels. We may approximate in L 1 γ-norm the functions f i, g j by convolution with a compactly supported continuous non-negative kernel, keeping 31 and meeting the assumptions of Proposition 12. We obtain thus, for a sequence of continuous approximations f i,k and g j,k, k N, the inequality q j= ν j log e g j,k dγ p i= µ i log e f i,k dγ. Some subsequences of the sequence of continuous approximations tend almost everywhere to f i, g j respectively, and the conclusion of the restricted problem follows by the dominated convergence theorem. 24

25 Claim 26. In Theorem 17, we can remove the assumptions that g t is l.s.c. and that f i or bounded from below, i =, 1. Sketch of proof. As in the proof of Theorem 4, we begin by replacing the Lebesgue measure in e fix dx and e gtx dx by a Gaussian probability measure γ. The reason is the same here for the g t side, but is easier for f i, i =, 1, a simple application of monotone convergence. Next, as in the proof of Corollary 7, one can replace g t by g t,n = ming t, N and f i by f i,n = maxf i, N: since g t,n increases to g t as N +, the argument for log e g t,n dγ dµ 2 t N log e gt dγ dµ 2 t Ω 2 Ω 2 is the same as at the end of the proof of Theorem 4, and for f i,n, decreasing to f i, the reason is monotone convergence again in e f i,n dγ, i =, 1. Then g t,n N and f,n, f 1,N N. As in the proof of Corollary 7, one can then assume that f,n, f 1,N 2N/β with β = minα, 1 α. Finally, approximation by convolution reduces to the case of bounded continuous functions, and one can conclude by applying Proposition 12. Proof of Lemma 11. We may assume that f. For some integer N, let ϕx = minfx, N. As in 6, define ϕ r by e ϕr = P T r e ϕ, r T and let Φ = ϕ, F = log P T e f Φ >. Note that ϕ implies ϕ r. Since ϕ is bounded and continuous, we may by Lemma 9 consider the optimal martingale corresponding to ϕ, r M r = ϕ r B r + u ρ dρ + 1 r r u ρ 2 dρ = M u ρ db ρ, r T, 2 with M = Φ, u r = ϕ r X r and X r = B r + r u ρ dρ. Define the square function S r r T of the martingale M r M r T by r 1/2. S r = u ρ dρ 2 We know by 8 that S r, X r and M r converge almost surely and in L 2 to S T, X T and M T. Observe that u r, S r and M r are bounded random variables for each fixed r < T. Consider the exponential martingale e λmr λ2 S 2 r /2 for λ = 1/2 and r < T, namely exp M r /2 S 2 r /8 = exp ϕ r X r /2 + S 2 r /4 S 2 r /8 = exp ϕ r X r /2 + S 2 r /8. By Fatou and the martingale property, we have that E exp S 2 T /8 E exp ϕx T /2 + S 2 T /8 lim r T E exp ϕ r X r /2 + S 2 r /8 = e Φ /2. Let Y T = T 1/2 X T T 1/2 B T +S T and observe that in, one has E e B 1 2 /4 = 2 n/2. For every λ >, using Cauchy Schwarz and the inequality 2στ cσ 2 + τ 2 /c when c, σ, τ >, we can write 2 E e λy T E e 2λT 1/2 B T E e 2λS T = E e 2λ B 1 E e 2λS T 25