Φ entropy inequalities and asymmetric covariance estimates for convex measures

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1 Φ entropy inequalities and asymmetric covariance estimates for convex measures arxiv: v1 [math.fa] 16 Oct 2018 Van Hoang Nguyen October 17, 2018 Abstract Inthispaper, weusethesemi-groupmethodandanadaptation ofthel 2 method of Hörmander to establish some Φ entropy inequalities and asymmetric covariance estimates for the strictly convex measures in. These inequalities extends the ones for the strictly log-concave measures to more general setting of convex measures. The Φ entropy inequalities are turned out to be sharp in the special case of Cauchy measures. Finally, we show that the similar inequalities for log-concave measures can be obtained from our results in the limiting case. 1 Introduction Letϕ : (0, beastrictlyconvex, C 2 smoothfunctionsuchthatϕ β isintegrablefor some β > 0. Bystrictly convex, we meanthat thehessian matrix, D 2 ϕ(x = ( 2 ijϕ(x n i,j=1, of ϕ is everywhere positive in the matrix sense. Let dµ ϕ,β denote the probability measure dµ ϕ,β = ϕ(x β Z ϕ,β dx, wherez ϕ,β isthenormalizationconstantwhichturnsµ ϕ,β intoaprobability. Themainaims of this paper is to establish several functional inequalities for the probability measure µ ϕ,β such as Φ entropy inequalities and asymmetric covariance estimates. These inequalities Institute of Research and Development, Duy Tan University, Da Nang, Vietnam. vanhoang0610@yahoo.com Mathematics Subject Classification: 26D10. Key words and phrases: Φ entropy inequalities, Poincaré type inequalities, Beckner type inequalities, semi-group, L 2 method of Hörmander, Brascamp Lieb type inequalities, asymmetric covariance estimates, convex measures. 1

2 extend the Φ entropy inequalities in [11] and the asymmetric covariance estimates in [13] for the log-concave measure to the context of convex measures. Let Φ : I R be a convex function on an interval I R and f : I be a measurable function such that f and Φ(f is integrable with respect to the probability measure µ ϕ,β, we define ( Ent Φ µ ϕ,β (f = Φ(fdµ ϕ,β Φ fdµ ϕ,β as the Φ entropy of f under the probability measure µ ϕ,β. For examples, if Φ(x = x 2 then we let Var µϕ,β (f = Ent Φ µ ϕ,β (f be the variance of f with respect to µ ϕ,β, and if Φ(x = xlnx on (0, then we let Ent µϕ,β (f = Ent Φ µ ϕ,β (f be the Boltzmann entropy of a positive function f with respect to µ ϕ,β. Notice that Ent Φ µ ϕ,β (f is always nonnegative quantity by Jensen s inequality. We are interested in to finding the upper bound for Ent Φ µ ϕ,β (f under some suitable conditions on ϕ, Φ and β. The first main result of this paper is the following theorem. Theorem 1.1. Let β > n+1 and Φ : I R be a convex function such that Φ (4 (tφ (t 1 (4β 5 2 +n 1 8(β 1(β n 1 (Φ(3 (t 2, (1.1 for any t I. Assume, in addition, that ϕ is uniformly convex in, i.e., D 2 ϕ(x ci n in the matrix sence for some c > 0. Then for any smooth function f with value in I, we have Ent Φ 1 µ ϕ,β (f Φ (f f 2 ϕdµ ϕ,β. (1.2 2c(β 1 Let us give some comments on Theorem 1.1. The Φ entropy inequalities have been proved in [11] for such function Φ under the curvature-dimension condition CD(ρ, (see also [15]. Let L be a differential operator of order 2 given by Lf(x = n 2 f D ij (x (x x i x j i,j=1 n a i (x f x i (x where D(x = (D ij (x 1 i,j n is a nonnegative symmetric n n matrix in the matrix sense with smooth entires and a(x = (a i (x 1 i n has smooth elements. Such an operator generates a semigroup P t acting on the smooth functions on such that L = ( P t t=0 t. The carré du champ operator (see [2] associated to L (or semigroup P t is defined by Γ(f,g = 1 (L(fg flg glf. 2 For simplicity, we write Γ(f = Γ(f,f. The Γ 2 operator is defined by Γ 2 (f = 1 2 (LΓ(f 2Γ(f,Lf 2

3 We say that the operator L (or semigroup P t satisfies the curvature-dimension condition CD(ρ, for some ρ R if Γ 2 (f ργ(f, for all function f. This condition is a special case of the curvature dimension condition CD(ρ,m with ρ R and m 1 introduced by Bakry and Émery [2]. Let dµ = e ψ dx be a probability measure in with ψ being a convex function such that D 2 ψ(x ρi n for any x for some ρ > 0, then the operator L defined by Lf(x = f(x ψ(x, f(x, where, denotes the scalar product in, satisfies the CD(ρ, condition. Indeed, it is easy to see that Γ(f,g = f, g and by Bochner Lichnerowicz formula Γ 2 (f = D 2 f 2 HS + D2 ϕ(x f(x, f(x, where HS denotes Hilbert-Schmidt norm on the space of symmetric matrices. It was proved by Bolley and Gentil [11] for such measures that the following Φ entropy inequality with Φ satisfying Φ (4 Φ 2(Φ (3 2 holds Ent Φ µ (f 1 2ρ Φ (f f 2 dµ. (1.3 It is interesting that the Φ entropy inequality (1.3 can be derived from Theorem 1.1 by an approximation process. This will be shown at the end of Sect. 2 below. Taking the function Φ = Φ p := t 2 p on (0,. The function Φp satisfies the condition (1.1 if 4(β 1(β n 1 1 p p β := 1+ < 2. (1.4 4(β (3n 2(β 1+n Thus, we obtain the following Beckner-type inequalities for the measures µ ϕ,β from Theorem 1.1. Corollary 1.2. Let β > n+1 and D 2 ϕ ci n for some c > 0. Then for any p [1,p β ] one has ( 2 f 2 dµ ϕ,β f p p 2 p dµ ϕ,β f 2 ϕdµ ϕ,β, (1.5 R c(β 1 n for any positive, smooth function f. If ϕ(x = 1+ x 2, then the probability dµ β = 1 Z β (1+ x 2 β, β > n is the generalized 2 Cauchymeasures. NoticethatD 2 ϕ(x = 2I n. FromCorollary(1.2,weobtainthefollowing Beckner type inequalities for the Cauchy measures µ β : let β > n+1 and p [1,p β ] then it holds 1 2 p ( ( f 2 dµ β f p dµ β 2 p 3 1 f 2 (1+ x 2 dµ β (1.6 2(β 1

4 for any positive, smooth function f. When writing this paper, I learned from the work of Bakry, Gentil and Scheffer [4] that the inequality (1.6 can be proved by a different method based on the harmonic extensions on the upper-half plane and probabilistic representation and curvature-dimension inequalities with some negative dimensions. This method was initially introduced by Scheffer [20]. It seems that the approach in [4] is special for the Cauchy distributions and can not be applied for more general convex measures. For p = 1 we obtain the sharp weighted Poincaré type inequality for Cauchy measures which was previously studied by Blanchet, Bonforte, Dolbeault, Grillo and Vazquez [6, 8] with applications to the asymptotics of the fast diffusion equations [7,8] (see also [1,9,10,19]: let β n+1, then it holds ( 2 f 2 dµ β fdµ β 1 f 2 (1+ x 2 dµ β 2(β 1 for any smooth function f. It is remarkable that the constant C p = 1 in (1.6 is sharp 2(β 1 in the sense that it can not be replaced by any smaller constant. To see this, let B p denote the sharp constant in (1.6, then obviously B p 1. For any smooth bounded function 2(β 1 g such that gdµ β = 0, applying (1.6 for 1+ǫg with ǫ > 0 small enough and expanding the obtained inequality in term ǫ 2, we get ǫ 2 g 2 dµ β +o(ǫ 2 B p ǫ 2 g 2 ϕdµ β, for ǫ > 0 small enough. Letting ǫ 0 we have g 2 dµ β B p g 2 ϕdµ β for any bounded smooth function g with gdµ β = 0. This implies B p B 1 = 1 2(β 1. Consequently, we get B p = 1 2(β 1. The last remark concerning to Corollary 1.2 is that p β < 2, hence we cannot let p 2 to obtain a weighted logarithmic Sobolev inequality for the convex measures µ ϕ,β (or Cauchy measure µ β with weighted ϕ. It s was shown in [10] that the weighted logarithmic Sobolev inequality forthecauchy measures holdstruewiththeweight w(x = (1+ x 2 2 ln(e+ x 2. In [14], by using Lyapunov method, Cattiaux, Guillin and Wu found the correct order of magnitude of the weight in this inequality as w(x = (1+ x 2 ln(e+ x 2. Finally, we have p β 2 as β, we can see that the logarithmic Sobolev inequality for the uniform logconcave measure can be obtained from (1.5. Indeed, suppose dµ = e ψ dx is a log-concave probability measure such that D 2 ψ ρi n for some ρ > 0. For each β > n+1, consider the function ϕ β = 1 + ψ and the probability measure µ β ϕ β,β. We have D 2 ϕ β c β := 2ρ. β For any positive smooth function f, we apply (1.5 for µ ϕβ,β, f and p = p β and then let β with remark that Z ϕβ,βϕ β β e ψ to obtain the following inequality ( f 2 lnf 2 dµ f 2 dµ ln 4 f 2 dµ 2 ρ f 2 dµ.

5 Especially, when ψ(x = x 2 /2 we obtain the famous Gross s logarithmic Sobolev inequality for Gaussian [16]. The second main result of this paper is the asymmetric covariance estimates for the convex measure µ ϕ,β. Let µ be a probability measure in. For any two real-valued function g,h L 2 (µ, the covariance of g and h is quantity ( ( cov µ (g,h = ghdµ gdµ hdµ. Notice that cov µ (g,g = Var µ (g. If µ is a log-concave measure, i.e., dµ = e V(x dx for some strictly convex function V on, the Brascamp Lieb inequality (see [12] asserts that Var µ (h (D 2 V 1 h, h dµ, h L 2 (µ. (1.7 Since (cov µ (g,h 2 Var µ (gvar µ (h, as an immediate consequence of (1.7, we have the following covariance estimate (cov µ (g,h 2 (D 2 V 1 g, g dµ (D 2 V 1 h, h dµ. (1.8 The one-dimensional variant of (1.8 was established by Menz and Otto [18] as follows cov µ (g,h g L 1 (µ (V 1 h L (µ = g h (x dµ sup x R V (x. (1.9 They call this inequality an asymmetric Brascamp Lieb inequality. Note that it is asymmetric in two respects: One respect is to take an L 1 norm of g and an L norm of h, instead of L 2 norm and L 2 norm. The second respect is that the L norm is weighted with (V (x 1 while the L 1 norm is not weighted. The higher dimension version of (1.9 was proved by Carlen, Cordero Erausquin and Lieb [13]. In fact, they established a more general estimate as follows: let λ min (x denotes the smallest eigenvalued of D 2 V(x then for any (locally Lipschitz functions f,g L 2 (µ and for any 2 p and q = p/(p 1 we have cov µ (g,h (D 2 2 p V 1 q g L q p (µ λmin (D2 V 1 p h L p (µ. (1.10 The inequality (1.10 is sharp in the sense that the constant 1 in the right hand side can not be replaced by any smaller constant. For p = 2 we recover (1.8 from (1.10. Since D 2 V λ min I n then (1.10 implies For p = and q = 1, we get cov µ (g,h λ 1 p min g L p (µ λ 1 q min h L q (µ. cov µ (g,h g L (µ λ 1 min h L 1 (µ. 5 R

6 In particular, if n = 1 we obtain the inequality (1.9 of Menz and Otto. In this paper, we extend the asymmetric covariance estimate (1.10 to the convex measure µ ϕ,β. For n 1 and β n+1, let us denote { if n = 1, p β,n = Our next result is the following theorem. 2(1+ (β 1(β n 1+((β 1(β 2(β n(β n 11 2 n 1 if n 2. Theorem 1.3. Let β n+1 and λ min denotes the smallest eigenvalue of D 2 ϕ(x. Then for any 2 p p β,n, q = p/(p 1 and any (locally Lipschitz functions g,h in L 2 (µ ϕ,β, we have cov µϕ,β (g,h 1 β 1 ( (D 2 ϕ 1 ( 1 p g q q ϕdµ ϕ,β Rn λ 2 p min (D2 ϕ 1 p h p ϕdµ ϕ,β 1 p. (1.11 It is interesting that Theorem 1.3 implies the asymmetric covariance estimates (1.10 of Carlen, Cordero-Erausquin and Lieb for log-concave measure by letting β. We will show this fact in Sect. 3 below. We conclude this introduction by giving some comments on the methods used to prove our Theorem 1.1 and Theorem 1.3. Theorem 1.1 is proved by using the semi-group method while Theorem 1.3 is proved by adapting the L 2 method of Hörmander [17] to the L p setting. Both the proofs concern to a differential operator L on L 2 (µ ϕ,β defined by Lf(x = ϕ(x f(x (β 1 ϕ(x, f(x. To prove Theorem 1.1, we consider the semi-group P t on L 2 (µ ϕ,β associated with L, and define the function α(t = Φ(P t fdµ ϕ,β, f L 2 (µ ϕ,β. Using the semi-group property of P t and the assumption on Φ, we will establish the following differential inequality α (t 2c(β 1α (t, t > 0, which leads to the Φ entropy inequalities. We notice that the semi-group method is an useful methods to prove the functional inequalities (especially in sharp form. We refer the readers to the paper [2,3,11] and references thereinformoredetailsaboutthismethodanitsapplications. TheL 2 approach of Hörmander [17] is based on the classical dual representation for the covariance to establish the spectral estimates. In [13], Carlen, Cordero Erausquin and Lieb adapted the L 2 approach of Hörmander to the L p setting to prove the inequality (1.10 for log-concave measure. Our proof of Theorem 1.3 is an adaptation of their method to the setting of convex measures. However, the computations in our situation are more complicated. The rest of this paper is organized as follows. In Sect. 2 we use the semi-group method to prove the Φ entropy inequality in Theorem 1.1 and show how derive the Φ entropy inequalities for uniform log-concave measures from Theorem 1.1. Sect. 3 is devoted to prove the asymmetric covariance estimates for convex measures in Theorem 1.3 and show how derive the inequality of Carlen, Cordero Erausquin and Lieb from this theorem. 6

7 2 Proof of Theorem 1.1 This section is devoted to prove Theorem 1.1. Assume that D 2 ϕ ci n for some c > 0 and β > n+1. As in the introduction, let us define a differential operator L of order 2 on C c ( by Lf(x = ϕ(x f(x (β 1 ϕ(x, f(x, f C c (. By integration by parts, we have (Lfgdµ ϕ,β = f, g ϕdµ ϕ,β, f,g Cc (. Since D 2 ϕ(x ci n, c > 0 then the following weighted Poincaré inequality holds (see [19]: 1 Var µϕ,β (f f 2 ϕ(xdµ ϕ,β, f Cc (. 2c(β 1 Hence the operator L is uniquely extended to a self-adjoint operator on L 2 (µ ϕ,β (we still denoted the extended operator by L with domain D(L. Notice that C c is dense in D(L under the norm ( f 2 L 2 (µ ϕ,β + Lf 2 L 2 (µ ϕ,β 1 2. Let P t denote the semi-group on L 2 (µ ϕ,β generated by L. For any f L 2 (µ ϕ,β then P t f D(L and satisfies the equation P t f t (x = LP tf(x, P 0 f(x = f(x. Moreover, P t f fdµ ϕ,β in L 2 (µ ϕ,β and µ ϕ,β a.e. in as t. With these preparations, we are now ready to prove Theorem 1.1. Proof of Theorem 1.1. Let f L 2 (µ ϕ,β such that f 2 ϕ(xdµ ϕ,β <. Define the function α(t = Φ(P t (fdµ ϕ,β. By integration by parts, we have the following expression for α (t α (t = Φ (P t flp t fdµ ϕ,β = Φ (P t f P t f 2 ϕdµ ϕ,β. (2.1 We next compute α (t. For simplicity, we denote g = P t f. It is easily to verify the following relation i (Lg = L( i g+ i ϕ g (β 1 n ij 2 ϕ jg, i = 1,2,...,n, (2.2 j=1 7

8 where i = x i and ij 2 = 2 x i x j. Using the relation (2.2 and integration by parts, we have α (t = Φ (3 (g g 2 Lgϕdµ ϕ,β +2 Φ (g g, Lg ϕdµ ϕ,β R n = (Φ (3 (g g 2 ϕ, g ϕdµ ϕ,β +2 Φ (g g,l( g ϕdµ ϕ,β R n +2 Φ (g g, ϕ gϕdµ ϕ,(β 1 Φ (g D 2 ϕ g, g ϕdµ ϕ,β, (2.3 here, for simplifying notation, we denote L( g = (L( 1 g,...,l( n g. It follows from intgeration by parts that Φ (g g,l( g ϕdµ ϕ,β n = i g, (Φ (g i gϕ ϕdµ ϕ,β R n = Φ (3 (g 2 g g, g ϕ 2 dµ ϕ,β Φ (g 2 g 2 HSϕ 2 dµ ϕ,β R n Φ (g 2 g g, ϕ ϕdµ ϕ,β. (2.4 Noting that (Φ (3 (g g 2 ϕ = Φ (4 (g g 2 ϕ g +2Φ (3 (gϕd 2 g g +Φ (3 (g g 2 ϕ. (2.5 Plugging (2.4 and (2.5 into (2.3 and using the uniform convexity assumption D 2 ϕ ci n,c > 0 of ϕ we obtain α (t 2c(β 1α (t Φ (4 (g g 4 ϕ 2 dµ ϕ,β 4 Φ (3 (g D 2 g g, g ϕ 2 dµ ϕ,β R n Φ (3 (g g 2 ϕ, g ϕdµ ϕ, Φ (g D 2 g 2 HS ϕ2 dµ ϕ,β R n 2 Φ (g D 2 g g, ϕ ϕdµ ϕ,β +2 Φ (g g, ϕ gϕdµ ϕ,β. (2.6 Using again integration by parts, we have Φ (3 (g g 2 ϕ, g ϕdµ ϕ,β = 1 Φ (3 (g g 2 g, ϕ β+2 dx R Z n ϕ,β = 1 Φ (4 (g g 4 ϕ 2 dµ ϕ,β + 2 Φ (3 (g D 2 g g, g ϕ 2 dµ ϕ,β R n + 1 Φ (3 (g g 2 gϕ 2 dµ ϕ,β, (2.7 8

9 Φ (g D 2 g g, ϕ ϕdµ ϕ,β = 1 = 1 and + 1 Φ (g g, ϕ gϕdµ ϕ,β = 1 = 1 Φ (g D 2 g g, ϕ β+2 dx R Z n ϕ,β Φ (3 (g D 2 g g, g ϕ 2 dµ ϕ,β Φ (g g, g ϕ 2 dµ ϕ,β Φ (g D 2 g 2 HS ϕ2 dµ ϕ,β, (2.8 Φ (g g g, ϕ β+2 dx R Z n ϕ,β Φ (3 (g g 2 gϕ 2 dµ ϕ,β + 1 Φ (g g, g ϕ 2 dµ ϕ,β Φ (g( g 2 ϕ 2 dµ ϕ,β. (2.9 Inserting (2.7, (2.8 and (2.9 into (2.6, we get α (t 2c(β 1α (t β 1 Φ (4 (g g 4 ϕ 2 dµ ϕ,β R ( n 4(β 1 Φ (3 (g R D2 g g, g 1 g 2 g ϕ 2 dµ ϕ,β n ( β 1 2 Φ (g R D2 g 2 HS 1 ( g2 ϕ 2 dµ ϕ,β. (2.10 n It is well known that ( g 2 n D 2 g 2 HS, then it holds β 1 D2 g 2 HS 1 ( g2 β n 1 D 2 g 2 HS. (2.11 Let λ 1,...,λ n denote the eigenvalue of D 2 g with respect to the eigenvector e 1,...,e n respectively such that e i = 1 for any i = 1,2,...,n. Denote a i = g,e i 2 then it holds g 2 a 1 + +a n = 1, a i 0 for i = 1,...,n. Using these notation, we have ( 4(β 1 D2 g g, g 1 g 2 g = g 2 4(β 1 n λ i a i 1 n λ i = g 2 n 4(β 1a i 1 λ i. 9

10 Using Cauchy-Schwartz inequality, we have ( n 2 ( n ( 2 4(β 1a i 1 4(β 1ai 1 λ i (λ λ2 n = (4(β 12 n a2 i 8(β 1+n D 2 g 2 ( 2 HS 16(β 12 8(β 1+n ( 2 D 2 g 2 HS, here we used n a i = 1, n a2 i 1 and D 2 g 2 HS = n λ2 i. Putting the previous estimates together, we get 4(β 1 D2 g g, g g 2 g ((4β 52 +n D 2 g HS g 2. (2.12 Plugging (2.11 and (2.12 into (2.10 and using Φ 0, we obtain α (t 2c(β 1α (t β 1 Φ (4 (g g 4 ϕ 2 dµ ϕ,β + ((4β 52 +n Φ (3 (g D 2 g HS g 2 ϕ 2 dµ ϕ,β 2 β n 1 Φ (g D 2 g 2 HS ϕ2 dµ ϕ,β. It follows from the assumption on Φ and Cauchy Schwartz inequality that β 1 Φ(4 (g g 4 +2 β n 1 Φ (g D 2 g 2 HS 2(β 1(β n 1Φ (4 (gφ 2 (g g 2 D 2 g HS Therefore, it is easy to check that ((4β 52 +n α (t 2c(β 1α (t, t > 0. Φ (3 (g g 2 D 2 g HS This differential inequality implies α (t e 2c(β 1t α (0. Integrating the latter inequality on (0, we obtain lim α(t α(0 1 t 2c(β 1 α (0 which yields the Φ entropy inequality (1.2 because α(0 = Φ(fdµ ϕ,β, α (0 = Φ (f f 2 ϕdµ ϕ,β, 10

11 and ( lim α(t = Φ fdµ ϕ,β t since P t f fdµ ϕ,β in L 2 (µ ϕ,β. The proof of Theorem 1.1 is then completely finished. We conclude this section by showing that the Φ entropy inequality (1.3 can be derived from our Theorem 1.1. Let ψ be a convex function on such that D 2 ψ ρi n for some ρ > 0 and e ψ dx = 1. Denote µ the measure on with density e ψ. For β > n+1, denote ϕ β = 1 + ψ. By the uniform convexity of ψ, we have ϕ β β > 0 on for β large enough and D 2 ϕ β β 1 ρi n. Denote Z ϕβ,β = ψ β β dx and µ ϕ β,β the probability measure with density Z 1 ϕ β,β ϕ β β. Our aim is to apply the Φ entropy inequality (1.2 for the measure µ ϕβ,β and then letting β to derive the inequality (1.3. However, there is a difficulty here that although but lim β 1 (4β 5 2 +n 1 8(β 1(β n 1 = 2, 1 (4β 5 2 +n 1 8(β 1(β n 1 > 2, for any β > n+1. Hence for a convex function Φ satisfying Φ Φ (4 2(Φ (3 2 we do not know whether or not it satisfies (1.1. To overcome this difficulty, we use a approximation process as follows. Denote by I the domain of Φ. Let I 0 = (a,b be a bounded interval in I such that Ī0 I. Denote M = sup I0 Φ (3 <. Notice that the function Ψ p (t = (t a+1 p for p (1,2 satisfies Ψ p Ψ(4 p = 3 p 2 p (Ψ(3 p 2 = γ p (Ψ (3 p 2, γ p = 3 p 2 p > 2. For ǫ > 0, consider the function Φ ǫ = Φ+ǫΨ p on I 0. By Cauchy-Schwartz inequality, we have Φ ǫφ (4 ǫ ( 2 Φ (3 + γ p ǫ Ψ (3 p 2 on I 0. Denote N = inf I0 Ψ (3 p > 0. It is easy to check that on I 0, for any ( 2 Φ (3 + γ p ǫ Ψ (3 p 2 δ( Φ (3 +ǫ Ψ (3 p 2, { } 2γp 2 < δ < min, 2M2 +γ p ǫ 2 N 2. M 2 +ǫ 2 N 2 Consequently, the function Φ ǫ satisfies the condition (1.1 on I 0 for β > 0 large enough. Applying the inequality (1.2 for the convex function Φ ǫ and for any smooth function f with value in I 0 and the probability measure µ ϕβ,β with β large enough, we have ( 1 Φ ǫ (fdµ ϕβ Φ ǫ fdµ ϕβ Φ ǫ R (β 1 (f f 2 ϕ β dµ ϕβ,β. n 11 2 ρ β

12 Notice that Z 1 ϕ β,β ϕ β β e ψ and ϕ β 1. Letting β and then letting ǫ 0, we get ( Φ(fdµ Φ fdµ 1 Φ (f f 2 dµ, (2.13 R 2ρ n for any smooth function f with value in I 0 and for any bounded interval I 0 I with Ī 0 I. Suppose I = (a,b, let (a n n,(b n n be two sequence such that a n a and b n b. For any smooth function f with value in I, define f n = max{a n,min{f,b n }}. Applying the inequality (2.13 for I n and f n and then letting n we obtain the inequality (1.3 for f. 3 Proof of Theorem 1.3 In this section, we prove the asymmetric covariance estimates given in Theorem 1.3. Our method is based on the L 2 method of Hörmander which turns out to be very useful to prove the Brascamp Lieb type and Poincaré type inequalities (see, e.g., [13, 19]. Again, let L denote the differential operator Lf(x = ϕ(x f(x (β 1 ϕ(x, f(x, f C c (. Note by integration by parts that glfdµ ϕ,β = g, f ϕdµ, f,g Cc (Rn. hence L is extended uniquely to self-adjoint operator in L 2 (µ ϕ,β (which we still denote by L. By approximation argument, we can assume that ϕ is uniform convex in. Consequently, if we denote P t the semi-group associated with L, then by the weighted Poincaré inequality, we see that P t h L 2 µϕ,β exponentially decays to 0 for any function h L 2 (µ ϕ,β with hdµ ϕ,β = 0. For such a function h, the integral u := exists and is in the domain of L, and satisfies Lu = h. Since ( cov µϕ,β (g,h = g(x h(x 0 P t hdt, (3.1 hdµ ϕ,β dµ ϕ,β, then cov µϕ,β (g,h + c = cov µϕ,β (g,h for any constant c. Whence we can assume that hdµ ϕ,β = 0. Let u define by (3.1. We have by integration by parts and approximation argument that cov µϕ,β (g,h = g(xh(xdµ ϕ,β = g(xlu(xdµ ϕ,β = g, u ϕdµ ϕ,β. (3.2 With these preparations, we are now ready to give the proof of Theorem

13 Proof of Theorem 1.3. We can assume hdµ ϕ,β = 0. Let u define by (3.1. Using (3.2 and Hölder inequality, we have cov µϕ,β (g,h = g, u ϕdµ ϕ,β = (D 2 ϕ 1 p g,(d 2 ϕ 1 p u ϕdµϕ,β R ( n 1 ( 1 (D 2 ϕ 1 p g q q ϕdµ ϕ,β (D 2 ϕ 1 p u p p ϕdµ ϕ,β, (3.3 Rn here recall q = p/(p 1. It remains to show that ( 1 ( 1 (D 2 ϕ 1 p u p p 1 ϕdµ ϕ,β λ 2 p min R β 1 (D2 ϕ 1 p h p p ϕdµ ϕ,β, (3.4 n where λ min is the smallest eigenvalue of D 2 ϕ. To prove (3.4, we first compute L( u p as follows L( u p = ϕ ( u p (β 1 ϕ, ( u p n = pϕ u p 2 D 2 u 2 HS +p u p 2 ϕ ( j u j u+p(p 2ϕ u p 4 D 2 u u 2 j=1 ( n n p(β 1 u p 2 i ϕ ij 2 u j u j=1 ( = p u p 2 L( u, u +ϕ D 2 u 2 HS +(p 2ϕ D2 u u 2, (3.5 u 2 here we use the notation L( u = (L( 1 u,...,l( n u. By integration by parts, we have L( u p ϕdµ ϕ,β = ( u p, ϕ ϕdµ ϕ,β = 1 ( u p, ϕ β+2 dx R Z n ϕ,β = 1 ( u p ϕ 2 dµ ϕ,β. (3.6 We are readily to check that ( u p = p ( u, u + D 2 u 2HS +(p 2 D2 u u 2 u p 2. u 2 Plugging the previous identity into (3.6, we arrive L( u p ϕdµ ϕ,β = p ( u, u + D 2 u 2HS u u 2 +(p 2 D2 u p 2 ϕ 2 dµ u 2 ϕ,β. (3.7 13

14 From (2.2, we have L( u = (Lu u ϕ+(β 1D 2 ϕ u. Using this commutation relation together with (3.5 and Lu = h, we get L( u p ϕdµ ϕ,β = p h, u u p 2 ϕdµ ϕ,β +p(β 1 D 2 ϕ u, u u p 2 ϕdµ ϕ,β R n +p ( D 2 u 2HS +(p 2 D2 u u 2 u p 2 ϕ 2 dµ R u n 2 ϕ,β p u u p 2 u, ϕ ϕdµ ϕ,β. (3.8 Using integration by parts, we have u u p 2 u, ϕ ϕdµ ϕ,β = 1 u u p 2 u, ϕ β+2 dx R Z n ϕ,β = 1 ( u, u +( u 2 +(p 2 u D2 u u, u u 2 Inserting the previous equality into (3.8 implies u p 2 ϕ 2 dµ ϕ,β. L( u p ϕdµ ϕ,β = p h, u u p 2 ϕdµ ϕ,β +p(β 1 D 2 ϕ u, u u p 2 ϕdµ ϕ,β R n +p ( D 2 u 2HS u u 2 +(p 2 D2 u p 2 ϕ 2 dµ R u n 2 ϕ,β p ( u, u +( u 2 +(p 2 u D2 u u, u u p 2 ϕ 2 dµ u 2 ϕ,β. (3.9 Combining (3.7 and (3.9, we get 0 = p h, u u p 2 ϕdµ ϕ,β +p(β 1 D 2 ϕ u, u u p 2 ϕdµ ϕ,β + p + p(p 2 ( (β 1 D 2 u 2 HS ( u 2 u p 2 ϕ 2 dµ ϕ,β ((β 1 D2 u u 2 u D2 u u, u u 2 u 2 14 u p 2 ϕ 2 dµ ϕ,β. (3.10

15 We next claim that if u > 0 then (β 1 D 2 u 2 HS ( u2 +(p 2 ((β 1 D2 u u 2 u 2 u D2 u u, u 0 (3.11 u 2 provided 2 p p β,n. Indeed, if n = 1 then the left hand side of (3.11 is equal to (β 2(p 1 u 2 and hence is non-negative. We next consider the case n 2. Let λ 1,...,λ n denote the eigenvalues of D 2 u with respect to the eigenvectors e 1,...,e n respectively such that e i = 1 for any i = 1,...,n. Denote a i = u,e i 2 [0,1]. We have u 2 a 1 + +a n = 1, u = n λ i, D 2 u 2 HS = n λ2 i, and D 2 u u 2 u 2 = n λ 2 i a i, D 2 u u, u u 2 = n λ i a i. Hence, the left hand side of (3.11 becomes ( n n 2 ( (β 1 λ 2 i λ i +(p 2 (β 1 ( n n λ 2 i a i λ i λ i a i. The set S := {x = (x 1,...,x n : x i 0, i = 1,...,n, n x i = 1} is a convex subset of with extreme points v i,i = 1,...,n such that the ith coordinate is 1 and other coordinates are 0. The function ( n n 2 ( F(x = (β 1 λ 2 i λ i +(p 2 (β 1 ( n n λ 2 ix i λ i λ i x i is affine on. Hence min S F is attained at a point v i for some i {1,...,n}. Let i 0 be such an index i. Note that a = (a 1,...,a n S, hence we have F(a ((p 1λ 2 i 0 +(β 1 ( 2 λ 2 i pλ i 0 λ i λ i i i 0 i i 0 i i 0 ( ((p 1λ 2 i 0 + β 1 2 ( 2 λ i pλ i0 λ i λ i n 1 i i 0 i i 0 i i 0 ( = ((p 1λ 2 i 0 + β n 2 λ i pλ i0 λ i n 1 i i 0 i i 0 ( 1 ((β n(p λi0 n 1 pλ i 0 i i 0 λ i i i 0 λ i here the second and fourth inequalities come from Cauchy Schwartz inequality. Therefore F(a 0 provided ( 1 ((β n(p p, n 1 15

16 for p 2. However, this condition is equivalent to our assumption 2 p p β,n. Hence, we have proved (β 1 ( n n 2 ( λ 2 i λ i +(p 2 (β 1 ( n n λ 2 i a i λ i λ i a i = F(a 0, for 2 p p β,n. This proves (3.11 when n 2. It follows from (3.10 and (3.11 that (D 2 ϕ 1 2 u 2 u p 2 ϕdµ ϕ,β = D 2 ϕ u, u u p 2 ϕdµ ϕ,β 1 β 1 = 1 β 1 1 β 1 h, u u p 2 ϕdµ ϕ,β (D 2 ϕ 1 p u,(d 2 ϕ 1 p h u p 2 ϕdµ ϕ,β (D 2 ϕ 1 p u (D 2 ϕ 1 p h u p 2 ϕdµ ϕ,β. (3.12 Let A be a positive n n matrix, and v be a vector in. It is well-known that for p 2. Moreover, it is obvious that A 1 p v p v p 2 A 1 2 v 2, (3.13 u λ 1 p min (D2 ϕ 1 p u. (3.14 Inserting the estimates (3.13 and (3.14 into (3.12 for A = D 2 ϕ and v = u with notice that p 2, we get (D 2 ϕ 1 p u p ϕdµ ϕ,β 1 (D 2 ϕ 1 p u p 1 λ p 2 p min R β 1 (D2 ϕ 1 p h ϕdµϕ,β. (3.15 n Applying Hölder inequality to the right hand side of (3.15 and simplifying the obtained inequality, we arrive (3.4. The proof of Theorem 1.3 is completed. We conclude this section by showing that the inequality (1.10 can be derived from our Theorem 1.3. Let ψ be a strictly convex function on such that e ψ dx = 1, and µ be the probability of density e ψ. Perturbing ψ by ǫ x 2 /2, we can assume that ψ is uniform convex on. Let ϕ β = 1+ ψ and µ β β be the probability measure of density Z 1 β ψ β β for β > n+1 where Z β is normalization constant. We have D 2 ϕ β = β 1 D 2 ψ. Denote λ min and λ min,β the smallest eigenvalue of D 2 ψ and D 2 ϕ β respectively, we have λ min,β = β 1 λ min. 16

17 Let g,hc c ( and 2 p <. We have p β,n as β, hence p β,n > p for β large enough. Applying Theorem 1.3 to g,h and for µ β with β large enough, we have cov µβ (g,h β ( 1 ( 1 (D 2 ψ 1 p g q q ϕ β dµ β λ 2 p min β 1 (D2 ψ 1 p h p p ϕ β dµ β. Rn Since ϕ β 1 and Z 1 β ψ β β e ψ as β, then by letting β in the preceding inequality, we obtain (1.10 for any function g,h Cc (. By standard approximation argument, we get (1.10 for any 2 p <. The case p = is obtained from the case p < by letting p. Acknowledgements This work was done when the author was PhD student at the Université Pierre et Marie Curie (Paris VI under the supervision of Prof. Dario Cordero Erausquin. The author would like to thank him for his help and advice. References [1] M. Arnaudon, M. Bonnefont, and A. Joulin, Intertwining and generalized Brascamp Lieb inequalities, Rev. Mat. Iberoam., 34 (2018, no. 3, [2] D. Bakry, and M. Émery, Diffusions hypercontractives (French [Hypercontractive diffusions], Séminaire de probabilités, XIX, 1983/84, , Lecture Notes in Math., 1123, Springer, Berlin, [3] D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 348. Springer, Cham, xx+552 pp. [4] D. Bakry, I. Gentil and G. Scheffer, Sharp Beckner type inequalities for Cauchy and spherical distributions, preprint, arxiv: [5] W. Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc., 105 (1989, no. 2, [6] A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo, and J. L. Vázquez, Hardy Poincaré inequalities and applications to nonlinear diffusions, C. R. Math. Acad. Sci. Paris, 344 (2007, no. 7, [7] A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo, and J. L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal., 191 (2009, no. 2,

18 [8] M. Bonforte, J. Dolbeault, G. Grillo, and J. L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proc. Natl. Acad. Sci. USA, 107 (2010, no. 38, [9] M. Bonnefont, A. Joulin, and Y. Ma, A note on spectral gap and weighted Poincaré inequalities for some one dimensional diffusions, ESAIM Probab. Stat., 20 ( [10] S. G. Bobkov, and M. Ledoux, Weighted Poincaré type inequalities for Cauchy and other convex measures, Ann. Probab., 37 (2009, no. 2, [11] F. Bolley, and I. Gentil, Phi-entropy inequalities for diffusion semigroups, J. Math. Pures Appl. (9, 93 (2010, no. 5, [12] H. J. Brascamp, and E. H. Lieb, On extensions of the Brunn Minkowski and Prékopa Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal., 22 (1976, no. 4, [13] E. Carlen, D. Cordero Erausquin, and E. H. Lieb, Asymmetric covariance estimates and related inequalities of Brascamp Lieb type for log-concave measures, Ann. Inst. H. Poincaré Probab. Statist., 49 ( [14] P. Cattiaux, A. Guillin, and L. M. Wu, Some remarks on weighted logarithmic Sobolev inequality, Indiana Univ. Math. J., 60 (2011, no. 6, [15] D. Chafaï, Etropies, convexity, and functional inequalities: on Φ entropy and Φ Sobolev inequalities, J. Math. Kyoto Univ., 44 (2004, no. 2, [16] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975, no. 4, [17] L. Hörmander, L 2 estimates and existence theorems for the operator, Acta Math., 113 ( [18] G. Menz, and F. Otto, Uniform logarithmic Sobolev inequalities for conservative spin systems with super quadratic single site potential, Ann. Probab., 41 (2013, no. 3B, [19] V. H. Nguyen, Dimensional variance inequalities of Brascamp Lieb type and a local approach to dimensional Prékopa s theorem, J. Funct. Anal., 266 (2014, no. 2, [20] G. Scheffer, Local Poincaré inequalities in non negative curvature and finite dimension, J. Funct. Anal., 198 (2003, no. 1,