Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality

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Journal of Functional Analysis 262 (2012) 2646–2676 www.elsevier.com/locate/jfa

Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality Fabrice Baudoin a,∗,1 , Michel Bonnefont b a Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA b Institut de Mathmatiques de Bordeaux, Université Bordeaux 1, 33405 Talence, France

Received 8 June 2011; accepted 19 December 2011 Available online 10 January 2012 Communicated by Cédric Villani

Abstract Let M be a smooth connected manifold endowed with a smooth measure μ and a smooth locally subelliptic diffusion operator L which is symmetric with respect to μ. We assume that L satisfies a generalized curvature dimension inequality as introduced by Baudoin and Garofalo (2009) [9]. Our goal is to discuss functional inequalities for μ like the Poincaré inequality, the log-Sobolev inequality or the Gaussian logarithmic isoperimetric inequality. © 2011 Elsevier Inc. All rights reserved. Keywords: Log-Sobolev inequality; Subelliptic diffusion operators; HWI inequality; Isoperimetric inequality

Contents 1. 2. 3.

Introduction, main results and examples . . . . . . . . . . . . . . . . . . Spectral gap and modified log-Sobolev inequalities . . . . . . . . . . . Wang inequality for the heat semigroup and log-Sobolev inequality 3.1. Reverse log-Sobolev inequalities . . . . . . . . . . . . . . . . . . . 3.2. Wang inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Log-Harnack inequality . . . . . . . . . . . . . . . . . . . . . . . . .

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* Corresponding author.

E-mail addresses: [email protected] (F. Baudoin), [email protected] (M. Bonnefont). 1 First author supported in part by NSF Grant DMS 0907326. 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.12.020

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3.4. Log-Sobolev inequality and exponential integrability of the square distance 3.5. A dimensional bound on the log-Sobolev constant . . . . . . . . . . . . . . . . . . 3.6. Log-Sobolev inequality and diameter bounds . . . . . . . . . . . . . . . . . . . . . 4. Log-Sobolev inequality and transportation cost inequalities . . . . . . . . . . . . . . . . 4.1. A bound of the entropy of the semigroup by the L2 -Wasserstein distance . . 4.2. Modified HWI inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Logarithmic isoperimetric inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction, main results and examples Logarithmic Sobolev inequalities, introduced and studied by L. Gross [20], are a major tool for the analysis of finite- or infinite-dimensional spaces, see for instance [2] and the references therein. The celebrated Bakry–Émery criterion [6] which is based on the so-called Γ2 calculus for diffusion operators provides a powerful way to establish such inequalities. However this criterion requires some ellipticity property from the diffusion operator and fails to hold even for simple subelliptic diffusion operators like the sub-Laplacian on the Heisenberg group (see [25]). However in the past few years, numerous works like [4,5,7,12,14,15,22,28–30,34,31] have shown on some examples that the heat semigroup associated with certain subelliptic operators may satisfy functional inequalities that were only known to hold in elliptic situations. Most of these examples have in common the property that the subelliptic diffusion operator satisfies the generalized curvature dimension inequality that was introduced in [9] in an abstract setting. As we will see in this work, this curvature dimension inequality may also be used to prove the Poincaré inequality, the log-Sobolev inequality or the Gaussian logarithmic isoperimetric inequality for the invariant measure of a subelliptic diffusion operator in some interesting new situations. Let us describe our framework and results in more details. In this paper, M will be a C ∞ connected finite-dimensional manifold endowed with a smooth measure μ and a second-order diffusion operator L on M, locally subelliptic in the sense of [16] (see also [24]), satisfying L1 = 0, 

 f Lg dμ = M

 f Lf dμ  0,

gLf dμ, M

M

for every f, g ∈ C0∞ (M). We indicate with Γ (f ) := Γ (f, f ) the carré du champ, that is the quadratic differential form defined by Γ (f, g) =

 1 L(fg) − f Lg − gLf , 2

f, g ∈ C ∞ (M).

(1.1)

An absolutely continuous curve γ : [0, T ] → M is said to√be subunit for the operator L if d f (γ (t))|  (Γf )(γ (t)). We then define the for every smooth function f : M → R we have | dt subunit length of γ as s (γ ) = T . Given x, y ∈ M, we indicate with    S(x, y) = γ : [0, T ] → M  γ is subunit for L, γ (0) = x, γ (T ) = y .

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In this paper we assume that S(x, y) = ∅,

for every x, y ∈ M.

Under such assumption it is easy to verify that    d(x, y) = inf s (γ )  γ ∈ S(x, y) ,

(1.2)

defines a true distance on M. Furthermore, it is known that      d(x, y) = sup f (x) − f (y)  f ∈ C ∞ (M), Γ (f )∞  1 ,

x, y ∈ M.

(1.3)

Throughout this paper we assume that the metric space (M, d) is complete. In addition to the differential form (1.1), we assume that M is endowed with another smooth symmetric bilinear differential form, indicated with Γ Z , satisfying for f, g ∈ C ∞ (M) Γ Z (fg, h) = f Γ Z (g, h) + gΓ Z (f, h), and Γ Z (f ) = Γ Z (f, f )  0. We make the following assumptions that will be in force throughout the paper: (H.1) There exists an increasing sequence hk ∈ C0∞ (M) such that hk  1 on M, and   Γ (hk )

∞

  + Γ Z (hk )∞ → 0,

as k → ∞.

(H.2) For any f ∈ C ∞ (M) one has     Γ f, Γ Z (f ) = Γ Z f, Γ (f ) . As it has been proved in [9], the assumption (H.1) which is of technical nature, implies in particular that L is essentially self-adjoint on C0∞ (M). The assumption (H.2) is more subtle and is crucial for the validity of most the subsequent results: It is discussed in details in [9] in several geometric examples. Let us consider

1 LΓ (f, g) − Γ (f, Lg) − Γ (g, Lf ) , 2

1 Γ2Z (f, g) = LΓ Z (f, g) − Γ Z (f, Lg) − Γ Z (g, Lf ) . 2 Γ2 (f, g) =

(1.4) (1.5)

As for Γ and Γ Z , we will freely use the notations Γ2 (f ) = Γ2 (f, f ), Γ2Z (f ) = Γ2Z (f, f ). Definition 1.1. We say that L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, d) if there exist constants ρ1 ∈ R, ρ2 > 0, κ  0, and 0 < d  ∞ such that the inequality  1 κ Z 2 Γ (f ) + ρ2 Γ Z (f ) Γ2 (f ) + νΓ2 (f )  (Lf ) + ρ1 − d ν holds for every f ∈ C ∞ (M) and every ν > 0, where Γ2 and Γ2Z are defined by (1.4) and (1.5).

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Remark 1.2. Of course, it is understood in the previous definition that CD(ρ1 , ρ2 , κ, ∞), means  κ Γ2 (f ) + νΓ2Z (f )  ρ1 − Γ (f ) + ρ2 Γ Z (f ). ν The purpose of our work is to understand the functional inequalities that are satisfied by the invariant measure μ under the assumption that the generalized curvature dimension inequality is satisfied. Let us observe that unlike [9,8], where the authors focused on functional inequalities involving in a crucial way the dimension d, here we shall mainly be interested in functional inequalities that are independent from the dimension d. The paper is organized as follows. The purpose of Section 2 is to prove the following theorem: Theorem 1.3. Assume that L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞) with ρ1 > 0, ρ2 > 0 and κ  0. • The measure μ is finite and the following Poincaré inequality holds 

 f 2 dμ − M

2 f dμ



M

κ + ρ2 ρ1 ρ2

 Γ (f ) dμ,

f ∈ D(L).

M

• If μ is a probability measure, that is μ(M) = 1, then for f ∈ C0 (M), 





f 2 ln f 2 dμ − M

f 2 dμ ln M

2(κ + ρ2 )  ρ1 ρ2

f 2 dμ M



κ + ρ2 Γ (f ) dμ + ρ1

M

Γ (f ) dμ .



Z

M

In Section 3, we will prove the following theorem which is a subelliptic analogue of a famous result due to F.Y. Wang [38]. Theorem 1.4. Assume that the measure μ is a probability measure and that L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞) for some ρ1 ∈ R, ρ2 > 0, κ  0. Assume moreover that  eλd

2 (x ,x) 0

dμ(x) < +∞,

M

for some x0 ∈ M and λ > C0∞ (M),

ρ1− 2 ,

then there is a constant ρ0 > 0 such that for every function f ∈ 

 f ln f dμ − 2

M

2

 2

f dμ ln M

2 f dμ  ρ0



2

M

Γ (f ) dμ. M

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In Section 4, adapting some methods of Bobkov, Gentil and Ledoux [11], we prove an analogue of an Otto–Villani theorem [32]. We recall that L2 -Wasserstein distance of two measures ν1 and ν2 on M is defined by  W2 (ν1 , ν2 ) = inf 2

d 2 (x, y) dΠ(x, y)

Π

M

where the infimum is taken over all coupling of ν1 and ν2 that is on all probability measures Π on M × M whose marginals are respectively ν1 and ν2 . Theorem 1.5. Assume that the measure μ is a probability measure and that L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞) for some ρ1 ∈ R, ρ2 > 0, κ  0. If the quadratic transportation cost inequality  W2 (μ, ν)  c Entμ 2

dν dμ

(1.6)

is satisfied for every absolutely continuous probability measure ν with a constant c < the following modified log-Sobolev inequality  Entμ (f )  C1 M

Γ (f ) dμ + C2 f

 M

2 , ρ1−

then

Γ Z (f ) dμ f

holds for some constants C1 and C2 depending only on c, ρ1 , κ, ρ2 . Finally, the goal of the Section 5 is to study isoperimetric inequalities. We will prove the following result which is a subelliptic generalization of a theorem due to Ledoux [26]: Theorem 1.6. Assume that the measure μ is a probability measure, that L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞) for some ρ1 ∈ R, ρ2 > 0, κ  0 and that μ satisfies the log-Sobolev inequality: 

 f 2 ln f 2 dμ − M

 f 2 dμ 

f 2 dμ ln M

M

2 ρ0

 Γ (f ) dμ,

f ∈ C0∞ (M)

(1.7)

M

for all smooth functions f ∈ C0∞ (M). Let A be a set of the manifold M which has a finite perimeter P (A) and such that 0  μ(A)  12 , then   1 2 ρ0 1 √

ρ , . min P (A)  μ(A) ln 0 μ(A) 4(3 + 2κ ρ1− ρ2 ) ln 2

To conclude this introduction, let us now turn to the fundamental question of examples to which our above results apply. We refer the reader to [9] for more details about most of the examples we discuss below.

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A first observation is that if M is an n-dimensional complete Riemannian manifold and L is the Laplace–Beltrami operator, the assumptions (H.1) and (H.2) hold trivially with Γ Z = 0. Indeed, the assumption (H.1) is satisfied as a consequence of the completeness of M and the assumption (H.2) is trivially satisfied. In this example, the generalized curvature dimension inequality CD(ρ1 , 1, 0, n) is implied by (and it is in fact equivalent to) the assumption that the Ricci curvature of M satisfies the lower bound Ric  ρ1 . Besides Laplace–Beltrami operators on complete Riemannian manifolds with Ricci curvature bounded from below, a wide class of examples is given by sub-Laplacians on Sasakian manifolds. Let M be a complete strictly pseudo convex CR Sasakian manifold with real dimension 2n + 1. Let θ be a pseudo-hermitian form on M with respect to which the Levi form is positive definite. The kernel of θ determines a horizontal bundle H. Denote now by T the Reeb vector field on M, i.e., the characteristic direction of θ . We denote by ∇ the Tanaka–Webster connection of M. We recall that the CR manifold (M, θ ) is called Sasakian if the pseudo-hermitian torsion of ∇ vanishes, in the sense that T(T , X) = 0, for every X ∈ H. For instance the standard CR structures on the Heisenberg group H2n+1 and the sphere S2n+1 are Sasakian. On CR manifolds, there is a canonical subelliptic diffusion operator which is called the CR sub-Laplacian. It plays the same role in CR geometry as the Laplace–Beltrami operator does in Riemannian geometry. In this framework we have the following result that shows the relevance of the generalized curvature dimension inequality. Proposition 1.7. (See [9].) Let (M, θ ) be a CR manifold with real dimension 2n + 1 and vanishing Tanaka–Webster torsion, i.e., a Sasakian manifold. If for every x ∈ M the Tanaka–Webster Ricci tensor satisfies the bound Ricx (v, v)  ρ1 |v|2 , for every horizontal vector v ∈ Hx , then, for the CR sub-Laplacian of M, the curvature dimension inequality CD(ρ1 , d4 , 1, d) holds with d = 2n and Γ Z (f ) = (Tf )2 . In addition to sub-Laplacians on Heisenberg groups, more generally, the sub-Laplacian on any Carnot group of step 2 has been shown to satisfy the generalized curvature dimension inequality CD(0, ρ2 , κ, d), for some values of the parameters ρ2 and κ. Let us mention that recently, in [10], the authors study sub-Laplacians in infinite-dimensional Heisenberg type groups and show that a generalized curvature dimension inequality is satisfied with d = +∞. In that case the assumption (H.1) is of course not satisfied but is somehow replaced by the existence of nice and uniform finite-dimensional approximations, so that with suitable modifications the results of the present paper may be used. For infinite-dimensional situations, we also point out the reader to the work [23] that uses completely different methods. Another interesting example, which has recently been highlighted in different contexts by several works (see [40,41,44,21], see also [19]) is given by the Grushin operator on R2n . It is defined by

L=

n  2  ∂

x 2 ∂ 2 + 2 ∂yi2 ∂xi2 i=1

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where x 2 = x12 + · · · + xn2 for x = (x1 , . . . , xn ) ∈ Rn . This operator admits the Lebesgue measure λ as invariant and symmetric measure. If we set Xi = ∂x∂ i , Yi,j = xj ∂y∂ i and Zi = ∂y∂ i , we can write this operator as L=

n  i=1

Xi2 +

n 

2 Yi,j =−

i,j =1

n 

Xi∗ Xi −

i=1

n 

∗ Yi,,j Yi,j .

i,j =1

The only non-zero Lie bracket relations are [Xi , Yi,j ] = Zj

for 1  i, j  n.

This algebra structure is then exactly the one of a Carnot group of step 2 and the criterion  ∂f 2 CD(0, ρ2 , κ, n + n2 ) therefore holds with Γ Z (f, f ) = ni=1 ( ∂y ) and some constant ρ2 . Also it i is easy to see that assumptions (H.1) and (H.2) are satisfied in that case. Let us however observe that more general Grushin operators are considered in [44], and that they cannot be handled at the moment with our methods, since their Lie algebra correspond to a Carnot group of step higher than 2. Finally, we mention that some close results are obtained in [21] for Fokker–Planck type operators. In those examples, that typically do not satisfy the generalized curvature dimension inequality studied in this work, the hypoellipticity of the operator stems from its first order part; a situation radically different from the examples discussed above. 2. Spectral gap and modified log-Sobolev inequalities Throughout this section, we assume that the operator L satisfies the curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞) for some ρ1 > 0, ρ2 > 0, κ  0. The main tool to prove the fore mentioned theorems, is the heat semigroup Pt = etL , which is defined using the spectral theorem. Since L satisfies the curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞), this semigroup is stochastically complete (see [9]), i.e. Pt 1 = 1. Moreover, thanks to the hypoellipticity of L, for f ∈ Lp (M), 1  p  ∞, the function (t, x) → Pt f (x) is smooth on M × (0, ∞) and  Pt f (x) = p(x, y, t)f (y) dμ(y) M

where p(x, y, t) = p(y, x, t) > 0 is the so-called heat kernel associated to Pt . Henceforth, in all the paper, we denote Cb∞ (M) = C ∞ (M) ∩ L∞ (M). For ε > 0 we denote by Aε the set of functions f ∈ Cb∞ (M) such that f = g + ε,  √ for some ε > 0 and some g ∈ Cb∞ (M), g  0, such that g, Γ (g), Γ Z (g) ∈ L2 (M). As shown in [9], this set is stable under the action of Pt , i.e., if f ∈ Aε , then Pt f ∈ Aε . Our goal is to prove Theorem 1.3. In that direction, we first establish a useful gradient bound for Pt .

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Proposition 2.1. Let ε > 0 and f ∈ Aε . For x ∈ M, t  0 one has κ + ρ2 (Pt f )Γ Z (ln Pt f ) ρ1  ρ ρ t   κ + ρ2   −2 1 2  e κ+ρ2 Pt f Γ (ln f ) + Pt f Γ Z (ln f ) . ρ1

(Pt f )Γ (ln Pt f ) +

Proof. Let us fix T > 0 once time for all in the following proof. Given a function f ∈ Aε , for 0  t  T we introduce the entropy functionals φ1 (x, t) = (PT −t f )(x)Γ (ln PT −t f )(x), φ2 (x, t) = (PT −t f )(x)Γ Z (ln PT −t f )(x), which are defined on M × [0, T ]. As it has been proved in [9], a direct computation shows that Lφ1 +

∂φ1 = 2(PT −t f )Γ2 (ln PT −t f ), ∂t

Lφ2 +

∂φ2 = 2(PT −t f )Γ2Z (ln PT −t f ). ∂t

and

Let us observe that for the second equality the hypothesis (H.2) is used in a crucial way. Consider now the function φ(x, t) = a(t)φ1 (x, t) + b(t)φ2 (x, t) = a(t)(PT −t f )(x)Γ (ln PT −t f )(x) + b(t)(PT −t f )(x)Γ Z (ln PT −t f )(x), where a and b are two non-negative functions that will be chosen later. Applying the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞), we obtain Lφ +

∂φ = a (PT −t f )Γ (ln PT −t f ) + b (PT −t f )Γ Z (ln PT −t f ) ∂t + 2a(PT −t f )Γ2 (ln PT −t f ) + 2b(PT −t f )Γ2Z (ln PT −t f )  a2 (PT −t f )Γ (ln PT −t f )  a + 2ρ1 a − 2κ b   + b + 2ρ2 a (PT −t f )Γ Z (ln PT −t f ).

Let us now chose b(t) = e and

−

2ρ1 ρ2 t κ+ρ2

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a(t) = −

b (t) , 2ρ2

so that b + 2ρ2 a = 0 and a + 2ρ1 a − 2κ

a2 = 0. b

With this choice, we get Lφ +

∂φ  0, ∂t

and therefore from a comparison theorem for parabolic partial differential equations (see for instance p. 52 in [17] or Proposition 3.2 in [9]) we have   PT φ(·, T ) (x)  φ(x, 0). Since, φ(x, 0) = a(0)(PT f )(x)Γ (ln PT f )(x) + b(0)(PT f )(x)Γ Z (ln PT f )(x) and       PT φ(·, T ) (x) = a(T )PT f Γ (ln f ) (x) + b(T )PT f Γ Z (ln f ) (x), the proof is completed.

2

A similar proof as above also provides the following: Proposition 2.2. Let f ∈ L2 (M) such that f ∈ C ∞ (M) and Γ (f ), Γ Z (f ) ∈ L1 (M). For x ∈ M, t  0 one has  ρ ρ t   κ + ρ2  Z  κ + ρ2 Z −2 1 2 Γ (Pt f )  e κ+ρ2 Pt Γ (f ) + Pt Γ (f ) . Γ (Pt f ) + ρ1 ρ1 Proof. We introduce φ1 (x, t) = Γ (PT −t f )(x), φ2 (x, t) = Γ Z (PT −t f )(x), and observe that Lφ1 +

∂φ1 = 2Γ2 (PT −t f ), ∂t

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and Lφ2 +

∂φ2 = 2Γ2Z (PT −t f ). ∂t

The conclusion is then reached by following the lines of the proof of Proposition 2.1.

2

A first interesting consequence of the above functional inequalities is the fact that ρ1 > 0 implies that the invariant measure is finite. Corollary 2.3. The measure μ is finite, i.e. μ(M) < +∞ and for every x ∈ M, f ∈ L2 (M), Pt f (x) →t→+∞

1 μ(M)

 f dμ. M

Proof. Let f, g ∈ C0∞ (M), we have t  

 (Pt f − f )g dμ = M

0 M

t

t  ∂ Ps f g dμ ds = (LPs f )g dμ ds ∂s 0 M



=−

Γ (Ps f, g) dμ ds. 0 M

By means of Proposition 2.2, and Cauchy–Schwarz inequality, we find      (Pt f − f )g dμ   M

 t



e

ρ ρ s 2

1 2 − κ+ρ

0

 ds

  Γ (f )

 κ + ρ2  Γ Z (f ) + ∞ ∞ ρ1



1

Γ (g) 2 dμ.

(2.1)

M

Now it is seen from spectral theorem that in L2 (M) we have a convergence Pt f → P∞ f , where P∞ f belongs to the domain of L. Moreover LP∞ f = 0. By hypoellipticity of L we deduce that P∞ f is a smooth function. Since LP∞ f = 0, we have Γ (P∞ f ) = 0 and therefore P∞ f is constant. Let us now assume that μ(M) = +∞. This implies in particular that P∞ f = 0 because no constant besides 0 is in L2 (M). Using then (2.1) and letting t → +∞, we infer    +∞ ρ ρ s         1 2 1 −2 κ+ρ  f g dμ  Γ (f ) + κ + ρ2 Γ Z (f ) 2 ds e Γ (g) 2 dμ.   ∞ ∞ ρ1 M

0

M

Let us assume g  0, g = 0 and take for f the sequence hn from assumption (H.1). Letting n → ∞, we deduce

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 g dμ  0, M

which is clearly absurd. As a consequence μ(M) < +∞. The invariance of μ implies then 

 P∞ f dμ = M

f dμ, M

and thus 1 P∞ f = μ(M)

 f dμ. M

Finally, using the Cauchy–Schwarz inequality, we find that for x ∈ M, f ∈ L2 (M), s, t, τ  0,     Pt+τ f (x) − Ps+τ f (x) = Pτ (Pt f − Ps f )(x)     =  p(τ, x, y)(Pt f − Ps f )(y) μ(dy) M



 2 p(τ, x, y)2 μ(dy)Pt f − Ps f 2

 M

 2  p(2τ, x, x)Pt f − Ps f 2 . Thus, we also have Pt f (x) →t→+∞

1 μ(M)

 2

f dμ. M

We also deduce a spectral gap inequality: Corollary 2.4. For every f in the domain of L, 

 f 2 dμ − M

2 f dμ

M



κ + ρ2 ρ1 ρ2

 Γ (f ) dμ. M

Proof. We use an argument close to one found in [35]. Let f ∈ C ∞ (M) with a compact support. By Proposition 2.2, we have for t  0  Γ (Pt f, Pt f ) dμ  C(f )e M

−

2ρ1 ρ2 t κ+ρ2

,

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2657

with  C(f ) =

κ + ρ2 Z Γ (f, f ) dμ. ρ1

Γ (f, f ) + M

By the spectral theorem, one has 

∞ Γ (Pt f, Pt f ) dμ =

M

λe−2λt dEλ (f )

0

and ∞

 Γ (f, f ) dμ = M

λ dEλ (f ) 0

where dEλ is the spectral measure associated to −L. Thus, by Holder inequality, for 0  s  t 

∞ Γ (Ps , Ps f ) dμ =

M

λe

−2λs

 ∞ dEλ (f ) 

λe

0

−2λs

 s  ∞ dEλ (f )

0 s

 C(f ) t e

−

2ρ1 ρ2 s κ+ρ2

 t−s

t

t

λ dEλ (f ) 0



Γ (f, f ) dμ

t−s t

.

M

Letting t → ∞ gives  Γ (Ps , Ps f ) dμ  e

−

2ρ1 ρ2 s κ+ρ2

M

 Γ (f, f ) dμ M

for all C ∞ function with a compact support. Since this space is dense in the domain of the Dirichlet form, it implies the desired Poincaré inequality. 2 Finally, we also deduce a modified log-Sobolev inequality that involves a vertical term: Corollary 2.5. Let us assume μ(M) = 1. For f ∈ C0 (M), 

 f ln f dμ − 2

M

2

 2

f dμ ln M

2(κ + ρ2 ) f dμ  ρ1 ρ2



2

M

M

κ + ρ2 Γ (f ) dμ + ρ1



Γ (f ) dμ . Z

M

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Proof. Let g ∈ Aε . We have 

 g ln g dμ − M

 g dμ = −

gdμ ln M

+∞

M

∂ ∂t

0

 Pt g ln Pt g dμ dt M

+∞ =−

LPt g ln Pt g dμ dt M

0

+∞ = 0

M

Γ (Pt g) dμ dt Pt g

+∞ =

Pt gΓ (ln Pt g) dμ dt 0

M

+∞ 

e

ρ ρ t 2

1 2 −2 κ+ρ

0

κ + ρ2  2ρ1 ρ2

  M

  gΓ (ln g) +

dt M

κ + ρ2 gΓ Z (ln g) dμ ρ1

Γ (g) κ + ρ2 Γ Z (g) + dμ. g ρ1 g

Let now f ∈ C0 (M) and consider g = ε + f 2 ∈ Aε . Using the previous inequality and letting ε → 0, yields    f 2 ln f 2 dμ − f 2 dμ ln f 2 dμ M

M



2(κ + ρ2 ) ρ1 ρ2

M



Γ (f ) dμ + M

κ + ρ2 ρ1



Γ Z (f ) dμ .

2

M

3. Wang inequality for the heat semigroup and log-Sobolev inequality Throughout this section, we assume that the operator L satisfies the curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞) for some ρ1 ∈ R, ρ2 > 0, κ  0. We shall denote ρ1− = max(−ρ1 , 0). Our main goal is to prove Theorem 1.4. 3.1. Reverse log-Sobolev inequalities Proposition 3.1. Let ε > 0 and f ∈ Aε . For x ∈ M, t > 0 one has tPt f (x)Γ (ln Pt f )(x) + ρ2 t 2 Pt f (x)Γ Z (ln Pt f )(x) 

2κ −  1+ + 2ρ1 t Pt (f ln f )(x) − Pt f (x) ln Pt f (x) . ρ2

F. Baudoin, M. Bonnefont / Journal of Functional Analysis 262 (2012) 2646–2676

2659

Proof. We may assume ρ1  0. We proceed similarly to the proof of Proposition 2.1. Let f ∈ Aε , 0  t  T and φ1 (x, t) = (PT −t f )(x)Γ (ln PT −t f )(x), φ2 (x, t) = (PT −t f )(x)Γ Z (ln PT −t f )(x). As before, we consider the function φ(x, t) = a(t)φ1 (x, t) + b(t)φ2 (x, t) = a(t)(PT −t f )(x)Γ (ln PT −t f )(x) + b(t)(PT −t f )(x)Γ Z (ln PT −t f )(x), where a and b are to be later chosen. As already seen, applying the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞), we obtain Lφ +

 ∂φ a2  a + 2ρ1 a − 2κ (PT −t f )Γ (ln PT −t f ) ∂t b   + b + 2ρ2 a (PT −t f )Γ Z (ln PT −t f ).

The idea is now to chose the functions a and b in such a way that b + 2ρ2 a = 0 and a + 2ρ1 a − 2κ

a2 C b

where C is a constant independent from t. This leads to the candidates a(t) =

1 (T − t) ρ2

and b(t) = (T − t)2 , for which we obtain C=−

1 2κ 2ρ1 − 2+ T. ρ2 ρ2 ρ2

For this choice of a and b, we obtain Lφ +

∂φ  C(PT −t f )Γ (ln PT −t f ). ∂t

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The comparison principle for parabolic partial differential equations leads then to   PT φ(·, T ) (x)  φ(0, x) + C

T

  Pt (PT −t f )Γ (ln PT −t f ) (x) dt.

0

It is now seen that T

  Pt (PT −t f )Γ (ln PT −t f ) (x) dt = PT (f ln f )(x) − PT f (x) ln PT f (x),

0

which yields T PT f (x)Γ (ln PT f )(x) + ρ2 T 2 PT f (x)Γ Z (ln PT f )(x) 

2κ − + 2ρ1 T PT (f ln f )(x) − PT f (x) ln PT f (x) .  1+ ρ2

2

Using a similar argument, we may prove the following: Proposition 3.2. Let f ∈ C0∞ (M), then for x ∈ M, t > 0 one has 

2κ 1 −  2 1+ + 2ρ1 t Pt f (x) − Pt f (x)2 . tΓ (Pt f )(x) + ρ2 t Γ (Pt f )(x)  2 ρ2 2

Z

As a consequence, we get the following useful regularization bound that will be later used: Corollary 3.3. Let f ∈ C0∞ (M), then for all t > 0,    Γ (Pt f )

∞



1 + 2

κ ρ2

+ ρ1− t 2 1

t

f ∞ .

3.2. Wang inequality An important by-product of the reverse log-Sobolev inequality that was proved in the previous section (Proposition 3.1) is the following inequality that was first observed by F.Y. Wang [38] in a Riemannian framework. Proposition 3.4. Let α > 1. For f ∈ L∞ (M), f  0, t > 0, x, y ∈ M, 1 +   α α (Pt f ) (x)  Pt f (y) exp α−1 α

2κ ρ2

+ 2ρ1− t 2 d (x, y) . 4t

F. Baudoin, M. Bonnefont / Journal of Functional Analysis 262 (2012) 2646–2676

2661

Proof. We first assume f ∈ Aε . Consider a subunit curve γ : [0, T ] → M such that γ (0) = x, γ (T ) = y. Let α > 1 and β(s) = 1 + (α − 1) Ts , 0  s  T . Let φ(s) =

  α ln Pt f β(s) γ (s) , β(s)

0  s  T,

where t > 0 is fixed. Differentiating with respect to s and using then Proposition 3.1 yields

  α(α − 1) Pt (f β(s) ln f β(s) ) − (Pt f β(s) ) ln Pt f β(s) α − Γ ln Pt f β(s) 2 β(s) β(s) T β(s) Pt f

    α α(α − 1)t  Γ ln Pt f β(s) . Γ ln Pt f β(s) − − 2κ 2 β(s) T β(s) (1 + ρ2 + 2ρ1 t)

φ (s) 

Now, for every λ > 0,

   λ2 1  − Γ ln Pt f β(s)  − 2 Γ ln Pt f β(s) − . 2 2λ If we chose λ2 =

(1 +

2κ ρ2

+ 2ρ1− t)

2(α − 1)t

T β(s)

we infer φ (s)  −

α(1 +

2κ ρ2

+ 2ρ1− t)

4(α − 1)t

T.

Integrating from 0 to L yields − α(1 + 2κ   ρ2 + 2ρ1 t) 2 ln Pt f α (y) − ln(Pt f )α (x)  − T . 4(α − 1)t

Minimizing then T 2 over the set of subunit curves such that γ (0) = x and γ (T ) = y gives the claimed result. If f ∈ L∞ (M), f  0, then for ε > 0, n  0, and τ > 0, the function ε + hn Pτ f ∈ Aε , where hn ∈ C0∞ (M) is an increasing, non-negative, sequence that converges to 1. Letting then ε → 0, n → ∞ and τ → 0 proves that the inequality still holds for f ∈ L∞ (M). 2 3.3. Log-Harnack inequality An easy consequence of the Wang inequality of Proposition 3.4 is the following log-Harnack inequality.

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Proposition 3.5. For f ∈ L∞ (M), inf f > 0, t > 0, x, y ∈ M, Pt (ln f )(x)  ln Pt (f )(y) +

1 +

2κ ρ2

+ 2ρ1− t 4t

d 2 (x, y).

The proof of this result appears in Section 2 of [42] where a general study of these Harnack inequalities is done. For the sake of completeness, we reproduce the argument here. 1

Proof of Proposition 3.5. Applying Proposition 3.4 to the function f 2n for α = 2n , we get   −n   2−n exp Pt f 2 (x)  Pt (f )(y)

1 + 1 2n − 1

2κ ρ2

+ 2ρ1− t 4t

d 2 (x, y) .

Now, since 2−n → 0 as n → ∞, by the dominated convergence theorem,  Pt (ln f )(x) = lim Pt n→∞

  lim

−n f2 −1 (x) 2−n −n

(Pt f (y))2

1+ ρ2κ +2ρ1− t  2 )d 2 (x, y)) − 1 4t 2−n

exp( 2n1−1 (

n→∞



−n

2−n exp( 2n1−1 ( (Pt f (y))2 − 1  = lim + P f (y) t n→∞ 2−n  1 + 2κ + 2ρ − t 1 ρ2 d 2 (x, y). 2 = ln(Pt f )(y) + 4t

1+ ρ2κ +2ρ1− t  2 )d 2 (x, y)) − 1 4t 2−n

When μ is a probability measure, the above log-Harnack inequalities implies the following lower bound for the heat kernel. Corollary 3.6. Assume that μ is a probability measure, then for t > 0, x, y ∈ M,  1+ p2t (x, y)  exp −

2κ ρ2

+ 2ρ1− t 4t

d 2 (x, y) .

Proof. Again, we reproduce an argument of Wang [43]. By applying Proposition 3.5 to the function f (·) = pt (x, ·) and integrating over the manifold, one gets 

 pt (x, z) ln pt (x, z)dμ(z)  ln M

pt (y, z)pt (x, z) dμ(z) +

1+

2κ ρ2

4t

M

Now, by Jensen inequality,



M pt (x, z) ln pt (x, z) dμ(z)  0,

ln p2t (x, y)  −

1+

2κ ρ2

+ 2ρ1− t 4t

thus

d 2 (x, y).

+ 2ρ1− t

2

d 2 (x, y).

F. Baudoin, M. Bonnefont / Journal of Functional Analysis 262 (2012) 2646–2676

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3.4. Log-Sobolev inequality and exponential integrability of the square distance With Wang’s inequality in hands, we can prove a log-Sobolev inequality provided the square integrability of the distance function. This extends a well-known theorem of Wang (see [2,39]). Theorem 3.7. Assume that the measure μ is a probability measure and that L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞) for some ρ1 ∈ R, ρ2 > 0, κ  0. Assume moreover that  2 eλd (x0 ,x) dμ(x) < +∞, M

for some x0 ∈ M and λ > C0∞ (M),

ρ1− 2 ,

then there is a constant C > 0 such that for every function f ∈ 





f 2 ln f 2 dμ − M

M

 f 2 dμ  C

f 2 dμ ln M

Γ (f ) dμ. M

Proof. Let α > 1 and f ∈ L∞ (M), f  0. From Proposition 3.4, by integrating with respect to y, we have 

 f α (y) dμ(y)  (Pt f )α (x) M

M

1 +  α exp − α−1 

 (Pt f )α (x) B(x0 ,1)

2κ ρ2

+ 2ρ1− t 4t

1 +  α exp − α−1

2κ ρ2

d 2 (x, y) dμ(y)

+ 2ρ1− t 4t

1 +    α α  μ B(x0 , 1) (Pt f ) (x) exp − α−1

2κ ρ2

d 2 (x, y) dμ(y)

+ 2ρ1− t  2  d (x0 , x) + 1 . 4t

As a consequence, we get (Pt f )(x) 

1 1

 exp

μ(B(x0 , 1)) α

1 + 1 α−1

2κ ρ2

+ 2ρ1− t  4t

d (x0 , x) + 1 2

Therefore if  eλd

2 (x ,x) 0

dμ(x) < +∞,

M

for some x0 ∈ M and λ >

ρ1− 2 ,

then we can find 1 < α < β and t > 0 such that

Pt f Lβ  Cα,β f Lα ,



f Lα .

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for some constant Cα,β . This implies the supercontractivity of the semigroup (Pt )t0 and therefore from Gross’ theorem (see [3]), a defective logarithmic Sobolev inequality is satisfied, that is there exist two constants A, B > 0 such that 

 f 2 ln f 2 dμ −

M



M

 f 2 dμ  A

f 2 dμ ln M

 Γ (f ) dμ + B

M

f 2 dμ,

f ∈ C0∞ (M).

M

Now, since moreover the heat kernel is positive and the invariant measure a probability, we deduce from the uniform positivity improving property (see [1, Theorem 2.11]) that L admits a spectral gap. That is, a Poincaré inequality is satisfied. It is then classical (see [2]), that the conjunction of a spectral gap and a defective logarithmic Sobolev inequality implies the logSobolev inequality (i.e. we may actually take B = 0 in the above inequality). 2 3.5. A dimensional bound on the log-Sobolev constant If we take into account the dimension in the generalized curvature dimension inequality, we may obtain an upper bound for the log-Sobolev constant under the assumption that the curvature parameter ρ1 is positive. Theorem 3.8. Assume that the measure μ is a probability measure and that L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, d) for some ρ1 > 0, ρ2 > 0, κ  0 and d  1. For every function f ∈ C0∞ (M), 

 f 2 ln f 2 dμ − M



M

 f 2 dμ  C

f 2 dμ ln M

Γ (f ) dμ, M

with C=

   d 3κ 3(ρ2 + κ) 1+Φ , 1+ ρ1 ρ2 2 2ρ2

where Φ(x) = (1 + x) ln(1 + x) − x ln x. Proof. It is proved in [9] that the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, d) with ρ1 > 0, ρ2 > 0, κ  0 and d > 0 implies the following upper bound for the heat kernel: For x, y ∈ M and t > 0, 1

p(x, y, t)  (1 − e

2ρ ρ2 t − 3(ρ1 +κ) 2

d

)2

3κ (1+ 2ρ )

.

2

Therefore, from Davies’ theorem (Theorem 2.2.3 in [13]), for f ∈ C0∞ (M), we obtain the following defective log-Sobolev inequality which is valid for every t > 0,

F. Baudoin, M. Bonnefont / Journal of Functional Analysis 262 (2012) 2646–2676



 f ln f dμ − 2

2

M

  2t M

2665

 2

f 2 dμ

f dμ ln M

M

  2ρ ρ t  3κ − 1 2  ln 1 − e 3(ρ2 +κ) Γ (f ) dμ − d 1 + f 2 dμ. 2ρ2 M

The previous heat kernel upper bound also implies that −L has a spectral gap of size at least 2ρ1 ρ2 3(ρ2 +κ) . Therefore, the following Poincaré inequality holds 

 f dμ − 2

2 f dμ

M

M

3(ρ2 + κ)  2ρ1 ρ2

 Γ (f ) dμ. M

If we combine the two previous inequalities using Rothaus’ inequality (see Lemma 4.3.8 in [2]) and then chose the optimal t, we get the result. 2 Remark 3.9. It has been proved in [9] that if L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, d) with ρ1 > 0, then M needs to be compact. 3.6. Log-Sobolev inequality and diameter bounds We now generalize to a result due to Saloff-Coste [36]. Theorem 3.10. Assume that the measure μ is a probability measure and that L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, d) for some ρ1 ∈ R, ρ2 > 0, κ  0 and d > 0. • The metric space (M, d) is compact if and only if a log-Sobolev inequality 

 f 2 ln f 2 dμ − M



M

 f 2 dμ  C

f 2 dμ ln M

Γ (f ) dμ,

f ∈ C0∞ (M)

(3.1)

M

is satisfied for some C > 0. • Moreover, if (M, d) is compact with diameter D then, there is a constant C(ρ1 , ρ2 , κ, d) such that D where

2 ρ0

C(ρ1 , ρ2 , κ, d) min(1, ρ0 )

is the smallest constant C such that (3.1) is satisfied.

Proof. If M is compact, then  eλd M

2 (x ,x) 0

dμ(x) < +∞,

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for every x0 ∈ M and λ > fied. Let us now assume that

ρ1− 2 .



Therefore, from Theorem 3.7, a log-Sobolev inequality is satis-

 f 2 ln f 2 dμ −

M

 f 2 dμ 

f 2 dμ ln M

M



2 ρ0

Γ (f ) dμ,

f ∈ C0∞ (M)

M

is satisfied. Here we only sketch the proof, since we may actually follow quite closely an argument from Ledoux [27] to which we refer for more details and in particular for tracking the constants. The key is to note that the curvature dimension inequality CD(ρ1 , ρ2 , κ, d) for some ρ1 ∈ R, ρ2 > 0, κ  0 and d > 0 implies a Li–Yau type inequality (see Theorem 5.1 in [9]). In particular for 0 < t  1 and a positive function f 0A

B LPt f + Pt f t

where A and B are some explicit positive constants depending only on ρ1 , ρ2 , κ, d. Since ∂t ln Pt f , integrating between t and 1 yields, with γ = Pt f 

1 P1 f tγ

B A,

LPt f Pt f

=

for all 0 < t  1.

Using now the equivalence between the log-Sobolev inequality and the hypercontractivity of the heat semigroup due to Gross, we find that for 1 < p < q < ∞

Pt f q  f p as soon as e2ρ0 t 

q−1 p−1 .

Therefore, for t = 1, p = 2 and q = 1 + e2ρ0 ,

Pt f q 

1 1

P1 f q  γ f 2 γ t t

for 0 < t  1.

Such a semigroup estimate implies a Sobolev inequality  2  

f 2r  8 f 22 +  Γ (f, f )2 for some r > 2 (see [37]). Finally, the conjunction of the logarithmic Sobolev inequality and of the above Sobolev inequality implies an entropy-energy inequality that may be used to prove that the diameter is bounded (see [3]). Carefully tracking the constants leads to the desired bound for the diameter (see [27]). 2

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4. Log-Sobolev inequality and transportation cost inequalities In this section, we shall examine the links between the log-Sobolev inequality and some transportations cost inequalities. First, it is well known that the log-Sobolev inequality implies some transportation inequalities in a general “metric” setting. Conversely, on a weighted Riemannian manifold, under the hypothesis that the Bakry–Émery curvature is bounded from below, the converse implication holds true (see [11,32]). Therefore, in this section we shall study how some transportation inequalities can, if the generalized curvature dimension inequality is satisfied, imply a log-Sobolev inequality. Unfortunately, we were only able to establish a partial converse in the sense that the log-Sobolev inequality we obtain involves a term with Γ Z . Throughout the section, we assume μ(M) = 1. Let us begin with some notations. For a positive function f on M, we write  Entμ (f ) =

 f ln f dμ −

M

 f dμ ln

M

f dμ. M

We recall that the L2 -Wasserstein distance of two measures ν1 and ν2 on M is given by  W2 (ν1 , ν2 )2 = inf

d 2 (x, y) dΠ(x, y)

Π

M

where the infimum is taken over all coupling of ν1 and ν2 that is on all probability measures Π on M × M whose marginals are respectively ν1 and ν2 . 4.1. A bound of the entropy of the semigroup by the L2 -Wasserstein distance First we show how to bound the entropy of the semigroup by this L2 -Wasserstein distance. The following result is a generalization in our setting of Lemma 4.2 in [11] (see also [33] for an alternative proof which is more PDE oriented). Proposition 4.1. Assume that L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2, κ, ∞) for some ρ1 ∈ R, ρ2 > 0, κ  0. Let f be a non-negative function on M such that M f dμ = 1 and set dν = f dμ. Then, for any t > 0, Entμ (Pt f ) 

1 +

2κ ρ2

+ 2ρ1− t 4t

W2 (μ, ν)2 .

 Proof. Let t > 0 and f be a positive function on M such that M f dμ = 1. The log-Harnack inequality of Proposition 3.5 applied to the function Pt f gives then 1 Pt (ln Pt f )(x)  ln P2t (f )(y) + d 2 (x, y), s

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with s=

4t 1+

2κ ρ2

+ 2ρ1− t

.

For x fixed, by taking the infimum with respect to y on the right-hand side of the last inequality, we obtain Pt (ln Pt f )(x)  Qs (ln P2t f )(x) where Qs is the infimum-convolution semigroup:   1 Qs (φ)(x) = inf φ(y) + d(x, y)2 . y∈M 2s Setting φ = ln P2t f , by Jensen inequality    φ dμ = ln P2t f dμ  ln P2t f dμ = 0, M

M

M

thus

 Pt (ln Pt f )(x) = Qs (φ)(x) −

φ dμ. M

Since by symmetry:  Entμ (Pt f ) =

f Pt (ln Pt f ) dμ, M

one finally gets    Entμ (ft )  sup Qs (ψ)(x)dν − ψ dμ ψ

M

M

where the supremum is taken over all bounded measurable functions ψ and where the measure dν = f . By Monge–Kantorovich duality, ν is defined by dμ     sup Qs (ψ)(x) − ψ dμ = inf T (x, y) dΠ(x, y) ψ

Π

M

M

where the infimum is taken over all coupling of μ and ν and where the cost T is just T (x, y) =

1 2 d (x, y). s

Therefore the latter infimum is equal to 1s W2 (μ, ν)2 .

2

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4.2. Modified HWI inequality The following lemma may be proved in the very same way as Proposition 2.1. Lemma 4.2. Assume that L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞) for some ρ1 ∈ R, ρ2 > 0, κ  0. Let ε > 0 and f ∈ Aε . For x ∈ M, t  0 one has      Pt f Γ (ln Pt f ) + Pt f Γ Z (ln Pt f )  e2αt Pt f Γ (ln f ) + Pt f Γ Z (ln f ) ,

t  0,

where α = − min(ρ2 , ρ1 − κ, 0). We may now prove: Theorem 4.3. Assume that L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞) for some ρ1 ∈ R, ρ2 > 0, κ  0. If the quadratic transportation cost inequality  W2 (μ, ν)  c Entμ 2

dν dμ

(4.1)

is satisfied for every absolutely continuous probability measure ν with a constant c < the following modified log-Sobolev inequality  Entμ (f )  C1 M

Γ (f ) dμ + C2 f

 M

Γ Z (f ) dμ, f

f ∈ Aε , ε > 0,

holds for some constants C1 and C2 depending only on c, ρ1 , κ, ρ2 . Proof. Let f ∈ Aε such that



Mf

dμ = 1, by the diffusion property, we have d Entμ (Pt f ) = −I (Pt f ) dt

with  I (Pt f ) = M

Γ (Pt f ) dμ. Pt f

From Lemma 4.2, we have      Γ (Pt f )  e2αt Pt f Γ (ln f ) + Pt f Γ Z (ln f ) , Pt f which implies, by integration over the manifold M,  I (Pt f )  e2αt M

Γ (f ) dμ + f

 M

Γ Z (f ) dμ . f

2 , ρ1−

then

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As a consequence, T Entμ (f ) 

I (Pt f ) dt + Entμ (PT f ) 0

 T 

  e

2αt

dt M

0

Γ (f ) dμ + f

Γ Z (f ) dμ + Entμ (PT f ). f

 M

We now use Proposition 4.1 and infer  T Entμ (f ) 

  e

2αt

dt M

0

+

1 +

Γ (f ) dμ + f

2κ ρ2

+ 2ρ1− T 4T

 M

Γ Z (f ) dμ f



W2 (μ, ν)2 ,

where dν = f dμ. Using the assumption W2 (μ, ν)2  c Entμ (f ) and choosing T big enough finishes the proof. 2 5. Logarithmic isoperimetric inequality In this section, we assume that the measure μ is a probability measure, that is μ(M) = 1, and we show how the curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞) together with a log-Sobolev inequality implies a logarithmic isoperimetric inequality of Gaussian type. The method used here is very close from the one in Ledoux [26]. We first need to precise what we mean by the perimeter of a set in our subelliptic setting: This is essentially done in [18]. Let us first observe that, given any point x ∈ M there exists an open set x ∈ U ⊂ M in which the operator L can be written as L=−

m 

Xi∗ Xi ,

(5.1)

i=1

where the vector fields Xi have Lipschitz continuous coefficients in U , and Xi∗ indicates the formal adjoint of Xi in L2 (M, dμ). We indicate with F (M) the set of C 1 vector fields which are subunit for L. Given a function f ∈ L1loc (M), which is supported in U we define the horizontal total variation of f as 

 Var(f ) = sup

φ∈F (M)

m

f U

m 

 Xi∗ φi

dμ,

i=1

where on U , φ = i=1 φi Xi . For functions not supported in U , Var(f ) may be defined by using a partition of unity. The space

F. Baudoin, M. Bonnefont / Journal of Functional Analysis 262 (2012) 2646–2676

2671

   BV(M) = f ∈ L1 (M)  Var(f ) < ∞ , endowed with the norm

f BV(M) = f L1 (M) + Var(f ), is a Banach space. It is well known that W 1,1 (M) = {f ∈ L1 (M) | subspace of BV(M) and when f ∈ W 1,1 (M) one has in fact

√ Γf ∈ L1 (M)} is a strict

  Var(f ) =  Γ (f )L1 (M) . Given a measurable set E ⊂ M we say that it has finite perimeter if 1E ∈ BV(M). In such case the perimeter of E is by definition P (E) = Var(1E ). We will need the following approximation result, see Theorem 1.14 in [18]. Lemma 5.1. Let f ∈ BV(M), then there exists a sequence {fn }n∈N of functions in C0∞ (M) such that: (i) fn − f L1 (M) → 0;  √ (ii) M Γ (fn ) dμ → Var(f ). After this digression, we now state the main theorem of this section. Theorem 5.2. Assume that L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞) and that μ satisfies the log-Sobolev inequality: 

 f ln f dμ − 2

2

M

 2

f dμ ln M

2 f dμ  ρ0



2

M

Γ (f ) dμ,

(5.2)

M

for all smooth functions f ∈ C0∞ (M). Let A be a set of the manifold M which has a finite perimeter P (A) and such that 0  μ(A)  12 , then  1  2 1 ρ0 √ μ(A) ln ρ0, . min P (A)  2κ μ(A) 4(3 + ρ2 ) ρ1− ln 2

Remark 5.3. The constant

ln 2 4(3+ ρ2κ ) 2

is of course not optimal but unlike the result stated in [26],

it does not depend on the dimension. This comes from the fact that we use the reverse Poincaré inequality of Proposition 3.2 instead of a Li–Yau type inequality to obtain Lemma 5.4 below.

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Lemma 5.4. Assume that L satisfies the generalized curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞), let f ∈ C0∞ (M), then for all t > 0 

f − Pt f 1 

 √  κ 1 + + ρ1− t t  Γ (f )1 . 2 ρ2

(5.3)

Proof. First, since the curvature dimension inequality CD(ρ1 , ρ2 , κ, ∞) holds true, by Corollary 3.3, for all g ∈ C0∞ (M) and for all 0 < t  t0 ,    Γ (Pt g)

∞



1 + 2

κ ρ2

+ ρ1− t0 2 1

t

g ∞ .

Therefore, by duality, for every positive and smooth function f , every smooth function g such that g ∞  1 and all 0 < t  t0 , t 

 g(f − Pt f ) dμ = −

gLPs f dμ ds

M

0 M

t



=

Γ (Ps g, f ) dμ ds 0 M

t     Γ (Ps g) ds 1 ∞

    Γ (f )   which ends the proof.

0

 √  κ 1 − + + ρ 1 t0 t  Γ (f )1 2 ρ2

2

Now we can turn to the proof of the theorem. Proof of Theorem 5.2. Let A be a set with finite perimeter. Applying Lemma 5.4 to smooth functions approximating the characteristic function 1A as in Lemma 5.1 gives 

1A − Pt 1A 1 

√ κ 1 − + + ρ1 t tP (A). 2 ρ2

By symmetry and stochastic completeness of the semigroup, 

1A − Pt 1A 1 =

 (1 − Pt 1A ) dμ +

A



=

Pt (1A ) dμ Ac



(1 − Pt 1A ) dμ + A

(Pt 1Ac ) dμ A

F. Baudoin, M. Bonnefont / Journal of Functional Analysis 262 (2012) 2646–2676

2673

  = 2 μ(A) − Pt (1A ) dμ A

2    = 2 μ(A) − P t (1A )2 . 2

Now we can use the hypercontractivity constant to bound P t (1A ) 22 . Indeed, from Gross’ 2 theorem it is well known that the logarithmic Sobolev inequality 

 f 2 ln f 2 dμ −

M

 f 2 dμ 

f 2 dμ ln M

M

2 ρ0

 Γ (f ) dμ,

f ∈ C0∞ (M),

M

is equivalent to hypercontractivity property

Pt f q  f p for all f in Lp (M) whenever 1 < p < q < ∞ and eρ0 t  Therefore, with p(t) = 1 + e−ρ0 t < 2, we get, 

q−1 p−1 .

√ 2   1 κ − + + ρ1 t tP (A)  2 μ(A) − μ(A) p(t) 2 ρ2 1−e−ρ0 t    2μ(A) 1 − μ(A) 1+e−ρ0 t .

Since for x > 0   1 − e−x x 1 x 1 and , 1 − e  min ,  min , 2 2 1 + e−x 4 2   1−e−ρ0 t 1 ρ0 t 1 −ρ0 t 1+e μ(A) , ln ,  exp − min 4 2 μ(A)   1−e−ρ0 t 1 1 ρ0 t 1 −ρ0 t 1+e 1 − μ(A) , ln , .  min min 8 4 μ(A) 2 −x

Therefore for all t > 0, P (A) 

  1 1 ρ0 t 1 , ln , . μ(A) min min √ 8 4 μ(A) 2 + ρ1− t) t 2

( 12 +

κ ρ2

With t0 = min( ρ10 , ρ1− ), for 0 < t  t0 , we have 1

P (A) 

 1 1 ρ0 t ln , . √ μ(A) min 8 μ(A) 2 + 1) t

2 ( 12 +

κ ρ2

(5.4)

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Now, if μ(A) is small enough, i.e. μ(A)  e−4 , we can chose t = 1 , 12 ) = min( ρ80 t ln μ(A)

ρ0 t0 2

4t0 1 ln μ(A)

 t0 so that

and then get

P (A) 

√ 1 1 ρ0 t0 μ(A)(ln μ(A) )2 3+

2κ ρ2

.

1 When 0  μ(A)  12 , we can apply (5.4) with t = t0 and since ln μ(A)  ln 2,

 ρ 0 t0 ρ0 t0 ln 2 1 1 min  ln , 8 μ(A) 2 8 and thus P (A) 

√ ln 2ρ0 t0 μ(A) 2(3 +

2κ ρ2 )

.

1 Noticing ln μ(A)  4 if μ(A)  e−4 , we obtain that for every A with 0  μ(A)  12 ,

P (A) 

√ 1 1 ρ0 t0 μ(A)(ln μ(A) ) 2 ln 2 4(3 +

2κ ρ2 )

Keeping in mind that t0 = min( ρ10 , ρ1− ) ends the proof. 1

.

2

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