Refined convex Sobolev inequalities

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Anton ARNOLD and Jean DOLBEAULT

1 _{Institut f¨}_{ur Numerische und Angewandte Mathematik, Universit¨}_{at M¨}_unster

Einsteinstr. 62, D-48149 M¨unster, Germany E-mail: anton.arnold@math.uni-muenster.de

2 _{CEREMADE, U.M.R. C.N.R.S. no. 7534, Universit´}_{e Paris IX-Dauphine}

Pl. du Mar´echal de Lattre de Tassigny, 75775 Paris C´edex 16, France E-mail: dolbeaul@ceremade.dauphine.fr

May 12, 2004

Abstract

This paper is devoted to refinements of convex Sobolev inequalities in the case of power law relative entropies: a nonlinear entropy–entropy production relation im-proves the known inequalities of this type. The corresponding generalized Poincar´e type inequalities with weights are derived. Best constants are compared to the usual Poincar´e constant.

Key words and phrases: Sobolev inequality – Poincar´e inequality – entropy method – diffusion – logarithmic Sobolev inequality – spectral gap

AMS 2000 Mathematics Subject Classification: 35K10, 46E35, 35J20

1 Introduction and main results

In this paper, we consider convex Sobolev inequalities relating a (non-negative) convex entropy functional eψ(ρ|ρ∞) := Z Rn ψ ρ ρ∞ dρ∞

to an entropy production functional I_ψ(ρ|ρ∞) := − Z Rn ψ00 ρ ρ∞ D ∇ ρ ρ∞ 2 dρ∞, (1.1)

where ρ and ρ∞ belong to L1+(Rn, dx) and satisfy kρkL1_(Rn₎ = kρ_∞k_L1_(Rn₎ = M > 0.

Here we use the notation dρ∞ = ρ∞(x) dx. The generating function ψ : R+0 → R+0 of

the relative entropy is strictly convex and satisfies ψ(1) = 0.

A very efficient method to prove convex Sobolev inequalities has been developped by D. Bakry and M. Emery [3, 4] in probability theory and by A. Arnold, P. Markowich, G. Toscani, A. Unterreiter [2] in the context of partial differential equations. See [1] for a recent review. The main idea goes as follows: for any solution ρ(x, t) of

∂ρ ∂t = div D ρ∞∇ ρ ρ∞ , x ∈ Rn, t > 0 , (1.2)

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the time evolution of the relative entropy is given by the entropy production: d

dteψ(ρ(t)|ρ∞) = Iψ(ρ(t)|ρ∞) ≤ 0 .

In (1.1) and (1.2) D = D(x) denotes a (positive) scalar diffusion coefficient, and we assume D ∈ W_loc2,∞(Rn). It is also clear that ρ∞(x) is a steady state solution of (1.2).

For D ≡ 1, the main assumption is that A := − log ρ∞is a uniformly convex function,

i.e. (A1) λ1 := inf x∈Rn ξ∈Sn−1 ξ,∂ 2_A ∂x2(x) ξ > 0 .

For D 6≡ 1 the corresponding assumption reads: (A2) ∃ λ1 > 0 such that for any x ∈ Rn

1 2 − n 4 1 D∇D ⊗ ∇D + 1 2(∆D − ∇D · ∇A)II +D∂ 2_A ∂x2 + 1 2 ∇A ⊗ ∇D + ∇D ⊗ ∇A−∂ 2_D ∂x2 ≥ λ1II

(in the sense of positive definite matrices). Here II denotes the identity matrix. In these two cases, one can prove the convex Sobolev inequality

Rψ(ρ(t)|ρ∞) ≥ 0 . (1.4)

Integrating this differential inequality from t to ∞ then yields (1.3).

Actually, these calculations can only be carried out only for admissible relative en-tropies where ψ ∈ C4_(R+_{) has to satisfy}

2 (ψ000)2≤ ψ00ψIV on R+.

Typical and the most important – for practical applications – examples are generating functions of the form

ψp(σ) = σp− 1 − p (σ − 1) for p ∈ (1, 2] , (1.5)

and

˜

ψ1(σ) = σ log σ − σ + 1 ,

which corresponds to the limiting case of ψp/(p − 1) as p → 1. With ψ = ˜ψ1, Inequality

(1.3) is exactly the logarithmic Sobolev inequality found by L. Gross [8, 9], and generalized by many authors later on.

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Analyzing the precise form of Rψ(ρ|ρ∞) allows to identify cases of optimality of (1.3)

under the assumption D ≡ 1. For p = 1 or 2, and for potentials A that are quadratic in at least one coordinate direction (with convexity λ1) there exist extremal functions

ρ = ρex 6= ρ∞ such that (1.3) becomes an equality, cf. [2]. Some of these optimality

results were already noted by E. Carlen [6], M. Ledoux [11], and G. Toscani [13]. The non-optimality of the other cases may have two reasons: either λ1 from (A1),

(A2) is not the sharp convex Sobolev constant (an example for this is A(x) = x4, x ∈ R: see §3.3 of [2]), or there exists no extremal function to saturate (1.3), even for the sharp constant λ1. This happens for the entropies with p ∈ (1, 2), and it is due to the fact

that the linear relationship of |I_ψ| and e_ψ is then not optimal.

The refinement of (1.3) for p ∈ (1, 2) is the topic of this paper. In this case, the non-optimality of (1.3) stems from the fact that, for any fixed D and ρ∞,

J(e, e0, M ) := inf

Iψ(ρ|ρ∞) = e0, eψ(ρ|ρ∞) = e

ρ ∈ L1₊(Rn), kρk_L1_(Rn_)=M

Rψ(ρ|ρ∞)

is a positive quantity for e > 0 and e0 _{≤ −2λ}

1e. Here, the t-derivatives entering in Rψ

are defined via (1.2). Our main result is based on a lower bound for J(e, e0, M ): J(e, e0, M ) ≥ 2 − p

p · |e0|2 M + e ,

which yields an improvement of (1.3). Finding the minimizers of J (if they exist) is probably difficult.

Theorem 1 Let ρ∞ satisfy (A2) for some λ1> 0, and take ψ = ψp for some p ∈ (1, 2).

Then _Z Rn ψ00 ρ ρ∞ D ∇ ρ ρ∞ 2 dρ∞≥ k Z Rn ψ ρ ρ∞ dρ∞ (1.6) holds for any ρ ∈ L1

+(Rn) with R Rnρ dx = R Rnρ∞dx = M , where k(e) := 2λ1M 1 − κ 1 + e M − 1 + e M κ , κ = 2 − p p .

We will show that there are still no extremal functions to saturate the refined convex Sobolev inequality (1.6). Therefore it is not yet known whether the above functional dependence of |Iψ| and eψ is optimal. But it improves (1.3) since we have

k(e) > 2 λ1e , ∀ e > 0 , (1.7)

and the best possible constants λ1 are shown to be independent of p (see Theorem 4

and Corollary 6).

Also, the presented method can be extended to the case λ1 = 0 (see Proposition 3

below), thus giving a decay rate of t 7→ Iψ(ρ(t)|ρ∞) for any solution ρ of (1.2), even

if A is not uniformly convex. We remark that nonlinear entropy–entropy production inequalities, or “defective logarithmic Sobolev inequalities,” have been derived for the logarithmic entropy (i.e. ψ = ˜ψ1) and Gaussian measures ρ∞ (cf. §1.3, §4.3 of [12]).

Next we consider reformulations of the convex Sobolev inequalities (1.3) and (1.6). We assume M = 1 and substitute

ρ ρ∞ = |f | 2 p R Rn|f | 2 p _dρ ∞ (1.8)

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in (1.3) to obtain the generalized Poincar´e inequalities derived by W. Beckner for Gaus-sian measures ρ∞ in [5] and generalized in [2] for log-convex measures:

p p − 1 Z Rn f2dρ∞− Z Rn |f |2/pdρ∞ p ≤ 2 λ1 Z Rn D|∇f |2 dρ∞ (1.9) for all f ∈ L2/p_(dρ

∞), 1 < p ≤ 2. In [11] this interpolation is discussed for the

Orn-stein–Uhlenbeck process on Rn and for the heat semigroup on spheres. Note that in the limit p → 1, (1.9) yields the logarithmic Sobolev inequality:

Z Rn f2ln f 2 kf k2 L2_(dρ_∞₎ ! dρ∞≤ 2 λ1 Z Rn D|∇f |2 dρ∞ (1.10) for all f ∈ L2(dρ∞).

Using the transformation (1.8) on the refined Sobolev inequality (1.6) directly yields a refinement of (1.9), which is nothing else than a reformulation of (1.6):

Theorem 2 Let ρ∞ satisfy (A2) for some λ1> 0 and assume that RRndρ∞= 1. Then

1 2 p p − 1 2"Z Rn f2dρ∞− Z Rn |f |2/p dρ∞ 2(p−1) · Z Rn f2dρ∞ 2 p−1# ≤ 2 λ1 Z Rn D|∇f |2dρ∞ (1.11)

holds for all f ∈ L2/p_(dρ

∞), 1 < p ≤ 2 and the limit p → 1 again yields (1.10).

Note that the left hand sides of (1.9) and (1.11) are related by 1 2 p p − 1 2"Z Rn f2dρ∞− Z Rn |f |2/pdρ∞ 2(p−1) · Z Rn f2 dρ∞ 2 p−1# ≥ p p − 1 Z Rn f2dρ∞− Z Rn |f |2/pdρ∞ p (1.12) as a consequence of (1.7) and (1.8). This can of course be recovered using H¨older’s

inequality: _Z Rn |f |2/pdρ∞ p ≤ Z Rn |f |2 dρ∞ (1.13)

and the identity: 1₂_p−1p (1 − t2p(p−1)_{) ≥ 1 − t for any t ∈ [0, 1], p ∈ (1, 2]. Note that the}

equality holds in (1.12) if and only if 1 = t = RRn|f |2/p dρ∞ p R

Rn|f |2 dρ∞−1, i.e. if

f is a constant.

For p = 2, (1.9) and (1.11) hold without absolute values, cf. [2], provided ψ2(σ) =

σ2_{− 1 − 2(σ − 1) is defined over the whole real line. In that case, ρ is allowed to take}

negative values.

In the next section, we shall prove Theorems 1 and 2 and exploit the method in the case λ1 = 0. Further results on best constants, perturbations and connections with

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2 Convex Sobolev inequalities for power law entropies

In the following, we shall assume for simplicity that Z

Rn

ρ∞dx = M = 1 .

The general case M > 0 then immediately follows by scaling.

Proof of Theorem 1. Since the first part of the proof is identical to §2.3 of [2] we shall not go into details here. After a sequence of integrations by parts, dIψ/dt can be

written as d dtIψ(ρ(t)|ρ∞) = 2 Z Rn ψ00(µ)D " u>∇ ⊗ (∇AD − ∇D)u +1 2∆ D|u| 2₋1 2|u| 2_{∇D · ∇A +} 1 D 2 − n 4 (u · ∇D) 2 # dρ∞ (2.1) + Z Rn " ψIV(µ)D2|u|4+ ψ000(µ)(4D2u>∂u ∂xu + 2D|u| 2_{∇µ · ∇D)} + 2 ψ00(µ)X i,j D ∂ 2_µ ∂xi∂xj +1 2 ∂D ∂xi ∂µ ∂xj +1 2 ∂µ ∂xi ∂D ∂xj −1 2δij∇D · ∇µ 2# dρ∞,

where we used the notation µ = _ρρ

∞ and u = ∇µ. Using (A2), the first integral of (2.1)

can be estimated below by −2λ1Iψ(ρ(t)|ρ∞). In the second integral, we now insert ψp

from (1.5) and write it as a sum of squares. This is the key step in our analysis, where we deviate from the strategy of [2] by using a sharper estimate:

d dtIψ(ρ(t)|ρ∞) ≥ −2λ1Iψ(ρ(t)|ρ∞) + p(p − 1) 2_{(2 − p)}Z Rn µp−4D2|u|4 dρ∞ + 2p(p − 1) Z Rn µp−2X i,j p − 2 µ D ∂µ ∂xi ∂µ ∂xj (2.2) + D ∂ 2_µ ∂xi∂xj +1 2 ∂D ∂xi ∂µ ∂xj +1 2 ∂µ ∂xi ∂D ∂xj −1 2δij∇D·∇µ 2 dρ∞ ≥ −2λ1Iψ(ρ(t)|ρ∞) + p(p − 1)2(2 − p) Z Rn µp−4D2|u|4 dρ∞.

In the two limiting cases p = 1 (replace ψp by ˜ψ1) and p = 2, the second term on the

r.h.s. of (2.2) disappears. In [2], this term was always disregarded. For 1 < p < 2, however, it makes it possible to improve (1.3).

Using the Cauchy-Schwarz inequality, we have the estimate Z Rn µp−2D|u|2 dρ∞ 2 ≤ Z Rn µp−4D2|u|4dρ∞· Z Rn µp dρ∞ = Z Rn µp−4D2|u|4 dρ∞· Z Rn [ψ(µ) + 1] dρ∞

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and hence _Z Rn µp−4D2|u|4 dρ∞≥ Iψ(ρ|ρ∞) p(p − 1) 2 · [eψ(ρ|ρ∞) + 1)]−1 .

With the notation e(t) = eψ(ρ(t)|ρ∞), we get from (2.2)

e00≥ −2λ1e0+ κ

|e0_|2

1 + e . (2.3)

From (2.3) we shall now derive

|e0| = −e0≥ k(e) , (2.4) which is the assertion of Theorem 1. We first note that both Iψ(ρ(t)|ρ∞) and eψ(ρ(t)|ρ∞)

decay exponentially with the rate −2λ1. This follows, respectively, from (1.4) and from

the usual convex Sobolev inequality (1.3). The function k(e) = 2λ1 1 − κ 1 + e − (1 + e)κ is the solution of k0 = 2λ1+ κ k(e) 1 + e , k(0) = 0 . Let y(t) =e0(t) + k(e(t))· e−κ Rt 0 e0(s) 1+e(s)ds _.

For any t ≥ 0, we calculate y0(t) = e00(t) + 2λ1e0(t) − κ |e0(t)|2 1 + e(t) · e−κ Rt 0 e0(s) 1+e(s)ds _. Since

as t → +∞, we conclude that y(t) ≤ 0, which proves (2.4). As we had to expect, one recovers the usual convex Sobolev inequality (1.3) in the limiting cases p = 1 (take the limit p → 1 after dividing (1.6) by p − 1) and p = 2 (this gives κ = 0 and k(e) = 2λ1e).

For 1 < p < 2, we notice that _1−κ1 2λ1e > k(e) > 2λ1e for any e > 0, but

lim

e→0+

k(e)

e = 2λ1 .

Hence, the estimate of Theorem 1 does not improve the asymptotic convergence rate of the solution of Equation (1.2) except for λ1 = 0:

Proposition 3 With the above notations, let λ1 = 0 and 1 < p < 2. Any solution of

Equation (1.2) satisfies

|Iψ(ρ(t)|ρ∞)| ≤

I0

1 + αt ∀ t > 0 with I0= |Iψ(ρ(0)|ρ∞)| and α = κ_1+e_ψ_(ρI0(0)|ρ∞).

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Proof. Inequality (2.3) can be rewritten in the form −|e 0_|0 |e0_|2 ≥ κ 1 + e ≥ κ 1 + e(0) ,

thus proving the result.

Next we address the question of saturation of the refined convex Sobolev inequality (1.6), for simplicity only for the case D ≡ 1. Using the strategy from [2] we rewrite (2.1) as

e00 = −2λ1e0+ κ

|e0_|2

1 + e + rψ(ρ(t)) , where the remainder term is

rψ(ρ(t)) = 2 Z Rn ψ00(µ)u> ∂2A ∂x2 − λ1II u dρ∞ + 2 p(p − 1) Z Rn µ2−pX i,j ∂zi ∂xj 2 dρ∞ (2.5) +p(p − 1) 2_{(2 − p)} e + 1 · Z Rn µpdρ∞· Z Rn µp−4|u|4dρ∞− Z Rn µp−2|u|2dρ∞ 2 ≥ 0 , with the notation z = µp−2∇µ. Using the notation from the proof of Theorem 1, we have y0(t) = rψ(ρ(t)) e−κ Rt 0 e0(s) 1+e(s) ds _,

and an integration with respect to t gives −y(0) = |e0(0)| − k(e(0)) = Z ∞ 0 rψ(ρ(t)) e−κ Rt 0 e0(s) 1+e(s)ds _{dt ≥ 0 .}

Hence we conclude that (2.4) becomes an equality, for ρ = ρ(0), if and only if the remainder vanishes along the whole trajectory of ρ(t), i.e.

rψ(ρ(t)) = 0 , t ∈ R+a.e.

However, no extremal function can simultaneously annihilate the second integral and the square bracket of (2.5): to make the second integral vanish, the function µ has to be of the form µ(x) = (C1+ C2 · x)

1

p−1 (whenever µ(x) 6= 0), and for the last term it

would have to be µ(x) = eC1+C2·x_{. Hence, (2.4) does not admit extremal functions.}

3 Further results and comments

In this section, we shall derive estimates for the sharp convex Sobolev constants in the Inequalities (1.9) and (1.11). These estimates are only based on the Poincar´e inequality. For p ∈ (1, 2) (and D ≡ 1) the inequalities of type (1.3), (1.6), (1.9), and (1.11) hence hold as soon as ρ∞ gives rise to a (classical) Poincar´e inequality (cf. (3.1) below). Note

that this condition is much weaker than the assumption (A1). We shall also derive a perturbation lemma, which allows to consider cases where assumptions (A1) or (A2) are violated.

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3.1 Spectral gap, Poincar´e and convex Sobolev inequalities

Consider the Poincar´e constant:

Λ1 := inf w ∈ D(Rn) w 6≡ 0, RRnw dρ∞= 0 R Rn|∇w|2 dρ∞ R Rn|w|2 dρ∞ . (3.1)

Theorem 4 Let D ≡ 1 and assume that ρ∞∈ L1+(Rn) is such that Λ1 > 0. Define

K(e) := M 1 − κ 1 + e M − 1 + e M κ

with κ = 2−p_p , M =RRnρ∞dx. Then, for any p ∈ (1, 2], one has the estimate

4 p − 1 p 2 Λ1 ≤ 1 2 ρ∞6≡ρ∈L1inf₊₍Rn₎ R Rn ρ dx=M Iψp(ρ|ρ∞) K eψp(ρ|ρ∞) ≤ Λ1. (3.2)

A method related to our approach has been introduced in [7] to prove the existence of a positive constant in convex Sobolev inequalities.

Proof of Theorem 4. The case p = 2 is trivial. For p ∈ (1, 2), consider Gλ(f ) := −Iψp(ρ|ρ∞)−2λ K eψp(ρ|ρ∞)

and Hλ(f ) := −Iψp(ρ|ρ∞)−2λ eψp(ρ|ρ∞) ,

where f and ρ are related by (1.8): _ρρ

∞ = |f |

2/p R

Rn|f |2/pdρ∞ −1

. From now on, we assume for simplicity that M =RRnρ dx =

R

Rnρ∞dx = 1. According to (1.12), Gλ(f ) ≤

Hλ(f ) for any f ∈ H1(dρ∞) and λ ≥ 0. Define Λ := sup{λ ∈ R : Gλ(f ) ≥ 0 ∀f ∈

H1(dρ∞)}. By continuity of Gλ, this supremum is actually a maximum:

GΛ(f ) ≥ 0 ∀ f ∈ H1(dρ∞) .

Also notice that L2_(dρ

∞) ⊂ L2/p(dρ∞) by H¨older’s inequality (1.13). As a first step of

the proof of Theorem 4, we can give a slightly more detailed statement. Lemma 5 Let p ∈ (1, 2). With the above notations,

(i) For any λ ∈ R, Gλ(1) = 0, and for any λ < Λ, fλ ≡ 1 is the unique minimizer of

Gλ in H1(dρ∞), up to a multiplicative constant.

(ii) For any λ ≤ Λ, fλ ≡ 1 is the unique minimizer of Hλ in H1(dρ∞), up to a

multiplicative constant.

(iii) For any λ ≤ ¯Λ := 4 (p−1_p )2Λ1, any minimizing sequence (fn)n∈N of Gλ satisfying

the constraint RRn|fn|2/pdρ∞= M = 1 strongly converges to 1 in H1(dρ∞). As a

consequence, Λ ≥ ¯Λ.

Proof. It is straightforward to verify that Gλ(1) = 0 for any λ ∈ R, so that f ≡ 1 is the

unique minimizer of Gλ (up to a multiplicative constant) for any λ < Λ and a minimizer

for λ = Λ. For λ < Λ we have indeed

Gλ(f ) = GΛ(f ) + 2(Λ − λ) K eψp(ρ|ρ∞)

,

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and both terms of the r.h.s. are nonnegative, so that eψp(ρ|ρ∞) = 0 implies ρ ≡ ρ∞, i.e.

f ≡ 1 a.e. This proves (i). The equality case in (1.12) means f ≡ 1. This proves (ii) including when λ = Λ since

HΛ(f ) − GΛ(f ) = 2Λ K eψp(ρ|ρ∞)

− eψp(ρ|ρ∞)

and K(e) ≥ e for any e ≥ 0, with equality if and only if e = 0.

Let λ ∈ (−∞, ¯Λ) and consider a minimizing sequence (fn)n∈N. Since

|fn|

R

Rn|fn|2/pdρ∞

p/2

is also a minimizing sequence, we may assume fn ≥ 0 a.e. and RRn|fn|2/p dρ∞ = 1

without restriction. Then Gλ(fn) takes the form

Gλ(fn) = 4 p − 1 p Z Rn |∇fn|2dρ∞− λ p p − 1 xn− x 2 p−1 n , with xn:= Z Rn |fn|2 dρ∞.

Gλ(fn) ≥ 4 p − 1 p Λ1 xn− ( ¯fn) 2_{− λ} p p − 1 xn− x 2 p−1 n . Note that by H¨older’s inequality

1 = Z Rn |fn|2/p dρ∞≤ Z Rn |fn|2dρ∞ 1/pZ Rn ρ∞dx 1−1/p = x1/p_n , ¯ fn= Z Rn fnρ∞p/2· ρ∞1−p/2dx ≤ Z Rn f_n2/pdρ∞ p/2Z Rn ρ∞dx = 1 .

Hence, xn≥ 1, ¯fn≤ 1 and, with gλ(x) :=

4p−1_p Λ1− λ_p−1p x + λ_p−1p x2p−1− 4p−1 p Λ1, Gλ(fn) ≥ gλ(xn)

is certainly bounded from below if 0 ≤ 4p − 1 p Λ1− λ p p − 1 ⇐⇒ λ ≤ 4 p − 1 p 2 Λ1= ¯Λ .

The function gλ is then monotone increasing on (1, +∞):

gλ(xn) ≥ gλ(1) = 0 ,

and as a consequence, xn =

R

Rn|fn|2 dρ∞ → 1 as n → +∞. Thus ∇fn and fn− ¯fn

strongly converge to 0 in L2(dρ∞), and ¯fn → 1 because of (3.3). Hence, fn → 1 in

H1_(dρ

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To complete the proof of Theorem 4, we have to prove that: Λ ≤ Λ1. Consider the

second variation of GΛ: D2GΛ(1) is a bilinear form, namely

(ϕ, χ) 7→ D2GΛ(1) (ϕ, χ) = 8 p − 1 p Z Rn ∇ϕ · ∇χ dρ∞− Λ Z Rn ϕχ dρ∞

for any ϕ, χ ∈ H1(dρ∞) such that RRnϕ dρ∞ = RRnχ dρ∞ = 0. The Poincar´e

con-stant Λ1 corresponds to the value of Λ in the limit case p = 2 (for which K(e) = e):

Z Rn |∇g|2 dρ∞− Λ1 Z Rn |g|2 dρ∞≥ 0 ∀ g ∈ H1(dρ∞) such that Z Rn g dρ∞= 0 .

Assume that Λ > Λ1 and consider λ ∈ (Λ1, Λ), and φ such thatRRnφ dρ∞= 0 and

Z Rn |∇φ|2dρ∞≤ 1 2(λ + Λ1) Z Rn |φ|2 dρ∞.

For ε > 0, small enough, we then have Gλ(1 + εφ) ≤ 2 p − 1 p ε 2_(Λ 1− λ) Z Rn |φ|2 dρ∞(1 + o(1)) < 0 ,

which is a contradiction to (ii) of Lemma 5. A result similar to the one of Theorem 4 holds if we replace Gλ by Hλ.

Corollary 6 Under the conditions of Theorem 4, we have for any p ∈ (1, 2): 2p − 1 p Λ1 ≤ 1 2 ρ∞6≡ρ∈L1inf₊₍Rn) R Rn ρ dx=M Iψp(ρ|ρ∞) eψp(ρ|ρ∞) ≤ Λ1 . (3.4)

Proof.With the conventions M = 1 and ρ = |f |2/pρ∞, thenRRn|f |2/p dρ∞= 1, so that

Hλ(f ) = −Iψp(ρ|ρ∞) − 2λ eψp(ρ|ρ∞) can be rewritten as

Hλ(f ) = 4 p − 1 p Z Rn |∇f |2dρ∞− 2λ Z Rn |f |2− 1 dρ∞.

As in the proof of Theorem 4 we have on the one hand Hλ(f ) ≥ 2 2p − 1 p Λ1− λ Z Rn |f |2− 1 dρ∞.

On the other hand, a development of Hλ in a neighbourhood of 1 gives a contradiction

if Λ1 < λ < ˜Λ := sup{λ ∈ R : Hλ(f ) ≥ 0 ∀ f ∈ H1(dρ∞)}.

Remark 7 (i) Note that the upper bound in (3.2) (resp. the lower bound in (3.4)) is a consequence of the upper bound in (3.4) (resp. the lower bound in (3.2)) using K(e) ≥ e (resp. using K(e) ≤ _1−κe ).

(ii) If f ≡ 1 is the unique minimizer of GΛ, if there exists a unique minimizer fλ of Gλ

for λ − Λ > 0, small, and if fλ → 1 in H1(dρ∞) as λ & Λ, then Λ = Λ1, as can

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3.2 Holley-Stroock type perturbations

In Section 1 we presented the refined convex Sobolev inequality (1.6) for steady state measures ρ∞= e−A(x), whose potential A(x) satisfies the Bakry–Emery condition (A2).

We shall now extend that inequality for potentials eA(x) that are bounded perturbations of such a potential A(x). Our result generalizes the perturbation lemma of Holley and Stroock (cf. [10] for the logarithmic entropy ψ1 and §3.3 of [2] for general admissible

entropies).

For our subsequent calculations it is convenient to rewrite (1.6) as k Z Rn ψ f 2 kf k2 L2_(dρ_∞₎ ! dρ∞ ! ≤ 4 Z Rn f2 kf k4 L2_(dρ_∞₎ ψ00 f 2 kf k2 L2_(dρ_∞₎ ! D|∇f |2 dρ∞, (3.5) where we substituted ρ ρ∞ = f 2 R Rnf2dρ∞ .

Theorem 8 Assume that ψ = ψp with some 1 < p < 2 is a fixed entropy generator. Let

ρ∞(x) = e−A(x), fρ∞(x) = e− eA(x) ∈ L1+(Rn) with

R Rnρ∞dx = R Rnρf∞dx = M and e A(x) = A(x) + v(x) , 0 < a ≤ e−v(x) ≤ b < ∞ , x ∈ Rn. (3.6) Let the given diffusion D(x) be such that the convex Sobolev inequality (3.5) holds for all f ∈ L2(dρ∞). Then a convex Sobolev inequality also holds for the perturbed measure fρ∞:

1 ap−1k ap b Z Rn ψ f2 kf k2_L2 d fρ∞ ≤ 4 Z Rn f2 kf k4_L2 ψ00 f2 kf k2_L2 D|∇f |2_{d f}ρ∞ (3.7)

for all nontrivial f ∈ L2_{(d f}ρ∞) = L2(dρ∞). Here kf k_L22 stands for kf k2_L2_(d_ρ_f_∞₎.

Note that the normalization of ρ∞ and fρ∞ implies a ≤ 1 and b ≥ 1.

Proof. First we introduce the notations χ(x) := f 2_(x) kf k2 L2_(dρ_∞₎ , χ(x) :=˜ f 2_(x) kf k2 L2_(d_ρ_f_∞₎ , γ := χ ˜ χ = kf k2_L2_(d_ρ_f_∞₎ kf k2 L2_(dρ_∞₎ , and because of (3.6) we have a ≤ γ ≤ b.

We adapt the idea of [10, 2] and define for a fixed f ∈ L2(dρ∞) the function

g(s) := sp Z Rn ψ f2 s d fρ∞.

Since g attains its minimum at s = kf k2 L2_(d_ρ_f ∞), by differentiating w.r.t. s, we have kf k2p_L2_(d_ρ_f_∞₎ Z Rn ψ( ˜_{χ) d f}ρ∞= g(kf k2_L2_(d_ρ_f_∞₎) ≤ g(kf k2_L2_(dρ_∞₎) ≤ b kf k2p_L2_(dρ_∞₎ Z Rn ψ(χ) dρ∞,

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Using the monotonicity of k and Assumption (3.5), this yields: k γp b Z Rn ψ( ˜_{χ) d f}ρ∞ ≤ k Z Rn ψ(χ) dρ∞ ≤ 4 Z Rn f2 kf k4 L2_(dρ_∞₎ ψ00(χ)D|∇f |2 dρ∞ (3.8) ≤ 4γ p a Z Rn f2 kf k4 L2_(d_ρ_f_∞₎ ψ00( ˜χ)D|∇f |2 _{d f}ρ∞ ,

where we again used (3.6) in the last estimate.

Since γ/a ≥ 1, the convexity of k and k(0) = 0 imply: γp ap k ap b Z Rn ψ( ˜_{χ) d f}ρ∞ ≤ k γp b Z Rn ψ( ˜_{χ) d f}ρ∞ .

Together with (3.8), this finishes the proof. Note that a Holley–Stroock perturbation of the usual convex Sobolev inequality (1.3) would lead – under the assumptions of Theorem 8 – to the inequality

a b2λ1e ≤ |e 0_| _(3.9) (cf. [2]). Since a b 2λ1e < 1 ap−1k ap b e ∀ a p b e > 0 , Inequality (3.7) certainly improves (3.9).

Acknowledgments

The first author acknowledges support by the bilateral DAAD-Procope Program, the DFG under Grant-No. MA 1662/1-3, and the ENS Cachan. Both authors were sup-ported by the EU financed networks ERBFMRXCT970157 and HPRN-CT-2002-00282.

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