On Gaussian interpolation inequalities

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Sur des inégalités d’interpolation Gaussiennes

Giovanni Brigati ^a, Jean Dolbeault ^band Nikita Simonov ^c

aCEREMADE (CNRS UMR n^◦7534), PSL University, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France

bCEREMADE (CNRS UMR n^◦7534), PSL University, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France

cLJLL (CNRS UMR n^◦7598) Sorbonne Université, 4 place Jussieu, 75005 Paris, France

E-mails:[email protected] (G. Brigati), [email protected] (J. Dolbeault), [email protected] (N. Simonov)

Abstract.This paper is devoted to Gaussian interpolation inequalities with endpoint cases corresponding to the Gaussian Poincaré and the logarithmic Sobolev inequalities, seen as limits in large dimensions of Gagliardo-Nirenberg-Sobolev inequalities on spheres. Entropy methods are investigated using not only heat flow techniques but also nonlinear diffusion equations as on spheres. A new stability result is established for the Gaussian measure, which is directly inspired by recent results for spheres.

Résumé.Cet article est consacré à des inégalités d’interpolation Gaussiennes, avec comme cas extrêmes l’inégalité de Poincaré Gaussienne et l’inégalité de Sobolev logarithmique, vues comme limites en grandes dimensions des inégalités de Gagliardo-Nirenberg-Sobolev sur les sphères. Les méthodes d’entropie sont abordées en utilisant non seulement des techniques basée sur l’équation de la chaleur mais aussi sur des équations de diffusion non-linéaires, comme pour les sphères. Un nouveau résultat de stabilité est établi pour les mesures Gaussiennes, qui s’inspire directement de résultats récents sur les sphères.

Keywords.logarithmic Sobolev inequality, Gagliardo-Nirenberg-Sobolev inequalities, Gaussian Poincaré inequality, sphere, spectral decomposition, entropy methods, Ornstein-Uhlenbeck operator, nonlinear diffu- sions, improved inequalities, stability.

2020 Mathematics Subject Classification.26D10, 46T12, 46E35, 39B62, 43A90, 49J40, 58E35.

Date:8 February 2023.File:BDS2023-PLS.tex

1. Introduction and main results

Let us consider theGagliardo-Nirenberg-Sobolev inequalitieson the unitd-dimensional sphere k∇uk²_L2(S^d,dµ_d)≥ d

p−2

³

kuk²_Lp(S^d,dµ_d)− kuk²_L2(S^d,dµ_d)

´

∀u∈H¹(S^d,dµd) (1) for anyp∈[1,2)∪(2,+∞) ifd=1, 2, and for anyp∈[1,2)∪(2,2^∗] ifd≥3. Heredµ_d denotes the uniform probability measure onS^d⊂R^d+1and, ifd≥3, 2^∗=2d/(d−2) is the critical Sobolev

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exponent. By convention, we take 2^∗= +∞ifd=1 or 2. The purpose of this paper is to clarify the links of these interpolation inequalities with the family ofGaussian interpolation inequalities

k∇vk²_L2(Rⁿ,dγ)≥ 1 2−p

³

kvk²_L2(Rⁿ,dγ)− kvk²_L^p₍Rⁿ,dγ)

´

∀v∈H¹(Rⁿ,dγ) (2) where the exponentpis taken in the range 1≤p<2. Inequality (2) is intermediate between the Poincaré inequality corresponding top=1 and theGaussian logarithmic Sobolev inequality

k∇vk²_L2(Rⁿ,dγ)≥1 2

Z

Rⁿ|v|²log



 |v|² kvk²_L2(Rⁿ,dγ)



dγ ∀v∈H¹(Rⁿ,dγ)

obtained as a limit case of (2) asp→2₋. Heredγ(y) :=(2π)⁻^n/2e⁻¹²^|^y^|²d ydenotes the centred normalized Gaussian probability measure and the dimensionnis any positive integer.

It is somewhat classical that, if we consider the Sobolev inequality on the sphere,i.e., Inequal- ity (1) withp =2^∗andd≥3, rescale it to the sphere of radiusp

d and fix a function depending onn variables only on this sphere, since the curvature tends to 0 at the right order as one takes the limit asd→ +∞, then the sphere tends to the flat space with Gaussian measure. For instance, we read (with adapted notations) from [9, p. 4818] thatif we rescale this inequality so as to be on a sphere of radiusp

d and take the limit d→ ∞or p→2we obtain in the Poincaré limit the Gross logarithmic inequality for the Gaussian measure since−(1/d)∆onS^d goes to−∆+x· ∇ on the infinite-dimensional limit.The last part of the sentence refers to a result known as the Maxwell–Poincaré lemma : see [37] and [39, Remark 4, p. 254] for some historical comments. The statement of [9] has been made more precise later in [11, 12] using a slightly different limit. How- ever, to our knowledge, the infinite-dimensional limit has not been considered in the subcritical rangep<2^∗.

On the sphere, Inequality (1) follows from [4, 6] for anyp≥1 ifd=1 and anyp∈[1,2^#] with 2^#:=(2d²+1)/(d−1)²ifd ≥2. The proof in the rangep∈(2^#,+∞) ifd=2 andp∈(2^#,2^∗] if d≥3 can be found in [15, Corollary 6.1], [16] and [10]. Also see [35] in the casep=2^∗. In the case of the Gaussian measure, we refer to [8] (also see [32]) for a first proof of Inequality (2). The formal analogy of (1) and (2) is striking. Although computations are somewhat standard, our first purpose is to make this point rigorous and recover (2) as a special limit of (1) asd→ +∞.

Theorem 1. Let n be a positive integer, p∈[1,2)and consider a function v ∈H¹(Rⁿ,dγ)with compact support. For any d≥n, large enough, if ud(ω)=v¡

ω1/p d,ω2/p

d,...,ωn/p d¢

where ω=(ω1,ω2,...,ωd+1)∈R^d+1⊃S^dis such that|ω| =1, then

d→+∞lim d µ

k∇udk²_L2(S^d,dµd)− d 2−p

³

kudk²_L2(S^d,dµd)− kudk²_Lp(S^d,dµd)

´¶

= k∇vk²_L2(Rⁿ,dγ)− 1 2−p

³

´. The carré du champmethod has frequently been applied to prove Gaussian interpolation inequalities ranging between the logarithmic Sobolev inequality and the Poincaré inequality like (2) using the linear flow associated with the Ornstein-Uhlenbeck operator; see [4], and [7]

for an overview. Still in the case of the Gaussian measure, we adopt here a new point of view by usingnonlinear diffusion equationsin order to prove the same inequalities, but with different remainder terms. This is a very natural point of view when dealing with inequalities like (1) on the sphere, as shown for instance in [24] (see earlier references therein). In that case, linear flows are indeed limited to exponentsp≤2^#ifd≥2. To overcome this difficulty if eitherd =2 and p>2^#ord≥3 andp∈(2^#,2^∗], one has to consider fast diffusion flows. Before explaining the results for the Gaussian measure, let us summarize the main known results on the sphere.

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OnS^d, let us consider a positive solutionuof

∂u

∂t =u⁻^p⁽¹⁻^m) µ

∆u+(m p−1)|∇u|² u

¶

(3) where∆denotes the Laplace-Beltrami operator onS^d. In the casem=1,u^p solves the heat equation and for this reason, we shall call it thelinear case. Otherwiseu^p solves a nonlinear diffusion equation corresponding either to a fast diffusion flow withm<1 or to a porous media equation withm>1. In any case, we claim that

d

d tkuk²_Lp(S^d)=0 and d

d t kuk²_L2(S^d)=2(p−2) Z

S^d

u⁻^p⁽¹⁻^m)|∇u|²dµd. Let us define

m_±(d,p) := 1 (d+2)p

µ

d p+2± q

d(p−1)¡

2d−(d−2)p¢

¶

. (4)

The following result can be found in [20, 24] with additional details in [17, 22, 25].

Proposition 2 ([20, 24]). Assume that d ≥1, with either p ∈[1,2)∪(2,+∞) if d =2 or p ∈ [1,2)∪(2,2^∗]if d≥3, and let m∈[m₋(d,p),m₊(d,p)]. If u>0solves(3)with an initial datum inL²∩L^p(S^d,dµd), then

d d t

µ

k∇uk²_L2(S^d,dµd)− d p−2

³

kuk²_Lp(S^d,dµd)− kuk²_L2(S^d,dµ_d)

´¶

≤0 ∀t>0.

The limit, ast→ +∞, of any solution of (3) is a constant. This means that thedeficit, that is, the difference of the two sides in Inequality (1), converges to 0. Then, by Proposition 2, it follows that the deficit is non-negative, which directly proves (1). The same monotonicity property applies to the deficit of thelogarithmic Sobolev inequality on the sphere

k∇uk²_L2(S^d,dµd)≥d 2

Z

S^d|u|²log Ã |u|²

kuk²_L2(S^d)

!

dµd ∀u∈H¹(S^d,dµd)

in the limit case corresponding top=2. The admissible values of the parameters are limited to m₋(d,p)≤m≤m₊(d,p) and 1≤p≤2^∗ifd≥3. Moreover, at the endpoints, we havem_±(d,1)=1 andm_±(d,2^∗)=(d−1)/d ifd≥3, whilem₊(d,2^#)=1 so thatm=1 is admissible if and only if 1≤p≤2^#when d ≥2. See Fig. 1. For appropriate initial data, it is shown in [25] that the monotonicity property of the deficit along the flow of (3) is violated for anyp∈[2,2^∗) orp=2^∗if d≥3 as soon as eitherm<m₋(d,p) orm>m₊(d,p).

1 2 3 4

0.5 1.0 1.5

Figure 1. Cased=5. The admissible parametersp andmcorrespond to the grey area.

The boundary of the admissible set is tangent to the vertical linesp=1 at (m,p)=(1,1) and p=2^∗=10/3 at (m,p)=(4/5,10/3). Qualitatively, this figure does not change asdincreases but gets squeezed in the interval 1≤p≤2 asd→ +∞.

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In view of the results of Theorem 1, it is natural to ask whether there is also a monotonicity property of the deficit associated to the Gaussian interpolation inequalities (2) when we rely on a nonlinear diffusion flow onRⁿ. Letm_±(p) :=limd→+∞m_±(d,p) and notice that

m_±(p)=1±1 p

p(p−1)(2−p). (5)

The diffusion operator associated to the Gaussian measure is theOrnstein-Uhlenbeck operator L=∆−x· ∇and we consider now the nonlinear parabolic equation

∂v

∂t =v⁻^p⁽¹⁻^m) µ

Lv+(m p−1)|∇v|² v

¶

. (6)

In the definition ofL,∆denotes the standard Laplacian onRⁿ. The following result is new for m6=1 while the casem=1 follows from the method of thecarré du champdeveloped by Bakry and Emery in [4].

Theorem 3. Assume that n≥1, p∈[1,2). If v>0solves(6)with m∈[m₋(p),m₊(p)]for an initial datum inL²∩L^p(Rⁿ,dγ), then

d d t

µ

k∇vk²_L2(Rⁿ,dγ)− 1 p−2

³

kvk²_Lp(Rⁿ,dγ)− kvk²_L2(Rⁿ,dγ)

´¶

≤0 ∀t>0.

The limiting casep=2 corresponding to the Gaussian logarithmic Sobolev inequality is also covered but it is obtained as a standard application of the linearcarré du champmethod known from [4] becausem_±(2)=1. See Fig. 2.

0.5 1.0 1.5 2.0

0.4 0.6 0.8 1.0 1.2 1.4

Figure 2. The admissible parameterspandmcorrespond to the grey area and are independent of the dimensionn. The boundary of the admissible set is tangent to the vertical linesp=1 at (m,p)=(1,1) andp=2 at (m,p)=(1,2). It is the limit set of the admissible parameters for Proposition 2 asd→ +∞.

Said in simple words, the result of Theorem 3 is that the admissible range of exponents of the nonlinear flow, for which the deficit associated to (2) is monotone non-increasing, is obtained as the limit of the range of the corresponding exponents on the sphere, in the large dimensions limit. Moreover,p=2 appears as acritical exponentfor the Gaussian measure.

Let us now focus on stability results. The main result of [17] for the sphere is a constructive stability estimate for Inequality (1), limited to the subcritical rangep∈(1,2^∗), which measures the distance to optimal functions, and distinguishes the subspaces generated by constant functions, spherical harmonic functions associated to the first positive eigenvalue of the Laplace-Beltrami operator, and the orthogonal directions. Optimal exponents in the stability estimate measuring the distance to the set of optimal functions differ, depending on the directions. Here we have the exact counterpart in the Gaussian case. LetΠ1denote the orthogonal projection of L²(Rⁿ,dγ) onto the (n+1)-dimensional function space generated by 1 andxi withi=1, 2, . . . ,n.

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Theorem 4. For all n≥1, and all p∈(1,2),there is an explicit constant cn,p>0such that, for all v∈H¹(dγ), it holds

k∇vk²_L²₍Rⁿ,dγ)− 1 p−2

³

kvk²_L^p₍Rⁿ,dγ)− kvk²_L²₍Rⁿ,dγ)

´

≥cn,p



k∇(Id−Π1)vk²_L2(Rⁿ,dγ)+ k∇Π1vk⁴_L2(Rⁿ,dγ)

k∇vk²_L2(Rⁿ,dγ)+ kvk²_L2(Rⁿ,dγ)



. Exponents 2 and 4 which appear in the right-hand side of the inequality are sharp and the constantcn,p has an explicit although complicated expression given in the proof. Ifp=1, the last term drops and the distance to optimal functions is measured only by

k∇(Id−Π1)vk²_L2(Rⁿ,dγ),

and a decomposition on Hermite polynomials shows that the estimate cannot be improved. If p=2^∗andn≥3, the recent stability result on Sobolev’s inequality onRⁿof [23], which quantifies the estimate of Bianchi and Egnell in [14] can be translated into a stability result for (1) onS^d, that can be recast in the form of [17, Theorem 6]. By a large dimension argument, a stability on the Gaussian logarithmic Sobolev inequality is also shown in [23], although the distance is measured only by an L²(Rⁿ,dγ) norm. Whether a stronger estimate can be obtained in the limiting case p=2, eventually under some restriction, is therefore so far an open question.

This paper is organized as follows. In Section 2, we give a new proof of Inequality (2) as a consequence of Inequality (1) by taking a large dimensions limit, applied to the inequality written for a function depending only on a fixed numbernof real variables. To our knowledge, this is new except for the limit casep=2 of the logarithmic Sobolev inequality. Section 3 is devoted to the proof of Theorem 3: we characterize the nonlinear diffusion flows of porous medium or fast diffusion type such that the deficit is monotone non-increasing and recover the picture known on the sphere in the large dimensions limit. Moreover, by thecarré du champmethod, we establish improved inequalities that provide us with first stability results. The stability result of Theorem 4 for the Gaussian measure is proved in Section 4 using a detailed Taylor expansion and the improved inequalities of Section 3.

2. From subcritical interpolation inequalities on the sphere to Gaussian interpolation In this section we explain how Inequality (2) can be seen as the limit of Inequality (1) in the large dimensions limit, that is, asd→ +∞, and prove Theorem 1. Comments on the limit casep=2 can be found at the end of this section.

The unit sphereS^dis parametrized in terms of thestereographic coordinatesby ωj= 2xj

1+ |x|² if 1≤j≤d and ωd+1=1− |x|² 1+ |x|²

whereω=(ω1,ω2,...,ωd+1) denote the coordinates in R^d+1 ⊃S^d and x =(x1,x2,...,xd) are Cartesian coordinates inR^d. To a functionuonS^d, we associate a functionwonR^d using the stereographic projection such that

µ 2 1+ |x|²

¶^d⁻₂²

u(ω)=w(x) ∀x∈R^d. It is a standard result that

Z

S^d|u|^pdµ_d=2^δ(p)² ¯

¯S^d¯

¯⁻¹ Z

Rⁿ〈x〉⁻^δ(p)|w|^pd x

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and that Z

S^d|∇u|²dµd+¹₄d(d−2) Z

S^d|u|²dµd=¯

¯S^d¯

¯⁻¹ Z

Rⁿ|∇w|²d x. where〈x〉:=p

1+ |x|²andδ(p) :=2d−p(d−2). Using the stereographic projection, Inequality (1) can be written on the Euclidean spaceR^das the weighted interpolation inequality

Z

Rⁿ|∇w|²d x+dδ(p) p−2

Z

Rⁿ

|w|²

〈x〉⁴d x≥ C_d,p p−2

µZ

Rⁿ

|w|^p

〈x〉^δ(p)d x

¶²_p

with C_d,p=2^δ(p)^p d¯

¯S^d¯

¯¹⁻

2 p. (7) See [22] for details. Equality is achieved by the Aubin-Talenti functionw_⋆(x) := 〈x〉²⁻^d. Assume thatd≥4. Let us considerf =w/w_⋆and notice that the inequality rewritten in terms off is

Z

Rⁿ|∇f|²w_⋆²d x+ 4d p−2

Z

Rⁿ|f|²w_⋆²^∗d x≥ C_d,p p−2

µZ

Rⁿ|f|^pw_⋆²^∗d x

¶²

p .

At this point, we may notice that for anyd≥3, we can choose freely anyp∈[1,2)∪(2,2^∗] ifd≥3.

Since we are interested in the limit asd→ +∞, in order to consider a fixed, given value ofp, we have no choice but to restrictpto the case 1≤p<2.

Withg(x)=f¡ x/p

d¢

, we obtain after changing variables that 1

4 Z

R^d|∇g|² d x

¡1+_d¹|x|²¢d−2+ 1 p−2

Z

R^d|g|² d x

¡1+¹_d|x|²¢d ≥ C_d,pd^d

p−2 2p

4d(p−2)



 Z

R^d|g|^p d x

¡1+_d¹|x|²¢d





2 p

. Let us assume thatn≥1 is a given integer and taked>max{n,3}. Withx=(y,z)∈Rⁿ×R^d−n≈R^d, we also assume that the functiong depends only ony. In other words, we writeg=gd where gd(y,z)=v(y) for some functionvdefined onRⁿ, that is,

gd(y,z)=v(y) ∀(y,z)∈Rⁿ×R^d⁻ⁿ (8) Here we use an indexdin order to emphasize thatgdhas to be considered as a function onR^d. Let us define

cd:=(dπ)^d² Γ(d/2) Γ(d) .

Lemma 5. Let n be a positive integer, p∈[1,2), consider a function v∈H¹(Rⁿ,dγ)with compact support and define gdaccording to(8). Then we have

d→+∞lim 1 4cd

Z

R^d|∇gd|² d x

¡1+_d¹|x|²¢d−2

+ 1 2−p lim

d→+∞

1 cd







C_d,pd^d^p−

2 2p

4d



 Z

R^d|gd|^p d x

¡1+_d¹|x|²¢d





2 p

− Z

R^d|gd|² d x

¡1+_d¹|x|²¢d







= k∇vk²_L2(Rⁿ,dγ)− 1 2−p

³

´. In other words, we prove that the infinite dimensional limit of (7), for functions depending only on a finite numbernof real variables, is (2). The assumption of compact support can be removed ifgdis square integrable with respect to the measure¡

1+¹_d|x|²¢₋d

d xand∇gdis square integrable with respect to the measure¡

1+_d¹|x|²¢2−d

d x, at least for somed∈Nlarge enough.

Proof. Using

¡1+_d¹|x|²¢2−d

=³ 1+_d¹¡

|y|²+ |z|²¢´2−d

=¡

1+¹_d|y|²¢2−d¡

1+_d¹|ζ|²¢2−d

with ζ= z

q1+_d¹|y|² ,

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we can integrate with respect tozand obtain Z

Rd−n

¡1+_d¹|x|²¢2−d

d z=¡

1+_d¹|y|²¢2−^d+n2

Z

Rd−n

¡1+_d¹|ζ|²¢2−d

dζ.

Letdγ(y) :=(2π)⁻^n/2e⁻¹²^|^y^|²d y be the centred normalized Gaussian probability measure. We recall that|S^k−1| =2π^k/2/Γ(k/2) for anyk∈N\ {0} and

Z_+∞

0 r^a⁻¹¡

1+¹_dr²¢₋b

d r=d^a² Γ¡_a

2

¢Γ¡ b−^a₂¢ 2Γ(b)

if 0<a<2b. Applying these formulas witha=k=d−nandb=d−2>2−n, we find that Z

Rd−n

¡1+_d¹|ζ|²¢2−d

dζ=(dπ)^d⁻²ⁿ Γ³

d+n 2 −2´ Γ(d−2) . Applying these formulas witha=b=k=d≥2, we find that

Z

Rⁿ

¡1+_d¹|x|²¢₋d

d x=cd

and, as a consequence

dlim→+∞

1 cd

Z

R^d−n

¡1+_d¹|ζ|²¢2−d

dζ= 4 (2π)^n/2 using Stirling’s formula. Since

d→+∞lim

¡1+_d¹|y|²¢2−^d+n2

=e⁻¹²^|^y^|², we obtain

dlim→+∞

1 cd

Z

Rⁿ|∇gd(y)|²¡

1+_d¹|x|²¢2−d

d x=4 Z

Rⁿ|∇v|²dγ.

Similar computations show that Z

Rⁿ|gd(y)|²¡

1+¹_d|x|²¢₋d

d x= Z

Rⁿ|gd(y)|²¡

1+_d¹|y|²¢₋^d⁺₂ⁿ d y

Z

Rd−n

¡1+_d¹|ζ|²¢₋d

dζ,

dlim→+∞

1 cd

Z

Rⁿ|gd(y)|²¡

1+_d¹|x|²¢₋d

d x= Z

Rⁿ|v|²dγ, and

Z

Rⁿ|gd(y)|^p¡

1+_d¹|x|²¢₋d

d x= Z

Rⁿ|gd(y)|^p¡

1+_d¹|y|²¢₋^d+n₂ d y

Z

Rd−n

¡1+_d¹|ζ|²¢₋d

dζ,

dlim→+∞

C_d_,pd^d

p−2 2p

4d cd

µZ

Rⁿ|gd(y)|^p¡

1+¹_d|x|²¢₋d

d x

¶_p²

= µZ

Rⁿ|v|^pdγ

¶_p² .

This completes the proof of Lemma 5.

Proof of Theorem 1. Applied to the functionud, Inequality (1) is transformed into Inequality (7) applied to

gd(x)=ud

Ã 2y 1+_d¹|x|²

!

∀x=(y,z)∈Rⁿ×R^d⁻ⁿ.

Since the right-hand side uniformly converges tov(y) for any smooth and compactly supported functionv, the same conclusion holds for Theorem 1 as for Lemma 5.

It is a natural question to ask what happens in (1) to the marginals depending only on a finite numbern of variables ifp=2 or in the case 2<p≤2^∗=2d/(d−2). We may notice that limd→+∞2d/(d−2)=2 and it is known, for instance from [9], that one recovers theGaussian logarithmic Sobolev inequalityas a limit case of Sobolev’s inequality onS^d corresponding to p=2^∗when d→ +∞. This is also true if we consider a sequence (pd)d∈Nwith 1<pd <2^∗,

(8)

depending ond, if its limit is also 2, as shown next. By convention, whenpd=2, we consider the Gaussian logarithmic Sobolev inequality instead of (2).

Proposition 6. Let n be a positive integer and consider a function v ∈H¹(Rⁿ,dγ)with com- pact support. For any d≥n, large enough, let ud(ω)=v¡

ω1/p d,ω2/p

d,...,ωn/p d¢

whereω= (ω1,ω2,...,ωd)∈R^d+1⊃S^dis such that|ω| =1, as in Theorem 1. Then we have

d→+∞lim d Ã

k∇udk²_L2(S^d,dµ_d)−d 2

Z

S^d|ud|²log

Ã |ud|² kudk²_L2(S^d)

! dµd

!

= k∇vk²_L2(Rⁿ,dγ)−1 2

Z

Rⁿ|v|²log





|v|² kvk²_L2(Rⁿ,dγ)



dγ.

If(pd)d≥3is a sequence of real numbers such that pd∈(1,2)∪(2,2^∗)andlimd→+∞pd=2, then

d→+∞lim d µ

k∇udk²_L2(S^d,dµ_d)− d 2−pd

³

kudk²_L2(S^d,dµ_d)− kudk²_L_pd₍_Sd,dµ_d)

´¶

= k∇vk²_L2(Rⁿ,dγ)−1 2

Z

Rⁿ|v|²log





dγ.

Proof. The proof is an adaptation of the proof of Theorem 1 and, in the casepd6=2, relies on the standard observation that

plim→2

kvk²_Lp(Rⁿ,dγ)− kvk²_L2(Rⁿ,dγ)

p−2 =1

2 Z

Rⁿ|v|²log





dγ.

As this computation raises no special difficulty, details are omitted.

3. Entropy methods and nonlinear flows for Gaussian measures

In this section, we prove the result of Theorem 3 for the Gaussian measuredγand extend it to the slightly more general framework of a uniformly strictly log-concave measuredµ, before drawing some consequences. Most of the results are similar to computations usually done on the sphere, but we are not aware of the use of nonlinear flows (m6=1) in the context of Gaussian measures.

This approach is very natural in the perspective of spheres in the large-dimensional limit.

3.1. Gaussian interpolation inequalities: a proof by thecarré du champmethod OnRⁿ, let us consider the probability measure

dµ=Z⁻¹e⁻^φd y with Z= Z

Rⁿe⁻^φd y (9)

and redefine theOrnstein-Uhlenbeck operatorby

L:=∆− ∇φ· ∇ (10) on L²(Rⁿ,dµ). This generalizes the case of the harmonic potentialφ(y)=¹₂|y|²considered in the introduction. We assume thatφsatisfies theBakry-Emery condition

Hessφ≥λ_⋆Id a.e. (11)

for someλ_⋆>0. The harmonic potential corresponds to the equality case withλ_⋆=1. Under Assumption (11), it is well known (see for instance [7, Section 7.6.2]) that, withλ=λ_⋆and for any p∈[1,2),

k∇fk²_L2(Rⁿ,dµ)≥ λ 2−p

³

kfk²_L2(Rⁿ,dµ)− kfk²_L^p₍Rⁿ,dµ)

´

∀f ∈H¹(Rⁿ,dµ) (12)

(9)

and also, by taking the limit asp→2₋, that k∇fk²_L2(Rⁿ,dµ)≥λ

2 Z

Rⁿ|f|²log





|f|² kfk²_L2(Rⁿ,dµ)



dµ ∀f ∈H¹(Rⁿ,dµ).

The classical proof by the carré du champ, as in [3–5], relies on theOrnstein-Uhlenbeck flow

∂tρ=Lρapplied to the solution with initial datumρ(t=0,·)= |f|^p. Here we consider a more general strategy and compute as in the carré du champ method using thenonlinear diffusion

flow ∂ρ

∂t = 1

mLρ^m ∀t≥0, ρ(t=0,·)= |f|^p. (13) Our goal is to understand the range ofmfor which we have a monotonicity property of the deficit as in the casem=1. Letm_±(p) be defined as in (5).

Theorem 7. Assume that n≥1, p∈[1,2)and m∈[m₋(p),m₊(p)]. We consider the measure dµas in(9)such that(11)holds for someλ_⋆>0. Ifρ>0solves(13)withLdefined by(10)for an initial datumρ(t=0,·)= |f|^pinL^2/p∩L¹(Rⁿ,dµ), then

d d t

µ

k∇ρ^1/pk²_L2(Rⁿ,dµ)− λ_⋆ 2−p

³

kρ^1/pk²_L2(Rⁿ,dµ)− kρk^2/p_L1(Rⁿ,dµ)

´¶

≤0 ∀t>0.

The result of Theorem 7 is new form6=1. Theorem 3 corresponds to the special case of the harmonic potential φ(y)= ¹₂|y|² with v =ρ^1/p in Theorem 7. The range of the admissible parameters (m,p) is the same in Theorem 7 and shown in Fig. 2. With the additional observation thatρ(t,·) converges to a constant ast→ +∞so that the limit of the deficit is 0, the monotonicity of the deficit of Theorem 7 provides us with a proof of (12).

Proof of Theorem 7. In order to do computations, a very convenient reformulation is obtained with the flow

∂w

∂t =w²⁻^2β µ

Lw+κ|∇w|² w

¶

(14) for anyt≥0, with initial datumw(t=0,·)= |f|^1/β, where

κ:=β(p−2)+1 and β= 2 2−p(1−m). A first computation shows thatR

Rⁿw^βpdµ=R

Rⁿ|f|^pdµis independent oftbecause d

d t Z

Rⁿ

w^βpdµ=βp Z

Rⁿ

w^κ µ

Lw+κ|∇w|² w

¶ dµ=0.

A second useful computation goes as follows:

− 1 2β²

d d t

Z

Rⁿ

µ

¯

¯∇w^β¯

¯²+ λ_⋆ p−2w^2β

¶ dµ

= Z

Rⁿ

µ

Lw+(β−1)|∇w|²

w − λ_⋆w β(p−2)

¶ µ

Lw+κ|∇w|² w

¶ dµ

= Z

Rⁿ

(Lw)²dµ+(κ+β−1) Z

Rⁿ

(Lw)|∇w|²

w dµ+κ(β−1) Z

Rⁿ

|∇w|⁴

w² dµ−λ_⋆ Z

Rⁿ|∇w|²dµ.

(10)

The purely technical purpose of introducing the exponentβis to make the last line of the above computation 2-homogeneous inw, which makes the discussion easier to read. Inserting the two following estimates,

Z

Rⁿ

(Lw)²dµ= − Z

Rⁿ∇w· ∇(Lw)dµ= − Z

Rⁿ∇w·(L∇w)dµ+ Z

Rⁿ∇w·[L,∇]w dµ

= Z

RⁿkHesswk²dµ+ Z

Rⁿ∇w·[L,∇]w dµ

= Z

Rⁿ

kHesswk²dµ+ Z

Rⁿ

Hessφ:∇w⊗ ∇w dµ

≥ Z

RⁿkHesswk²dµ+λ_⋆ Z

Rⁿ|∇w|²dµ

and Z

Rⁿ(Lw)|∇w|²

w dµ= −2 Z

RⁿHessw:∇w⊗ ∇w

w dµ+

Z

Rⁿ

|∇w|⁴ w² dµ, we obtain that

d d t

Z

Rⁿ

µ

¯

¯∇w^β¯

¯²+ λ_⋆ p−2w^2β

¶ dµ≤0 if, for any functionw, we have

Q_β[w] := Z

RⁿkHesswk²dµ−2(κ+β−1) Z

Rⁿ

Hessw:∇w⊗ ∇w

w dµ

+¡

κ(β−1)+κ+β−1¢ Z

Rⁿ

|∇w|⁴ w² dµ≥0.

A sufficient condition is obtained if the reduced discriminant is negative, that is, if (κ+β−1)²−¡

κ(β−1)+κ+β−1¢

≤0.

Altogether, this gives the condition

β₋(p)≤β≤β₊(p) with β_±(p) :=1±p

(p−1)(2−p)

1−(p−1)(2−p) . (15)

See Fig. 3.

0.5 1.0 1.5 2.0

Figure 3. The admissible parameterspandβcorrespond to the grey area and are independent of the dimensionn. The boundary of the admissible set is tangent to the vertical linesp=1 at (β,p)=(1,1) andp=2 at (β,p)=(1,2). This figure corresponds to Fig. 2 up to the transformation ofm→β=2/¡

2−p(1−m)¢ .

Equivalently, written in terms ofm, the condition ism₋(p)≤m≤m₊(p) withm_±(p) defined by (5). Summarizing our computations, we learn that

d d t

µ

k∇wk²_L2(Rⁿ,dµ)− λ_⋆ p−2

³

kwk²_L^p₍Rⁿ,dµ)− kwk²_L2(Rⁿ,dµ)

´¶

= −2β²Q_β[w]≤0

(11)

becauseQ_β[w] is non-negative under Condition (15).

To make these computations rigorous, one has to justify all integrations by parts. This is by now rather standard and can be done using the following scheme.

x∈Rⁿ :φ(x)<hª

. We can consider the evolution equation restricted toΩh, with no flux boundary conditions. Then apply the carré du champmethod and keep track of the boundary terms. SinceΩh is a bounded convex domain, these terms have a sign due to Grisvard’s lemma: see for instance [27, Lemma 5.2], [36, Proposition 4.2], [21, 29] and [31, Lemma A.3].

(2) Extend Inequality (12) written onΩhtoR^dby taking the limit ash→ +∞and then argue by density.

This completes the proof of Theorem 7.

Remark 8. For anyp∈[1,2), if the conditionm₋(p)≤m≤m₊(p) is not satisfied, then one can find a positive initial datum such that the solutionvof (6) is such that

d d t

µ

k∇vk²_L2(Rⁿ,dµ)− d p−2

³

kvk²_L^p₍Rⁿ,dµ)− kvk²_L2(Rⁿ,dµ)

´¶

|t=0>0.

See [25] for a similar statement on the sphere and its proof.

3.2. Improved inequalities based on thecarré du champmethod

In the proof of Theorem 7, using onlyQ_β[w]≥0 is a crude estimate. Let us explain how one can obtain improved estimates by making a better use ofQ_β[w]≥0. Under Condition (15), we can indeed rewriteQ_β[w] as an integral of a sum of squares,

Q_β[w]= Z

Rⁿ

°

°Hessw−(κ+β−1)∇w⊗ ∇w w

°

2

dµ+δ Z

Rⁿ

|∇w|⁴

w² dµ (16)

with

δ:=κ(β−1)+κ+β−1−(κ+β−1)²=¡

β−β₋(p)¢ ¡

β₊(p)−β¢

>0. (17) As in [2, Theorem 2], let us consider the special casem=β=1 of the linear flow. Let us define the entropyand theFisher informationby

E[w] := 1 2−p

³

kwk²_L2(Rⁿ,dµ)− kwk²_L^p₍Rⁿ,dµ)

´ and I[w] := k∇wk²_L2(Rⁿ,dµ). Inequality (12) amounts simply to

I[w]−λ_⋆E[w]≥0.

We can now state a firstimproved entropy–entropy production inequality.

Proposition 9. Let n be any positive integer. We consider the measure dµas in(9)such that(11) holds for someλ_⋆>0. For any p∈(1,2), letϕ(s) :=1+s−(1+s)^p⁻¹. With the above notations, we have

I[f]≥ λ_⋆

(2−p)²kfk²_L^p₍Rⁿ,dµ)ϕ

Ã(2−p)E[f] kfk²_Lp(Rⁿ,dµ)

!

∀f ∈H¹(Rⁿ,dµ). (18) Sincep∈(1,2), the functionϕis convex withϕ(0)=0 andϕ^′(0)=2−p, which implies in particular thatϕ(s)≥(2−p)s for anys ≥0, so that Inequality (18) is stronger than (12). Inequality (18) amounts to

k∇fk²_L2(Rⁿ,dµ)≥ λ_⋆ (2−p)²

³

kfk²_L2(Rⁿ,dµ)− kfk^2(p_L2(R⁻ⁿ¹⁾,dµ)kfk²⁽²_Lp(⁻Rⁿ^p),dµ)

´