Rigidity phenomenons for an infinite dimension diffusion operator and cases of near equality in the Bakry--Ledoux isoperimetric comparison Theorem
aa r X i v : . [ m a t h . P R ] A ug Rigidity phenomenons for an infinite dimension diffusion operatorand cases of near equality in the Bakry–Ledoux isoperimetriccomparison Theorem.
Raphaël Bouyrie
University of Marne–la–Vallée, France
Abstract
We study rigidity phenomenons for infinite dimension diffusion operators of positive curva-ture using semigroup interpolations. In particular, for such diffusions, an analogous statement ofObata’s theorem is established. Moreover, the same rigidity holds for the Bakry–Ledoux isoperi-metric comparison Theorem - a result due to Franck Morgan. Recently, Mossel and Neemanhave exploited the semigroup proof of Bakry and Ledoux to derive dimension free bounds forthe Gaussian isoperimetry. We extend theirs arguments to obtain in particular new quantitativebounds on the spherical isoperimetric inequality in large dimension.
In this paper, we study rigidity phenomenons for spaces endowed with an infinite dimension diffusionoperators. The natural setting consists of abstract Markov triples ( E, µ, Γ) , which belongs to theframework initiated by the seminal paper of Bakry and Émery [B-E]. For a complete account aboutMarkov diffusion operator, we refer the reader to the recent monograph [B-G-L]. This setting encir-cles as an important illustration the examples of weighted Riemannian manifolds with generalizedRicci curvature bounded from below and without any condition on the dimension, justifying theterm “infinite dimension”.In such framework, sharp geometric and functional inequalities has been established using theprinciple of monotonicity along the heat flow attached to the subsequent diffusion. In the caseof Riemannian manifolds of finite dimension n with positive Ricci curvature some rigidity occursfor inequalities such as spectral gap (a result due to Obata), Myers’ maximal diameter theorem(known as Topogonov–Cheng theorem) or Lévy–Gromov isoperimetric comparison theorem (see[Bay], [Mi1]). That is, in those examples, a Riemannian manifold achieving equality is isometricto the model spaces, i.e. the Euclidean spheres. In two remarkable recent papers ([C-M1], [C-M2])Cavalletti and Mondino have proved analogous statements of these important results in the generalframework of metric measure spaces with an appropriate notion of positive curvature and finitedimension. Theirs results goes by needle decompositions of metric measure spaces inspired bythe work of Klartag ([Kla]) in Riemannian geometry. Moreover similar rigidity phenomenons hasbeen established by the authors, where in this more general setting the model spaces are sphericalsuspensions. 1he infinite dimensional case presents in various aspects less rigidity. The analogous modelsspaces are given by the Gaussian spaces of fixed variance. For example there is no longer boundednesson the diameter of manifolds (as immediately checked by considering Gaussian space). However, itis a natural guess that semigroup tools are well suited to investigate rigidity phenomenons in thissetting : indeed since such inequalities are established by monotoncity along the heat flow, equalitycases correspond to a constant evolution along such heat flow.The aim of this paper is to investigate this idea more precisely. A common and classical schemeis to integrate in space and use a commutation property between the gradients and the underlyingsemigroups. Any function saturating these functional inequalities satisfies therefore equality in thesescommutations. We will see that it implies that such function is necessary an eigenfunction of theunderlying diffusion operator, a specific property of the Gaussian space. Before stating our results,we briefly recall the geometric and functional inequalities subsequent to this work.The isoperimetric problem consists in minimizing the boundary of sets given a fixed volume.Isoperimetrical problems has a long history, and make sense in very general metric measure spaces ( E, d, µ ) . Indeed, one only need a measure to define the “volume” and a distance to define the“boundary”. In this context, denote the outer Minkowski content by µ + ( A ) = lim inf r → µ ( A r ) − µ ( A ) r , with A r = { y ∈ E, d ( x, A ) ≤ r } , where d ( x, A ) = inf { d ( x, y ) , y ∈ A } . Define then the isoperimetricprofile of the set ( E, d, µ ) by I ( E,d,µ ) ( v ) = inf { µ + ( A ) , A ⊂ E, µ ( A ) = v } . The isoperimetric problem is two-fold : one aims to find the function I ( E,d,µ ) as well as describing theoptimal sets. In general such a problem is very difficult to solve. Even the existence of optimal sets,called isoperimetric sets, is far from being guaranteed. Throughout this note, we will consider theisoperimetric problem over a Riemannian manifold M with some probability measure µ of the form e − ψ d vol , where vol designs the canonical Riemannan volume. We will consider the usual distanceinduced by the metric g . As a consequence, I ( M,d,µ ) is defined on [0 , and symmetric with respectto / .The Euclidean spheres are among the few examples in which the isoperimetric problem is com-pletely solved. This goes back to the early XXth century by Lévy and Schmidt. The extremalsets are, perhaps unsurprisingly, the spherical caps. Later on, Gromov have extended the proof tomore general Riemannian manifold with Ricci curvature bounded form below. The Lévy–Gromovcomparison theorem can be then stated as follows. Theorem 1.1 (Lévy–Gromov) . Let ( M, g, µ ) be a compact Riemannian manifold of dimension n ≥ with Ricci curvature bounded from below by κ > (in the sense that Ric ( M,g ) ≥ κg ), with µ = volvol ( M ) its normalized measure.Then its isoperimetric profile satisfies I ( M,g,µ ) ≥ I ( S nκ ,g κ ,σ κ,n ) , where S nκ denotes the n dimensionalsphere whose Ricci curvature is equal to κ - that is of radius q n − κ - equipped with its normalizedsurface measure σ κ,n . The above statement links curvature and dimension. Curvature and dimension are also linkedby Lichnerowicz’ minoration of the first eigenvalue of the Laplace Beltrami operator λ . In theabove setting, Lichnerowicz theorem express that λ ≥ nκn − . Actually, this constant arises in2lassical functional inequalities that will be recalled below. First of all, Lichnerowicz’ minoration isequivalent to the following spectral gap inequality (by classical integration by parts): ∀ f ∈ H ( M ) , Var µ ( f ) = Z M f dµ − (cid:18) Z M f dµ (cid:19) ≤ n − nκ Z M |∇ f | dµ, where we have denoted H ( M ) = { f : M → R , R M f dµ + R M |∇ f | dµ < + ∞} . A further important functional inequality is given by the following Sobolev logarithmic inequality.For all f ∈ L log L , Ent µ ( f ) = Z M f log f dµ − (cid:18) Z M f dµ (cid:19) log (cid:18) Z M f dµ (cid:19) ≤ n − nκ Z M |∇ f | dµ. The optimal constant nκn − in this inequality is somewhat challenging to reach. For instance, inthe case of the Spheres S nκ , n ≥ , it has been established by Mueller and Weissler [M-W] only fewyears before [B-E].For the further purposes, as a consequence of an inequality linking spectral gap constant andSobolev logarithmic constant originally proven by Rothaus [Rot], the Obata rigidity theorem holdsfor the Sobolev logarithmic inequality (see also [B-G-L]). That is any n dimensional Riemannianmanifold with Ricci curvature bounded from below by κ such that its Sobolev logarithmic constantis equal to nκn − is isometric to S nκ .It is an observation actually going back to Poincaré that the Gaussian space ( R n , γ n ) where γ n ( dx ) = √ π e −| x | / dx designs the standard Gaussian measure can be seen as the limit in N of N dimensional Euclidean spheres of radius √ N − . More presicely, the uniform measure µ N on thesespheres projected on a fixed subsace R n converges to the measure γ n . This observation combinedwith the Lévy–Gromov’s theorem is at the root of the Gaussian isoperimetry, proved in the mid 70’sindependently by Borell and Sudiakov–Tsirelson. The isoperimetric sets are the half spaces. Thatis, if A ⊂ R n and H is an half space such that γ n ( A ) = γ n ( H ) , then γ + ( A ) ≥ γ + ( H ) = ϕ (Φ − ( γ ( A ))) . Here, and throughout this paper, ϕ designs the density of γ and Φ( x ) = R x −∞ ϕ ( t ) dt its cummulativedistribution function.It is an immediate consequence that the isoperimetric profile I γ of the Gaussian space is inde-pendant of n and given by the function ϕ ◦ Φ − . Around twenty years later, a proof of Bobkov of afunctional version led to an extension of Lévy–Gromov comparison theorem by Bakry–Ledoux, in afairly abstract setting that rely on a curvature dimension condition denoted by CD ( κ, ∞ ) , κ ∈ R , and that will be described in Section . In particular, for Riemannian manifolds, the result is thefollowing analogous of Lévy–Gromov comparison theorem. Theorem 1.2 (Bakry–Ledoux) . Let ( M, g, e − ψ d vol) be a weighted Riemannian manifold such that Ric g + ∇ ψ ≥ κg (unformly as symmetric tensors), with κ > . Then its isoperimetric profilesatisfies I µ ≥ √ κ I γ = I γ κ where γ κ is the Gaussian measure on R of variance κ − . In addition of the proof of Bakry–Ledoux, two others proofs of this theorem are known. Oneby Bobkov for log concave measures on R n using localization techniques and one by Franck Mor-gan ([Mor]) for manifold with densities using more geometric arguments. Notice also that the3akry–Ledoux setting is the more general, as it allows for more general structure than Riemmanianmanifolds.Concerning functionnal inequalities, in the setting of the above theorem, sharp spectral gap andSobolev inequalities has been established by Bakry and Émery (see [B-E]). It holds ∀ f ∈ H ( M ) , Var µ ( f ) ≤ κ Z M |∇ f | dµ, and ∀ f ∈ H ( M ) , Ent µ ( f ) ≤ κ Z M |∇ f | dµ. Moreover, the constant κ is sharp and attained for the Gaussian space ( R , | · | , γ κ ) .Towards rigidity, for the isoperimetric comparison a statement follows from the proof of FranckMorgan. The result is the following : if there exists v ∈ (0 , such that I µ ( v ) = I γ ( v ) , then themanifold splits as ( M ′ , g ′ , µ ′ ) × ( R , | · | , γ k ) where ( M ′ , g ′ , µ ′ ) is such that Ric ′ g + ∇ ψ ′ ≥ κg ′ if µ ′ = e − ψ ′ d vol ′ . Such rigidity can be seen as an infinite dimensionnal analogous statement of theresults of Bayle and Milman. It is a natural guess that the same splitting phenomenon holds for thefunctionnal inequalities, i.e. if the manifold ( M, g, e − ψ d vol) reach the sharp κ spectral or Sobolevconstant. In this paper, we establish some rigidity statements in the infinite dimension setting (i.e. spacessatisfying appropriate curvature dimension conditions - see next section for a precise definition). Inparticular, following the result of Klartag [Kla] about needles decomposition and the ideas of [C-M2],we establish an analogous statement of Obata’s theorem for manifolds with densities.
Theorem 1.3.
Let ( M, g, e − ψ d vol) a Riemannian manifold with the following condition Ric g + ∇ ψ ≥ κg . Assume that κ is either the spectral gap constant or the Sobolev logarithmic constant.Then there exists another manifold ( M ′ , g ′ , e − ψ ′ d vol ′ ) with the same condition Ric g ′ + ∇ ψ ′ ≥ κ suchthat M splits as ( M, g, e − ψ d vol) = ( R , | · | , dγ κ ) × ( M ′ , g ′ , e − ψ ′ d vol ′ ) , where dγ κ = e − κx / dx is the Gaussian measure on the real line with variance /κ . Moreover the eigenfunctions associated to the constant κ are linear functions in the Gaussiandirection. The same conclusion holds for the optimal Sobolev logarithmic constant and in the samemanner, any non constant function achieving equality in the Sobolev logarithmic inequality is of thesame form.In particular, the last assertion is an extension of Carlen results about characterization of theequality cases in Sobolev logarithmic inequality.Concerning the isoperimetric problem, Carlen and Kerce [C-K] have used the Bakry and Ledouxsemigroup proof to characterize the isoperimetric minimizers of the Gaussian space without conditionof smoothness. Extending the ideas of [C-K], we re derive the rigidity result of Franck Morgan. Theorem 1.4.
Let ( M, d, µ ) a weighted Riemmannian manifold of CD ( κ, ∞ ) class. If there existsa v ∈ (0 , such that I ( M,µ ) ( v ) = I ( R ,γ κ ) ( v ) then ( M, g, µ ) = ( R , | · | , γ κ ) × ( M ′ , g ′ , µ ′ ) . Moreoverthe isoperimetric minimizers are of the form ( −∞ , Φ − κ ( v )) × M ′ . κ = 1 ) Gaussian isoperimetric profile is its inde-pendence with respect to the dimension. Above this rigidity theorem, a more difficult question isto address the cases of near equality while keeping dimension free bounds. In order to investigatecase of near equality in isoperimetric inequality, the semigroup proof of Bakry and Ledoux seemswell-suited as it describe a monotone evolution along the heat-flow. This idea has been put in shapeby Mossel and Neeman [M-N], who where the first to derive dimension free bounds for the Gaussianisoperimetry deficit δ , with a dependance of order log − / ( δ − ) . Notice that the optimal dependancein δ for the Gaussian space is √ δ and has been established using different techniques since then in[B-B-J] (see also [Eld]). Within the scheme of proof of [M-N], it does not seem to be possible to evenreduce the dependance to a power of δ . However there is - we believe - two main advantages of [M-N]with respect to [Eld] and [B-B-J]. First, the proof is somewhat simpler and more importantly, mostof the arguments of [M-N] are valid in general CD (1 , ∞ ) spaces. In the last section, we extend theproof of Mossen and Neeamn to obtain a quantitative and dimension free estimates for the Bobkov’sinequality in the case of high dimensional Euclidean spheres of radius √ n − . Our main result canbe stated as follows. Theorem 1.5.
Let ( S n , µ ) be the Euclidean sphere of radius √ n − endowed with the uniformprobability measure µ . Let A be a (Borel measurable) subset of S n such that µ + ( A ) ≤ I γ ( µ ( A )) + δ. Then, denoting δ n = max( δ, /n ) , there exists a spherical cap H and a positive constant c ∈ (0 . , / such that µ ( A ∆ H ) ≤ O ( | log δ n | − c ) . As we shall show in the last section, this theorem implies some results about the spherical deficitisoperimetry himself. Since the scheme of proof is of abstract flavor, we will also discuss the case oflog-concave probability measures.This paper is organized as follows. In the next section we briefly recall the general setting inwhich we will work. In Section 3 we make some comments about the infinite dimensional setting.Then we recall the functional and geometric inequalities. In Section 5 we prove Theorem 1.3. InSection 6 we show how to deduce a rigidity statement for the isoperimetric problem and Theorem1.4. Then the last two sections are devoted to the deficit problem in the Bakry–Ledoux comparisonTheorem and the proof of Theorem 1.5.
This section aims at presenting the abstract framework that is subsequent to the synthetic notiondue to Bakry–Émery [B-E] of Curvature–Dimension condition. For a comprehensive introductionto the topic, we refer the reader to [B-G-L], or [Le1] for a shorter - but nonetheless complete -overview. Recall that curvature and dimension are linked in a Riemmanian setting by Lichnerowicz’sminoration of the first eigenvalue of the Laplace operator. In a complete Riemannian manifold, theBochner’s formula expresses that
12 ∆( |∇ f | ) − ∇ ∆ f · ∇ f = Ric g ( ∇ f, ∇ f ) + k∇ ( f ) k HS . If the manifold is n dimensional, Schwarz’s inequality ensures that k∇ ( f ) k HS ≥ n (∆ f ) . It impliesby a standard integration by parts Lichnerowicz’ minoration of the first eigenvalue of the Laplace5eltrami operator. The Bochner’s formula is a key observation that led the authors of [B-E] to definea generalized curvature-dimension criterion CD ( κ, n ) , κ ≥ , n ∈ [0 , ∞ ] for a large class of spaces.This criterion has proven to be very useful to reach sharp geometric and functional inequalities. Webriefly recall a few basic definitions.Let ( E, µ ) be a probability space equipped with L a diffusion operator acting on a domain A ,that is such that for each ψ : R k → R with ψ (0) = 0 , and f , . . . , f k , ∈ A ,Lψ ( f , . . . , f k ) = k X i =1 ∂ i ψ ( f , . . . , f k ) Lf i + X ≤ i,j ≤ k ∂ ij ψ ( f , . . . , f k )Γ( f i , f j ) . A diffusion L is generating a semigroup - that is a family of operators ( P t ) t ≥ such that P = Id and P t + s = P t ◦ P s - by taking ( P t := e tL )) t ≥ . These semigroups ( P t ) t ≥ are Markov that is ∀ t ≥ , P t = , where is the constant function defined on E equal to . We shall assume moreover that themeasures µ are reversible with respect to L or ( P t ) t ≥ that is ∀ f, g ∈ L ( µ ) , Z E f Lg dµ = Z E gLf dµ, and invariants with respect to ( P t ) t ≥ that is ∀ f ∈ L ( µ ) , Z E P t f dµ = Z E f dµ. Another important property is the ergodicity of these semigroups that is lim s →∞ P s f = Z E f dµ, for every function f (say integrable with respect to µ ).The Laplace Beltrami operator over a Riemmanian manifold is a relevant example of diffusion,which generates the heat semigroup ( e t ∆ ) t ≥ . Following Paul André Meyer, Bakry and Emery definethe “carré du champ” operator Γ ( Γ below) and its iterations (Γ n ) n ≥ by setting Γ ( f, g ) = f g andthen by induction ∀ n ≥ , Γ n ( f, g ) = 12 (cid:18) L (Γ n − ( f, g )) − Γ n − ( f, Lg ) − Γ n − ( g, Lf ) (cid:19) . The diffusion property implies the following chain rule formulas Γ( u ( f )) = u ′ ( f ) Γ( f ) , Γ ( u ( f )) = u ′ ( f ) Γ ( f ) + u ′ ( f ) u ′′ ( f )Γ( f, Γ( f )) + u ′′ ( f ) Γ( f ) . When ( M, g, vol) is a Riemannian manifold and the diffusion is the Laplace Beltrami operator ∆ ,the carré du champ Γ( f ) is equal to |∇ f | , and Bochner’s formula can then be written as Γ ( f ) = Ric g ( ∇ f, ∇ f ) + k∇ f k HS . As a consequence, if the manifold is n dimensional with Ricci curvature bounded below by κ , theBochner’s formula implies Γ ( f ) ≥ κ Γ( f ) + 1 n ( Lf ) , which is a good formulation for a general curvature-dimension definition.6 efinition 2.1. Say that ( E, µ, Γ) satisfy the curvature-dimension criterion CD ( κ, n ) if ∀ f ∈ A , Γ ( f ) ≥ κ Γ( f ) + 1 n ( Lf ) . Obviously, it is clear that CD ( κ, n ) ⇒ CD ( κ ′ , n ) for κ ′ ≤ κ and CD ( κ, n ) ⇒ CD ( κ, m ) for n ≤ m . The curvature-dimension condition CD ( κ, n ) is therefore a condition of curvature boundedfrom below by κ and dimension bounded from above by n .If n ∈ N and κ > , the model spaces for this condition are naturally given by the EuclideanSpheres S n of radius q n − κ , and moreover Bochner’s formula indicates that Γ ( f ) = κ Γ( f ) + k∇ ( f ) k HS .More generally one can take weighted Riemannian manifold, which is Riemannian manifold withdensity dµ = e − V d vol with the associated diffusion operator L = ∆ − ∇ V · ∇ , called the Witten-Laplacian. Then the “carré du champ” Γ( f ) is still given by |∇ f | and by the Bochner’s formula,the CD ( κ, ∞ ) condition Γ ( f ) ≥ κ Γ( f ) turns out to be true whenever ∇ V + Ric g ≥ κ (uniformlyas symmetric tensors). This tensor is called Bakry-Emery tensor .The Gaussian space ( R n , γ n ) , n ≥ , is an example of a space with infinite dimension andcurvature (as immediately checked). Therefore, in this definition, the dimension do not necessarycoincide with the topological dimension. A good explanation of this fact is given by the Poincaréobservation. Recall indeed that the Gaussian space can be seen as limit of n -dimensional spheres ofradius √ n − , that is of objects of class CD (1 , n ) . This is also a good explanation of the fact thatthe geometric and functional inequalities in Gaussian space are dimension free .The underlying semigroup in Gaussian space ( Q t ) t ≥ is the Ornstein-Uhlenbeck semigroup. Itacts on functions f : R n → R of the Dirichlet domain, by x ∈ R n Q t f ( x ) = Z R n f ( e − t x + p − e − t y ) dγ ( y ) = Z R n f ( y ) q t ( x, y ) dγ ( y ) , with q t the Mehler kernel associated to the Ornstein-Uhlenbeck semigroup.Since R n is flat, ( R n , µ ( dx ) = e − V ( x ) dx ) satisfies CD ( κ, ∞ ) whenever ∇ V ≥ κ , which corre-sponds to the class of strictly log-concave measures. Indeed, the iterated “carré du champ” Γ takesthe following form : Γ ( f ) = ∇ V ( ∇ f, ∇ f ) + k∇ f k HS . Of course, this curvature dimension criterion in its abstract formulation covers a wider settingthan Riemannian manifolds. One still need a diffusion operator, hence some smooth structure.The principal strength of this abstract formulation is that it has been proven to be a veryefficient criterion to established sharp geometric and functional inequalities. In this context, saythat ( E, µ, Γ) satisfies SG ( λ ) (spectral gap, or Poincaré, inequality with constant λ ) if λ is the best(i.e. largest) positive constant such that for all f ∈ H ( M ) , Var µ ( f ) ≤ λ Z M |∇ f | dµ. In the same manner, say that ( E, µ, Γ) satisfies LS ( ρ ) (Sobolev logarithmic inequality with constant ρ ) if ρ is the best (i.e. largest) positive constant such that for all f ∈ H ( M ) , Ent µ ( f ) ≤ ρ Z M |∇ f | dµ.
7n [B-E], the authors show that the CD ( κ, n ) -condition implies sharp Sobolev logarithmic in-equalities, i.e with ρ = nκn − ( = κ if n = ∞ ), equivalent to the important property of hypercon-tractivity of the underlying semigroups. We confer to reader to [Bak], [B-G-L] for the analogousdimensional Sobolev inequalities in the case n < ∞ .In the case n = ∞ , referred as infinite dimensional setting, a common scheme of proof toestablished these inequalities is to interpolate along the corresponding semigroups and to showmonotony along these semigroups. We will illustrate the later in Section 4. It is therefore wellsuited to investigate cases of equality. Before doing so, in the next section, we discuss the analogousrigidity statements in the case of weighted Riemannian manifolds. We now focus on the infinite dimensional setting corresponding to the CD ( κ, ∞ ) condition. Recallthat in this context, when κ > , the model space is the Gaussian space ( R n , γ n,κ ) , where γ n,κ = γ ⊗ nκ , n ≥ and that it satisfies LS ( κ ) and SG ( κ ) for all n ≥ . Theses important inequalities has beenestablished well before [B-E], in the early XXth century for the spectral gap inequality and in the70’s for Sobolev logarithmic inequalities.An analogous of Obata’s theorem is the following statement : if a weighted manifold ( M, g, µ ) have the optimal constant κ for the spectral gap inequality, then the manifold splits as ( M, g, µ ) =( R , | | , γ κ ) × ( M ′ , g ′ , µ ′ ) . As ρ ≥ θκ + (1 − θ ) λ for some θ ∈ (0 , the same conclusion would holdfor the Sobolev logarithmic inequality. We will prove this conclusion whenever there is non constantextremal functions for this inequalities. Since equality for the spectral gap inequality is attain foreigenfunctions of diffusion operator L = ∆ − ∇ ψ · ∇ associated to the eignenvalue κ , this impliesthe statement. Furthermore, we can establish characterization of these extremal functions. We notethat this conclusion overlaps with the work of Bentaleb [Ben], who uses very similar tools as ours(i.e. interpolations along heat flows). For the reader’s convenience, we will still give the completeproofs in the next sections.Concerning the isoperimetric problem, a rigidity theorem for the Bakry–Ledoux’s comparisontheorem has been established by Franck Morgan for manifolds with densities. The proof relies ondeep results from geometric measure theory which ensure that in such setting there is always a regularset that minimize the isoperimetric problem. Note that for the real line, Bobkov [Bo1] has provedthat minimizers exist and are half-lines in the weaker class of log concave measures (not strictly andwithout regularity assumptions on the potential). We will make the more general observation thatif there is a non trivial minimizer in an abstract Markov triple, then some kind of rigidity occurs.In particular, we recover Morgan’s theorem for isoperimetric problem with characterization of theoptimal sets in the setting of a weighted Riemannian Manifold.The starting point is a functional form of the Gaussian isoperimetry. In a remarkable work [Bo2],Bobkov showed that the Gaussian isoperimetry is equivalent to the following functional inequality I γ (cid:18) Z R n f dγ n (cid:19) ≤ Z R n q I γ ( f ) + |∇ f | dγ n (1)for each function f : R n → [0 , locally Lipschitz. Using this inequality to a sequence of Lipschitzfunctions ( f ε ) ε ≥ approaching A as ε goes to implies back I γ ( γ ( A )) ≤ γ + ( A ) . Bobkov’s originalproof is build in three steps. Firstly, one proves the inequality on the two point space { , } , theusing tensorization property of the inequality one establishes it on the discrete cubes { , } n , n ≥ before proving it on the Gaussian space by means of a central limit argument.8oticing that by ergodicity, R R n f dγ n = lim t →∞ Q t f , and that f = Q f , Bakry and Ledoux’sidea [B-L1] is to consider the function ψ : s ∈ [0 , ∞ ) Z R n q I γ ( Q s f ) + |∇ Q s f | dγ n and showing that it is non increasing, recovering therefore Bobkov inequality.A main advantage of this proof and its tools is that it can be easily adapted on a Markov triple ( E, µ, Γ) with the condition CD ( κ, ∞ ) described in the previous section. The result is the following I γ (cid:18) Z E f dµ (cid:19) ≤ Z E r I γ ( f ) + 1 κ Γ( f ) dµ (2)for each function f : E → [0 , locally Lipschitz (in an appropriate sense). Since for weightedRiemannian manifolds, Γ( f ) = |∇ f | and µ + ( A ) = lim k →∞ k∇ f k k L ( µ ) for a sequence of functions ( f k ) k ≥ approaching A , it yields the comparison theorem announced in the introduction.To close the picture over a Riemmanian manifold ( M, g, µ = e − V d vol) , Barthe–Maurey [B-M]have shown the equivalence between a comparison theorem over isoperimetric profiles ∀ A ⊂ M, µ + ( A ) ≥ κ I γ ( µ ( A )) and the Bobkov inequality (2) for all Lipschitz functions mapping to [0 , . For the further purposes,let us discuss the case of the Euclidean Spheres S n . Standard arguments (see e.g. [B-M]) imply that inf a ∈ (0 , I ( S n ,g,µ ) ( a ) I γ ( a ) = I ( S n ,g,µ ) (1 / I γ (1 /
2) = √ π vol n − ( S n − )vol n ( S n ) = √ n +12 )Γ( n ) = c n , and the latter is always larger than √ n − , and therefore always less than √ n . This can be seenas a consequence of log-convexity of the Gamma function. The optimal Bobkov inequality cantherefore be written as I γ (cid:18) Z S n f dµ (cid:19) ≤ Z S n s I γ ( f ) + 1 c n |∇ f | dµ. As we will see below, the isoperimetric problem admits a minimizer for every manifold withdensity. Towards rigidity, whenever I ( M,g,µ ) = I γ κ , this will imply that there is always a nonconstant extremal function for the Bobkov’s inequality. This in turn implies the splitting theoremof Franck Morgan. In this section, we recall how to reach sharp geometric and functional inequalities in the abstractframework of CD ( κ, ∞ ) spaces described in Section . The starting point is that the curvaturedimension criterion of Bakry–Emery implies a commutation between Γ and the semigroup ( P t ) t ≥ in the following sense. Lemma 4.1.
Under the CD ( κ, ∞ ) condition, κ ∈ R , we have for all t ≥ , Γ( P t f ) ≤ e − κt P t (Γ( f )) . (3) Indeed by log-convexity of the Gamma function, n − Γ( n ) ≤ n − Γ( n − )Γ( n +12 ) = Γ( n +12 ) , which is equivalentto the claim. Proof.
Define ψ t : s e − κs P s (Γ( P t − s f )) . Then using the semigroup property and the fact that Γ is bilinear, ψ ′ t ( s ) = e − κs [ P s (2Γ ( P t − s f ) − κP s (Γ( P t − s f ))] = 2 e − κs [ P s (Γ ( P t − s f ) − κ Γ( P t − s f ))] ≥ , by the CD ( κ, ∞ ) condition. But the statement of the lemma reads as ψ t (0) ≤ ψ t ( t ) .For the further purposes, the proof implies that there is equality in (3) if and only if for all s ∈ (0 , t ) , Γ ( P t − s f ) = κ Γ( P t − s f ) so that taking the limit s → t , Γ ( f ) = κ Γ( f ) .It is relevant to point out that in the case of the heat semigroup ( P t ) t ≥ on the Euclideanspaces and the Ornstein–Uhlenbeck semigroup ( Q t ) t ≥ on the Gaussian space we have an exactcommutation, which follows immediately from the explicit formulas of such semigroups. That is, forsmooth functions f of the respective Dirichlet domains, ∇ P t f = P t ∇ f and ∇ Q t f = e − t Q t ∇ f . Using the above lemma, we reach the following propositions (see e.g. [Le1], [B-G-L] - actually forthe second one, the a priori stronger, but in fact equivalent, property p Γ( P t f ) ≤ e − κt P t ( p Γ( f )) is needed). Proposition 4.2.
Let ( E, µ, Γ) be a probability space satisfying CD ( κ, ∞ ) . Then, ∀ f ∈ A , C ( κ, t )Γ( P t f ) ≤ P t ( f ) − ( P t f ) ≤ D ( κ, t ) P t (Γ( f )) , where C ( κ, t ) = 2 R t e κs ds and D ( κ, t ) = 2 R t e − κu du . In particular, if κ > , ( E, µ, Γ) satisfies SG ( κ ) . Proposition 4.3.
Let ( E, µ, Γ) be a probability space satisfying CD ( κ, ∞ ) . Then, ∀ f ∈ A , C ( κ, t ) Γ( P t f ) P t f ≤ P t ( f log f ) − P t f log P t f ≤ D ( κ, t ) P t (cid:18) Γ( f ) f (cid:19) , where again C ( κ, t ) = 2 R t e κs ds and D ( κ, t ) = 2 R t e − κs ds. In particular, if κ > , ( E, µ, Γ) satisfies LS ( κ ) . Let us discuss the reverse form of these inequalities. The reverse Poincaré inequality implies thateach bounded (by one) function is / p C ( κ, t ) -Lipschitz in the sense that Γ( P t f ) ≤ /C ( κ, t ) . It isalso known that it implies the following lemma (see [Le1]). Lemma 4.4.
Let ( E, µ ) satisfying the CD (0 , ∞ ) condition. Then ∀ f ∈ L ( E ) , k f − P t f k ≤ √ t k p Γ( f ) k . (4)Turning to the reverse Sobolev Logarithmic inequality, it implies that whenever ≤ f ≤ , C ( κ, t )Γ(log P t f ) ≤ − log( P t f ) so that φ = ( − log P t f ) / is √ C ( κ,t ) -Lipschitz in the sense that Γ( φ ) ≤ C ( κ,t ) .10 .2 Bakry–Ledoux comparison theorem and reverse isoperimetric inequality As discussed in Section 3, given a Markov triple ( E, µ, Γ) with the condition CD ( κ, ∞ ) , κ > , Bakryand Ledoux gave a semigroup proof of Bobkov’s inequality. Using this proof, Carlen and Kerce [C-K]has characterized cases of equality. While some of the computations arising are somewhat tedious,the following proposition is a consequence of [B-L1], which implies the Bobkov inequality. Proposition 4.5.
Let ( E, µ, Γ) satisfying the CD ( κ, ∞ ) condition and f : E [0 , . Set ψ : s ∈ (0 , ∞ ) Z E q I γ ( P s f ) + κ − Γ( P s f ) dµ. Then − ψ ′ ( s ) ≥ Z E I γ ( P s f )(Γ − κ Γ)(Φ − ◦ P s f )(1 + Γ(Φ − ◦ P s f )) / dµ. Proof.
The proof relies on the work of Bakry–Ledoux. Denoting P s f = f s , it is shown in [B-L1] that − ψ ′ ( s ) ≥ Z E I γ ( f s )(Γ − κ Γ)( f s ) − I γ ( f s ) I ′ γ ( f s )Γ( f s , Γ( f s )) + I ′ γ ( f s ) Γ( f s ) ( I γ ( f s ) + Γ( f s ) / ) dµ. Following the idea of Carlen and Kerce, we express the derivative ψ ′ in terms of Φ − ◦ f s . Thereis actually some geometric intuition behind this change of variable taking roots in the Bobkovinequality himself. Using the chain rule formulas Γ( u ( f )) = u ′ ( f ) Γ( f ) , Γ ( u ( f )) = u ′ ( f ) Γ ( f ) + u ′ ( f ) u ′′ ( f )Γ( f, Γ( f )) + u ′′ ( f ) Γ( f ) , with u = Φ − , since (Φ − ) ′ = I γ and (Φ − ) ′′ = − I ′ γ I γ we have ( I γ ( f s ) + Γ( f s )) / = I γ ( f s )(1 + Γ(Φ − ◦ f s )) / and I γ ( f s )(Γ − κ Γ)( f s ) − I γ ( f s ) I ′ γ ( f s )Γ( f s , Γ( f s )) + I ′ γ ( f s ) Γ( f s ) = I γ ( f s )(Γ − κ Γ)(Φ − ◦ f s ) , which concludes the proposition.Of course, since its derivative is non positive, ψ is decreasing which implies by ergodicity of thesemigroup ( P s ) s ≥ the Bobkov’s inequality, and thus the isoperimetric conclusion.For the further purposes, it has been established in [B-L1] by the same scheme of proof a reverseisoperimetric inequality in the following form. Proposition 4.6.
Let ( E, µ, Γ) be a probability space satisfying CD ( κ, ∞ ) . Then, for all functions f : E → [0 , , C ( κ, t )Γ( P t f ) ≤ ( I γ ( P t f )) − ( P t ( I γ ( f ))) . By the chain-rule formula, this inequality expresses that Φ − ( P t f ) is Lipschitz in the sense that Γ(Φ − ( P t f )) ≤ C ( κ,t ) .One can prove that it implies in the Gaussian space a reverse Bobkov inequality (cf [B-L1])).Indeed (say with κ = 1) , using the exact commutation e t ∇ Q t f = Q t ∇ f and letting t goes to infinity,it holds by ergodicity (cid:12)(cid:12)(cid:12)(cid:12) Z R n ∇ f dγ n (cid:12)(cid:12)(cid:12)(cid:12) ≤ I γ (cid:18) Z R n f dγ n (cid:19) − (cid:18) Z R n I γ ( f ) dγ n (cid:19) ,
11r equivalently, s(cid:12)(cid:12)(cid:12)(cid:12) Z R n ∇ f dγ n (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) Z R n I γ ( f ) dγ n (cid:19) ≤ I γ (cid:18) Z R n f dγ n (cid:19) . (5) The aim of this section is to put forward a rigidity phenomenon that occurs when there exists a nonconstant function achieving equality in the above geometric and functional inequalities. A commonfactor that appears in the proofs is the non negative quantity Γ − κ Γ . In order to have equality, isis necessary that this quantity is equal to .Without loss of generality, we can assume that in any of the above inequalities, R E f dµ = 0 .Let us take a (smooth) function achieving inequality in spectral gap inequality, Sobolev logarithmicor Bobkov inequality. Then, there is equality in the commutation (3), which implies the equality Γ = κ Γ . More precisely, Γ ( f ) = κ Γ( f ) for the spectral gap inequality, Γ (log f ) = κ Γ(log f ) forthe Sobolev Logarithmic inequality and Γ (Φ − ◦ f ) = κ Γ(Φ − ◦ f ) for the Bobkov’s inequality.Besides, we have the following Stein’s lemma. Lemma 5.1.
Assume that ( E, Γ , µ ) satisfies CD ( κ, ∞ ) and that there exists f non constant suchthat Γ ( f ) = κ Γ( f ) . Then Lf = − κf . In particular the law of f is Gaussian.Proof. Without loss of generality, we can make the assumption that R E f dµ = 0 . Then by Cauchy-Schwarz’ inequality and by the spectral gap inequality, κ Z E Γ( f ) dµ = κ Z E f ( − Lf ) dµ ≤ κ (cid:18) Z Ω f dµ Z E ( Lf ) dµ (cid:19) / ≤ (cid:18) κ Z E Γ( f ) dµ Z E ( Lf ) dµ (cid:19) / , so that κ Z E Γ( f ) dµ ≤ Z E ( Lf ) dµ = Z E Γ ( f ) dµ. Since by hypothesis Γ ( f ) = κ Γ( f ) , there is equality everywhere. Then, there exists a negative realnumber k such that Lf = kf and necessary k = − κ .For the next statement we confer to [Le2].Let ( E, µ, Γ) be a CD ( κ, n ) space, with n < ∞ . It is important to point out that since the firsteigenvalue of L is given by nκn − there is no function achieving Γ ( f ) = κ Γ( f ) . In fact this propertyis true only for “real” CD ( κ, ∞ ) spaces, that is not satisfying CD ( κ, n ) for any finite n .When ( M, g, e − ψ d vol) is a weighted Riemannian manifold, as a consequence of the Bochner’sformula, Γ ( f ) = Γ( f ) ⇐⇒ Ric g ( ∇ f, ∇ f ) + ( ∇ ψ − κId )( ∇ f, ∇ f ) + k∇ f k = 0 , and the conclusion of Stein’s lemma reads ∆ g f − ∇ ψ · ∇ f = − κf. When the manifold is R n , it easily implies that there is a Gaussian marginal and f is an eigenvectorof L in the Gaussian direction. In general, we can make use of the needle decompostions of [Kla],so that it implies the following theorem. 12 heorem 5.2. Let ( M, g, e − ψ d vol) be a CD ( κ, ∞ ) manifold. If there exists a non constant extrem-izer f in one of the inequality of Section 4, then ( M, g, e − ψ d vol) = ( R , | · | , γ κ ) × ( M ′ , g ′ , e − ψ ′ d vol) , where ( M ′ , g ′ , e − ψ ′ d vol) is a CD ( κ, ∞ ) manifold and f , log f or Φ − ( f ) is an eigenfunction of theOrnstein-Uhlenbeck semigroup associated to the eigenvalue κ , that is a linear function. In particular if the spectral gap constant is equal to κ , an eigenfunction associated to κ satisfiesthe condition of the above theorem. Recall that, if ρ designs the Sobolev logarithmic constant and λ the spectral gap constant, then ρ ≥ θκ + (1 − θ ) λ with θ ∈ (0 , . Thus, it implies Theorem1.3 announced in the introduction, which can be viewed as an infinite dimensional analogous of theObata’s theorem.In particular, the Gaussian space is the only weighted manifold for which there is non trivialextremizers. The linear functions are the only non constant functions achieving equality in thespectral gap inequality and the exponential functions are the only non constant functions achievingequality in Sobolev logarithmic inequality. The same conclusion holds for the reverse spectral gapand Sobolev logarithmic inequalities.For the isoperimetric inequality, we are interested in finding extremal sets. Actually, the factthat half spaces are the only isoperimetric minimizers is a consequence of the last assertion of theproposition. This characterization is due to Carlen and Kerce and the purpose of the next sectionis the generalize this fact to general CD ( κ, ∞ ) weighted Riemannian manifolds. Notice that for thereverse Bobkov inequality (5) there is equality if and only if Φ − ( f ) is linear. This conclusion hasalready been obtained in [B-C-F].To conclude this section, the condition that there exists a non constant function f such that Γ ( f ) = κ Γ( f ) implies ∆ g f = 0 , so that ∇ ( ∇ ψ · ∇ f ) = ( ∇ ψ )( ∇ f ) = κ ∇ f . Thus, there exists a ∈ T x M such that ∇ f = a and moreover Ric g ( a, a ) = 0 . One wonder if one can prove that itimplies in a more direct way (i.e. wihout relying on needles decomposition) that the manifold isa Cartesian product with a one dimensional Gaussian space. Notice that if M contains a line, asplitting Theorem has been established in [W-W]. We now turn to the isoperimetric rigidity statement. Intuitively, lim t → = R M |∇ P t A | dµ is a goodcandidate for the boundary measure of A as P t A is a smooth function whenever t > and P t A converges to A as t goes to . However we have defined the isoperimetric problem relatively to theMinkownski content µ + ( A ) = lim inf ε → µ ( A ε ) − µ ( A ) ε . In [C-K], the isoperimetric problem is givenwith respect to another definition of boundary, the weighted relative perimeter, that will be recalledbelow. Still, it is known for a while that in Gaussian space the isoperimetric problem can be takenwith respect to any kind of definition of perimeters. The remark in [C-K] is that our naive definitionof boundary using the semigroup agrees with the weighted relative perimeter. Actually, as we shallshow, this fact is more general.Let ( M, g, d vol) be a Riemmanian manifold. We denote by | Du | the variation, in the sense of DeGiorgi [DG], of u ∈ L ( M ) defined by | Du | ( M ) = sup (cid:26) Z M u div ϕd vol , ϕ ∈ C c ( M, R ) & k ϕ k ∞ ≤ . (cid:27) u is the indicator function of a set A , | Du | is called perimeter of A . It has been establishedin [C-M] that the variation of u is linked with the standard heat kernel ( P t ) t ≥ = ( e t ∆ ) t ≥ in thefollowing sense. Proposition 6.1.
For all u ∈ L ( M ) , | Du | = lim t → Z M |∇ P t u | d vol . When ( M, g, µ ) is a weighted Riemannian manifold, one naturally extends the definition of DeGiorgi and define the weighted-perimeter as follows. Definition 6.2.
Let ( M, g, µ = e − ψ vol) a weighted Riemannian manifold with ( P t ) t ≥ its relativesemigroup. Then we define the relative perimeter of A ⊂ M by Per ψ ( A ) = sup (cid:26) Z A (div ϕ − ∇ ψ · ϕ ) dµ, ϕ ∈ C c ( M, R ) & k ϕ k ∞ ≤ (cid:27) . The proof of [C-M] can be easily adapted as below to reach the following conclusion. Actually,in the Gaussian space, a similar proof appears in [C-K].
Proposition 6.3.
For all A ⊂ M , Per ψ ( A ) = lim t → Z M |∇ P t A | dµ. (6) Proof.
First, as shown in [C-M], one have
Per ψ ( A ) = sup (cid:26) Z A (div ϕ − ∇ ψ · ϕ ) dµ, ϕ ∈ C ( M, R ) & k ϕ k ∞ ≤ (cid:27) . Then one have, for all ϕ ∈ C ( M, R ) with k ϕ k ∞ ≤ , by integration by parts, Z A (div ϕ − ∇ ψ · ϕ ) dµ = lim t → Z M P t A (div ϕ − ∇ ψ · ϕ ) dµ = lim t → Z M h∇ P t A , ϕ i dµ ≤ lim t → Z M |∇ P t A | dµ. Conversely, for all ϕ ∈ C ( M, R ) with k ϕ k ∞ ≤ , by integrations by parts, and using the fact that ( P t ) t ≥ is self adjoint, Z M h∇ P t A , ϕ i dµ = Z A P t (div ϕ − ∇ ψ · ϕ ) dµ = Z A (div P t ( ϕ ) − ∇ ψ · P t ( ϕ )) dµ ≤ Per ψ ( A ) , the last inequality following from the fact that k P t ϕ k ∞ ≤ . Since Z M |∇ P t A | dµ = sup ϕ, k ϕ k L ∞ ≤ Z M h∇ P t A , ϕ i dµ = sup ϕ ∈ C ( M, R ) , k ϕ k L ∞ ≤ Z M h∇ P t A , ϕ i dµ, it concludes the proof. 14s an immediate consequence, this definition of boundary is the limit in Bobkov semigroupfunctional, i.e. √ κ Per ψ ( A ) = lim t → Z M r I γ ( P t A ) + 1 κ |∇ P t A | dµ. Now define the isoperimetric problem with respect to this definition of boundary, i.e. define thefollowing isoperimtric profile ˜ I ( M,g,µ ) ( v ) = inf { Per ψ ( A ) , A ⊂ M, µ ( A ) = v } . It is known that in general, for any Borel set A ⊂ M , µ + ( A ) ≥ Per ψ ( A ) so that this isoperimetricproblem is a priori weaker than the one we have formulated. However, with respect to this newnotion of boundary, for any weighted manifold endowed with probability (finite) measure the infimumis a minimum. This fact is not hard to prove, see [R-R]. Since for each v there exists an extremalset of measure v , say A , by the above fact, t P t A has to be constant along Bobkov functional.This in turns implies rigidity by the preceding section.It is well known that when the set A is “regular enough” (say with C smooth boundary ∂A ), itsrelative perimeter agrees with its Minkowski content and both are equal to R ∂A ψ ( x ) d H n − ( x ) . In[C-K], the authors define indeed the Gaussian isoperimetric problem with the function ˜ I ( R n ,γ n ) ( v ) .In this space, the isoperimetric problems with respect to any kind of boundary agree. This propertyhas been used by Carlen and Kerce, although somewhat hidden in [C-K]. This enables them toestablish full characterisation of isoperimetric sets, without any smoothness assumptions.Actually the conclusion remains true in the more general setting of weighted Riemannian mani-fold. That is, in weighted Riemannian manifold there is a minimizer such that Per ψ ( A ) agrees withthe Minkowski content µ + ( A ) and moreover both are equal to R ∂A ψ ( x ) d H n − ( x ) . In fact, thereexists a minimizer A whose boundary ∂A is the union of a regular part ∂ r A and a closed set ofsingularity ∂ s A , empty is the dimension n is less or equal than , and of Hausdorff dimension atmost n − if n ≥ . This follows from the works on geometric measure theory initiated by Almgren,Morgan and others (see [Mi1], section . , for a comprehensive discussion on this topic). We referto [Mi1] for a complete bibliography. As a result, in this setting we have that ˜ I ( M,g,µ ) = I ( M,g,µ ) .The above discussion implies the following fact. Let ( M, g, e − ψ d vol) a weighted Riemmannianmanifold of CD ( κ, ∞ ) class, that is Ric g + ∇ ψ ≥ κ . Assume that there is equality in one point v ∈ (0 , in Bakry–Ledoux comparison theorem. Then there is a set A ⊂ M of measure v achievingequality in Bobkov inequality. The previous sections imply therefore a splitting theorem. Moreover,in the Gaussian direction, Φ − ◦ Q t A is a linear function. Following [C-K], this implies that A mustbe an half space. In order to be self contained, let us recall their proof that also appears in [M-N]. Lemma 6.4.
Let A be a subset of R such that Φ − ◦ Q t A is a linear function. Then A is anhalf-space.Proof. Let f t to be Q t A . Then by hypothesis f t = Φ( h a t , x + b t i ) . Now let H = { x ∈ R n , h a, x i + b ≥ } with a ∈ S n − , b ∈ R such that γ ( H ) = γ ( A ) . Define k t = √ − e − t . Then a fairly easy calculationshows that Q t H = Φ( h k t a, x + b i ) . If k t > | a t | , we can find a s > such that k t + s = | a t | andthen there is an half space H such that Q t + s H = Q t A so that Q s H = A . This is impossiblesince Q t f is smooth for every t > . Therefore k t ≤ | a t | and for the same smoothness assumptionsnecessary k t = | a t | . Then A = H since the semigroup ( Q t ) t ≥ is one-to-one.Summarizing, we have the following theorem.15 heorem 6.5. Let ( M, g, µ ) a weighted Riemmannian manifold of CD ( κ, ∞ ) class. If there existsa v ∈ (0 , such that I ( M,µ ) ( v ) = I ( R ,γ κ ) ( v ) then ( M, g, µ ) = ( R , | · | , γ κ ) × ( M ′ , g ′ , µ ′ ) . Moreover theisoperimetric minimizers are of the form ( −∞ , Φ − κ ( v )) × M ′ . This theorem is not new : indeed, it has been already proved by Franck Morgan [Mor]) usingmore geometrical arguments. This proof only use semigroup arguments.Notice also that in the Gaussian space, the above lemma proves in the same manner that half-spaces are the only minimizers of the reverse Bobkov inequality (5). However, by integration byparts, (5) expresses that halfspaces maximize the Euclidean norm of barycenters of sets of givenmeasure. This fact is far more general and rather easy to prove (see [B-C-F]).
One can notice than in each of the proofs of Section 4, arises the common non negative factor Γ − κ Γ . Thus, in view of studying robustness of geometric or functional inequalities, it would beworth to give a lower bound of the quantity R E (Γ − κ Γ)( f ) dµ in terms of a distance to extremalfunctions.In this section, we derive a second order Poincaré type inequality for the spheres and a subclassesof log-concave measures. For sake of simplicity, we consider CD (1 , ∞ ) spaces although it works moregenerally for CD ( κ, ∞ ) , κ > .We have seen in Section 5 that the equality Γ ( f ) = Γ( f ) imply Lf + f = 0 . With little additionaleffort, we can give a lower bound on R E (Γ − Γ)( f ) dµ in term of the L distance of the operator L + Id . Indeed, let ( E, µ, Γ) be a CD (1 , ∞ ) space. Fix a centered function f on the Dirichletdomain. By the condition Γ ≥ Γ and the spectral gap inequality, we get Z E ( Lf ) dµ = Z E Γ ( f ) dµ ≥ Z E Γ( f ) dµ ≥ Z E f dµ. Thus Z E (Γ − Γ)( f ) dµ = Z E ( Lf ) dµ − Z E f ( − Lf ) dµ ≥ (cid:18) Z E ( Lf ) dµ − Z E f ( − Lf ) + Z E f dµ (cid:19) = 12 k f + Lf k L ( µ ) . As a standard example, let ( E, µ, Γ) be the Gaussian space. Then, the eigenvectors of theoperator − L are given by the Hermite polynomials h α associated to the eigenvalues | α | = α + · · · + α n .If f k = P α, | α | = k h f, h α i h α designs the chaos of order k , k ∈ N , we have that k f + Lf k L ( µ ) = X k ≥ ( k − f k ≥ X k ≥ f k = k f − Π f k , with Π being the projection on linear functions. Since in the Gaussian space, Γ − Γ is simply theHilbert-Schmidt norm of the Hessian, this implies in particular a second order Poincaré inequality.This inequality appears in a sharp form in [M-N] (i.e. without the factor ) and is used by theauthors to prove robust dimension free Gaussian isoperimetry.16ecent results ([CE], [Mi2]) allow us to extend this second order Poincaré inequality for moregeneral log-concave measures with Hessian bounded from below and above i.e. ≤ ∇ V ≤ K .Indeed following E. Milman’s result, the spectrum ( λ k ) k ≥ of the operator L = ∆ − ∇ V · ∇ satisfies λ k ≥ k for all k ∈ N . As a result, ( λ k − ≥ λ k for all k ≥ and so we have k f + Lf k = X k ≥ ( λ k − f k ≥ X k ≥ λ k f k = 12 (cid:18)Z R n |∇ f | dµ − Var µ ( f ) (cid:19) . The right hand-side is simply the spectral gap deficit, say δ SG . As a consequence of a more generalresult concerning variance Brascamp–Lieb inequality of Cordero-Erausquin [CE], there exists C = C ( K ) > v ∈ R n and t ∈ R such that δ SG ≥ C k f − ∇ V · v + t k and so Z R n ( k∇ f k HS + ( ∇ V − Id )( ∇ f, ∇ f )) e − V ( x ) dx = Z R n (Γ − Γ)( f ) dµ ≥ C k f − ∇ V · v + t k . Here t = − R R n f dµ = − E µ ( f ) and v = R R n x ( f ( x ) − E µ ( f )) dµ ( x ) . This can be seen as a generalizedsecond order Poincaré inequality.Other interesting instances are given by the Euclidean spheres ( S n , g ) satisfying the CD (1 , n ) condition (that is of radius √ n − with this normalization) with the associated Laplace–Beltramioperator ∆ . The eigenvalues of − ∆ are given by λ k = k ( n + k − / ( n − and it is clear thatwhenever k ≥ , ( λ k − ≥ . Thus Z S n (Γ − Γ)( f ) dµ = Z S n k∇ f k HS dµ ≥ k f + ∆ f k ≥ X k ≥ f k . Besides, it is well known that the corresponding eigenvectors ( Z k ) | k |≥ are given by the sphericalharmonics (see e.g. [M-W]). The eigenvectors associated to λ are simply given by the coordinatesfunctions. As a consequence, similarly as in the Gaussian space, the right hand side corresponds to k f − Π f k where Π f is the projection of f on linear functions.This extention of the Gaussian case to more general log-concave measures and the Euclideanspheres is important toward the generalization of the main result of Mossel and Neeman [M-N] aboutquantitative dimension free isoperimetry for the Gaussian measure. The next section is devoted tothis problem. Mossel and Neeman’s scheme of proof is directly inspired by the heat-flow proof of Bobkov’s inequal-ity of Bakry and Ledoux. In Section 4, we have build the first step in expressing the derivative ofthe Bobkov functionnal for general spaces satisfying the CD (1 , ∞ ) condition in the same manneras in the Gaussian space. One can therefore ask if is it possible to push the arguments of [M-N]to more general CD (1 , ∞ ) -spaces. On those spaces, cases of near equality have to imply that bothnear extremal sets are close to half spaces and that the subsequent space is close to the Gaussianspace in an appropriate sense. There is some vagueness in this affirmation because it seems thatthere is no canonical way to define a good notion of closeness with respect to the Gaussian space.In the case of log-concave probability measures, the isoperimetric deficit on the real line (whereby the Bobkov’s result extremals sets are half lines) appears in a sharp form in the work of de Castro17DC] for more general measures than the ones satisfying the CD (1 , ∞ ) condition. On R n ( n > )however, for other log-concave measures than the Gaussian, dimension free estimates are left open.As we saw in the preceding sections, when equality holds in the Bobkov inequality, we have that forevery t > , (Γ − Γ)(Φ − ◦ P t A ) = 0 so that it forces A to be an half space. Besides, this impliesthat µ must have a one dimensional Gaussian marginal. It is a very natural guess that in the abovesetting, a subset A ⊂ R n such that µ + ( A ) = I γ ( µ ( A )) + δ must be “close” to an half space when δ is (very) small.An other interesting instance is given by the n dimensional Euclidean spheres of radius √ n − - denoted by S n - equipped with the uniform measure for large values of n . Indeed, although thereis no equality cases in the Bobkov inequality, the Poincaré lemma indicates that these spaces areapproximately Gaussian. Moreover, the isoperimetric sets are spherical caps and therefore of thesame shape as the isoperimetric sets for the Gaussian space.It is a consequence of [Bar], Proposition 11, that, point-wise, I nn − { S n ,g ,µ } ≤ I γ . Therefore we getthe estimates valid for all v ∈ (0 , , I nn − { S n ,g ,µ } ( v ) ≤ I γ ( v ) ≤ I { S n ,g ,µ } ( v ) , and since for all t ∈ (0 , , t − t nn − = O ( n − ) , it implies that for all v ∈ (0 , , I { S n ,g ,µ } ( v ) − I γ ( v ) = O ( n − ) . Toward robust isoperimetry, this is a good insight that one can consider it on the spheres S n forsome large n .It is well known that for small sets, the spherical isoperimetric profile is equivalent to the Eu-clidean one, i.e. to the map x x ( n − /n (up to some constant a n ), whereas the Gaussian isoperi-metric profile behaves as the map x x √− x . This indicates that the Bobkov inequality isa pretty bad approximation for the spherical isoperimetry for small (or large) sets. The optimalBobkov inequality over S n , as recalled in Section 3 (and after rescaling), is I γ (cid:18) Z S n f dµ (cid:19) ≤ Z S n s I γ ( f ) + n − c n |∇ f | dµ, where c n = √ Γ( n +12 )Γ( n ) . Using Stirling’s formula, it is easily seen that n − c n = 1 − (2 n ) − + o ( n − ) ,or equivalently I γ (1 /
2) = (1 − (2 n ) − + o ( n − )) I { S n ,g ,µ } (1 / . Therefore, the preceding estimate I { S n ,g ,µ } ( v ) − I γ ( v ) = O ( n − ) is tight.In what follow, let ( E, µ, Γ) be a CD (1 , ∞ ) space, and let ( P t ) t ≥ be its underlying semigroup. Hypothesis 8.1.
Assume that there exists some positive constants
C, η, t and ε ∈ (0 , / suchthat, for all t ∈ (0 , t ) , ε ∈ (0 , ε ) , the following upper-bound holds : Z { P t f ≤ ε } (Γ − Γ)(Φ − ( P t f )) dµ ≤ Ct η Z { P t f ≤ ε } (Φ − ( P t f )) dµ. This inequality may appear weird, but this is a kind of reverse isoperimetric inequality of secondorder. On the Gaussian space the point-wise estimate (Γ − Γ)(Φ − ( P t f )) ≤ C (Φ − ( P t f )) t η
18s satisfied with exponent η = 2 on the set { P t f ≤ / } . Two different proofs has been given byMossel and Neeman. Either directly by using the expression of the Ornstein-Uhlenbeck kernel (see[M-N]), either by pushing to the second order the arguments of the proof that amounts to the reverseSobolev logarithmic inequality (see [Nee]). However in the first case computations are specifics tothe Ornstein-Uhlenbeck semigroup. In the second case, one need to use an exact commutationbetween P t and ∇ , which is no longer true for any other CD (1 , ∞ ) space than the Gaussian space.Concerning the log-concave setting ( R n , µ ) , dµ ( x ) = e − V ( x ) dx , ≤ ∇ V ≤ K , even under additionalrestrictions (such as Γ ≥ ), it seems that we can reach a similar bound but crucially C dependson the dimension n .We will make use to the fact that t goes to , proving a similar point-wise upper bound - andtherefore the validity of the hypothesis - for some t and η = 4 in the case of the Euclidean sphere S n endowed with uniform measure µ . For that we will use short-times estimates on the spacesderivatives of the heat-kernel (see [M-S], [Eng]). Such bounds are not yet proven in the non compactcase. For sake of clarity we postpone the proof of this technical lemma to the appendix.We now state the main result of this section. Theorem 8.2.
Let ( S n , µ ) with µ the uniform probability measure. For each measurable subset A of S n such that µ + ( A ) = I γ ( µ ( A )) + δ, there exists a spherical cap H and a positive constant c ∈ (0 . , / such that µ ( A ∆ H ) ≤ O ( | log δ | − c ) . Recall that I γ and I { S n ,g ,µ } differs only by O ( n − ) . Therefore Theorem 8.2 implies to followingfollowing corollary for the deficit on the isoperimetric problem over the Euclidean spheres. Corollary 8.3.
Let ( S n , µ ) with µ the uniform probability measure. Then for all measurable subset A of S n such that µ + ( A ) − I ( S n ,g ,µ ) ( µ ( A )) = δ, there exists a spherical cap H and a positive constant c ∈ (0 . , / such that µ ( A ∆ H ) ≤ O ( | log δ n | − c ) , where δ n = max( δ, n − ) . The proof of this theorem is directly inspired from the work of [M-N]. We will use the generalnotation ( E, µ, Γ) , keeping in mind that most of the arguments are valid for general spaces. Specificbounds to the Euclidean spheres will be explicited in the process of the proof. Throughout the proof, c, C will denote numerical constants ( a priori explicits) with c ∈ (0 , and C ≥ which may changefrom a line to another. Proof of Theorem 8.2.
The starting point is the semigroup proof of [B-L1] of the functional versionof the Bobkov inequality. We recall that if f is the characteristic function of a measurable set A ,then δ = Ψ(0) − lim t →∞ Ψ( t ) = R ∞ − Ψ ′ ( s ) ds , where Ψ : t ∈ [0 , ∞ ) R E q I γ ( P t f ) + Γ( P t f ) dµ. Moreover, by the remark of Section 4, for all t > , − Ψ ′ ( t ) ≥ Z E I γ ( P t f )(Γ − Γ)(Φ − ◦ P t f )(1 + Γ(Φ − ◦ P t f )) / dµ,
19o that δ ≥ µ + ( A ) − I γ ( µ ( A )) ≥ Z ∞ Z E I γ ( P t A )(Γ − Γ)(Φ − ◦ P t A )(1 + Γ(Φ − ◦ P t A )) / dµ dt. (7)Recall that, since ( P t ) t ≥ is mass preserving, and since the associated kernel p t is a positive function,for all t > , P t A is a smooth function that takes value in (0 , . Let now a smooth function f : E → (0 , (that will be P s A for some positive s ) and δ to be the deficit in the Bobkovinequality associated to f .In the preceding section, we gave a lower bound of R E (Γ − Γ)( f ) dµ in terms of projection overlinear functions in the case of Euclidean spheres S n or function of the type x
7→ h a, ∇ V ( x ) + b i forlog-concave measures on R n . In order to make use of this lower bound, the first task is to removethe quotient in the integrand of (7). For that, we make use of the reverse isoperimetric inequalitygiven by Proposition 4.6. Recall that for all function f : E → [0 , and for all t > , C ( t )Γ(Φ − ◦ P t f )) ≤ , where C ( t ) = 2 R t e s ds = e t − . Therefore, this implies (1 + Γ(Φ − ◦ P t f )) − / ≥ (1 + 1 /C ( t )) − / and since (1 + 1 /C ( t )) − / ∼ (2 t ) − / as t goes to , for all u > and for some numerical constant c , δ ≥ cu − / Z uu Z E I γ ( P t f )(Γ − Γ)(Φ − ◦ P t f ) dµ dt. As a consequence, by the mean value theorem, there exists a t ∈ ( u, u ) such that δ ≥ cu / Z E I γ ( P t f )(Γ − Γ)(Φ − ◦ P t f ) dµ ≥ − / ct / Z E I γ ( P t f )(Γ − Γ)(Φ − ◦ P t f ) dµ. (8)All the remaining work is devoted to take account of the extra term I γ ( P t f ) (which is small whenever P t f is close to and ).To get rid of this extra term, Mossel and Neeman use the reverse Hölder’s inequality. After somework, this implies that Φ − ( P t f ) is close to a linear function for some t ∈ ( c, c + 1) when c > is universal. In order to conclude, the authors need a “time reversal” argument. For that they usespectral estimates that seem to be rather specific to the Gaussian space - even though it seems thatit should work on the spheres. Our approach avoids this use by choosing an appropriate small t .Then the “time reversal” argument follows easily. Indeed, since we are close to achieve equality, k p Γ( f ) k is finite and so the reverse Poincaré inequality ensures that f and P t f are close in L distance. Besides, Φ is a Lipschitz map, so that Φ − ( P t f ) and Φ − ( f ) are as least that close. Moreprecisely, our proof continues as follows.Since x
7→ I γ ( x ) is increasing on (0 , / and symmetric with respect to / on (0 , , for ε < / we get that Z E I γ ( P t f )(Γ − Γ)(Φ − ◦ P t f ) dµ ≥ Z { ε ≤ P t f ≤ − ε } I γ ( P t f )(Γ − Γ)(Φ − ◦ P t f ) dµ ≥ I γ ( ε ) Z { ε ≤ P t f ≤ − ε } (Γ − Γ)(Φ − ◦ P t f ) dµ. Recalling (8), we deduce that δ ≥ ct / I γ ( ε ) Z { ε ≤ P t f ≤ − ε } (Γ − Γ)(Φ − ◦ P t f ) dµ. (9)20oreover, denoting h t = Φ − ◦ P t f , Z { ε ≤ P t f ≤ − ε } (Γ − Γ)( h t ) dµ = Z E (Γ − Γ)( h t ) dµ − Z { P t f ≤ ε } (Γ − Γ)( h t ) dµ − Z { P t f ≥ − ε } (Γ − Γ)( h t ) dµ. (10)The first term is precisely what we are looking for, as we proved a lower bound of it in previoussection. We therefore need to show that, for an appropriate choice of ε , we do not lose too muchwhile subtracting the integral on the sets { P t f ≤ ε } and { P t f ≥ − ε } . That is why we made thehypothesis ( H ) . Recall that in the case of Euclidean spheres, ( H ) is satisfied and it holds, for some t ∈ (0 , , ∀ ε ≤ / , ∀ t ≤ t , Z { P t f ≤ ε } (Γ − Γ)( h t ) dµ ≤ Ct Z { P t f ≤ ε } h t dµ, so that by symmetry ∀ ε ≤ / , ∀ t ≤ t , Z { P t f ≥ − ε } (Γ − Γ)( h t ) dµ ≤ Ct Z { P t f ≥ − ε } (Φ − (1 − P t f )) dµ. Both integrals are bounded in the same manner. We deal with the first one. Recall that as aconsequence of the reverse isoperimetric inequality h t is (2 t ) − / -Lipschitz and so by classical con-centration for Lipschitz maps, µ ( {− h t ≥ u } ) ≤ e − tu . Since Z { P t f ≤ ε } h t dµ = Z {− h t ≥− Φ − ( ε ) } h t dµ, by the layer-cake representation, Z {− h t ≥− Φ − ( ε ) } h t dµ = 2 Z ∞− Φ − ( ε ) uµ ( {− h t ≥ u } ) du ≤ Z ∞− Φ − ( ε ) ue − tu du = e − t (Φ − ( ε )) t . Thus Z { P t f ≤ ε } (Γ − Γ)( h t ) dµ + Z { P t f ≥ − ε } (Γ − Γ)( h t ) dµ ≤ Ct e − t (Φ − ( ε )) . (11)Therefore, using (9), (10), and (11), we get δ ≥ ct / I γ ( ε ) (cid:18) Z E (Γ − Γ)( h t ) dµ − t e − t (Φ − ( ε )) (cid:19) , so that Z E (Γ − Γ)( h t ) dµ ≤ C (cid:18) δt / I γ ( ε ) + 1 t e − t (Φ − ( ε )) (cid:19) . Besides, recall that the Poincaré inequality of second order for ( E, µ,
Γ) = ( S n , µ ) implies that k h t − Π h t k ≤ Z E (Γ − Γ)( h t ) dµ, where Π is the projection on linear functions. In this case, the two preceding bounds imply then k h t − Π h t k ≤ C (cid:18) δ I γ ( ε ) t / + 1 t e − t (Φ − ( ε )) (cid:19) . (12)21ow, we recall that as a consequence of the reverse Poincaré inequality, (4) holds, i.e. k f − P t f k ≤ √ t k p Γ( f ) k . Since we are close to a case of equality, we can make the assumption that k p Γ( f ) k ≤ I γ ( E f ) + δ ≤ otherwise δ is larger than / − (2 π ) − / and the announced claim is still true. This implies that k f − P t f k ≤ √ t/ . Using the triangular inequality, we thus get k f − Φ(Π h t ) k L ( µ ) ≤ √ √ t + k P t f − Φ(Π h t ) k . (13)Besides, since Φ is -Lipschitz, for every norm k · k , it holds k P t f − Φ(Π h t ) k = k Φ( h t ) − Φ(Π h t ) k ≤ k h t − Π h t k , and therefore k P t f − Φ(Π h t ) k ≤ k P t f − Φ(Π h t ) k ≤ k h t − Π h t k . (14)Using (12), (13) and (14) together, we obtain k f − Φ(Π h t ) k ≤ √ √ t + C (cid:18) δt / I γ ( ε ) + 1 t e − t (Φ − ( ε )) (cid:19) / . Recall that ε ∈ (0 , / and t ∈ (0 , t ) are arbitrary. It remains to optimize over t and ε . We usethe well known asymptotic estimates I γ ( x ) ∼ x √− x and I ′ γ ( x ) = Φ − ( x ) ∼ −√− x , when x goes to .Therefore, we can take t = | log( δ ) | − c , for some c ∈ (0 . , / , and ε = δ / . The second termbetween the brackets decays at faster rate in δ than the first one so that it yields the existence of apositive constant δ such that ∀ δ ≤ δ , k f − Φ(Π h t ) k L ( µ ) ≤ | log δ | − c . (15)To recover the statement of Theorem 8.5, we now need to go back to sets. The followingarguments are directly taken from [M-N].First, we apply this result for some smooth approximation f of A . Using strong continuity of ( P t ) t ≥ , we can find a small s > such that for every g ∈ L ( E ) , k A − g k ≤ k P s A − g k + δ. Since P s A is smooth for every s ≥ , we can apply (15) for f = P s A so that ∀ δ ≤ δ , k A − Φ(Π h t ) k ≤ k f − Φ(Π h t ) k + δ ≤ | log δ | − c . In order to conclude, we use the following lemma which is a straightforward generalization of Lemma5.2 from [M-N] - it consists of rounding
Φ(Π h t ) to { , } . Lemma 8.4.
Let A be measurable set and g = Φ(Π h t ) . Let H to be the spherical cap { Π h t ≥ } .Then µ ( A ∆ H ) = Z E | A − H | dµ ≤ Z E | A − g | dµ. δ ≥ δ , µ ( A ∆ H ) ≤ | log δ | c | log δ | − c so that for every δ ≥ , µ ( A ∆ H ) ≤ O ( | log δ | − c ) . Theorem 8.2 is thus established.
Remark : Concerning the log-concave case, we have seen that the second order Poincaré’s inequalityreads as Z R n (Γ − Γ)( f ) dµ ≥ C k f − ∇ V · v + t k , with t = − R R n f dµ = − E f and v = R R n x ( f ( x ) − E f ) dµ ( x ) . If Hypothesis (8.1) holds (with C independent on n ), it would imply by a completely similar proof the following Theorem. Theorem 8.5.
Let ( R n , dµ ( x ) = e − V ( x ) dx ) where µ is such that ≤ ∇ V ≤ K . Let A be a set suchthat µ + ( A ) ≤ I γ ( µ ( A )) + δ. Then there exists a set B with of the form B = { x ∈ R n , ∇ V ( x ) · v + x ≥ } , with some v ∈ R n and x ∈ R , and a positive constant c such that µ ( A ∆ B ) ≤ O ( | log δ | − c ) . In this appendix, we prove the following technical lemma, in the case of Euclidean spheres S n . Lemma 9.1.
There exists some positive constants t ∈ (0 , , C such that, for all < t ≤ t andall ε ∈ (0 , / , Z { P t f ≤ ε } (Γ − Γ)(Φ − ( P t f )) dµ ≤ Ct Z { P t f ≤ ε } (Φ − ( P t f )) dµ. We shall show that the upper bound holds point-wise on the set { P t f ≤ ε } for sufficiently small t and ε . We will write as a short-hand f t for P t f and h t for Φ − ( P t f ) and stick to abstract notations ( E, µ, Γ) since most of the arguments are valids in general. Proof of Lemma 9.1.
Recall that for Euclidean Spheres, (Γ − Γ)( h t ) = k∇ h t k HS . (16)First, we expand the hessian thanks to the chain rule formula. k∇ h t k HS = (cid:13)(cid:13)(cid:13)(cid:13) ∇ f t I γ ( f t ) + I ′ γ ( f t ) ∇ f t T ∇ f t I γ ( f t ) (cid:13)(cid:13)(cid:13)(cid:13) HS = (cid:13)(cid:13)(cid:13)(cid:13) ∇ f t I γ ( f t ) + I ′ γ ( f t ) ∇ h t T ∇ h t (cid:13)(cid:13)(cid:13)(cid:13) HS . ( a + b ) ≤ a + b ) , we get k∇ h t k HS ≤ k∇ f t k HS I γ ( f t ) + 2( I ′ γ ( f t )) k∇ h t T ∇ h t k HS . (17)We will bound the two terms separately.Recall that the reverse isoperimetric inequality reads as C ( t )Γ( P t f ) ≤ I γ ( P t f ) , or equivalently bythe chain rule formula, C ( t )Γ( h t ) ≤ . (18)Since t is small, we can simply take C ( t ) = 2 t as C ( t ) ∼ t when t goes to 0. Thus, (18) impliesthat ( I ′ γ ( f t )) k∇ h t T ∇ h t k HS = ( I ′ γ ( f t )) (Γ( h t )) ≤ ( I ′ γ ( f t )) t = h t t . (19)Recalling (16) and (17), by (19), in view of Lemma 9.1, we will prove ∃ t ∈ (0 , , ∀ t ∈ (0 , t ) , k∇ f t k HS I γ ( f t ) ≤ Ch t t on the set { f t ≤ / } .We therefore proceed to show that actually a stronger estimates holds. It is known indeed thatthe following estimates on the derivatives of the heat kernel p t ( x, y ) associated to the heat semi-grouphold - these estimates are valid for compact Riemannian manifolds with Ricci curvature boundedfrom below (see e.g [M-S], [Eng]). Lemma 9.2.
There exists a positive constant C such that ∀ t ≤ , |∇ log p t ( x, y ) | ≤ C (1 + d ( x, y )) t , k∇ log p t ( x, y ) k HS ≤ C (1 + d ( x, y )) t . (20)Such bounds in ( R n , µ ) satisfying CD ( κ, ∞ ) are not proven yet in general (see [Li] for relatedwork). Actually, as stated in [M-S], [Eng], C depends on n . However, looking closely at the proofs,the dependance on n for general compact Riemannian manifolds, comes from the lower and upperbounds on the heat kernel of the form c q µ ( B ( x, √ t )) µ ( B ( y, √ t )) e − C d ( x,y )2 t ≤ p t ( x, y ) ≤ C q µ ( B ( x, √ t )) µ ( B ( y, √ t )) e − c d ( x,y )2 t , (21)where B ( · , r ) designs the geodesic ball centered in · of radius r . In general, if the curvature is simplybounded from below, µ ( B ( · , √ t )) is estimated as approximately t n/ as in the Euclidean space. Yet,in the cases of S n , the uniform distribution µ is approximately the Gaussian measure γ (by thePoincaré lemma). More precisely, by rotational invariance of the measure µ and the Pythagoreantheorem, ∀ θ ∈ S n , µ ( B ( θ, √ t )) = vol n − S n − vol n S n ( n − n/ Z √ t ( n − − u ) n − du q − u n − = 1 √ π c n √ n − Z √ t (cid:18) − u n − (cid:19) n − du.
24t is not hard to check that, uniformly on u > , √ π (cid:12)(cid:12)(cid:12)(cid:12) c n √ n − (cid:18) − u n − (cid:19) n − − e − u / (cid:12)(cid:12)(cid:12)(cid:12) ≤ O ( n − ) so that by integration | µ ( B ( · , √ t )) − γ ([0 , √ t ]) | = O ( √ tn − ) . Thus the fractions in the estimatesof (21) can be lower or upper bounded by dimension free ones. As a result (see [Eng] for the detailsof how to reach from (21) the inequalities of Lemma 9.2), the dependance on the dimension in C can be remove. Expanding the derivatives, Lemma 9.2 can be restated as X ≤ i,j ≤ n (cid:18) ∂ x i ,x j p t ( x, y ) p t ( x, y ) − ∂ x i p t ( x, y ) ∂ x j p t ( x, y ) p t ( x, y ) (cid:19) ≤ C (1 + d ( x, y )) t and X ≤ i,j ≤ n (cid:18) ∂ x i p t ( x, y ) ∂ x j p t ( x, y ) p t ( x, y ) (cid:19) = (cid:18) n X i =1 ( ∂ x i p t ( x, y )) p t ( x, y ) (cid:19) ≤ (cid:18) C (1 + d ( x, y )) t (cid:19) . These two bounds imply, using the inequality A ≤ A − B ) + B ] , X ≤ i,j ≤ n (cid:18) ∂ x i ,x j p t ( x, y ) p t ( x, y ) (cid:19) ≤ C (1 + d ( x, y ) + d ( x, y )) t . (22)Denoting d ˜ µ ( y ) = f ( y ) p t ( x,y ) dµ ( y ) R E f ( y ) p t ( x,y ) dµ = f ( y ) p t ( x,y ) dµ ( y ) f t ( x ) , by Jensen’s inequality, we have that f t ( x ) X ≤ i,j ≤ n (cid:18) Z E f ( y ) ∂ x i ,x j p t ( x, y ) dµ (cid:19) = X ≤ i,j ≤ n (cid:18) Z E ∂ x i ,x j p t ( x, y ) p t ( x, y ) d ˜ µ (cid:19) ≤ X ≤ i,j ≤ n Z E (cid:18) ∂ x i ,x j p t ( x, y ) p t ( x, y ) (cid:19) d ˜ µ. Besides, (22) implies that X ≤ i,j ≤ n Z E (cid:18) ∂ x i ,x j p t ( x, y ) p t ( x, y ) (cid:19) d ˜ µ ≤ Ct (cid:18) Z E d ( x, y ) (1 + d ( x, y ) ) d ˜ µ (cid:19) . Moreover, Z E d ( x, y ) (1 + d ( x, y ) ) d ˜ µ = 1 f t Z E f ( y ) d ( x, y ) (1 + d ( x, y ) ) p t ( x, y ) dµ ( y ) , and since when t goes to , p t ( x, y ) is approaching the Dirac mass δ x ( y ) , this integral goes to as t goes to . Therefore, there exists T > such that for all t ≤ T , R E d ( x, y ) (1 + d ( x, y ) ) d ˜ µ ≤ ,and therefore for all t ≤ t = min(1 , T ) and all x ∈ S n , f t ( x ) X ≤ i,j ≤ n (cid:18) Z E f ( y ) ∂ x i ,x j p t ( x, y ) dµ (cid:19) ≤ Ct , Notice that in the log-concave case, even if an analogous statement of Lemma 9.2 would be established, it wouldnot be possible to remove the dependance on the dimension. Indeed, in the simplest case of the Ornstein–Uhlenbeckkernel, it is immediately checked by the Mehler’s representation formula that these constants are of the form Cn ,where n is the dimension. k∇ f t k HS ≤ Cf t t . Then the point-wise estimates holds ∀ t ∈ (0 , t ) , k∇ f t k HS I γ ( f t ) ≤ Cf t t I γ ( f t ) . Notice that, on the set { f t ≤ ε } , f t I γ ( f t ) goes to as ε goes to so that it actually holds a (much)better estimate than the one needed for Lemma 9.1. Anyway, since h t ≥ as soon as P t f ≤ Φ( − and since Φ( − ≥ / , it proves that for all ε ≤ / , (cid:13)(cid:13)(cid:13)(cid:13) ∇ f t I γ ( f t ) (cid:13)(cid:13)(cid:13)(cid:13) HS ≤ Ch t t . Thus, recalling (17) and (19), the following point-wise upper bound ∀ t ∈ (0 , t ) , ∀ ε ∈ (0 , / , (Γ − Γ)( h t ) ≤ Ch t t holds and after integration it yields Lemma 9.1 for ( S n , µ ) . Acknowledgment. The main part of this work has been completed when I made my Ph.D at theUniversity of Toulouse. I thank my Ph.D advisor Michel Ledoux for many valuable exchanges.
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J. Diff.Geom. 83, no. 2, 337–405 (2009)
Raphaël Bouyrie,
Laboratoire d’Analyse de Mathématiques Appliqués, UMR 8050 du CNRS, Université Paris-Est Marne-la-Vallée, 5 Bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex,France
E-mail address: [email protected]@gmail.com