Lyapunov conditions for logarithmic Sobolev and Super Poincaré inequality
aa r X i v : . [ m a t h . P R ] D ec LYAPUNOV CONDITIONS FOR LOGARITHMIC SOBOLEV ANDSUPER POINCAR´E INEQUALITY
PATRICK CATTIAUX ♠ , ARNAUD GUILLIN ♦ , FENG-YU WANG ∗ ,AND LIMING WU ♥ ♠ Universit´e de Toulouse ♦ Ecole Centrale Marseille and Universit´e de Provence ∗ Swansea University ♥ Universit´e Blaise Pascal and Wuhan University
Abstract.
We show how to use Lyapunov functions to obtain functional inequalities whichare stronger than Poincar´e inequality (for instance logarithmic Sobolev or F -Sobolev). Thecase of Poincar´e and weak Poincar´e inequalities was studied in [2]. This approach allows usto recover and extend in an unified way some known criteria in the euclidean case (Bakry-Emery, Wang, Kusuoka-Stroock ...). Key words :
Ergodic processes, Lyapunov functions, Poincar´e inequalities, Super Poincar´einequalities, logarithmic Sobolev inequalities.
MSC 2000 : 26D10, 47D07, 60G10, 60J60. Introduction.
During the last thirty years, a lot of attention has been devoted to the study of variousfunctional inequalities and among them a lot of efforts were consecrated to the logarithmicSobolev inequality. Our goal here will be to give a new and practical condition to provelogarithmic Sobolev inequality in a general setting. Our method being general, we will beable to get also conditions for Super-Poincar´e, and by incidence to various inequalities as F -Sobolev or general Beckner inequalities. Our assumptions are based mainly on a Lyapunovtype condition as well as a Nash inequality (for example valid in R d ). But let us make precisethe objects and inequalities we are interested in.Let ( X , F , µ ) be a probability space and L a self adjoint operator on L ( µ ), with domain D ( L ), such that P t = e t L is a Markov semigroup. Consider then the Dirichlet form associatedto L E ( f, f ) := h−L f, f i µ f ∈ D ( L ) Date : November 4, 2018. with domain D ( E ). Throughout the paper, all test functions in an inequality will belong to D ( L ). It is well known that L possesses a spectral gap if and only if the following Poincar´einequality holds (for all nice f ’s)(1.1) Var µ ( f ) := Z f dµ − (cid:18)Z f dµ (cid:19) ≤ C P E ( f, f )where C − P is the spectral gap. Note that such an inequality is also equivalent to the expo-nential decay in L ( µ ) of P t .A defective logarithmic Sobolev inequality (say DLSI) is satisfied if for all nice f ’s(1.2) Ent µ ( f ) := Z f log f dµ − Z f dµ log (cid:18)Z f dµ (cid:19) ≤ C LS E ( f, f ) + D LS Z f dµ. When D LS = 0 the inequality is said to be tight or we simply say that a logarithmic Sobolevinequality is verified (for short (LSI)). Dimension free gaussian concentration, hypercon-tractivity and exponential decay of entropy are directly deduced from such an inequalityexplaining the huge interest in it. Note that if a Poincar´e inequality is valid, a defectiveDLSI, via Rothaus’s lemma, can be transformed into a (tight) LSI. For all this we refer to[1] or [29].Recently, Wang [27] introduced so called Super-Poincar´e inequality (say SPI) to study theessential spectrum: there exists a non-increasing β ∈ C (0 , ∞ ), all nice f and all r > µ ( f ) ≤ r E ( f, f ) + β ( r ) µ ( | f | ) . Wang moreover establishes a correspondence between this SPI and defective F -Sobolev in-equality (F-Sob) for a proper choice of increasing F ∈ [0 , ∞ [ with lim ∞ F = ∞ , i.e. for allnice f with µ ( f ) = 1(1.4) µ ( f F ( f )) ≤ c E ( f, f ) + c . More precisely, if (1.4)holds for some increasing function F satisfying lim u → + ∞ F ( u ) = + ∞ and sup (cid:18) u − β ( u ) ut (cid:19) , a (F-Sob) inequality holds with F ( u ) = C u Z u ξ ( t/ dt − C for some well chosen C and C . For details see [29] Theorem 3.3.1 and Theorem 3.3.3. Notethat these results are still available when µ is a non-negative possibly non-bounded measure.In particular an inequality (DLSI) is equivalent to a (SPI) inequality with β ( u ) = ce c ′ /u .These inequalities and their consequences (concentration of measure, isoperimetry, rate ofconvergence to equilibrium) have been studied for diffusions and jump processes by variousauthors [27, 3, 4, 24, 10] under various conditions.In this paper we shall use Lyapunov type conditions. These conditions are well known tofurnish some results on the long time behavior of the laws of Markov processes (see e.g.[15, 17, 14]). The relationship between Lyapunov conditions and functional inequalities of RIFT AND SUPER POINCAR´E 3
Poincar´e type (ordinary or weak Poincar´e introduced in [23]) is studied in details in the recentwork [2]. The present paper is thus a complement of [2] for the study of stronger inequalitiesthan Poincar´e inequality.We will therefore suppose that ( X , d ) is a Polish space (actually a Riemannian manifold).Namely we will assume (L) there is a function W ≥
1, a positive function φ > φ > b > r such that(1.5) L WW ≤ − φ + b B ( o,r ) where B ( o, r ) is a ball, w.r.t. d , with center o and radius r .The main idea of the paper is the following one: in order to get some super Poincar´e inequalityfor µ it is enough that µ satisfies some (SPI) locally and that there exists some Lyapunovfunction. In other words the Lyapunov function is useful to extend (SPI) on (say) balls tothe whole space. General statements are given in section 2.In particular on nice manifolds the riemanian measure satisfies locally some (SPI), so that anabsolutely continuous probability measure will also satisfy a local (SPI) in most cases. Theexistence of a Lyapunov function allows us to get some (SPI) on the whole manifold.The aim of sections 3 and 4 is to show how one can build such Lyapunov functions, eitheras a function of the log-density or as a function of the riemanian distance. In the firstcase we improve upon previous results in [21, 9, 3, 4] among others. In the second casewe (partly) recover and extend some celebrated results: Bakry-Emery criterion for the log-Sobolev inequality, Wang’s result on the converse Herbst argument. In particular we thusobtain similar results as Wang’s one, but for measures satisfying sub-gaussian concentrationphenomenon. This kind of new result can be compared to the recent [5].The main interest of this approach (despite the new results we obtain) is that it providesus with a drastically simple method of proof for many results. The price to pay is that theexplicit constants we obtain are far to be optimal.2. A general result.
Diffusion case.
To simplify we will deal here with the diffusion case: we assume that X = M is a d -dimensional connected complete Riemannian manifold, possibly with boundary ∂M . We denote by dx the Riemannian volume element and ρ ( x ) = ρ ( x, o ) the Riemanniandistance function from a fixed point o . Let L = ∆ − ∇ V. ∇ for some V ∈ W , loc such that Z = R e − V dλ < ∞ , and L is self adjoint in L ( µ ) where dµ = Z − e − V dx . Note that in thiscase, we are in the symmetric diffusion case and the Dirichlet form is given by E ( f, f ) = Z |∇ f | dµ. We shall obtain (SPI) by perturbing a known super Poincar´e inequality.
Theorem 2.1.
Suppose that the Lyapunov condition ( L ) is verified for some function φ suchthat φ ( x ) → ∞ as ρ ( x, o ) → ∞ . Assume also that there exists T locally Lipschitz continuouson M such that dλ = exp( − T ) dx satisfies a (SPI) (1.3) with function β .Then (SPI) holds for µ and some α : (0 , ∞ ) → (0 , ∞ ) in place of β . P. CATTIAUX, A. GUILLIN, F.Y. WANG, AND L. WU
More precisely, for a family of compact sets { A r ⊃ B ( o, r ) } r ≥ such that A r ↑ M as r ↑ ∞ ,define for r > : Φ( r ) := inf A cr φ, Φ − ( r ) := inf { s ≥ s ) ≥ r } ,g ( r ) := sup ρ ( · ,A r ) ≤ | V − T | , G ( r ) := sup ρ ( · ,A r ) ≤ |∇ ( V − T ) | , H ( r ) = Osc ρ ( · , A r ) ≤ (V − T) . Then we may choose for s > , either (1) α ( s ) := inf ε ∈ (0 , ( ε β εs ∧ ε ∧ − ε ) G ◦ Φ − ( bε ∨ sε ) ! exp (cid:18) g ◦ Φ − ( 4 bε ∨ sε ) (cid:19)) , or (2) α ( s ) := 2 exp h H (cid:16) r ∨ Φ − (cid:0) s ∨ bs (cid:1)(cid:17)i β (cid:0) s e − H ◦ Φ − ( s ∨ bs ) (cid:1) . Proof.
For r > r it holds Z f dµ = Z A cr f dµ + Z A r f dµ = Z A cr f φφ dµ + Z A r f dµ ≤ r ) Z f φdµ + Z A r f dµ ≤ r ) Z f (cid:18) −L WW (cid:19) dµ + (cid:18) b Φ( r ) + 1 (cid:19) Z A r f dµ using our assumption ( L ).The proof turns then to the estimation of the two terms in the right hand side of the latterinequality, a global term and a local one. For the first term remark, by our assumption on L that Z f (cid:18) −L WW (cid:19) dµ = Z ∇ (cid:18) f W (cid:19) . ∇ W dµ = 2 Z fW ∇ f. ∇ W dµ − Z f |∇ W | W dµ ≤ Z |∇ f | dµ − Z (cid:12)(cid:12)(cid:12)(cid:12) ∇ f − fW ∇ W (cid:12)(cid:12)(cid:12)(cid:12) dµ which leads to(2.2) Z f (cid:18) −L WW (cid:19) dµ ≤ Z |∇ f | dµ. For the local term we will localize the (SPI) for the measure λ . To this end, let ψ be aLipschitz function defined on M such that 1I A r ≤ ψ ( u ) ≤ ρ ( .,A r ) ≤ and |∇ ψ | ≤
1. Writing
RIFT AND SUPER POINCAR´E 5 (SPI) for the function f ψ we get that for all s > Z A r f dλ ≤ Z f ψ dλ (2.3) ≤ s Z |∇ f | ρ ( .,A r ) ≤ dλ + 2 s Z f ρ ( .,A r ) ≤ dλ + β ( s ) (cid:18)Z | f | ρ ( .,A r ) ≤ dλ (cid:19) . To deduce a similar local inequality for µ we have two methods. For the first one we applythis inequality to f e − V/ T/ . It yields Z A r f dµ = Z A r f e − V + T dλ ≤ s Z |∇ f | ρ ( .,A r ) ≤ dµ + s Z f |∇ ( V − T ) | ρ ( .,A r ) ≤ dµ +2 s Z f ρ ( .,A r ) ≤ dµ + β ( s ) (cid:18)Z | f | e ( V − T ) / ρ ( .,A r ) ≤ dµ (cid:19) so that if we choose s small enough so that sG ( r ) ≤ − ε ), we get Z A r f dµ ≤ Z s |∇ f | dµ + (1 − ε ) Z f dµ + 2 s Z f dµ (2.4) + β ( s ) exp ( g ( r )) (cid:18)Z | f | dµ (cid:19) . Now combine (2.2) and (2.4). On the left hand side we get (cid:18) − (cid:18) b Φ( r ) + 1 (cid:19) (1 − ε + 2 s ) (cid:19) Z f dµ . For the coefficient to be larger than ε/ s ≤ ε/
16 and Φ( r ) ≥ b/ε . Assumingthis in addition to sG ( r ) ≤ − ε ) we obtain that for such s > r , µ ( f ) ≤ ε (cid:18) r ) + 5 s/ (cid:19) µ ( |∇ f | ) + 52 ε β ( s ) exp ( g ( r )) µ ( | f | ) . If t is given, it remains to choose first r = Φ − (cid:16) bε ∨ εt (cid:17) , and then s = εt ∧ ε ∧ − ε ) G ( r ) , to get the first α ( t ).The second method is more naive but do not introduce any condition on the gradient of V . P. CATTIAUX, A. GUILLIN, F.Y. WANG, AND L. WU
Start with Z f A r dµ = Z f e − V + T A r dλ ≤ e − inf Ar ( V − T ) Z f A r dλ ≤ e − inf Ar ( V − T ) (cid:16) s Z |∇ f | ρ ( .,A r ) ≤ dλ ++2 s Z f ρ ( .,A r ) ≤ dλ + β ( s ) (cid:18)Z | f | ρ ( .,A r ) ≤ dλ (cid:19) (cid:17) ≤ e − inf Ar ( V − T ) e sup ρ ( .,Ar ) ≤ ( V − T ) (cid:16) s Z |∇ f | ρ ( .,A r ) ≤ dµ ++2 s Z f ρ ( .,A r ) ≤ dµ + β ( s ) e sup ρ ( .,Ar ) ≤ ( V − T ) (cid:18)Z | f | ρ ( .,A r ) ≤ dµ (cid:19) (cid:17) ≤ e Osc ρ ( .,Ar ) ≤ ( V − T ) (cid:18) s Z |∇ f | dµ + 2 s Z f dµ (cid:19) ++ e Osc ρ ( .,Ar ) ≤ ( V − T ) β ( s ) (cid:18)Z | f | dµ (cid:19) . If we combine the latter inequality with (2.2) and denote s ′ = 2 s e Osc ρ ( .,Ar ) ≤ ( V − T ) we obtain (cid:18) − bs ′ Φ( r ) (cid:19) Z f dµ ≤≤ (cid:18) s ′ + 1Φ( r ) (cid:19) Z |∇ f | dµ + e Osc ρ ( .,Ar ) ≤ ( V − T ) β ( s ′ e − Osc ρ ( .,Ar ) ≤ ( V − T ) / (cid:18)Z | f | dµ (cid:19) . Hence, if we choose , r = Φ − ( s ∨ bs ) and s ′ = s/ α ( s ). (cid:3) Remark 2.5. (1) The previous proof extends immediately to the general case of a “diffusion”process with a “carr´e du champ” which is a derivation, i.e. if E ( f, f ) = R Γ( f, f ) dµ fora symmetric Γ such that Γ( f g, h ) = f Γ( g, h ) + g Γ( f, h ) (see [2] for more details on thisframework).(2) For a general diffusion process, say with a non constant diffusion term, as noted in theprevious remark we have to modify the energy term so that it is no further difficulty andthere are numerous examples where condition (L) is verified, i.e. consider L = a ( x )∆ − x. ∇ where a is uniformly elliptic and bounded (consider W = e a | x | so φ ( x ) = c | x | ). But ourmethod as expressed here relies crucially on the explicit knowledge of V . Note however, thatfor the second approach, only an upper bound on the behavior of V over, say, balls is needed,which can be made explicit in some cases. ♦ Remark 2.6.
We may for instance choose A r = ¯ V r := { x ; | V − T | ( x ) < r } (i.e. a level setof | V − T | ) provided | V − T | ( x ) → + ∞ as ρ ( o, x ) → + ∞ . However we have to look at anenlargement ¯ V r +2 = { x ; , ρ ( x, ¯ V r ) < } (not the level set of level r + 2).If we want to replace ¯ V r +2 by the level set ¯ V r +2 we have to modify the proof, choosing somead-hoc function ψ which is no more 1-Lipschitz. It is not difficult to see that we have to RIFT AND SUPER POINCAR´E 7 modify (2.3) and what follows, replacing 1 (the 1 of 1-Lipschitz) by sup ¯ V r +2 |∇ ( V − T ) | . Sowe have to modify the condition on s in (1) of the previous theorem, i.e.(2.7) 2inf ( ¯ V r ) c φ ≤ ε s ≤ ( ¯ V r ) c φ + 2 (1 − ε )sup ¯ V r +2 |∇ ( V − T ) | , i.e. we get the same result as (1) but with Φ( r ) = inf ( ¯ V r ) c φ , g ( r ) = r + 2 and G ( r ) =sup ¯ V r +2 |∇ ( V − T ) | .The second case (2) cannot (easily) be extended in this direction. ♦ Actually one can derive a lot of results following the lines of the proof, provided some “local”(SPI) is satisfied. Here is the more general result in this direction.
Theorem 2.8.
In theorem 2.1 define λ A r ( f ) = λ ( f A r ) where A r is an increasing familyof open sets such that S r A r = M . Given two such families A r ⊆ B r , assume that for all r large enough the following local (SPI) holds, (2.9) λ A r ( f ) ≤ sλ B r ( |∇ f | ) + β r ( s ) ( λ B r ( | f | )) . Then the conclusions of theorem 2.1 are still true if we replace ρ ( ., A r ) ≤ by B r and β ( s ) by β r ( s ) ( s ) with r ( s ) = Φ − (cid:16) bε ∨ εs (cid:17) for each given ε in case (1) and r ( s ) = Φ − ( s ∨ bs ) incase (2). General case.
We consider here the case of general Markov processes on a Manifold M , with a particular care to jump processes. Indeed, a crucial step in the previous proofis to prove (2.2) and it has been made directly taking profit of the gradient structure, butit can be proved in greater generality. However the second part relying on a perturbationapproach seems more difficult. We therefore introduce a local Super-Poincar´e inequality. Theorem 2.10.
Suppose that the Lyapunov condition ( L ) is verified for some function φ suchthat φ ( x ) → ∞ as ρ ( x, o ) → ∞ . Assume also the following family of local Super Poincar´einequality holds for µ : for a family of compact sets { A r ⊃ B ( o, r ) } r ≥ such that A r ↑ M as r ↑ ∞ , there exists β ( r, · ) such that for all nice f and s > µ ( f A r ) ≤ s E ( f, f ) + β ( r, s ) µ ( | f | ) . Then, denoting Φ( r ) := inf A cr φ, Φ − ( r ) := inf { s ≥ s ) ≥ r } ,µ verifies a Super Poincar´e inequality with function α ( s ) = β (Φ − (2 /s ) , s/ . The proof relies on a simple optimization procedure between the weighted energy term andthe local Super Poincar´e inequality. We then only have to prove (2.2) which is done by thefollowing large deviations argument.
P. CATTIAUX, A. GUILLIN, F.Y. WANG, AND L. WU
Lemma 2.12.
For every continuous function U ≥ such that −L U/U is bounded frombelow, (2.13) Z − L UU f dµ ≤ E ( f, f ) , ∀ f ∈ D ( E ) . Proof.
Remark that N t = U ( X t ) exp (cid:18) − Z t L UU ( X s ) ds (cid:19) is a P µ -local martingale. Indeed, let A t := exp (cid:16) − R t L UU ( X s ) ds (cid:17) , we have by Ito’s formula, dN t = A t [ dM t ( U ) + L U ( X t ) dt ] − L UU ( X t ) A t U ( X t ) dt = A t dM t ( U ) . Now let β := (1 + U ) − dµ/Z ( Z being the normalization constant). ( N t ) is also localmartingale, then a super-martingale w.r.t. P β . We so get E β exp (cid:18) − Z t L UU ( X s ) ds (cid:19) ≤ E β N t ≤ β ( U ) < + ∞ . Let u n := min {−L U/U, n } . The estimation above implies F ( u n ) := lim sup t →∞ t log E ν exp (cid:18) − Z t u n ( X s ) ds (cid:19) ≤ . On the other hand by the lower bound of large deviation in [31, Theorem B.1, CorollaryB.11] and Varadhan’s Laplace principle, defining I ( ν | µ ) = E ( p dν/dµ, p dν/dµ ) F ( u n ) ≥ sup { ν ( u n ) − I ( ν | µ ); ν ∈ M ( E ) } . Thus R u n dν ≤ I ( ν | µ ), which yields to (by letting n → ∞ and monotone convergence)(2.14) Z − L UU dν ≤ I ( ν | µ ) , ∀ ν ∈ M ( E ) . That is equivalent to (2.13) by the fact that E ( | f | , | f | ) ≤ E ( f, f ) for all f ∈ D ( E ). (cid:3) We will discuss examples on jump processes in future research, see however [29, Th. 3.4.2]for results in this direction. 3.
Examples in R n . We use the setting of the subsection 2.1 (or of the remark 2.5) but in the euclidean case M = R n for simplicity. Hence in this section λ is the Lebesgue measure, i.e we have T =0. Recall that dµ = Z − e − V dx . It is well known that λ satisfies a (SPI) with β ( s ) = c + c s − n/ . However it is interesting to have some hints on the constants (in particulardimension dependence). It is also interesting (in view of Theorem 2.8) to prove (SPI) forsubsets of R n .Hence we shall first discuss the (SPI) for λ and its restriction to subsets. Since we want toshow that the Lyapunov method is also quite quick and simple in many cases, we shall alsorecall the quickest way to recover these (SPI) results. RIFT AND SUPER POINCAR´E 9
Nash inequalities for the Lebesgue measure.
Let A be an open connected domainwith a smooth boundary. For simplicity we assume that A = { ψ ( x ) ≤ } for some C function ψ such that |∇ ψ | ( x ) ≥ a > x ∈ ∂A = { ψ = 0 } . It is then known that onecan build a Brownian motion reflected at ∂A , corresponding to the heat semi-group withNeumann condition. Let P Nt denote this semi-group, and denote by p Nt its kernel. Recallthe following Proposition 3.1.
The following statements are equivalent (3.1.1) for all < t ≤ and all f ∈ L ( A, dx ) , k P Nt f k ∞ ≤ C t − n/ k f k L ( A,dx ) , (3.1.2) (provided n > ) for all f ∈ C ∞ ( ¯ A ) , k f k L n/n − ( A,dx ) ≤ C (cid:18)Z A |∇ f | dx + Z A f dx (cid:19) , (3.1.3) for all f ∈ C ∞ ( ¯ A ) , k f k /n L ( A,dx ) ≤ C (cid:18)Z A |∇ f | dx + Z A f dx (cid:19) k f k /n L ( A,dx ) , (3.1.4) the (SPI) inequality Z A f dx ≤ s Z A |∇ f | dx + β ( s ) (cid:18)Z A | f | dx (cid:19) holds with β ( s ) = C ( s − n/ + 1) .Furthermore any constant C i can be expressed in terms of any other C j and the dimension n . These results are well known. they are due to Nash, Carlen-Kusuoka-Stroock ([8]) and Davies,and can be found in [13] section 2.4 or [25]. Generalizations to other situations (includinggeneral forms of rate functions β ) can be found in [29] section 3.3.If A = R n (3.1.1) holds (for all t ) with C = (2 π ) − n/ and α = n/
2, yielding a (SPI) inequalitywith(3.2) β ( s ) = c n s − n/ = (cid:18) π (cid:19) n/ s − n/ , which is equivalent, after optimizing in s , to the Nash inequality(3.3) k f k /n ≤ C n (cid:18)Z |∇ f | dx (cid:19) k f k /n , with C n = 2(1 + 2 /n ) (1 + n/ /n (1 / π ) n/ .For nice open bounded domains in R n , as we consider here, (3.1.2) is a well known consequenceof the Sobolev inequality in R n (see e.g. [13] Lemma 1.7.11 and note that the particular cases n = 1 , A is the level set ¯ V r wewould like to know how β r depends on r . Remark 3.4. If n = 1, we have an explicit expression for p Nt when A = (0 , r ), namely p Nt ( x, y ) = (2 πt ) − n/ X k ≥ (cid:18) exp (cid:18) − ( x − y − kr ) t (cid:19) + exp (cid:18) − ( x + y + 2 kr ) t (cid:19)(cid:19) . It immediately follows that(3.5)sup x,y ∈ (0 ,r ) p Nt ( x, y ) ≤ (2 πt ) − n/ X k ≥ (cid:18) exp (cid:18) − ((2 k − r ) t (cid:19) + exp (cid:18) − (2 kr ) t (cid:19)(cid:19) , so that, using translation invariance, for any interval A of length r > r and for 0 < t ≤ x,y ∈ A p Nt ( x, y ) ≤ c ( r ) (2 πt ) − n/ . Hence (3.1.1) is satisfied, and a (SPI) inequality holds in A with the same function β r ( s ) = c B ( s − n/ + 1) independently on r > r . By tensorization,the result extends to any cube or parallelepiped in R n with edges of length larger than r . ♦ If we replace cubes by other domains, the situation is more intricate. However in some casesone can use some homogeneity property. For instance, for n > C (for n = 2 we may add a dimension and consider acylinder B (0 , ⊗ R as in [13] theorem 2.4.4). But a change of variables yields k f k L n/n − ( B (0 ,r ) ,dx ) ≤ C Z B (0 ,r ) |∇ f | dx + r − Z B (0 ,r ) f dx ! , so that for r ≥ r with a constant C independent of r .The previous argument extends to A = ¯ V r provided for r ≥ r , ¯ V r is star-shaped, in particularit holds if V is convex at infinity. This is a direct consequence of the coarea formula (see e.g[16] proposition 3 p.118). Indeed if f has his support in an annulus r < r < V ( x ) < r thesurface measure on the level sets ¯ V r is an image of the surface measure on the unit sphere.This is immediate since the application x ( V ( x ) , x | x | ) is a diffeomorphism in this annulus.Hence for such f ’s the previous homogeneity property can be used. For a given r > r largeenough, it remains to cover ¯ V r by such an annulus and a large ball (such that the ball contains¯ V r and is included in ¯ V r ) and to use a partition of unity related to this recovering. We thusget as before that for r large enough, C can be chosen independent of r .For general domains A , recall that (3.1.2) holds true if A satisfies the “extension property” ofthe boundary, i.e. the existence of a continuous extension operator E : W , ( A ) → W , ( R n ).If this extension property is true, (3.1.2) is true in A with a constant C depending only on n and the operator norm of E (see [13] proposition 1.7.11).If A = ¯ V r is bounded, as soon as ∇ V does not vanish on ∂A , the implicit function theoremtells us that for all x ∈ ∂A one can find an open neighborhood v x of x , an index i x and a2-Lipschitz function φ x defined on v x such that v x ∩ A = v x ∩ { φ ( y , ..., y i x − , y i x +1 , ..., y n ) < y i x } . To this end choose i x such that | ∂ i x V | ( x ) ≥ | ∂ j V | ( x ) for all j = 1 , ..., n , so that, for y ∈ ∂A neighboring x , 2 | ∂ i x V | ( y ) ≥ | ∂ j V | ( y ), and the partial derivative of the implicit function φ given by the ratio ∂ j V ( y ) /∂ i x V ( y ) is less than 2 in absolute value. RIFT AND SUPER POINCAR´E 11
By compactness we may choose a finite number Q of points such that S j =1 ,...,Q v x j ⊃ ∂A .Hence we are in the situation of [13] proposition 1.7.9. This property implies the extensionproperty but with some extension operator E whose norm depends on two quantities : firstthe maximal ε > x ∈ ∂A , B ( x, ε ) ⊆ v x j for some j = 1 , ..., Q ; second, themaximal integer N such that any x ∈ ∂A belongs to at most N such v x j ’s. This is shown in[26] p.180-192.Actually an accurate study of Stein’s proof (p. 190 and 191) shows that k E k≤ C ( n ) ( N/ε )(recall that we have chosen φ R > v > k ∈ N such that for | x | ≥ R , |∇ V ( x ) | ≥ v > . Then it is easy to check that for A = ¯ V r it holds ε ≤ ε = c ( v, R, n ) θ − ( r ) with(3.7) θ ( r ) = sup x ∈ ∂ ¯ V r max i,j =1 ,...,n (cid:12)(cid:12)(cid:12)(cid:12) ∂ V∂x i ∂x j ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . But ε being given, it is well known that one can find a covering of A by balls of radius ε / x ∈ ¯ V r belongs to at most N = c n such balls for some universal c largeenough. Hence N can be chosen as a constant depending on the dimension only. It followsthat Proposition 3.8.
If (3.6) is satisfied, the (SPI) (3.1.4) holds with A = ¯ V r , θ defined by(3.7) and β r ( s ) = C ( n ) θ n ( r ) (1 + s − n/ ) . For the computation of β r we used [13] lemma 1.7.11 which says that C = c ( n ) k E k and[13] proof of theorem 2.4.2 p.77 which yields a logarithmic Sobolev inequality with β ( ε ) = − ( n/
4) log ε + ( n/
4) log( C n/
4) together with [13] corollary 2.2.8 which gives C = c ( n ) C n/ .Finally the proof of [13] theorem 2.4.6 gives β ( s ) = C (1 + ( s/ − n/ ).Proposition 3.8 gives of course the worse result and in many cases one can expect a muchbetter behavior of β r as a function of r . In particular in the homogeneous case we know theresult with a constant independent of r . Remark 3.9.
Another possibility to get (SPI) in some domain A , is to directly prove theNash inequality (3.1.3). One possible way to get such a Nash inequality is to prove somePoincar´e-Sobolev inequality. The case of euclidean balls is well known.According to [25] theorem 1.5.2, for n >
2, with p = 2 and s = 2 n/ ( n −
2) = 2 ∗ therein,for all r > B r with radius r , if λ r is the Lebesgue measure on B r and ¯ f r =(1 /V ol ( B r )) R B r f dx we have(3.10) λ r (cid:16) | f − ¯ f r | nn − (cid:17) n − n ≤ C n λ r (cid:0) |∇ f | (cid:1) , so that using first Minkowski, we have(3.11) λ r (cid:16) | f | nn − (cid:17) n − n ≤ C n λ r (cid:0) |∇ f | (cid:1) + 1 V ol ( B r ) λ r ( | f | ) , and finally using H¨older inequality and Cauchy-Schwarz inequality we get the local Nashinequality λ r ( | f | ) ≤ ( λ r ( | f | )) / ( n +2) (cid:18) C n λ r (cid:0) |∇ f | (cid:1) + 1 V ol ( B r ) λ r ( | f | ) (cid:19) n/ ( n +2) , (3.12) ≤ ( λ r ( | f | )) / ( n +2) C n λ r (cid:0) |∇ f | (cid:1) + 1 p V ol ( B r ) λ r (cid:0) | f | (cid:1) ! n/ ( n +2) . Again, for r > r we get a Nash inequality hence a (SPI) inequality independent of r with β r ( s ) = c n (1 + s − n/ ).Notice that (3.10) is scale invariant, i.e. if it holds for some subset A , it holds for thehomotetic rA ( r >
0) with the same constants. That is why the constants do not dependon the radius for balls. If we replace a ball by a convex set, the classical method of proofusing Riesz potentials (see e.g. [25] or [13] lemma 1.7.3) yields a similar results but with anadditional constant, namely diam n ( A ) /V ol ( A ), so that if V is a convex function the constantwe obtain with this method in (3.10) for ¯ V r may depend on r .Actually the Sobolev-Poincar´e inequality (3.10) extends to any John domain with a constant C depending on the dimension n and on the John constant of the domain. This result is dueto Bojarski [6] (also see [19] for another proof and [7] for a converse statement). Actually aJohn domain satisfies some chaining (by cubes or balls) condition which is the key for theresult (see the quoted papers for the definition of a John domain and the chaining condition).But an explicit calculation of the John constant is not easy. ♦ Typical Lyapunov functions and applications.
We here specify classes of naturalLyapunov function: function of the potential or of the distance. As will be seen, it gives newpractical conditions for super-Poincar´e inequality and for logarithmic Sobolev inequality.First, since W ≥ W = e U so that condition (L) becomes(3.13) ∆ U + |∇ U | − ∇ U. ∇ V + φ ≤ Lyapunov function e aV . Test functions e aV for a < µ ( e aV ) is finite if and only if a < L W ≤ − λW + b C formally implies by integration by µ , that µ ( W ) is finite. So in a sense, e aV are the “largest”possible Lyapunov functions.Hence, if W = e aV , L WW = a (cid:0) ∆ V − (1 − a ) |∇ V | (cid:1) . Introduce the following conditions(3.14.1) V ( x ) → + ∞ as | x | → + ∞ ,(3.14.2) there exist 0 < a <
1, a non-decreasing function η with η ( u ) → + ∞ as u → + ∞ and a constant b such that(1 − a ) |∇ V | − ∆ V ≥ η ( V ) + b | x | Then for 0 < a < a condition (L) is satisfied with φ = a ( a − a ) |∇ V | + aη ( V ) . In addition inf ( ¯ V r ) c φ ( V ) ≤ c sup ¯ V r +2 |∇ V | .Following remark 2.6 (we choose arbitrarily ε = 1 / c (3.15) Z f dµ ≤ s Z |∇ f | dµ + c ¯ V η − c/s ) |∇ V | n/ e η − ( c/s ) (cid:18)Z | f | dµ (cid:19) . We thus clearly see that to get an explicit (SPI) we need to control the gradient ∇ V on thelevel sets of V .If instead of using theorem 2.1.(1) we want to use theorem 2.1.(2) or more precisely theorem2.8 we have to use proposition 3.8. Hence since (3.6) is satisfied we obtain for s small enough(3.16) Z f dµ ≤ s Z |∇ f | dµ + C θ n ( η − ( c/s )) e η − ( c/s ) (cid:16) s − n/ e nη − ( c/s ) / (cid:17) (cid:18)Z | f | dµ (cid:19) . We have obtained Theorem 3.17. Assume that (3.14.1), (3.14.2), (3.14.3) are satisfied. Then µ will satisfya (SPI) inequality with function β in one of the following cases (3.17.1) for | x | large enough, |∇ V | ( x ) ≤ γ ( V ( x )) and β ( s ) = C (1 + e η − ( c/s ) γ n ( η − ( c/s ))) , (3.17.2) for | x | large enough (cid:12)(cid:12)(cid:12) ∂ V∂x i ∂x j ( x ) (cid:12)(cid:12)(cid:12) ≤ θ ( V ( x )) and β ( s ) = C (cid:16) θ n ( η − ( c/s )) s − n/ e ( n +4) η − ( c/s ) / (cid:17) . Remark 3.18. If η ( u ) = u we thus obtain that µ satisfies a (defective) logarithmic Sobolevinequality provided either γ ( u ) ≤ e Ku or θ ( u ) ≤ e Ku . But (3.14.1) and (3.14.2) imply that µ satisfies a Poincar´e inequality (see e.g. [2] corollary 4.1). Hence using Rothaus lemma weget that µ satisfies a (tight) logarithmic Sobolev inequality.Conditions (3.14.1) and (3.14.2), with η ( u ) = u , appear in [21] where the authors show thatthey imply the hypercontractivity of the associated symmetric semi-group, hence a logarith-mic Sobolev inequality by using Gross theorem. In particular the additional assumptions onthe first or the second derivatives do not seem to be useful. Another approach using Girsanovtransformation was proposed in [9] for a = 1 / 2, again without the technical assumptions onthe derivatives. This approach extends to more general processes with a “carr´e du champ”.Here we directly get the logarithmic Sobolev inequality without using Gross theorem, butwith some conditions on V .The advantage of theorem 3.17 is that it furnishes an unified approach of various inequalitiesof F -Sobolev type. In [3] conditions (3.14.1) and (3.14.2) are used (for particular η ’s) to getthe Orlicz-hypercontractivity of the semi-group hence a F -Sobolev inequality thanks to theGross-Orlicz theorem proved therein. The use of this theorem requires some quite stringentconditions on η but covers the case η ( u ) = u α for 1 < α < 2, yielding a F -Sobolev inequalityfor F ( u ) = log α + ( u ) (more general F are also studied in [4] section 7). Note that in theorem α < 2, but we need some control on the growth on γ or θ , namely we need again γ ( u ) ≤ e Ku (the same for θ ).We also obtain a larger class of F -Sobolev inequalities thanks to the correspondence between F -Sobolev and (SPI) recalled in the introduction. The reader is referred to [29] section 5.7for related results in the ultracontractive case. ♦ Lyapunov function e a | x | b . If we try to use W = e a | x | b we are led to choose φ ( x ) = ab | x | b − ψ ( x )with ψ ( x ) = x. ∇ V − (cid:16) n + ( b − 2) + ab | x | b (cid:17) provided the latter quantities are bounded from below by a positive constant for | x | largeenough.Introduce now the standard curvature assumption(3.19) for all x , HessV ( x ) ≥ c Id for some ρ ∈ R . This assumption allows to get some control on x. ∇ V namely Lemma 3.20. If (3.19) holds, x. ∇ V ( x ) ≥ V ( x ) − V (0) + c | x | / .Proof. Introduce the function g ( t ) = t x. ∇ V ( tx ) defined for t ∈ [0 , g ′ ( t ) ≥ x. ∇ V ( tx ) + tc | x | and the result follows by integrating the latter inequality between0 and 1. (cid:3) We may thus state Proposition 3.21. Assume that (3.19) is satisfied. Then one can find positive constants c, C such that µ satisfies some (SPI) with function β (given below for s small enough) in thefollowing cases (3.21.1) c ≥ , V ( x ) ≥ c ′ | x | b for | x | large enough some c ′ > and b > , β ( s ) = C e c (1 /s ) b b − ∧ . (3.21.2) c ≥ , d ′ | x | b ′ ≥ V ( x ) ≥ c ′ | x | b for | x | large enough some d ′ , c ′ > and b ′ ≥ b > , β ( s ) = C e c (1 /s ) b ′ b ′ + b − . (3.21.3) c ≤ , for | x | large enough, V ( x ) ≥ ( ε − c / | x | for some ε > , and β ( s ) = C e c (1 /s ) . (3.21.4) c ≤ , for | x | large enough, d ′ | x | b ′ ≥ V ( x ) ≥ c ′ | x | b and β as in (3.21.2).Proof. In all the proof D will be an arbitrary positive constant whose value may change fromplace to place. All the calculations are assuming that | x | is large enough.Consider first case 1. Choosing a small enough and using lemma 3.20, we see that φ ( x ) ≥ D | x | b − V ( x ). If b ≥ φ ( x ) ≥ D V ( x ) while for b < φ ( x ) ≥ D V b − b ( x ) forlarge | x | according to the hypothesis. For φ to go to infinity at infinity, b > V r we have either φ ( x ) ≥ D r or φ ( x ) ≥ D r b − /b . RIFT AND SUPER POINCAR´E 15 Now since the level sets ¯ V r are convex, we know that some Nash inequality holds on ¯ V r according to the discussion in the previous subsection. We may thus use theorem 2.8 in thesituation (2) of theorem 2.1. Choosing s = d/r or s = d/r b − b for some well chosen d yields the result with an extra factor s − k for some k > 0. This extra term can be skippedjust changing the constants in the exponential term.Case 2 is similar but improving the lower bound for φ . Indeed since D | x | ≥ V /b ′ ( x ), φ ( x ) ≥ D V b ′ + b − b ′ ( x ). It allows us to improve β .Let us now consider Case 3. Since b = 2, our hypothesis implies that for 2 a < ε , φ ≥ DV .But the curvature assumption implies that the level sets of x H ( x ) = V ( x ) + c | x | / V ( x ) ≥ D | x | , one has cr ≤ V ( x ) ≤ r if x ∈ ¯ H r . We may thus mimic case 1,just replacing ¯ V r by ¯ H r . Case 4 is similar to the previous one just improving the bound on φ as in case 2. (cid:3) Corollary 3.22. (1) If (3.21.3) holds, µ satisfies a logarithmic Sobolev inequality. (2) If (3.21.1) holds with b = 2 , µ satisfies a logarithmic Sobolev inequality. In partic-ular if ρ > , µ satisfies a logarithmic Sobolev inequality (Bakry-Emery criterion). (3) If (3.21.1) holds for some < b < , µ satisfies a F -Sobolev inequality with F ( u ) =log − (1 /b ))+ ( u ) . The first statement of the theorem is reminiscent to Wang’s improvement of the Bakry-Emery criterion, namely if R R e ( − ρ + ε ) | x − y | µ ( dx ) µ ( dy ) < + ∞ , µ satisfies a logarithmicSobolev inequality. Our statement is weaker since we are assuming some uniform behavior.The third statement can thus be seen as an extension of Wang’s result to the case of F -Sobolev inequalities interpolating between Poincar´e inequality and log-Sobolev inequality.These inequalities are related to the Latala-Oleskiewicz interpolating inequalities [22], see [3]for a complete description.It should be interesting to improve (3) in the spirit of Wang’s concentration result. See[20, 5] for a tentative involving modified log-Sobolev inequalities introduced in [18] and masstransport. 4. The general manifold case In fact as one guesses, the main point is to get the additional Super Poincar´e inequality,local as developed in Section 3.1, or global (and then using the localization technique alreadymentioned). It is of course a fundamental field of research which encompasses the scope ofthe present paper. We may however use our main results Theorem 2.1 and Theorem 2.8,with the same Lyapunov functionals as developed in Sections 3.2.1 and 3.2.2, replacing ofcourse the euclidean distance by the Riemannian distance (w.r.t a fixed point), at least intwo main cases.According to [12], if the injectivity radius of M is positive then (1.3) holds for T = 0 and β ( s ) = c + c s − d/ for some constants c , c > 0; if in particular the injectivity is infinite,then one may take c = 0, [27] page 225. Next, if the Ricci curvature of M is bounded below, then by [27] Theorem 7.1, there exists c , c > T = c ρ and β ( s ) = c s − d/ . For simplicity, throughoutthis section we assume that( H ) The injectivity radius of M is positive. Lyapunov condition e aV . In this context, one may readily generalizes the result ofTheorem (3.17) for the first case (3.17.1), with the euclidean distance replaced by the Rie-mannian one, assuming (3.14.1), (3.14.2) and (3.14.3). Theorem 4.1. Assume ( H ) and that (3.14.1), (3.14.2), (3.14.3) are satisfied. Supposemoreover that for large ρ , |∇ V | ( x ) ≤ γ ( V ( x )) . Then µ will satisfy a (SPI) inequality withfunction β given by β ( s ) = C (1 + e η − ( c/s ) γ n ( η − ( c/s ))) . The second point of Theorem 3.17 is more delicate as it relies on finer conditions on themanifold and the potential, it should however be possible to give mild additional assumptionsensuring such a result (for instance the so called “rolling ball condition”). Remark that itextends to the manifold case Kusuoka-Stroock’s result (giving life to Remark (2.49) in theirpaper).4.2. Lyapunov condition e aρ b . We suppose moreover here that M is a Cartan-Hadamardmanifold with lower bounded Ricci curvature.If we try to use W = e aρ b for ρ ≥ , since ∆ ρ is bounded above on { ρ ≥ } (see for exampleTh.0.4.10 in [29]), (L) holds for φ := abρ b − ψ with ψ := h∇ ρ , ∇ V i − (cid:16) c + abρ b (cid:17) for some constant c > ψ is positive for large ρ. We may then extend Lemma 3.20 in the manifold context. Lemma 4.2. If (3.19) holds, then ρ h∇ ρ, ∇ V i ≥ V − V ( o ) + c ρ / .Proof. For x ∈ M , let ξ : [0 , ρ ( x )] → M be the minimal geodesic from o to x . Let g ( t ) = t h∇ ρ, ∇ V i ( ξ t ) , t ≥ . We have g ′ ( t ) = h∇ ρ, ∇ V i ( ξ t ) + t Hess V ( ∇ ρ, ∇ ρ )( ξ t ) ≥ c t + dV( ξ t )dt . This implies the desired assertion by integrating both sides on [0 , ρ ( x )] . (cid:3) We may thus state Proposition 4.3. Let M be a Cartan-Hadamard manifold with Ricci curvature boundedbelow. Let V satisfy (3.19). Then one can find positive constants c, C such that µ satisfiessome (SPI) with function β (given below for s small enough) in the following cases RIFT AND SUPER POINCAR´E 17 (4.3.1) c ≥ , V ( x ) ≥ c ′ ρ b for ρ large enough some c ′ > and b > , β ( s ) = C e c (1 /s ) b b − ∧ . (4.3.2) c ≥ , d ′ ρ b ′ ≥ V ( x ) ≥ c ′ ρ b for ρ large enough some d ′ , c ′ > and b ′ ≥ b > , β ( s ) = C e c (1 /s ) b ′ b ′ + b − . (4.3.3) c ≤ , for ρ large enough, V ( x ) ≥ ( ε − c / ρ for some ε > , and β ( s ) = C e c (1 /s ) . (4.3.4) c ≤ , for ρ large enough, d ′ ρ b ′ ≥ V ( x ) ≥ c ′ ρ b and β as in (3.21.2). The first point of this proposition specialized tot the case c > F -Sobolev. Proof. The proof follows exactly the same line than in the flat case so that case 1 and case2 follows once it is noted that since Hess V ≥ V r , we know that some Nash inequality holds on ¯ V r according to the discussion in the previoussubsection and the boundedness of these level sets ensured by our hypotheses on V .Let us now consider Case 3. Since b = 2, our hypothesis implies that for 2 a < ε , φ ≥ DV .But (3.19) and Hess ρ ≥ x H = V + c ρ / V ≥ Dρ , one has cr ≤ V ≤ r on ¯ H r . We may thusmimic case 1, just replacing ¯ V r by ¯ H r . Case 4 is similar to the previous one just improvingthe bound on φ as in case 2. (cid:3) Remark 4.4. Remark that in full generality, according to [30] Theorem 1.2 and the recentpaper [11], there always exists T ∈ C ∞ ( M ) such that dλ := e − T ( x ) dx satisfies a logarithmicSobolev inequality hence (SPI) with β ( s ) = e s − . Of course for practical purposes, this verygeneral fact is not completely useful since T is unknown. References [1] C. An´e, S. 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Logarithmic Sobolev inequalities: conditions and counterexamples. J. Oper. Th. , 46:183–197,2001.[29] F. Y. Wang. Functional inequalities, Markov processes and Spectral theory . Science Press, Beijing, 2004.[30] F. Y. Wang. Functional inequalities on arbitrary Riemannian manifolds. J. Math. Anal. Appl. , 30:426–435,2004.[31] L. Wu. Uniformly integrable operators and large deviations for Markov processes. J. Funct. Anal. ,172(2):301–376, 2000. RIFT AND SUPER POINCAR´E 19 Patrick CATTIAUX ,, Universit´e Paul Sabatier Institut de Math´ematiques. Laboratoire deStatistique et Probabilit´es, UMR C 5583, 118 route de Narbonne, F-31062 Toulouse cedex 09. E-mail address : [email protected] Arnaud GUILLIN ,, Ecole Centrale Marseille et LATP Universit´e de Provence, TechnopoleChteau-Gombert, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13. E-mail address : [email protected] Feng-Yu WANG ,, Department of Mathematics, Swansea University, Singleton Park, SA2 8PP,Swansea UK E-mail address : [email protected] Liming WU ,, Laboratoire de Math´ematiques Appliqu´ees, CNRS-UMR 6620, Universit´e BlaisePascal, 63177 Aubi`ere, France. And Department of Mathematics, Wuhan University, 430072Hubei, China E-mail address ::