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Modified logarithmic Sobolev inequalities and
transportation inequalities
Ivan Gentil, Arnaud Guillin, Laurent Miclo
To cite this version:
Ivan Gentil, Arnaud Guillin, Laurent Miclo. Modified logarithmic Sobolev inequalities and
transporta-tion inequalities. Probability Theory and Related Fields, Springer Verlag, 2005, 133 (3), pp.409-436.
�hal-00001609�
inequalities
IvanGentil,Arnaud Guillin
Ceremade(UMRCNRS no. 7534),UniversiteParisIX-Dauphine,
Pla edeLattredeTassigny,75775ParisCedex16,Fran e
E-mail: fgentil,guilling eremade.dauphine.fr
Internet: http://www. eremade.dauphine.fr/ e
fgentil,guilling/
Laurent Mi lo
LaboratoiredeStatistiqueetProbabilites,(UMRCNRSno. 5583),
UniversitePaul-Sabatier,
118routedeNarbonne,31062Toulouse Cedex4,Fran e
E-mail: mi lomath.ups-tlse.fr
12th May 2004
Abstra t
We present a new lass of modi ed logarithmi Sobolev inequality, interpolating between Poin areandlogarithmi Sobolevinequalities,suitableformeasuresofthetypeexp( jxj
)or exp( jxj log
(2+jxj))( 2℄1;2[and 2R)whi hleadtonew on entrationinequalities. These modi ed inequalitiesshare ommon properties with usual logarithmi Sobolev inequalities, as tensorisation or perturbation, and imply as well Poin are inequality. We also study the link betweenthesenewmodi edlogarithmi Sobolevinequalitiesandtransportationinequalities.
1 Introdu tion
AprobabilitymeasureonR n
satis esalogarithmi SobolevinequalityifthereexistsC<1su h
that, forevery smoothenoughfun tionsf onR n , Ent f 2 C Z j rfj 2 d; (1) where Ent f 2 = Z f 2 logf 2 d Z f 2 dlog Z f 2 d
andwhere j rfjistheEu lideanlengthof thegradient rf off.
Grossin[Gro75 ℄de nesthisinequalityandshowsthatthe anoni alGaussianmeasurewithdensity (2) n=2 e jxj 2 =2
with respe t to the Lebesgue measure on R n
is the basi example of measure
satisfying (1) with C = 2. Sin e then, many results have presented measures satisfying su h an inequality, among them the famous Bakry-Emery
2
riterion, we refer to Bakry [Bak94℄ and Ledoux[Led99 ℄forfurtherreferen esand detailson various appli ationsofthese inequalities.
Let >1 and de netheprobabilitymeasure on R by (dx)= 1 Z e jxj dx; (2)
where Z
= e jxj
dx. It is well-known that the probability measure
satis es a logarithmi Sobolev inequality (1) if and only if > 2. But for 2 [1;2[, even if the measure
does not satisfy(1) , itsatis es a Poin are inequality(or spe tral gap inequality) whi h is forevery smooth
enoughfun tionf, Var (f)C Z jrfj 2 d ; (3) whereVar (f)= R f 2 d R fd 2 and C<1.
Re all, see for example Se tion 1.2.6 of [ABC +
00 ℄, that if a probability measure on R n
satis es a
logarithmi Sobolevinequalitywith onstantCthenitsatis esaPoin areinequalitywitha onstant lessthatC=2.
The problem is then to interpolate between logarithmi Sobolevand Poin are inequalities, whi h
willhelp us to study further properties, su h as on entration, of measures n for 2[1;2℄ and n2N .
A rst answer was brought by Lata la-Oleszkiewi z in [LO00 ℄ and re ently extended by
Barthe-Robertoin[BR03℄. LetbeaprobabilitymeasureonR n
,satis esinequalityI
(a)(fora2[0;1℄) with onstant C>0 ifforall p2[1;2[,
Z f 2 d Z f p d 2=p C(2 p) a Z jrfj 2 d: (4)
A signi ant result of [LO00 ℄ is that they prove that the measure n (for 2 [1;2℄, n 2 N ) satis es su h an inequality for a onstant C (independent of n) and with a = 2( 1)= . And
in[BR03℄the authorspresent a simpleproof of the result of Lata la-Oleszkiewi z and des ribe the measuresonthe linewhi henjoy thesame inequality.
Ourmainpurposeherewillbetoestablishanothertypeofinterpolationbetweenlogarithmi Sobolev
and Poin are inequalities, more dire tly linked to the stru ture of the usual logarithmi Sobolev inequalities,i.e. an inequality\entropy-energy" where we will modify the energy to enable us to onsider
measure. NotethatthispointofviewwastheoneusedbyBobkov-Ledoux[BL97℄when
onsidering doublesided exponential measure. Let us des ribe further these modi ed logarithmi Sobolevinequalities.
Let 2[1;2℄, a>0and >2 satisfying1= +1= =1,we note
H a; (x)= 8 > > > > < > > > > : x 2 2 ifjxja a 2 jxj +a 2 2 2 ifjxj>aand 6=1 +1 ifjxj>aand =1:
InSe tion2 we give de nitionand general properties ofthefollowing inequality
Ent f 2 C Z H a; rf f f 2 d: (LSI a; (C))
In parti ular we prove that inequality LSI a;
satis es some of the properties shared by Poin are orGross logarithmi Sobolevinequalities((1)or(3) ),namelytensorisation andperturbation. Note that in the ase = 1, it is exa tly the inequality used by Bobkov-Ledoux [BL97℄ and = 2 is
exa tlytheGross logarithmi Sobolevinequality.
We present also a on entration propertywhi h is adapted to this inequality. Morepre isely, ifa measure satis esthe inequalityLSI
a;
(C), we have that if f is a Lips hitz fun tionon R n
with
kfk Lip
1 (with respe t to the Eu lideanmetri ) then, there is B > 0 su h that for every >0 onehas f Z fd > exp Bmin ; 2 : (5)
Ledoux in [Mau91, BL97 ℄. Let us note that the ases > 2 are studied by Bobkov-Ledoux in [BL00℄, relying mainly on Brunn-Minkowski inequalities, and by Bobkov-Zegarlinski in [BZ04 ℄
whi h re ne theresults presenting,via Hardy's inequality,some ne essary and suÆ ient ondition formeasureson thereal line. Let usnoteto nishthatthey use, forthe ase >2, H
(x)=jxj
with1= +1= =1.
In Se tion 2.2, we extend Otto-Villani's theorem (see [OV00℄) for the relation with logarithmi Sobolevinequalityand transportation inequality. Letus de neL
a; byL a; =H a; ,theLegendre transformofH a;
. WeprovethatifaprobabilitymeasureonR n
satis estheinequalityLSI a;
(C) thentherearea
0
>0andD>0su hthatitsatis esalsoatransportationinequality: forallfun tion F onR
n
,densityof probabilitywithrespe tto ,
T L a 0 ; (Fd;d)DEnt ( F); (T a 0 ; (D)) where T L a 0 ; (Fd;d)=inf Z L a 0 ; ( x y)d(x;y) ;
wherethein mumis takenoverthesetofprobabilitiesmeasures onR n
R n
su hthat hastwo marginsFd and d. This inequalitywas introdu ed by Talagrand in[Tal96 ℄ forthe ase = 2 and =1. Letusnotethatthe ase =1wasalso studiedin[BGL01℄withexa tlythisformand
the ase >2wasstudiedin[Gen01 ℄.
In Se tion 3 we prove, as in [LO00 ℄, that the measure
de ned in (2) satis es the inequality
LSI a;
(C). More pre isely we prove that there is A;B > 0 su h that
satis es for all smooth fun tionsu h thatf >0and
R f 2 d =1, Ent f 2 AVar ( f)+B Z f>2 f 0 f f 2 d :
Dueto the fa tthat
enjoys Poin are inequality,
satis es also inequalityLSI a;
(C)for some onstantsC>0and a>0.
Our method relies ru ially on Hardy's inequality (see for example [ABC +
00 , BG99, BR03 ℄) we
re all now: let ; be Borel measures on R +
. Then the best onstant A so that every smooth fun tionf satis es Z 1 0 (f(x) f(0)) 2 d(x)A Z 1 0 f 02 d (6)
is niteifand onlyif
B=sup x>0 ([x;1[) Z x 0 d a d 1 dt (7) is nite, where a
is the absolutely ontinuous part of withrespe t to . Moreover, when A is
nitewehave
B A4B:
Finallyin Se tion 4 we will present some inequalities satis edby other measures. More pre isely, let'be twi e ontinuously di erentiableand notethe probabilitymeasure
' by, ' (dx)= 1 Z e '(x) dx: (8)
Amongthem is onsidered
'(x)=jxj
(log(2+jxj))
; with 2℄1;2[; 2R;
whi h exhibits a modi ed logarithmi Sobolev inequality of fun tion H (di erent in nature from
H a;
),and whi hisnot overed byLata la-Oleskiewi kzinequality. Wealsopresentexampleswhi h are unbounded perturbationof
. We then derive new on entration inequalities in the spiritof Maurey[Mau91℄orBobkov-Ledoux [BL97 ℄.
eral properties
2.1 De nitions and lassi alproperties
Let 2[1;2℄ and >2 satisfying1= +1= =1 and let a>0. Letde ne thefun tions L a; and H a; . If 2℄1;2℄we note L a; (x)= 8 > < > : x 2 2 ifjxja a 2 jxj +a 2 2 2 ifjxj>a and H a; (x)= 8 > > < > > : x 2 2 ifjxja a 2 jxj +a 2 2 2 ifjxj>a If =1 we note L a;1 (x)= 8 > < > : x 2 2 ifjxja ajxj a 2 2 ifjxj>a and H a;1 (x)= 8 < : x 2 2 ifjxja 1 ifjxj>a Letn2N and x=(x 1 ; ;x n )2R n ,wenote L (n) a; (x)= n X i=1 L a; (x i ) and H (n) a; (x)= n X i=1 H a; (x i ):
Note that when there is no ambiguity we will drop the dependen e inn and note L a; instead of L (n) a; .
Letusde nethelogarithmi Sobolevinequalityoffun tionH a;
.
De nition2.1 Let be a probability measure on R n
, satis es a logarithmi Sobolev inequality of fun tionH
a;
with onstant C, noted LSI a; (C), if for every C 1 and L 2 fun tion f on R n one has Ent f 2 C Z H a; rf f f 2 d; (LSI a; (C)) where Ent f 2 = Z f 2 log f 2 R f 2 d d and H a; rf f = n X i=1 H a; f x i 1 f : Itis supposed that 0=0 =1
We detailsome propertiesof L a;
andH a;
inthefollowinglemma.
Lemma2.2 Fun tions L a; and H a; satis es: i: If 2℄1;2℄, L a; and H a; areC 1 on R. ii: L a; = H a; , where L a;
is the Fen hel-Legendre transform of L a;
. Of ourse we have too H a; =L a; .
iii: For all t>0 one has for all x2R
L a; (tx)=t 2 L a t ; (x); H a; (tx)=t 2 H a t ; (x):
iv: Let 0aa, one has for all x2R L a; (x)L a 0 ; (x); H a 0 ; (x)H a; (x): v: If 2℄1;2℄, L a; and H a;
arestri tly onvex and
lim jxj!1 H a; (x) x = lim jxj!1 L a; (x) x =1:
Theassumptionsgivenon and aresigni antonlyfor onditioniv,and onditionvissigni ant forBrenier-M Cann-Gangbo's theorem,whi h is ru ialforthestudyof thelinkbetweenmodi ed logarithmi Sobolevinequalitiesand transportationinequalitiesof thenext se tion.
Herearesome properties oftheinequalityLSI a;
(C).
Proposition 2.3 i. This property is known underthe name of tensorisation.
Let 1
and 2
two probability measures on R n
1
and R n
2
. Suppose that 1
(resp. 2
) satis es
the inequality LSI a; (C 1 ) (resp. LSI a; (C 2
)) then the probability 1 2 on R n1+n2 , satis es inequality LSI a; (D), where D=maxfC 1 ;C 2 g .
ii. This property is known underthe name of perturbation.
Let a measure on R n
satisfying LSI a;
(C). Let h a bounded fun tion on R n and de ned ~ as d~= e h Z d; where Z= R e h d.
Then the measure ~ satis es the inequality LSI a;
(D) with D = Ce os (h)
, where os (h) = sup(h) inf(h).
iii. Link between LSI a;
(C) inequality with Poin are inequality.
Let a measure on R n
. If satis es LSI a;
(C), then satis es a Poin are inequality with
the onstant C=2. Letus re all that satis es a Poin are inequality with onstant C if
Var (f)C Z jrfj 2 d; (9)
for all smooth fun tionf.
Proof
C One an ndthedetailsof theproofof thepropertiesof tensorisationandperturbationand the impli ation of the Poin are inequality in the hapter 1 and 3 of [ABC
+
00 ℄ (Se tion 1.2.6., Theo-rem3.2.1 and Theorem3.4.3). B
Remark 2.4 We may of ourse de ne logarithmi Sobolev inequality of fun tion H, where H(x)
is quadrati for small values of jxj and with onvex, faster than quadrati , growth for large jxj. See Se tion 4 for su h examples. Note that Proposition 2.3 is of ourse still valid for this kind of inequality. These inequalitiesare also studied in a general ase in [Led99 ℄ in Proposition 2.9.
Asin[LO00,BL97℄,byusingtheargumentofHerbst, one an givepre ise estimatesabout on en-tration.
a; Let F be Lips hitz fun tionon R, then we getfor >0,
(j F (F)j>) 8 > > > < > > > : 2exp K ( aCkFk Lip (2 )) a 2 2 2 if > aCkFk Lip 2 ; 2exp 2 2 CkFk 2 Lip ! otherwise, where K = 2 ( 1) 1 a 2 C 1 kFk Lip . Considernow n and F :R n !R, C 1 , su h that P n i=1 F x i 2
1, then there exists ~ K
(indepen-dent of n) su h that
n F n (F) > 2exp ~ K min 2 ; : (10) Proof C Assumethat R
Fd=0. Letusre allbrie yHerbst'sargument(see[ABC +
00 ℄formoredetails).
Denote (t) = R
e tF
d, and remark that LSI a;
(C) appliedto f 2
=e tF
,usingbasi properties of H a; ,yieldsto t 0 (t) (t)log(t)CH a; tkFk Lip 2 (t) (11)
whi h,denotingK(t)=(1=t)log(t), entails
K 0 (t) C t 2 H a; tkFk Lip 2 :
Then,integrating, andusingK(0)= R Fd=0,we obtain (t)exp Ct Z t 0 1 s 2 H a; skFk Lip 2 ds : (12)
TheLapla etransform ofF isthenboundedby
(t) 8 > > > < > > > : exp Ct kFk Lip a 2 2 ( 1) +CtakFk Lip 2 2( 1) Ca 2 2 2 ift> 2a k Fk Lip ; exp C kFk 2 Lip t 2 8 ! if0t 2a kFk Lip :
Forthen-dimensionalextension, use thetensorisationpropertyof LSI a; and n X i=1 H a; t 2 F x i H a; t 2 :
Thenwe an use the ase ofdimension1. B
Remark 2.6 For general logarithmi Sobolev of fun tion H, we may obtain rude estimation of
the on entration, at least for large . Indeed, using inequality (12), we have dire tly that the on entrationbehaviorisgivenbytheFen hel-LegendretransformofHforlargevalues,seeSe tion4 for more details.
a;
De nition2.7 Let be a probability measure on R n
, satis es a transportation inequality of fun tion L
a;
with onstant C, noted T a;
(C), if for every fun tion F, density of probability with respe t to, one has
T L a; (Fd;d)CEnt (F); (T a; (C)) where T L a; ( Fd;)=inf Z L a; ( x y)d(x;y) ;
where thein mum istaken overthe set of probabilitiesmeasures on R n
R n
su h that has two margins Fd and.
Otto and Villani proved that a logarithmi Sobolev inequality impliesa transportation inequality witha quadrati ost (this isthe ase = =2), see[OV00,BGL01℄. Withthe notations of this
paperthey prove thatifsatis estheinequalityLSI ;2
(C), (when =2the onstant aisnotany more a parameter in this ase), then satis es the inequality T
;2
(4C). In [BGL01℄ another ase isstudied, when =1 and =1. In this rst theorem we give an extension forthe other ases,
where 2[1;2℄.
Theorem 2.8 Let be a probability measure on R n
and suppose that satis es the inequality LSI
a; (C).
Then satis es the transportation inequality TaC 2
; (C=4).
Proof
C As in [BGL01 ℄, we use Hamilton-Ja obi equations. Let f be a Lips hitz bounded fun tion on
R n ,and set Q t f(x)= inf y2R f(y)+tL aC 2 ; x y t ; t>0; x2R n ; (13) andQ 0 f =f. Thefun tionQ t
f isknownastheHopf-LaxsolutionoftheHamilton-Ja obiequation
( v t (t;x) =HaC 2 ; (rv) (t;x); t>0; x2R n v(0;x) =f(x); x2R n
seeforexample[Bar94 , Eva98 ℄.
Fort>0,de nethefun tion by
(t)= Z e 4t C Qtf d:
Sin ef isLips hitz and boundedfun tionone an prove that Q t
f is also aLips hitz and bounded
fun tionont foralmost every x2R n ,then is aC 1 fun tionon R + . One gets 0 (t)= Z 4 C Q t fe 4t C Q t f d Z 4t C H aC 2 ; ( rQ t f)e 4t C Q t f d = 1 t Ent e 4t C Qtf + 1 t (t)log (t) Z 4t C HaC 2 ; ( rQ t f)e 4t C Qtf d
LetuseinequalityLSI a;
(C) to thefun tionexp 2t C Q t f to get 0 (t) 1 t (t)log (t)+ C t Z H a; 2t C rQ t f e 4t C Q t f d Z 4t 2 C 2 H aC 2 ; ( rQ t f)e 4t C Q t f d :
Dueto thepropertyof H a; (see Lemma2.2), H a; 2t C rQ t f = 4t 2 C 2 HaC 2t ; ( rQ t f):
H a; 2t C rQ t f 4t 2 C 2 HaC 2 ; ( rQ t f): Then 8t2[0;1℄; t 0 (t) (t)log (t)0
Afterintegration on[0;1℄, we have
(1)exp 0 (0) (0) ; fromwhere Z e 4 C Q1f de R 4 C fd : (14) Sin e Ent ( F)=sup Z Fgd; Z e g d1 ; we havewith g= 4 C Q 1 f R 4 C fd, Z F Q 1 f Z fd d C 4 Ent (F):
Let take the supremum on the set of Lips hitz fun tion f, the Kantorovi h-Rubinstein's theorem
appliedto thedistan e T L
a;
(Fd;d), see[Vil03℄, impliesthat
T L aC 2 ; (Fd;d) C 4 Ent (F): B
As it is also the ase in quadrati ase, when the measure is log- on ave one an prove that a transportationinequalityimpliesalogarithmi Sobolevinequality.
For the next theorem we suppose that the fun tion of transport given by the theorem of
Brenier-Gangbo-M Cann is a C 2
fun tion. Su h a regularity result is outside the s ope of this paper and we refer to Villani [Vil03 ℄ for further dis ussions around this problem. However we show here, that on ethis result assumed, the methodology presented in Bobkov-Gentil-Ledoux [BGL01 ℄, for the
exponential measure, still works.
Theorem 2.9 Let be a probability measure on R n
. Assume that
(dx)=e '(x)
dx
where 'is a onvex fun tionon R n
.
If satis es the inequalityT a;
(C) then for all >C, satis es the logarithmi Sobolevinequality
LSI a 2 ; 4 2 C . Proof
C Letnote F densityof probabilitywith respe tto . Assumethat F is C 2
,thegeneral ase an resultbydensity.
BytheBrenier-Gangbo-M Cann'stheorem,see [Bre91 ,GM96℄,there existsafun tionsu hthat
S =Id rH a;
Z g(S)Fd= Z gd: The fun tion is a L a;
- on ave fun tion and if is C 2
, a lassi al argument of onvexity (see
hapter 2 of [Vil03℄), one has D[rH a;
Ær(x)℄ is diagonalizable with real eigenvalues, all less than1.
A ordingtotheassumptionmadeonfun tion,one an assumethatS issuÆ ientlysmoothand
we obtainfor x2R n , F(x)e '(x) =e 'ÆS(x) det(rS(x)) : (15)
Moreoverthisfun tion gives theoptimaltransport, i.e.
T L a; (Fd;d)= Z L a; (rH a; Ær)Fd:
Thenby(15) , one has forx2R n
,
logF(x)='(x) '(x rH a;
Ær(x))+logdet(Id D[ rH a;
Ær(x)℄ ):
Thensin eD[rH a;
Ær(x)℄is diagonalizablewithreal eigenvalues,all lessthan1,we get
logdet(Id D[rH a;
Ær(x)℄) div(rH a;
Ær(x)) :
Sin e'is onvexwe have '(x) '(x rH a; Ær(x))rH a; Ær(x)r'(x) and we obtain Ent ( F) Z f rH a; Ær(x)r'(x) div(rH a; Ær(x)) gF(x)d(x);
afterintegrationbyparts
Ent ( F) Z rF rH a; Ærd:
Let>0 and letuse Young inequalityforthe ombined fun tionsL a; and H a; rF F rH a; ÆrH a; rF F +L a; (rH a; Ær): Thus Ent (F) 1 Z H a; rF F Fd+ 1 Z L a; (rH a; Ær)Fd Z H a ; rF F Fd+ 1 T La; ( Fd;d) :
Thusifsatis es theinequalityT a;
(C) we getforall >C
Ent (F) 2 C Z H a ; rF F Fd:
Letusnotenowf 2 =F,we get Ent f 2 2 C Z H a ; 2 rf f f 2 d 4 2 C Z H a 2 ; rf f f 2 d:
Then satis es,forall >C inequalityLSI a 2 ; 4 2 C . B
tionof Theorem 2.9): LSI a; (C)!TaC 2 ; (C=4) T a; (C)! LSI a 2 ; 4 2 C >C :
Noti e,as it isthe asefor the traditional logarithmi Sobolevinequality,than thereisa losson the levelofthe onstantsinthedire tiontransportationinequalityimplieslogarithmi Sobolevinequality. When = =2, we getas in [OV00℄,T
;2
(C) !LSI ;2
(16C) . As in [OV00℄, Theorem 2.9 an be
modi ed in the aseHess(')>Id , where 2R.
Alsoletusnoti ethatasinthequadrati asewedonotknowifthesetwo inequalitiesareequivalent.
AsinProposition2.3, hereare some properties oftheinequalityT La;
(C).
Proposition 2.11 i. Let us re all Marton's theorem on on entration inequality.
Assume that satis es a transportation inequality T L
a;
(C) then satis es the following on entration inequality 8AR n ; with (A)> 1 2 ; (( A r ) )2e ( 1 C La; (r)) ; where (A r ) =f x2R n ; d(A;x)>rg.
ii. AsinProposition2.3,thepropertiesoftensorisationarealsovalidfortransportationinequality
T a; (C). Let 1 and 2
be two probability measures on R n 1 and R n 2 . Suppose that 1 (resp. 2 )
satis es the inequality T a; (C 1 ) (resp. T a; (C 2
)) then the probability 1 2 on R n1+n2 , satis es inequality T a; (D), whereD=maxf C 1 ;C 2 g.
iii. Ifthemeasureveri esT a;
(C),thensatis esaPoin areinequality (9)withthe onstantC.
Proof
C The demonstration of i, ii of these results is a simple adaptation of the traditional ase, we
returnto thereferen esforproofs(for example hapters3,7 and 8of [ABC +
00 ℄). The proof of iii is an adaptation of the quadrati ase. Supposethat satis es a T
a;
(C). By a lassi alargument of Bobkov-Gotze, themeasure satis es the dualform of T
a; (C) whi h is the inequality(14) , Z e 1 C Q 1 f de R 1 C fd ; (16) whereQ 1
f is de nedasin(13) withthefun tionL a;
. Letnotef =g with g,C
1
and bounded,weget
Q 1 f(x)=Q 1 (g)(x) = inf z2R n n g(x z)+L a ; (z) o =g(x) 2 2 jrgj 2 +o( 2 ) Thenweobtainby(16), 1+ C Z gd 2 2C Z j rgj 2 d+ 2 2C 2 Z g 2 d1+ C Z gd+ 2 2C 2 Z gd 2 +o( 2 ); implythat Var (g)C Z jrgj 2 d:
Unfortunately,asinthetraditional aseofthetransportationinequality,we donotknowifthisone haspropertyofperturbationasforinequalityLSI
a; (C).
3 An important example on R, the measure
Let >1 and de netheprobabilitymeasure on R by (dx)= 1 Z e jxj dx; whereZ = R e jxj dx.
Theorem 3.1 Let 2℄1;2℄. There existsA;B >0su h that the measure
satis es the following modi ed logarithmi Sobolev inequality, for any smooth fun tion f on R su h that f > 0 and R f 2 d =1 we have Ent f 2 AVar ( f)+B Z f>2 f 0 f f 2 d ; (17) where 1= +1= =1 and Ent f 2 = Z f 2 log f 2 R f 2 d d andVar ( f)= Z f 2 d Z fd 2 :
In the extreme ase, =1, we obtain the following inequality: for all f smooth enough su h that jf 0 j1, Ent f 2 AVar (f): (18)
Corollary 3.2 Assumethat f isa smooth fun tionon R. Thenwe obtainthe following estimation
Ent f 2 AVar (f)+B Z f 0 f f 2 d ; (19) where = ( f + >2 s Z f 2 + d ) [ ( f >2 s Z f 2 d ) ; f +
=max (f;0) and f =max( f;0).
Proof C We have f 2 =f 2 + +f 2 . Then Ent f 2 = sup Z f 2 gd with Z e g d 1 = sup Z f 2 + gd + Z f 2 gd with Z e g d 1 Ent f 2 + +Ent f 2 :
ByTheorem3.1 thereexists A;B >0 independent off su h that
Ent f 2 + AVar (f + )+B Z + f 0 + f + f 2 + d ;
Ent f 2 AVar (f )+B Z f 0 f f 2 d ; where + = n f + >2 q R f 2 + d o and = n f >2 q R f 2 d o .
To on lude, itisenough to noti ethat
Var (f + )+Var (f ) = Z f 2 d Z f + d 2 + Z f d 2 ! Var (f); and Z + f 0 + f + f 2 + d + Z f 0 f f 2 d = Z f 0 f f 2 d : B
Itimpliesthere are a
>0 andC
<1, su h that
satis es alogarithmi Sobolevinequalityof fun tionH
a ;
with onstant C
. Indeed, this is lear that
satis es a Poin are inequality, (see hapter 6 of [ABC + 00 ℄), with onstant <1, Var (f) Z f 02 d :
Then,by inequality(17) , we obtainforanysmoothfun tion f on R,
Ent f 2 A Z f 0 2 d +B Z f 0 f f 2 d :
LetusgiveafewhintontheproofoftheTheorem3.1,whi hwillenableustopresentkeyauxiliary lemmas. We rst usethe followinginequality
Z f 2 lnf 2 d 5 Z (f 1) 2 d + Z (f 2) 2 + ln(f 2) 2 + d (20)
whereitis obvious thattrun ationargumentsare ru ial. We willthenneedthefollowinglemma:
Lemma3.3 Letbeaprobability measureonR andletf >0su hthat R f 2 d=1then weobtain i: Z (f 1) 2 2Var (f): ii: Z f2 f 2 d8Var (f): iii: Z f>2 f 2 lnf 2 d ln4 ln4 1 Ent f 2 <4Ent f 2 : Proof C i. We have Z (f 1) 2 d=Var ( f)+ 1 Z fd 2 : R f 2 d=1 implythat0 R fd1,then R fd 2 R fd. Then Z (f 1) 2 dVar ( f)+(Var (f)) 2 ;
butsin e f 2 d=1, Var (f)1,then Z (f 1) 2 d2Var (f):
ii. One veri es triviallythat when x2,x 2
4(x 1) 2
and applyi.
iii. Let usgivethe proofgiven in[CG04 ℄.
Ifx>0 we have xlnx+1 x>0whi hyields
Z f2 f 2 lnf 2 d+(f 2) Z f2 f 2 d>0; hen e Ent f 2 > Z f>2 f 2 lnf 2 d Z f>2 f 2 d: Sin e Z f>2 f 2 d 1 ln4 Z f>2 f 2 lnf 2 d; we obtain Ent f 2 > 1 1 ln4 Z f>2 f 2 lnf 2 d: B
Re alltheHardy's inequalitypresentedinthe introdu tion: let ; be Borel measureson R +
, the best onstant Asothat every smoothfun tionf satis es
Z 1 0 (f(x) f(0)) 2 d(x)A Z 1 0 f 02 d (21)
is niteifand onlyif
B=sup x>0 ([x;1[) Z x 0 d a d 1 dt (22)
is niteand whenA is nitewe have
B A4B:
Wethen presentdi erent proof ofthedesired inequality,startingfrom (20), a ording tothe value of Ent
f
2
,in whi h Hardy'sinequalityplays a ru ial role. First, when theentropyis large we willneed
Lemma3.4 Let h de ned as follow.
h(x)= 1 if jxj1 jxj 2 if jxj>1
Thenthere exists C h
>0 su h that for every smooth fun tiong we have
Ent g 2 C h Z g 02 hd : (23)
C We use Theorem3 of [BR03℄whi h isa re nement of the riterion of a Bobkov-Gotze theorem (seeTheorem5.3 of[BG99℄). The onstant C h satis es max(b ;b + )C max (B ;B + ) where b + =sup x>0 ([x;+1[)log 1+ 1 2 ([x;+1[) Z x 0 Z e jtj h(t) dt; b =sup x<0 (℄ 1;x℄)log 1+ 1 2 (℄ 1;x℄) Z 0 x Z e jtj h(t) dt; B + =sup x>0 ([x;+1[)log 1+ e 2 ([x;+1[) Z x 0 Z e jtj h(t) dt; B =sup x<0 (℄ 1;x℄)log 1+ e 2 ([ 1;x[) Z 0 x Z e jtj h(t) dt:
Aneasy approximationprove thatforlargepositive x
([x;1[)= Z 1 x 1 Z e jtj dt 1 1 Z x 1 e x (24) Z x 0 Z e jtj h(t) dt 1 Z x e x ;
andone may prove same equivalentfornegative x. Asimple al ulation thenyieldsthat onstants
b +
,b ,B +
and B are niteand thelemma isproved. B
Notethatthefun tionh isthesmallestfun tionsu hthatthe onstantC h
intheinequality(23)is
nite,it\saturates" theinequalityon in nity.
Inthe ase ofsmallentropy,wewilluseso- alled -Sobolevinequalities(even ifour ontext is less
general),seeChafa[Cha04 ℄fora omprehensivereview,andBarthe-Cattiaux-Roberto[BCR04 ℄for ageneral approa h inthe ase of measure
.
Lemma3.5 Let g bede ned on [T;1[ with T 2[T 1
;T 2
[for some xed T 1 ;T 2 , g(T)=2; g>2 and Z 1 T g 2 d 13; then Z 1 T (g 2) 2 + (g 2 ) C g Z [T;1[\fg>2g g 0 2 d ; (25) where (x)=ln 2( 1) (x). The onstant C g
depend on the measure
but does not depend on the value of T 2[T 1 ;T 2 ℄. Proof
C Let use Hardy's inequality as explained in the introdu tion. We have g(T) = 2. We apply inequality(6)withthe fun tion(g 2)
+
and thefollowingmeasures
d= lng 2 2( 1) d and = :
Thenthe onstantC ininequality(25) is niteifand onlyif
B =sup x>T Z x T Z e jtj dt Z 1 x lng 2 2( 1) d ;
Sin e 2( 1) = <1 thefun tion x ! (lnx) 2( 1)
is on ave on [4;1[. By Jensen inequality we obtainforallx>T,
Z 1 x lng 2 2( 1) d ln 2( 1) R 1 x g 2 d ( [x;1[) ([x;1[) :
Thenbythepropertyof g we have
B sup x>T Z x T Z e jtj dtln 2( 1) 13 ([x;1[) ([x;1[) sup x>T 1 Z x T 1 Z e jtj dtln 2( 1) 13 ([x;1[) ( [x;1[) :
Usingtheapproximationgiven inequality(24)weprovethatB is nite,boundedbya onstantC g whi hdoesnotdependon T. B
Assaid before, we dividethe proof of Theorem 3.1 intwo parts: large and smallentropy, both in the aseof positive fun tion. Letusnowpresentthe proofinthe aseof largeentropy.
Large entropy ase.
Proposition 3.6 Suppose that 2℄1;2℄. There exists A;B >0 su h that for any fun tions f >0
satisfying Z f 2 d =1 andEnt f 2 >1 we have Ent f 2 AVar ( f)+B Z f>2 f 0 f f 2 d : (26) If =1, when jf 0 j1, then Ent f 2 AVar (f). Proof of Proposition 3.6 C Let f >0 satisfying R f 2 d =1.
A areful studyofthefun tion
x! x 2 lnx 2 +5(x 1) 2 +x 2 1+(x 2) 2 + ln(x 2) 2 +
provesthat forevery x2R
x 2 lnx 2 5(x 1) 2 +x 2 1+(x 2) 2 + ln(x 2) 2 + :
ThenweobtainbyLemma3.3.i, re alling that R f 2 d =1 andf >0, Z f 2 lnf 2 d 5 Z (f 1) 2 d + Z (f 2 1)d + Z (f 2) 2 + ln(f 2) 2 + d 10Var (f)+ Z (f 2) 2 + ln(f 2) 2 + d
whi his theannoun edstartingpoint inequality(20).
Sin e R f 2 d
=1,one an easilyprove that
Z (f 2) 2 + d 1;
then (f 2) 2 + ln(f 2) 2 + d Ent (f 2) 2 + ;and Ent f 2 10Var ( f)+Ent (f 2) 2 + :
Hardy'sinequalityofLemma 3.4withg=(f 2) + gives Ent (f 2) 2 + C h Z (f 2) 02 + hd =C h Z f>2 f 02 hd : (27)
For p 2℄1;2[ and q > 2 su h that and 1=p+1=q = 1 and we have for every x;y > 0 by Young inequality, xy x p p + y q q : (28) If =1,then ifjf 0
j1,thenthere existsC>0su hthat
C h Z f>2 f 02 hd CVar ( f)+ 1 2 Ent f 2
wherewe usedLemma 3.3.iiandthe largeentropy ase. We thendedu ethe resultwhen =1. Considerthen 2℄1;2℄and = =( 1). Letp= =2andq = =( 2). Let">0and letapply inequality(28)to theright termof (27),we obtain
1 " ( 2)= f 0 f 2 " ( 2)= h 2 " ( 2)=2 f 0 f + 2 "h =( 2) ; then Ent (f 2) + 2 2C h " ( 2)=2 Z f>2 f 0 f f 2 d + 2 C h " Z f>2 h =( 2) f 2 d
Letaprobabilitymeasure,thenwe haveforevery fun tionf su h that R
f 2
d=1,and forevery
measurablefun tiong Z f 2 gdEnt f 2 +log Z e g d:
Let>0 and we applythepreviousinequalitywithg=h =( 2) , Ent (f 2) + 2 2C h " ( 2)=2 Z f>2 f 0 f f 2 d + ( 2)C h " Ent f 2 +log Z exp h =( 2) d : (29)
Sin e = =( 1),let notethat h(x) =( 2)
=x
ifjxj>1. Then we x=1=2. Andnote
A=log Z exp 1 2 h =( 2) d <1:
Thenwenow x "=inff =(A( 2)4C h ); =(( 2)4C h )g . We obtain Ent (f 2) + 2 C h " ( 2)=2 Z f>2 f 0 f f 2 d + 1 4 Ent f 2 + 1 4 : Ent f 2 >1,implies Ent f 2 20Var ( f)+ 2C h " ( 2)=2 Z f>2 f 0 f f 2 d : B
B depends on the measure studied. We will see in se tion 4 that one an adapt the demonstration for other measures.
Remark 3.8 WiththesamemethodasdevelopedinProposition3.6we anprove theinequality (26)
withoutVar
( f). Supposethat 2℄1;2℄. ThereexistsA>0su hthatforanyfun tionsf satisfying
Z f 2 d =1 andEnt f 2 >1 we have Ent f 2 A Z f 0 f f 2 d :
Small entropy ase.
Letusnowgivethe resultwhen Ent
f 2
is small.
Proposition 3.9 Let 2[1;2℄. ThereexistsA;A 0
>0su h that forany fun tionsf >0satisfying
Z f 2 d =1 andEnt f 2 1 we have Ent f 2 34Var (f)+A Z f>2 f 0 f f 2 d :
when 2℄1;2℄, and if =1 we get
Ent f 2 A 0 Var ( f): Proof of Proposition 3.9 C Letf >0 satisfying R f 2 d
=1. Likein Proposition3.6, we start withinequality(20), whi h readilyimplies Ent f 2 = Z f 2 lnf 2 d 10Var (f)+ Z (f 2) 2 + lnf 2 d : (30)
We will now ontrol the se ond term of the right hand side of this last inequality via the use of -Sobolev inequalities, namely Lemma 3.5. Therefore we have to onstru t a fun tion g, greater
than2,whi h satis es(fora well hosen T), whenEnt f 2 1, (g1) Z 1 T g 2 d 12; (g2) Z 1 T (g 2) 2 + (g 2 )d C Z (f 2) 2 + lnf 2 d ; (g3) Z 1 T g 02 d C Z [T;1[[ff2g f 0 f f 2 d +DEnt f 2 , with(x)=ln 2( 1) (x), 0<D1=2 and (x)=x :
Letnowde neT 1 <0and T 2 >0su hthat (℄1;T 1 ℄)= 3 8 ; ( [T 1 ;T 2 ℄)= 1 4 and ([T 2 ;+1[)= 3 8 : Sin e R f 2 d =1there exists T 2[T 1 ;T 2 ℄ su hthat f(T)2.
1 g=2+(f 2) + ln f 2 on [T;1[; where =(2 )=(2 ).
Fun tiong satis esg(T)=2 andg(x)>2forall x>T. Letnow ompute R 1 T g 2 d . Wehave Z 1 T g 2 d 2 Z 1 T1 4d +2 Z 1 T1 (f 2) 2 + ln 2 f 2 d 4+2 Z [T 2 ;1[\ff>2g f 2 ln 2 f 2 d : Sin e2 2[0;1℄we have ln 2 f 2 lnf 2
on f f >2g . ThenweobtainbyLemma3.3.iii Z 1 T g 2 d 5+2 Z f>2 f 2 ln 2 f 2 d 5+8Ent f 2 13; sin eEnt f 2 1.
AssumptionsonLemma 3.5aresatis ed, we obtainby inequality(25)
Z 1 T (g 2) 2 + ln 2( 1) g 2 d C g Z [T;1[\fg>2g g 0 2 d :
Letus omparethe various termsnow.
Firstlyletnote u=2( 1)= , we have
(g 2) 2 + ln u g 2 =(f 2) 2 + ln 2 f 2 ln u 2+(f 2) + ln f 2 2 : On f f >2g we have 2+(f 2) + ln f 2 > 2+(f 2) + K ; where K = ln 4. Sin e K > 1 and u+2 =1 one has (g 2) 2 + ln u g 2 >(f 2) 2 + ln 2 +u f 2 =(f 2) 2 + lnf 2 : Thenweobtain Z 1 T (f 2) 2 + lnf 2 d Z 1 T (g 2) 2 + ln 2( 1) g 2 d : (31)
Se ondlyone hason f f >2g
g 0 =f 0 ln f 2 1+ 2 f 2 flnf 2 ; thenusinglnf 2 >ln4 one obtain g 0 2 f 0 2 ln 2 f 2 1+ 2 ln4 2 : LetnoteD=(1+ 2 =ln4) 2 ,one has Z [T;1[\ff>2g g 02 d D Z [T;1[\ff>2g f 02 ln 2 f 2 d ; (32) on[T;1[\f f >2g :
Then,usinginequalities(31)and (32) , there existsC >0(independentof T 2[T 1 ;T 2 ℄), su h that Z 1 T (f 2) 2 + lnf 2 d C Z [T;1[\ff>2g f 02 ln 2 f 2 d :
jf 0 j1,byC R f2 f 2 d
whi hisitselfbounded,byLemma3.3.ii,by8CVar
(f)whi h on ludes theproof inthis ase.
When 2℄1;2℄, we applyInequality(28) withq = =(2 )and p= =(2( 1)). We obtainfor every ">0, Z [T;1[\ff>2g f 0 f 2 ln 2 f 2 f 2 d 2( 1) " 2 2( 1) Z [T;1[\ff>2g f 0 f f 2 d +" 2 Z [T;1[\ff>2g f 2 lnf 2 d :
Fix"su h that"C 2
<1=16,then thereexists A>0su hthat
Z 1 T (f 2) 2 + lnf 2 d A Z [T;1[\ff>2g f 0 f f 2 d + 1 16 Z [T;1[\ff>2g f 2 lnf 2 d :
UsingLemma3.3.iiiwe have,
Z 1 T (f 2) 2 + lnf 2 d A Z [T;1[\ff>2g f 0 f f 2 d + 1 4 Ent f 2 :
Thesame method an be usedon ℄ 1;T℄and then,there is A 0 >0su hthat Z T 1 (f 2) 2 + lnf 2 d A 0 Z [ 1;T℄\ff>2g f 0 f f 2 d + 1 4 Ent f 2 :
Andthen, weget
Z (f 2) 2 + lnf 2 d (A+A 0 ) Z f>2 f 0 f f 2 d + 1 2 Ent f 2 :
Notethat onstants A and A 0
don't dependon T.
Then,by inequality(30) , Proposition3.9is proved. B
Letusgivenowa proof of thetheorem.
Proof of Theorem 3.1
C The proofof thetheorem is asimple onsequen e ofPropositions3.6and 3.9. B
4 Extension to other measures
We will present in this se tion modi ed logarithmi Sobolev inequality of fun tion H for more general measure than
whi h an be derived using the proof arried on in Se tion 3: the large entropy asewheretheoptimalHardyfun tionhisidenti edandusedtoderivetheoptimalH,and thesmallentropy ase where and g (used on the proof of Proposition3.8) have to be identi ed
leadingto thesame H fun tion.
Letus rst onsiderthe followingprobabilitymeasure ;
for 2[1;2℄and 2R de nedby
; (dx) = 1 Z e '(x) dx where'(x)=jxj (logjxj) forjxj1
; Sobolevinequality: for any smooth f on R su h that
R f 2 d =1 and f >0, we have Ent ; f 2 AVar ; ( f)+B Z f>2 H f 0 f f 2 d ; ; (33)
where H ispositive smooth and givenfor x2 by
H(x)= x 1 log 1 x if 2℄1;2[; 2R; H(x)=x 2 e x 1= if =1; 2R + and H(x)=x 2 log (x) if =2; 2R : Proof
C We willmimi loselythe proof given inthe
ase, onsidering large and smallentropy ase. We willnotpresent allthe al ulusbutgive theessentialarguments.
Letnowtreat the ase 2℄1;2[.
Large entropy. We will rst applyLemma 3.4to measure ;
,one hasthen that b +
, b , B +
,B are niteifone take h positive smooth
h(x)= x 2 log x jxj2:
Onehasthento determineH to onstru t su h thatthere exists>0with (h) exponentially integrable withrespe tto
; and H = (x 2 ) where
isthe Fen hel-Legendre transformof .
Consideringthe exponentialintegrability onditionleads usto onsider (x) behaving
asymptoti- allyasx 2 log 2 2
x. Onemay thusderive theasymptoti behaviorof
and nally H.
Small entropy. One desires here to apply Lemma 3.5, evaluating and then build the fun tion g satisfying onditions (g1), (g2) and (g3). By Hardy's inequalityand arguments in the proof of Lemma3.5, one may hoose forxlargeenough as
(x)=log 2 1 (x)(loglogx) 2 : Settingthen g=2+(f 2) + log 2 2 f 2 (loglogf 2 ) ;
onemaythenverify (g1), (g2)and (g3) with =H de nedinthe largeentropystep. Now if =1 and >0,thenthesame argumentsgivesthatfor largeenough x
(x)=xlog 2 x; (x)=xe x 1=(2 ) and H(x)=x 2 e x 1= :
If =2 and 0,we have forlargeenoughx.
(x)= 1 x e 2x 1= ; (x)=xlog x and H(x)=x 2 log x: B
Remark 4.2 i. Using on e again Herbst's argument, we may derive on entration properties
for the measure ;
of desired order, for every Lips hitz fun tion F with kFk l ip
1, there exists C >0 su h that, for all >0,
; ( jF ; (F)j)2e Cmin( log ; 2 ) :
family of measures ;
. Indeed, using Hardy's hara terization of this inequalities obtained by Barthe-Roberto [BR03 , Th. 13and Prop. 15℄, one mayshow that
;
satis es an I( =2)
inequality if 0 and an I( =2 ) ( being arbitrary small) for < 0, whi h entails onsequently notoptimal on entration properties.
iii. By the hara terizationof the spe tral gap propertyon R, one obtains that ea h measure ; satis es a Poin areinequality and thus a modi ed logarithmi Sobolev inequality.
Following the previous proof, we may generalize the family ;
adding an expli it multipli ative termto the potential jxj
log
jxj, as for exampleloglog
jxj whi h will give us new modi ed log-arithmi Sobolevinequality, butea h of this new measure has to be onsidered \one-by-one" (we
hope some general results for ' onvex). We may now state a result enabling us to get the sta-bilityof these modi edlogarithmi Sobolevinequalitybyaddition of an unboundedperturbation: onsiderthe measures
d (x)=exp jxj jxj 1 os (x) dx Z ; 2℄1;2℄; d ;b (x)=(1+x) b e x dx Z ;b 1 x>0 ; 2℄1;2℄;b2R:
Proposition 4.3 Thereexistsa>0su hthatthemeasures
and ;b
satisfyalogarithmi Sobolev
inequality of fun tionH a;
.
Proof
C Following the proof given in Se tion 3 ,one sees that the result hold true on e one mayverify thattheHardy'sinequalitiesofLemma 3.4andLemma 3.5holdwiththehand obtainedforthe
aseof
. It iseasily he ked on eremarked that
log d (x) dx 1 jxj and log d (x) dx 0 1 ( 1)jxj 1
andthe same for ;b
. B
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