The mother of most Gaussian and Euclidean inequalities

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1 The mother of most Gaussia ad Euclidea iequalities M. Agueh, N. Ghoussoub ad X. Kag December, 00 Pacific Istitute for the Mathematical Scieces ad Departmet of Mathematics, The Uiversity of British Columbia, Vacouver, B. C. V6T Z, Caada December, 00 Abstract A geeral iequality established i [] relatig the relative total eergy of probability desities, their Wasserstei distace ad their productio etropy fuctioal, is show to easily imply most kow geometrical Gaussia ad Euclidea iequalities. Some of the implicatios are kow but icluded here for pedagogical reasos. Itroductio Let F : [0, ) IR be a covex fuctio, V a real fuctioal o ad let be ope, bouded ad covex. The set of probability desities over is deoted by P a () = {ρ : IR; ρ 0 ad ρ(x)dx = }. The associated Free Eergy Fuctioal is defied o P a () as HV F (ρ) := (F (ρ) + ρv )dx, which is the sum of the Iteral Eergy H F (ρ) := F (ρ)dx, ad the Potetial Eergy H V (ρ) := ρv dx. Let HV F (ρ ρ) := HF V (ρ) HF V ( ρ) deote the relative eergy betwee two desities ρ ad ρ. I [], we established the followig geeral iequality relatig the relative total eergy of probability desities, their Wasserstei distace ad their productio etropy fuctioal. Theorem. Let be ope, bouded ad covex, let F : [0, ) IR be a differetiable fuctio o (0, ) such that F (0) = 0 ad x x F (x ) be covex ad o-icreasig, ad let P F (x) := xf (x) F (x) be its associated pressure fuctio. The, for ay strictly covex C -fuctio c : IR such that lim x c(x) x =, ad ay C -potetial V : IR with D V λ (where λ IR is ot ecessarily positive), This paper was doe while this author held a postdoctoral fellowship at UBC. The three authors were partially supported by a grat from NSERC. This paper is part of this author s PhD s thesis uder the supervisio of N. Ghoussoub.

2 we have for all probability desity fuctios ρ 0 ad ρ o, satisfyig supp ρ 0, ρ 0 > 0 a.e. o ad P F (ρ 0 ) W, (), HV F +c(ρ 0 ρ ) + λ W (ρ 0, ρ ) ρ 0 c ( (F ρ 0 + V ) ) dx + H P F c+ V.x (ρ 0), () where W is the Wasserstei distace ad where c deotes the Legedre cojugate of c defied by c (y) = sup z {y z c(z)}. Furthermore, equality holds i () wheever ρ 0 = ρ = ρ where the latter satisfies ( F (ρ (x) + V (x)) ) = c(x) a.e. () I particular, we have for ay probability desity ρ such that P F (ρ) W, (), H F +P F V x V (ρ) + λ W (ρ, ρ ) ρc ( (F ρ + V ) ) dx H P F (ρ ) + C (3) where C is the uique costat such that F (ρ ) + V + c = C while ρ =. (4) We shall see that this iequality easily implies most kow geometric iequalities. It provides a direct ad uified way for computig best costats as well as the extremals where they are attaied. The term H P F c+ V.x (ρ 0) should be see as a error term i (). It ca be itegrated i the etropy term which proves useful i the Gaussia case. If V is covex, the λ ca be take equal to 0 ad the Wasserstei distace disappears from the equatio. We the have the idetity V (x) x V (x) = V ( V (x) i such a way that a correctig momet appears i the iequality: H F +P F (ρ) ρc ( (F ρ + V ) ) dx H P F (ρ V ( V ) ) + C. (5) Also ote that the pressure P F is always positive which meas that we ca do away with the term H P F (ρ ) o the right had side. Fially, the case V = 0 amply covers the Euclidea case where the geeral iequality becomes the remarkably simple: H F +P F (ρ) ρc ( (F ρ) ) dx + C. (6) Geeralized HWI iequalities We first deduce the followig useful iequality that is relevat for the Gaussia case. Corollary. Uder the above hypothesis o ad F, let U : IR be a C - fuctio with D U µi where µ IR. The for ay σ > 0, we have for all probability desities ρ 0 ad ρ o, satisfyig supp ρ 0, ad P F (ρ 0 ) W, (), H F U (ρ 0 ρ ) + (µ σ )W (ρ 0, ρ ) σ ρ ( F ρ 0 + U ) dx. (7)

3 Proof: Use () with c(x) = σ x ad U = V + c, to obtai HU F (ρ 0 ) HU F (ρ ) + (µ σ )W (ρ 0, ρ ) (8) H P F c (U c) x (ρ 0) + ρ 0 c ( ( F ρ 0 + U c )) dx. By elemetary computatios, we have ρc ( ( F ρ + U c )) dx = σ ρ ( F ρ + U ) dx + σ ad ρ x dx ρx ( F ρ ) dx ρx U dx, H P F c (U c) x (ρ) = HP F (ρ) + ρx U dx x ρ dx. σ By combiig the last idetities, we ca rewrite the right had side of (8) as H P F c (U c) x (ρ) + ρc ( (F ρ + U c) ) dx = σ ρ ( F ρ + U ) dx ρx ( F ρ ) dx P F (ρ) dx = σ ρ ( F ρ + U ), dx + div (ρx)f (ρ) dx P F (ρ) dx = σ ρ ( F ρ + U ) dx + ρf (ρ) dx + x F (ρ) dx P F (ρ) dx = σ ρ ( F ρ + U ) dx + x F (ρ) dx + F ρ dx = σ ρ ( F ρ + U ) dx. (9) Isertig (9) ito (8), we coclude the proof. If U is uiformly covex (i.e., µ > 0) iequality (7) yields the followig iequality obtaied by Cordero et al. i [4] Corollary. (Geeralized Log Sobolev iequality) Uder the above hypothesis o ad F, let U : IR be a C -fuctio with D U µi where µ > 0. The for all probability desities ρ 0 ad ρ o, satisfyig supp ρ 0, ad P F (ρ 0 ) W, (), we have HU F (ρ 0 ρ ) (F ρ 0 + U) ρ 0 dx. (0) µ Oe ca also deduce the followig: Corollary.3 (Geeralized Talagrad Iequality) Uder the above hypothesis o ad F, let U : IR be a C -fuctio with D U µi where µ > 0. The for all probability desities ρ 0 ad ρ o, satisfyig supp ρ 0, ad P F (ρ 0 ) W, (), we have µ W (ρ ρ U ) HF U (ρ ρ U), () 3

4 where ρ U is the probability desity satisfyig ( F (ρ U ) + U ) = 0 a.e. () For that, it is sufficiet to take ρ 0 = ρ U i (7). We ow deduce the followig HWI iequalities first established by Otto-Villai [7] i the case of the classical etropy F (x) = x l x. Corollary.4 (Geeralized HWI-iequality) Uder the above hypothesis o ad F, let U : IR be a C -fuctio with D U µi where µ IR. The we have for all probability desities ρ 0 ad ρ o, satisfyig supp ρ 0, ad P F (ρ 0 ) W, (), HU F (ρ 0 ρ ) W (ρ 0, ρ ) I(ρ 0 ρ U ) µ W (ρ 0, ρ ) (3) where ad I(ρ 0 ρ U ) = σ ρ ( F ρ 0 + U ) dx, ( F (ρ U ) + U ) = 0 a.e. (4) Proof: It is sufficiet to rewrite (7) as H F U (ρ 0 ρ ) + µ W (ρ 0, ρ ) σ W (ρ 0, ρ ) + σ I(ρ 0 ρ U ), (5) the miimize the right had side over the variable σ > 0. The miimum is obviously achieved at σ = W (ρ 0,ρ ). I(ρ0 ρ U ) 3 Gaussia Iequalities Corollary. applied to F (x) = x l x yields the followig extesio of Gross Log Sobolev iequality established by Otto-Villai [7]. For ay fuctio U o, deote by σ U the itegral e U dx, ad by ρ U the ormalized fuctio e U σ U. Corollary 3. (Otto-Villai s HWI iequality) Let U : IR be a C -fuctio with D U µi where µ IR. The for ay σ > 0, the followig holds for ay oegative fuctio f such that fρ U W, ( ) ad fρ U dx =, f l(f) ρ U dx + (µ σ )W (fρ U, ρ U ) σ IR f ρ U dx. (6) f Corollary 3. (Origial Gross Log Sobolev iequality) If µ > 0 (i.e., U is uiformly covex) the for ay oegative fuctio f such that fρ U W, ( ) ad f ρ U dx =, we have f l(f ) ρ U dx µ f ρ U dx. (7) 4

5 Talagrad s iequality applied to the stadard Gaussia desity γ ad to a appropriate restrictio yields Corollary 3.3 (Cocetratio of measure iequality) For ay ɛ-eighborhood B ɛ of a measurable set B i, we have γ(b ɛ ) e ( ) ɛ l γ(b). (8) Ideed, if γ A deotes the ormalized stadard Gaussia measure restricted to a give measurable set A, the which yields (8). ɛ W (γ B ; γ \B ɛ ) l γ(b) + 4 Euclidea Log Sobolev Iequalities l γ(b ɛ ), (9) The folowig optimal Euclidea p-log Sobolev iequality was established by Becker [] i the case where p =, by Del Pio- Dolbeault [5] for < p < ad idepedetly by Getil for all p >. Corollary 4. (Geeral Euclidea Log-Sobolev iequality) Let be ope bouded ad covex, ad let c : IR be a Youg fuctioal such that its cojugate c is p- homogeeous for some p >. The, ρ l ρ dx p l ( p e p σ p/ c ( ρc ρ ρ ) ) dx, (0) for all probability desity fuctios ρ o, such that supp ρ ad ρ W, ( ). Moreover, equality holds i (0) if ρ(x) = K λ e λqc(x) for some λ > 0, where K λ = ( ) e λqc(x) dx ad q is the cojugate of p ( p + q = ). Proof: Use F (x) = x l(x) i (5). Note that here P F (x) = x which meas that H P F (ρ) = for ay ρ P a ( ). So, ρ (x) = e c(x) σ c. We the have for ρ P a ( ) ( ρ l ρ dx ρc ρ ) ( ) dx l e c(x) dx. () ρ with equality whe ρ = ρ. Now assume that c is p-homogeeous ad set Γ c ρ = ( ) ρc ρ ρ dx. c λ (x) := c(λx) i (), we get for λ > 0 that Usig ( ρ l ρ dx ρc ρ ) dx + l λ l σ c, () λρ 5

6 for all ρ P a ( ) satisfyig supp ρ ad ρ W, (). Equality holds i () if ( ) ρ λ (x) = e λqc(x) dx e λ qc(x). Hece ρ l ρ dx l σ c + if λ>0 (G ρ(λ)), where G ρ (λ) = l(λ) + ( λ p ρc ρ ) = l(λ) + Γc ρ ρ λ p. The ifimum of G ρ (λ) over λ > 0 is attaied at λ ( ) ρ = p /p. Γc ρ Hece ρ l ρ dx G ρ ( λ ρ ) l(σ c ) = ( ) p p l Γc ρ + p l(σ c) = ( ) p l p e p σc p/ Γ c ρ, for all probability desities ρ o, such that supp ρ, ad ρ W, ( ). Corollary 4. (Optimal Euclidea p-log Sobolev iequality) f p l( f p ) dx ) (C p l p f p dx, (3) holds for all p, ad for all f W,p ( ) such that f p =, where ( p ) ( ) [ ] p p p e π p Γ( +) Γ( q C p := +) if p >, [ π Γ( + )] if p =, ad q is the cojugate of p ( p + q = ). For p >, equality holds i (3) for f(x) = Ke λq x x q for some λ > 0 ad x, ( ) /p. where K = e (p ) λx q dx Proof: First assume that p >, ad set c(x) = (p ) x q ad ρ = f p i (0), where f Cc ( ) ad f p =. We have that c (x) = x p p, ad the, Γ c p ρ = f p dx. Therefore, (0) reads as f p l( f p ) dx ( ) p l p e p σc p/ f p dx. (5) Now it suffices to ote that σ c := (p ) x q e dx = 6 q ( ) π Γ q + (4) (p ) q Γ ( + ). (6)

7 To prove the case where p =, it is sufficiet to apply the above to p ɛ = + ɛ for some arbitrary ɛ > 0. Note that So that whe ɛ go to 0, we have ( ) ( ) + ɛ ɛ ɛ C pɛ = π +ɛ e [ Γ( + ) ] +ɛ Γ( ɛ +ɛ + ). lim C p ɛ = [ ( )] ɛ 0 Γ π + = C. 5 Gagliardo-Nireberg ad Sobolev Iequalities ( ] p Corollary 5. (Gagliardo-Nireberg) Let < p < ad r 0, p such that r p. Set γ := r + q, where p + q =. The, for ay f W,p ( ) we have where θ is give by p = p p f r C(p, r) f θ p f θ rγ, (7) r = θ p + θ rγ, (8) ad where the best costat C(p, r) > 0 ca be obtaied by scalig. Proof: Apply (5) with F (x) = xγ γ, where γ, which follows from the fact ( ] that p r 0,. Now, for this value of γ, the fuctio F satisfies the coditios p p of Theorem. Let c(x) = rγ q x q so that c (x) = p(rγ) p x p. Iequality (5) the gives where ρ = h r satisfies ( γ + ) f rγ rγ p f p H P F (ρ ) + C. (9) h (x) = x x q h r p (x) a.e., (30) ad where C isures that h r =. The costats o the right had side of (9) are ot easy to calculate, so oe ca obtai θ ad the best costat by a stadard scalig procedure. Namely, write (9) as rγ p f p p f p r ( ) f rγ γ + rγ f rγ r C, (3) for some costat C. The apply it to f λ (x) = f(λx) for λ > 0. A miimizatio over λ gives the required costat. The case where γ = gives the stadard Sobolev iequality. 7

8 Corollary 5. Let < p <, the we have for ay f W,p ( ), for some costat C(p, ) > 0. f p C(p, ) f p (3) By lettig p, oe the gets the isoperimetric iequality: For ay closed subset of, with σ deotig surface measure ad Lebesgue measure. σ( A) B A. (33) Similar results ca be established i the presece of a additioal covolutio operator. Refereces [] M. Agueh, N. Ghoussoub, X. Kag. Geometric iequalities via a duality betwee certai quasiliear PDEs ad Fokker-Plack equatios. Submitted, December, 00 assif/pims papers.html [] W. Becker. Geometric asymptotics ad the logarithmic Sobolev iequality. Forum Math. (999), No., [3] D. Cordero-Erausqui, B. Nazaret, ad C. Villai. A mass-trasportatio approach to sharp Sobolev ad Gagliardo-Nireberg iequalities. Preprit 00. [4] D. Cordero-Erausqui, W. Gagbo, ad C. Houdré. Iequalities for geeralized etropy ad optimal trasportatio. To appear i Proceedigs of the Workshop: Mass trasportatio Methods i Kietic Theory ad Hydrodyamics. [5] M. Del Pio, ad J. Dolbeault. The optimal euclidea L p -Sobolev logarithmic iequality. To appear i J. Fuct. Aal. (00). [6] R. McCa. A covexity priciple for iteractig gases. Adv. Math 8,, (997), [7] F. Otto ad C. Villai. Geeralizatio of a iequality by Talagrad, ad liks with the logarithmic Sobolev iequality. J. Fuct. Aal. 73, (000), [8] C. Villai. Topics i mass trasportatio. Lecture otes 00. 8