1/

l

C. R. Acad. Sci. Paris, Ser. I 340 (2005) 205–208http://france.elsevier.com/direct/CRASS

Partial Differential Equations

On logarithmic Sobolev inequalities for higher order fractionaderivatives

Athanase Cotsiolis, Nikolaos K. Tavoularis1

Department of Mathematics, University of Patras, Patras 26110, Greece

Received 18 October 2004; accepted 16 November 2004

Available online 11 January 2005

Presented by Thierry Aubin

Abstract

OnRn, we prove the existence of sharp logarithmic Sobolev inequalities with higher fractional derivatives. Lets be a positivereal number. Any functionf ∈ Hs(Rn) satisfies∫

Rn

∣∣f (x)∣∣2 ln

( |f (x)|2‖f ‖2

2

)dx +

(n + n

slnα + ln

s�(n2)

�( n2s

)

)‖f ‖2

2 � α2

πs

∥∥(−�)s/2f∥∥2

2

with α > 0 be any number and where the operators(−�)s/2 in Fourier spaces are defined by(−�)s/2f (k) := (2π |k|)s f (k).To cite this article: A. Cotsiolis, N.K. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 340 (2005). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Résumé

Sur les inégalités de Sobolev logarithmiques pour les dérivées fractionnelles d’ordre supérieur.Sur Rn, on établil’existence d’inégalités de Sobolev logarithmiques optimales pour les dérivées fractionnelles d’ordre supérieur. Soits etα deuxréels positifs. Pour toute fonctionf ∈ Hs(Rn), on établit l’inégalité suivante :∫

Rn

∣∣f (x)∣∣2 ln

( |f (x)|2‖f ‖2

2

)dx +

(n + n

slnα + ln

s�(n2)

�( n2s

)

)‖f ‖2

2 � α2

πs

∥∥(−�)s/2f∥∥2

2.

L’opérateur(−�)s/2 est defini dans les espaces de Fourier par(−�)s/2f (k) := (2π |k|)s f (k). Pour citer cet article : A. Cot-siolis, N.K. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 340 (2005). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.

E-mail addresses: cotsioli@math.upatras.gr (A. Cotsiolis), niktav@master.math.upatras.gr (N.K. Tavoularis).1 Supported by the Greek State Scholarship’s Foundation.

1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2004.11.030

206 A. Cotsiolis, N.K. Tavoularis / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 205–208

or

m

ables

1. Introduction

Logarithmic Sobolev inequalities have awide range of applications and have been extensively studied (see, fexample, [9,4,5,8,1,11] and the references therein).

Let � be the Laplacian inRn and letf (k) denote the Fourier transform off ∈ L1(Rn):

f (k) =∫Rn

e−2π ikxf (x)dx.

The operators(−�)s/2 are defined in Fourier spaces (i.e. in spaces with functions which have Fourier transforsuch asLp(Rn) for 1 � p � 2) as multiplication by(2π |k|)s , i.e.

(−�)s/2f (k) := (2π |k|)s f (k).

The spaceHs(Rn) is endowed with the inner product

(f, g)Hs(Rn) =∫Rn

¯f (k)g(k)

(1+ (

2π |k|)2s)dk

and

‖f ‖2Hs(Rn) :=

∫Rn

∣∣f (k)∣∣2(1+ (

2π |k|)2s)dk.

The original form of thelogarithmic Sobolev inequality (cf. [12]) is∫Rn

∣∣g(x)∣∣2 ln

( |g(x)|2‖g‖2

2

)dm � 1

π

∫Rn

∣∣∇g(x)∣∣2 dm,

where dm = e−π |x|2 dx is the Gauss measure and‖g‖2 is, of course, the norm inL2(Rn,dm).Choosingg(x) = eπ |x|2/2f (x) in the above inequality, and considering an homothetic change of vari

(α > 0), we find ([10], Th. 8.14, p. 223):∫Rn

∣∣f (x)∣∣2 ln

( |f (x)|2‖f ‖2

2

)dx + n(1+ lnα)‖f ‖2

2 � α2

π

∫Rn

∣∣∇f (x)∣∣2 dx, (1)

where theL2 norm is with respect to Lebesgue measure.The purpose of this Note is to give a generalization of inequality (1) with the operators(−�)s/2, s > 0.

Definition 1.1 [6]. We consider the operator semigroups e−t (−�)s , t > 0, defined by its Fourier transform:

(e−t (−�)sf )∧(k) = e−t (2π |k|)2s

f (k).

2. Logarithmic Sobolev inequalities

Theorem 2.1.Let f be any function in Hs(Rn) and let α > 0 be any real number. Then∫Rn

∣∣f (x)∣∣2 ln

( |f (x)|2‖f ‖2

2

)dx +

(n + n

slnα + ln

s�(n2)

�( n2s

)

)‖f ‖2

2 � α2

πs

∥∥(−�)s/2f∥∥2

2 (2)

A. Cotsiolis, N.K. Tavoularis / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 205–208 207

e

.

Proof. Let Fs(x) be the function with Fourier transformFs(k) = e−t (2π |k|)2s. Then e−t (−�)sf = Fs ∗ f . However,Fs(x) = Fs(−x) = Fs(x) becauseFs ∈ L2(Rn).

By Young’s inequality [3,10] we see that e−t (−�)s mapsLp(Rn) into Lq(Rn) providedp < q . Indeed,

∥∥e−t (−�)sf∥∥

q= ‖Fs ∗ f ‖q = ‖ Fs ∗ f ‖q �

(CrCp

Cq

)n

‖ Fs‖r‖f ‖p (3)

with 1+ 1q

= 1r

+ 1p

andC2p = p1/p/p′1/p′

(p′ denotes the dual index ofp).

‖ Fs‖r � Cnr ′ ‖Fs‖r ′ (4)

according to Hausdorff–Young’s inequality withr ′ = 11/p−1/q

= pqq−p

. Also

‖Fs‖r ′r ′ =

∫Rn

e−t (2π |k|)2s r ′dk = 2πn/2

�(n/2)

∞∫0

e−t (2πρ)2s r ′ρn−1 dρ

= 2πn/2

�(n2)

�(n+2s2s

)

n(2π)−n

(t

1/p − 1/q

)−n/2s

. (5)

So,

∥∥e−t (−�)sf∥∥

q�

(Cp

Cq

)n(tKs

1/p − 1/q

)− n2s ( 1

p − 1q )

‖f ‖p (6)

whereKs = (2π)2s[ 2πn/2

�(n/2)�(n+2s/2s)

n]−2s/n.

We setq = 2 and let

t = α2(

1

p− 1

2

)/πs → 0. (7)

From (6) and (7) we obtain the inequality:

‖f ‖22 − ‖f ‖2

p +(

1−((

Cp

C2

)n(Ks

πsα2

)−ntπs/2sα2)2)‖f ‖2

p � ‖f ‖22 − ∥∥e−t (−�)sf

∥∥22 (8)

Note that the right-hand side of (8), when divided by 2t , tends to‖(−�)s/2f ‖22 (Theorem announced in [6], th

proof is in the Appendix below).If f ∈ C∞

c (Rn) we have

d

dp‖f ‖2

p|p=2 = 1

2

∫Rn

∣∣f (x)∣∣2 ln

( |f (x)|2‖f ‖2

)dx.

A straightforward computation leads to

limp→2

1− ((Cp/C2)n(Ks/π

sα2)n(p−2)/4π)2

2− p= 1

2

(n + n

slnα + ln

s�(n2)

�( n2s

)

). (9)

Eqs. (8) and (9) prove inequality (2) forf ∈ Hs(Rn) by density [6,13]. Indeedf 2(x)(ln |f (x)|)+ is in-tegrable according to the Sobolev theorem [2]. Let{fi} ⊂ C∞

c (Rn) such thatfi → f in Hs(Rn) and a.e.∫f 2

i (x)(ln(fi(x))+ → ∫f 2(x)(ln(f (x))+ according to the Lebesgue theorem.�

Remark 1. The constants in inequality (2) are the best one because we use sharp inequalities in the proof

208 A. Cotsiolis, N.K. Tavoularis / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 205–208

s,

98)

02)

95

99.

,

Remark 2. We have equality in (2) only fors = 1. Since fors = 1 H–Y inequality (4) is strict.

Remark 3. For� ∈ N, we have seen (cf. [7]) that‖∇�f ‖2 = C‖(−�)�/2f ‖2 where

C = 2−�

�−1∏h=−�

(n + 2h)1/2[�((n − 2�)/2)

�((n + 2�)/2)

]1/2

.

Thus we can have the following logarithmic inequality:∫Rn

∣∣f (x)∣∣2 ln

( |f (x)|2‖f ‖2

2

)dx +

(n + n

�lnα + ln

��(n2)

�( n2�

)

)‖f ‖2

2 � α2

C2π�‖∇�f ‖2

2

with � ∈ N andC as given above.

Appendix. Proof of Theorem 1.2 announced in [6]

A function f is in Hs(Rn) if and only if it is in L2(Rn) andI ts (f ) = 1

t[(f,f ) − (f,e−t (−�)sf )] is uniformly

bounded and we have in which case supt>0 I ts (f ) = limt→0 I t

s (f ) = (f, (−�)sf ).

Proof.

I ts (f ) = 1

t

[(f,f ) − (

f,e−t (−�)sf)] = 1

t

[ ∫Rn

∣∣f (k)∣∣2(1− e−t (2π |k|)2s )

dk

].

When we pass to the limit

limt→0

I ts (f ) =

∫Rn

(2π |k|)2s∣∣f (k)

∣∣2 dk = ∥∥(−�)s/2f∥∥2

2 (10)

so if f ∈ Hs(Rn), the limit exists and (10) holds. If the limit exists for somef ∈ L2(Rn), thenf ∈ Hs(Rn).Moreover, supt>0 I t

s (f ) = limt→0 I ts (f ). �

References

[1] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu,C. Roberto, G. Scheffer, Sur les inégalités de Sobolev logarithmique10, Société Mathématique de France, Paris, 2000.

[2] Th. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer, 1998.[3] W. Beckner, Inequalities in Fourier analysis, Ann. Math. 102 (1975) 159–182.[4] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Ann. Math. 138 (2) (1993) 213–243.[5] W. Beckner, M. Pearson, On sharp Sobolev embedding and thelogarithmic Sobolev inequalities, Bull. London Math. Soc. 30 (19

80–84.[6] A. Cotsiolis, N.K. Tavoularis, Sharp Sobolev type inequalities forhigher fractional derivatives, C. R. Acad. Sci. Paris, Ser. I 335 (20

801–804.[7] A. Cotsiolis, N.K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl. 2

(2004) 225–236.[8] R. Gilles, Une initiation aux inégalités de Sobolev logarithmiques,Cours Spécialisés, 5, Société Mathématique de France, Paris, 19[9] L. Gross, Logarithmic Sobolev inequalities, Am. J. Math. 97 (761) (1975) 1061–1083.

[10] E. Lieb, M. Loss, Analysis, second ed., Amer. Math. Soc., 2001.[11] M. Del Pino, J. Dolbeault, I. Gentil, Nonlinear diffusions, hypercontractivity and the optimalLp -Euclidean logarithmic Sobolev inequality

J. Math. Anal. Appl. 293 (2) (2004) 375–388.[12] A.J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Inform. Control 2 (1959) 255–269.[13] N.K. Tavoularis, Thèse de Doctorat de l’Université de Patras, Spécialité Mathématiques, 2004.