Athanase Cotsiolis

Nikolaos Con. Tavoularis

Cotsiolis, Athanase

Tavoularis, Nikolaos Con.

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804Équations aux dérivées partielles/Partial Differential EquationsSharp Sobolev type inequalities for higher fractionalderivativesAthanase Cotsiolis, Nikolaos Con. Tavoularis 1Department of Mathematics, University of Patras, Patras 26110, GreeceReceived and accepted 1 October 2002Note presented by Thierry Aubin.Abstract On Rn, n 1 and n = 2, we prove the existence of a sharp constant for Sobolev inequalitieswith higher fractional derivatives. Let s be a positive real number. For n > 2s and q = 2nn−2sany function f ∈ Hs (Rn) satisfies‖f ‖2q  Sn,s∥∥(−)s/2f∥∥22,where the operator (−)s in Fourier spaces is defined by ̂(−)sf (k) := (2π |k|)2s f̂ (k).To cite this article: A. Cotsiolis, N.C. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 335 (2002)801–804. 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SASInégalités optimales de type Sobolev pour les dérivées fractionellesd’ordre supérieurRésumé Sur Rn, n 1 et n = 2, on établit l’existence de meilleurs constantes dans les inégalités deSobolev pour les dérivées fractionelles d’ordre supérieur. Soit s un reel positif. Pour n > 2set q = 2nn−2s toute fonction f ∈ Hs(Rn) vérifie l’inégalité suivante‖f ‖2q  Sn,s‖(−)s/2f ‖22,où Sn,s est la meilleure constante. L’opérateur (−)s est defini dans les espaces de Fourierpar ̂(−)sf (k) := (2π |k|)2s f̂ (k). Pour citer cet article : A. Cotsiolis, N.C. Tavoularis,C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804. 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS1. PreliminariesThe Sobolev space Hl (Rn) is endowed with the norm ‖f ‖2Hl (Rn) =∑0αl ‖Dαf ‖2L2(Rn) for f : Rn →C with l a positive integer. This norm is equivalent to the norm ‖f ‖′2Hl (Rn) :=∫Rn|f̂ (k)|2(1 + (2π |k|)2l)dk(thanks to the Plancherel formula) where |k| = (∑ni=1 k2i )1/2 and f̂ (k) := ∫Rn e−2π ikxf (x)dx is the Fouriertransform of the function f (kx := ∑ni=1 kixi). We set (f )∧(−x)= f̂ (−x)= f ∨(x). So f = (f̂ )∨.E-mail addresses: cotsioli@math.upatras.gr (A. Cotsiolis); niktav@master.math.upatras.gr (N.C. Tavoularis). 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservésS1631-073X(02)02576-1/FLA 801A. Cotsiolis, N.C. Tavoularis / C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804DEFINITION 1.1. – A function f ∈ L2(Rn) is said to be in Hs(Rn) if and only if‖f ‖2Hs (Rn) :=∫Rn∣∣f̂ (k)∣∣2(1 + (2π |k|)2s)dk <∞.The space Hs (Rn) is endowed with the inner product(f, g)Hs (Rn) =∫Rnf̂ (k)ĝ(k)(1 + (2π |k|)2s)dk.DEFINITION 1.2. – The operator (−)s is defined in Fourier spaces (i.e., in spaces with functions whichhave Fourier transform such as Lp(Rn) for 1  p  2, see [5]) as multiplication by (2π |k|)2s , i.e.,̂(−)sf (k) := (2π |k|)2s f̂ (k).If f and g are in Hs(Rn) then the sesquilinear form(g, (−)sf ) = ∫Rnĝ(k)f̂ (k)(1 + (2π |k|)2s)dkmakes sense by Hölder inequality. Also (f, (−)sf )= ‖(−)s/2f ‖22.PROPOSITION 1.1. – (i) We have H[s]+1(Rn)⊂ Hs(Rn)⊆ H[s](Rn) where [s] is the integer part of s.(ii) For 0< δ < γ < 1, Hγ (Rn)⊂ Hδ(Rn).PROPOSITION 1.2. – If f is in Hl (Rn) then D̂αf (k)= (2π ik)αf̂ (k) with |α|  l, l an integer.PROPOSITION 1.3. – If f ∈ Hs(Rn) then |f | ∈ Hs(Rn).THEOREM 1.1. – (i) If f is in Hl (Rn), then there exists a sequence of functions in C∞c (Rn) such that‖f m − f ‖Hl (Rn) → 0 as m→ ∞.(ii) If f is in Hs(Rn) then there exists a sequence of functions in C∞c (Rn) such that ‖f m − f ‖Hs (Rn) → 0as m→ ∞.Remark 1.1. – These results are in [5] for the spaces H1(Rn) and H1/2(Rn).DEFINITION 1.3. – Set Fs(k)= e−t (2π |k|)2s . When t > 0, we define the operator e−t (−)s on functions fin Lp(Rn) (1  p  2) by (e−t (−)sf)∧(k)= e−t (2π |k|)2s f̂ (k).Remark 1.2. – The functionFs(k)= e−t (2π |k|)2s is in Lp(Rn) for p  1. By Hausdorff–Young inequality(see [5]) we have that the function F̂s is in Lp′(Rn) with 1/p+ 1/p′ = 1, so we can define its convolutionF̂s ∗ f with f ∈ Lp(Rn).THEOREM 1.2. – A function f is in Hs (Rn) if and only if it is in L2(Rn) andI ts (f )=1t[(f,f )− (f, e−t (−)sf )]is uniformly bounded and we have in which casesupt>0I ts (f )= limt→0 Its (f )=(f, (−)sf ).802To cite this article: A. Cotsiolis, N.C. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804Remark 1.3. – For the cases s = 1 and s = 1/2 Theorem 1.2 is given by [5].2. Sharp Sobolev type inequalities (see [3])The following inequality (n > 2s, s a positive real)∣∣(f, g)∣∣2  2−2sπ−n/2((n− 2s)/2)(s)(f, (−)sf )(g, |x|2s−n ∗ g)is valid for f ∈ Hs (Rn) and g ∈ Lp(Rn)THEOREM 2.1. – For n > 2s, let f ∈ Hs(Rn) and q = 2nn−2s . Then the following inequality holds:‖f ‖2q  Sn,s∥∥(−)s/2f ∥∥22,whereSn,s = 2−2s/nπ−s(n+1)/n ((n− 2s)/2)((n+ 2s)/2)[(n+ 12)]2s/nand  denotes the Gamma function. There is equality in the inequality above if and only if f (x) is a multipleof the function (µ2 + (x − α)2)−(n−2s)/2 with µ> 0 and α ∈ Rn.Remark 2.1. – (i) For s = 1 the best constant of Theorem 2.1 is given in [1,2] and [7].(ii) For s = 1/2 the best constant of Theorem 2.1 is given in [5].(iii) For s = 2 the best constant of Theorem 2.1 is given in [4] and [8].Finally we give an inequality for the space Hs (R).THEOREM 2.2. – For f ∈ Hs (R) the inequality∥∥(−)s/2f ∥∥22 + ‖f ‖22  S1,q,s‖f ‖2qholds for all 2< q <∞ and 2 qq−2s > 1 with a constant S1,q,s that satisfiesS1,q,s > (q − 1)1−1/qq−1+2/q[(1 + 1/(2s))(−1/(2s)+ q/(q − 2))π(q/(q − 2))]−(q−2)/q.Remark 2.2. – For the spaces H1/2(R) and H1(R) Lieb’s results are in [5] and [6] respectively.1 Partially supported by State Scholarship’s Foundation.References[1] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer, 1998.[2] T. Aubin, Problèmes isopèrimétriques et espaces de Sobolev, J. Differential Geom. 11 (1976) 573–598.[3] A. Cotsiolis, N. Tavoularis, Best constant for the embedding of the space Hs (Rn) into L2n/(n−2s)(Rn) andapplications, in preparation.[4] Z. Djadli, E. Hebey, M. Ledoux, Paneitz type operators and applications, Duke Math. J. 104 (1) (2000) 129–169.[5] E. Lieb, M. Loss, Analysis, 2nd edition, American Mathematical Society, 2001.[6] E. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. Math. 118 (1983) 349–374.803A. Cotsiolis, N.C. Tavoularis / C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804[7] G. Talenti, Best constants in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976) 353–372.[8] R.C.A.M. van der Vorst, Best constant for the embedding of the space H2 ∩ H10( ) into L2n/(n−4)( ), DifferentialIntegral Equations 2 (2) (1993) 259–270.804

Sharp Sobolev type inequalities for higher fractional derivatives

Comptes Rendus Mathématique

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/S1631-073X(02)02576-1.pdf

Sharp Sobolev type inequalities for higher fractional derivatives

Abstract

Similar works

Full text

Available Versions

Numérisation de Documents Anciens Mathématiques

Comptes Rendus Mathématique