The Euclidean Onofri inequality in higher dimensions Manuel del Pino1 and Jean Dolbeault2 1 Departamento de Ingenier´ıa Matem´atica and CMM, UMI CNRS nr. 2807, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile, E-mail: [email protected], 2 Ceremade, UMR CNRS nr. 7534, Universit´e Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France, E-mail: [email protected]. Correspondence to be sent to: [email protected] 2 1 0 2 Theclassical Onofriinequalityinthetwo-dimensionalsphereassumesanaturalform intheplanewhentransformed via n a stereographic projection. Weestablish an optimal version of a generalization of this inequality in thed-dimensional J Euclidean space for any d≥2, byconsidering theendpoint of a family of optimal Gagliardo-Nirenberg interpolation 0 inequalities. Unlikethe two-dimensional case, this extension involvesa rather unexpectedSobolev-Orlicz norm, as well 1 as a probability measure no longer related to stereographic projection. ] Keywords: Sobolevinequality;logarithmicSobolevinequality;Onofriinequalities;Gagliardo-Nirenberginequal- P ities; interpolation; extremal functions; optimal constants; stereographic projection A Mathematics Subject Classification (2010): 26D10; 46E35;58E35 . h t a m 1 Introduction and main result [ 1 The Onofri inequality as stated in Onofri [1982] asserts that v 2 1 16 log(cid:18)ZS2evdσ2(cid:19)−ZS2v dσ2 ≤ 4k∇vkL2 2(S2,dσ2) (1) 2 . for any function v ∈H1(S2,dσ2). Here dσ2 denotes the standard surface measure on the two-dimensional unit 1 sphere S2 ⊂R3, up to a normalization factor 1 so that 1dσ =1. 0 4π S2 2 2 Using stereographic projection from S2 onto R2, thaRt is defining u by 1 v: 2x1 2x2 1−|x|2 u(x)=v(y) with y =(y ,y ,y ), y = , y = , y = i 1 2 3 1 1+|x|2 2 1+|x|2 3 1+|x|2 X ar for any x=(x1,x2)∈R2, then (1) can be reformulated into the Euclidean Onofri inequality, namely 1 log eudµ − udµ ≤ k∇uk2 (2) (cid:18)ZR2 2(cid:19) ZR2 2 16π L 2(R2,dx) for any u∈L 1(R2,dµ ) such that ∇u∈L 2(R2,dx), where 2 dx dµ (x):= 2 π(1+|x|2)2 is again a probability measure. Thepurposeofthisnoteistoobtainan(optimal)extensionofinequality(2)toanyspacedimension.There is a vast literature on Onofri’s inequality, and we shall only mention a few works relevant to our main result January11,2012 (cid:13)c 2012bytheauthors. Thispapermaybereproduced,initsentirety,fornon-commercialpurposes. 2 del Pino, M., and Dolbeault, J. below. Onofri’s inequality with a non-optimal constant was first established by J. Moser in Moser [1970/71], a work prior to that of E. Onofri, Onofri [1982]. For this reason, the inequality is sometimes called the Moser- Onofri inequality.We alsopointoutthat Onofri’spaperis basedonanearlierresultofT. Aubin,Aubin [1979]. WerefertheinterestedreadertoGhigi[2005]forarecentaccountontheMoser-Onofriinequality.Theinequality has an interesting version in the cylinder R×S1, see Dolbeault et al. [2008], which is however out of the scope of the present work. In this note, we will establish that the Euclidean version of Onofri’s inequality (2) can be extended to an arbitrary dimension d≥3 in the following manner. Let us consider the probability measure d dx dµ (x):= . d |Sd−1| d d 1+|x|d−1 (cid:16) (cid:17) Let us denote R (X,Y):=|X +Y|d−|X|d−d|X|d−2X·Y , (X,Y)∈Rd×Rd, d which is a polynomial if d is even. We define H (x,p):=R −d|x|−dd−−21 x,d−1p , (x,p)∈Rd×Rd, d d d d 1+|x|d−1 (cid:18) (cid:19) and H (x,∇u)dx Q [u]:= Rd d . d log eudµ − udµ RRd d Rd d The following is our main result. (cid:0)R (cid:1) R Theorem 1.1. With the above notation, for any smooth compactly supported function u, we have log eudµ − udµ ≤α H (x,∇u)dx . (3) d d d d (cid:18)ZRd (cid:19) ZRd ZRd The optimal constant α is explicit and given by d d1−dΓ(d/2) α = . d 2(d−1)πd/2 Small multiples of the function x·e v(x)= −d (4) d−2 d |x|d−1 1+|x|d−1 (cid:16) (cid:17) for a unit vector e are approximate extremals of (3) in the sense that 1 limQ [εv]= . d ε→0 αd Aratherunexpectedfeatureofinequality(3)whencomparedwithOnofri’sinequality(2),isthatitinvolves an inhomogeneous Sobolev-Orlicz type norm. As we will see below, as a by-product of the proof we obtain a new Poincar´einequality in entire space, (7) below, of which the function v defined by (4) is an extremal. Example1.2. Ifd=2, H (x,∇u)dx= 1 |∇u|2dxandwerecoverOnofri’sinequality(3)asinDolbeault Rd 2 4 R2 [2011], with optimal constant 1/α =4π. On the other hand, if for instance d=4, we find that H (x,∇u) is 2 4 R R a fourth order polynomial in the partial derivatives of u, since R (X,Y)=4(X·Y)2+|Y|2(|Y|2+4X·Y + 4 2|X|2). Extensions of inequality (2) to higher dimensions were already obtained long ago. Inequality (1) was generalized to the d-dimensional sphere in Beckner [1993], Carlen and Loss [1992], where natural conformally invariant,non-localgeneralizationsofthe Laplacianwereused. Thoseoperatorsareof differentnature thanthe ones in Theorem 1.1. Indeed, no clear connection through, for instance, stereographic projection is present. See also Kim [2000], Kawohl and Lucia [2008] in which bounded domains are considered. The Euclidean Onofri inequality in higher dimensions 3 Inequality(3)determinesanaturalSobolevspaceinwhichitholds.Indeed,aclassicalcompletionargument with respectto a norm correspondingto the integralsdefined in both sides of the inequality determines a space on which the inequality still holds. This space can be identified with the set of all functions u∈L 1(Rd,dµ ) d such that the distribution ∇u is a square integrablefunction. To avoidtechnicalities, computations will only be done for smooth, compactly supported functions. Our strategy is to consider the Euclidean inequality of Theorem 1.1 as the endpoint of a fam- ily of optimal interpolation inequalities discovered in Del Pino and Dolbeault [2002b] and then extended in Del Pino and Dolbeault [2002a]. These inequalities can be stated as follows. Theorem1.3. Letp∈(1,d],a>1suchthata≤ p(d−1) ifp<d,andb=pa−1.Thereexistsapositiveconstant d−p p−1 C such that, for any function f ∈L a(Rd,dx) with ∇f ∈L p(Rd,dx), we have p,a kfk ≤C k∇fkθ kfk1−θ withθ = (a−p)d (5) L b(Rd) p,a L p(Rd) L a(Rd) (a−1)(dp−(d−p)a) if a>p. A similar inequality also holds if a<p, namely kfkL a(Rd) ≤Cp,ak∇fkLθ p(Rd) kfkL1 −b(θRd) withθ = a(d(p−(pa−)+a)pd(a−1)) . In both cases, equality holds for any function taking the form f(x)=A 1+B|x−x0|p−p1 −ap−−p1 ∀x∈Rd + (cid:16) (cid:17) for some (A, B, x )∈R×R×Rd, where B has the sign of a−p. 0 While in Del Pino and Dolbeault [2002a], only the case p<d was considered, the proof there actually applies to also cover the case p=d, for any a∈(1,∞). Fora=p,inequality(5)degeneratesintoanequality.Bysubstractingittotheinequality,dividingbya−p and taking the limit as a→p , we obtain an optimal Euclidean L p-Sobolev logarithmic inequality which goes + as follows. Assume that 1<p≤d. Then for any u∈W1,p(Rd) with |u|pdx=1 we have Rd R p d p p−1 p−1 1 Γ d+1 d ZRd|u|plog|u|pdx≤ p log(cid:20)βp,dZRd|∇u|pdx(cid:21) , where βp,d := d (cid:18) e (cid:19) πp2 Γ(cid:18)d(cid:16)p2−p1+(cid:17)1(cid:19)   is the optimal constant. Equality holds if and only if for some σ >0 and x ∈Rd 0 1 1 p Γ d p dp−1 p p u(x)= 2 p e−σ1|x−x0|p−1 ∀x∈Rd. "2πd2 p−1Γ d(cid:0)p−p(cid:1)1 (cid:16)σ(cid:17) # This inequality has been established in D(cid:0)el Pino(cid:1) and Dolbeault [2003] when p<d and in general in Gentil [2003]; see also Cordero-Erausquinet al. [2004], Del Pino et al. [2004]. When p<d, the endpoint a= p(d−1) corresponds to the usual optimal Sobolev inequality, for which the d−p extremal functions were already known from the celebrated papers by T. Aubin and G. Talenti, Aubin [1976], Talenti [1976]. See also Bliss [1930], Rosen [1971] for earlier related computations, which provided the value of some of the best constants. When p=d, Theorem 1.1 will also be obtained by passing to a limit, namely as a→+∞. In this way, the d-dmensional Onofri inequality corresponds to nothing but a natural extension of the optimal Sobolev’s inequality.Indimensiond=2,withp=2,a=q+1>2andb=2q,ithasbeenrecentlyobservedinDolbeault [2011] that q−1 q+1 1≤ql→im∞C2,q+1 k∇fqkL k22fq(qRk2L) k2qf(qRk2L )2qq+1(R2) = e161πR2RRe2u|∇duµ|22dx if fq =(1+|x|2)−q−11 (1+ 2uq) and R2udµ2 =0. In that sense, Onofri’sRinequality in dimension d=2 replaces Sobolev’s inequality in higher dimensions as an endpoint of the family of Gagliardo-Nirenberginequalities R kfk ≤C k∇fkθ kfk1−θ L 2q(Rd) 2,q+1 L 2(Rd) L q+1(Rd) with θ = q−1 d . In dimension d≥3,we will see below that (3) canalso be seen as an endpoint of (5). q d+2−q(d−2) 4 del Pino, M., and Dolbeault, J. 2 Proof of Theorem 1.1 Assume that u∈D(Rd) is such that udµ =0 and let Rd d R f :=F 1+ d−1u , a a da where F is defined by (cid:0) (cid:1) a Fa(x)= 1+|x|d−d1 −ad−−d1 ∀x∈Rd. (6) (cid:16) (cid:17) From Theorem 1.3, Inequality (5), we know that k∇f kθ kf k1−θ 1≤ lim C a L d(Rd) a L a(Rd) d,a a→+∞ kfakL b(Rd) if p=d. Our goal is to identify the right hand side in terms of u. We recall that b= d(a−1) and θ = a−d . d−1 d(a−1) Using the fact that F is an optimal function, we can then rewrite (5) with f =f as a a Rd|fa|d(da−−11) dx ≤ Rd|∇fa|ddx da(d−−d1) Rd|fa|adx d(a−1) |∇F |ddx |F |adx RRd|Fa| d−1 dx (cid:18)RRd a (cid:19) RRd a R R and observe that: R (i) lima→+∞ Rd|Fa|d(da−−11) dx= Rd 1+|x|d−d1 −ddx= d1|Sd−1| and a→liRm+∞ZRd|fa|d(da−−11) dxR=(cid:0)a→lim+∞ZRdF(cid:1)ad(da−−11) (1+ dd−a1u)d(da−−11) dx=ZRd 1+|exu|d−d1 d dx, (cid:16) (cid:17) so that d(a−1) lim Rd|fa| d−1 dx = eudµ . a→+∞ RRd|Fa|d(da−−11) dx ZRd d R (ii) As a→+∞, 2aπd/2 |F |adx≈ , lim |f |adx=∞, ZRd a d2Γ(d/2) a→+∞ZRd a and |f |adx lim Rd a =1. a→+∞ RRd|Fa|adx (iii) Finally, as a→+∞, we also find that R Rd|∇fa|ddx d(ad−−d1) ≈ 1+ d(d−1)α H (x,∇u)dx d(ad−−d1) ≈exp α H (x,∇u)dx . (cid:18)RRd|∇Fa|ddx(cid:19) (cid:18) a dZRd d (cid:19) (cid:18) dZRd d (cid:19) HereandRaboveℓ (a)≈ℓ (a)meansthatlim ℓ (a)/ℓ (a)=1.Fact(iii)requiressomecomputationswhich 1 2 a→+∞ 1 2 we make explicit next. First of all, we have 2dd−2πd/2 |∇F |ddx= a1−d. a Γ(d/2) ZRd With X := 1+ d−1u ∇F and Y := d−1F ∇u, we can write, using the definition of R , that a da a a da a d (cid:0) |∇f |d =(cid:1)|∇F |d 1+ d−1u d+F |∇F |d−2∇F ·∇ 1+ d−1u d+R (X ,Y ). a a da a a a da d a a Consider the second term of the(cid:0)right han(cid:1)d side and integrate by pa(cid:0)rts. A stra(cid:1)ightforward computation shows that F |∇F |d−2∇F ·∇ 1+ d−1u d dx=− |∇F |d 1+ d−1u d dx− F ∆ F 1+ d−1u d dx a a a da a da a d a da ZRd ZRd ZRd (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) The Euclidean Onofri inequality in higher dimensions 5 where ∆ F =∇·(|∇F |p−2∇F ) is computed for p=d. Collecting terms, we get p a a a |∇f |ddx=− F ∆ F 1+ d−1u d dx+ R (X ,Y )dx. a a d a da d a a ZRd ZRd ZRd (cid:0) (cid:1) We may next observe that a∇Fa(x)=−ad−ad|x|−dd−−21 x 1+|x|d−d1 −aa−−d1 →−d |x|−dd−−12dx a.e. as a→+∞, + 1+|x|d−1 (cid:16) (cid:17) while a∇ 1+ d−1u = d−1∇u, so that both X = 1+ d−1u ∇F and Y = d−1F ∇u in R (X ,Y ) are of da d a da a a da a d a a the order of 1/a. By homogeneity, it follows that (cid:0) (cid:1) (cid:0) (cid:1) adR (X ,Y )→R −d|x|−dd−−21 x,d−1∇u =H (x,∇u) as a→+∞, d a a d 1+|x|d−d1 d d (cid:18) (cid:19) by definition of H . Hence we have established the fact that d |∇f |ddx=− F ∆ F 1+ d−1u d dx+ R (X ,Y )dx a a d a da d a a ZRd ZRd ZRd (cid:0) (cid:1) =− F ∆ F 1+dd−1u+o(a−1) dx+a−d H (x,∇u)dx a d a da d ZRd ZRd (cid:0) (cid:1) Next we can observe that − F ∆ F dx= |∇F |ddx, while −lim ad−1F ∆ F =dd−1|Sd−1|µ , Rd a d a Rd a a→+∞ a d a d so that R R − F ∆ F udx=a1−ddd−1|Sd−1| udµ +o(a1−d)=o(a1−d) as a→+∞ a d a d ZRd ZRd by the assumption that udµ =0. Altogether, this means that Rd d Rd|∇fa|ddx da(Rd−−d1) ≈ 1+ RdHd(x,∇u)dx d(ad−−d1) ≈ 1+ d(d−1)α H (x,∇u)dx d(ad−−d1) (cid:18)RRd|∇Fa|ddx(cid:19) (cid:18) Rad Rd|∇Fa|ddx(cid:19) (cid:18) a dZRd d (cid:19) as a→R+∞, which concludes the proof oRf (iii). Before proving the optimality of the constant α , let us state an intermediate result which of interest in d itself. Let us assume that d≥2 and define Q as d d2 Q (X,Y):=2 limε−2R (X,εY)= |X +tY|d =d|X|d−4 (d−2)(X ·Y)2+|X|2|Y|2 . d ε→0 d dt2 t=0 (cid:2) (cid:3) We also define (cid:12) (cid:12) G (x,p):=Q −d|x|−dd−−21 x,d−1p , (x,p)∈Rd×Rd. d d 1+|x|d−d1 d (cid:18) (cid:19) Corollary 2.1. With α as in Theorem 1.1, we have d |v−v|2dµ ≤α G (x,∇v)dx with v = v dµ , (7) d d d d ZRd ZRd ZRd for any v ∈L 1(Rd,dµ ) such that ∇v ∈L 2(Rd,dx). d ThisinequalityisaPoincar´einequality,whichisremarkable.Indeed,ifweprovethattheoptimalconstantin(7) is equal to α , then α is also optimal in Theorem 1.1, Inequality (3). We will see below that this is the case. d d Proof of Corollary 2.1. Inequality (7) is a straightforwardconsequence of (3),written with u replacedby εv. In the limit ε→0, both sides of the inequality are of order ε2. Details are left to the reader. To conclude the proof of Theorem 1.1, let us check that there is a nontrivial function v which achieves equality in (7). Since F is optimal for (3), we can write that a log |Fa|d(da−−11) dx =logCd,a+ d(ad−−d1) log |∇Fa|d dx +log |Fa|a dx . (cid:18)ZRd (cid:19) (cid:18)ZRd (cid:19) (cid:18)ZRd (cid:19) 6 del Pino, M., and Dolbeault, J. However, equality also holds true if we replace F by F with F (x):=F (x+εe), for an arbitrary given a a,ε a,ε a e∈Sd−1, and it is clear that one can differentiate twice with respect to ε at ε=0. Hence, for any a>d, we have d(a−1) d(a−1) −1 Rd|Fa|d(da−−11) |va|2 dx = a−d RdQd(Xa,d−d1Ya)dx +a(a−1) Rd|Fa|a|va|2 dx (8) d−1 d−1 d(a−1) d(d−1) |∇F |d dx |F |a dx (cid:16) (cid:17)R Rd|Fa| d−1 dx R Rd a R Rd a R R R with X =∇F , Y = d F ∇v and v :=e·∇logF , that is a a a d−1 a a a a d x·e v (x)=− . a a−d d−2 d |x|d−1 1+|x|d−1 (cid:16) (cid:17) Hence, if φ is a radial function, we may notice that φv dx=0 and Rd a R lim a2 φ|v |2 dx=d2 φ(x) |x|d−21−2(x·e)2 dx=d φ(x) |x|d−21 dx. a→+∞ ZRd a ZRd 1+|x|d−d1 2 ZRd 1+|x|d−d1 2 (cid:16) (cid:17) (cid:16) (cid:17) Since Rd|Fa|d(da−−11) dx=o Rd|Fa|a dx , the last term in (8) is negligible comparedto the other ones. Passing to the limit as a→+∞, with v :=lim av , we find that v is given by (4) and a→+∞ a R (cid:0)R (cid:1) d 2 |v|2 dµ =α Q − d|x|−dd−−12 x,d−1Y dx d−1 d d d d d ZRd ZRd (cid:16) 1+|x|d−1 (cid:17) (cid:0) (cid:1) where Y := d−1∇v and where we have used the fact that d d(d−1)α lim ad |∇F |d dx=1. d a a→+∞ ZRd Since the function Q is quadratic, we obtain that d d 2 |v|2 dµ =α G (x, d ∇v)dx=α d 2 G (x,∇v)dx, d−1 d d d d−1 d d−1 d ZRd ZRd ZRd (cid:0) (cid:1) (cid:0) (cid:1) which corresponds precisely to equality in (7) since v given by (4) is such that v =0. Equality in (3) is achieved by constants. The optimality of α amounts to establish that in the inequality d 1 Q [u]≥ , d α d equality can be achieved along a minimizing sequence. Notice that H (x,∇u)dx Q [u]= Rd d if udµ =0, d d Rlog Rdeudµd ZRd (cid:0)R (cid:1) The reader is invited to check that lim Q [εv]= 1 . In dimension d=2, v is an eigenfunction associatedto ε→0 d αd the eigenvalue problem: −∆v =λ vµ , corresponding to the lowest positive eigenvalue, λ . The generalization 1 2 1 to higherdimensions is givenby (4).Notice thatthe function v is aneigenfunctionofthe linearformassociated to G , in the space L2(Rd,dµ ). This concludes the proof of Theorem 1.1. d d Whether there are non-trivial optimal functions, that is, whether there exists a non-constant function u such that Q [u]= 1 , is an open question. At least the proof of Theorem 1.1 shows that there is a loss of d αd compactness in the sense that the limit of εv, i.e. 0, is not an admissible function for Q . d Acknowlegments.J.D.hasbeensupportedbytheprojectsCBDif andEVOLoftheFrenchNationalResearchAgency (ANR). M.D. has been supported by grants Fondecyt 1110181 and Fondo Basal CMM. Both authors are participating to theMathAmSud network NAPDE. The Euclidean Onofri inequality in higher dimensions 7 References Thierry Aubin. Probl`emes isop´erim´etriques et espaces de Sobolev. J. Differential Geometry, 11(4):573–598, 1976. ISSN 0022-040X. Thierry Aubin. Meilleures constantes dans le th´eor`eme d’inclusion de Sobolev et un th´eor`eme de Fredholm non lin´eaire pour la transformation conforme de la courbure scalaire. J. Funct. Anal., 32(2):148–174, 1979. ISSN 0022-1236. doi: 10.1016/0022-1236(79)90052-1. URL https://proxy.bu.dauphine.fr:443/http/dx.doi.org/10.1016/0022-1236(79)90052-1. William Beckner. Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequal- ity. Ann. of Math. (2), 138(1):213–242, 1993. ISSN 0003-486X. doi: 10.2307/2946638. URL http://dx.doi.org/10.2307/2946638. GA Bliss. An integral inequality. Journal of the London Mathematical Society, 1(1):40, 1930. ISSN 0024-6107. E. Carlen and M. Loss. Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on Sn. Geom. Funct. Anal., 2(1):90–104, 1992. ISSN 1016-443X. doi: 10.1007/BF01895706. URL http://dx.doi.org/10.1007/BF01895706. Dario Cordero-Erausquin, Wilfrid Gangbo, and Christian Houdr´e. Inequalities for generalized entropy and optimal transportation. In Recent advances in the theory and applications of mass transport, volume 353 of Contemp. Math., pages 73–94.Amer. Math. Soc., Providence, RI, 2004. Manuel Del Pino and Jean Dolbeault. Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the p-Laplacian. C. R. Math. Acad. Sci. Paris, 334(5):365–370, 2002a. ISSN 1631-073X. Manuel Del Pino and Jean Dolbeault. Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. (9), 81(9):847–875,2002b. ISSN 0021-7824. Manuel Del Pino and Jean Dolbeault. The optimal Euclidean Lp-Sobolev logarithmic inequality. J. Funct. Anal., 197(1):151–161,2003. ISSN 0022-1236. Manuel Del Pino, Jean Dolbeault, and Ivan Gentil. Nonlinear diffusions, hypercontractivity and the optimal Lp-Euclidean logarithmic Sobolev inequality. J. Math. Anal. Appl., 293(2):375–388,2004. ISSN 0022-247X. Jean Dolbeault. Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion. To appear in Math. Res. Lett., 2011. JeanDolbeault,MariaJ.Esteban,andGabriellaTarantello.TheroleofOnofritypeinequalitiesinthesymmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7(2):313–341,2008. ISSN 0391-173X. Ivan Gentil. The general optimal Lp-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations. J. Funct. Anal., 202(2):591–599, 2003. ISSN 0022-1236. doi: 10.1016/S0022-1236(03)00047-8. URL https://proxy.bu.dauphine.fr:443/http/dx.doi.org/10.1016/S0022-1236(03)00047-8. AlessandroGhigi. Onthe Moser-OnofriandPr´ekopa-Leindlerinequalities. Collect. Math., 56(2):143–156,2005. ISSN 0010-0757. Bernd Kawohl and Marcello Lucia. Best constants in some exponential Sobolev inequalities. Indi- ana Univ. Math. J., 57(4):1907–1927, 2008. ISSN 0022-2518. doi: 10.1512/iumj.2008.57.3307. URL http://dx.doi.org/10.1512/iumj.2008.57.3307. M Kim. Linearised Moser-Trudinger inequality. Bull. Austral. Math. Soc., 62:445–457,2000. J.Moser. AsharpformofaninequalitybyN.Trudinger. Indiana Univ. Math. J.,20:1077–1092,1970/71. ISSN 0022-2518. E. Onofri. On the positivity of the effective action in a theory of random surfaces. Comm. Math. Phys., 86(3): 321–326,1982. ISSN 0010-3616. 8 del Pino, M., and Dolbeault, J. GeraldRosen. Minimum value for c inthe Sobolev inequality kφ3k≤ck∇φk3. SIAM J. Appl. Math., 21:30–32, 1971. ISSN 0036-1399. Giorgio Talenti. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4), 110:353–372, 1976. ISSN 0003-4622.