1 Introduction and main result

Manuel del Pino

and Jean Dolbeault

1

Departamento de Ingenier´ıa Matem´ atica and CMM, UMI CNRS nr. 2807, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile, E-mail: delpino@dim.uchile.cl,

2

Ceremade, UMR CNRS nr. 7534, Universit´ e Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France, E-mail: dolbeaul@ceremade.dauphine.fr.

Correspondence to be sent to: delpino@dim.uchile.cl

The classical Onofri inequality in the two-dimensional sphere assumes a natural form in the plane when transformed via stereographic projection. We establish an optimal version of a generalization of this inequality in thed-dimensional Euclidean space for anyd≥2, by considering the endpoint of a family of optimal Gagliardo-Nirenberg interpolation inequalities. Unlike the two-dimensional case, this extension involves a rather unexpected Sobolev-Orlicz norm, as well as a probability measure no longer related to stereographic projection.

Keywords:Sobolev inequality; logarithmic Sobolev inequality; Onofri inequalities; Gagliardo-Nirenberg inequal- ities; interpolation; extremal functions; optimal constants; stereographic projection

Mathematics Subject Classification (2010):26D10; 46E35; 58E35

1 Introduction and main result

TheOnofri inequality as stated in [18] asserts that

log Z

S²

e^vdσ2

− Z

S²

v dσ2≤1

4k∇vk²_L2(S²,dσ2) (1)

for any functionv∈H¹(S², dσ2). Here dσ2 denotes the standard surface measure on the two-dimensional unit sphereS²⊂R³, up to a normalization factor ₄¹_π so thatR

S²1dσ₂= 1.

Using stereographic projection fromS² ontoR², that is defininguby

u(x) =v(y) with y= (y1, y2, y3), y1= 2x₁

1 +|x|² , y2= 2x₂

1 +|x|² , y3= 1− |x|² 1 +|x|² for anyx= (x₁, x₂)∈R², then (1) can be reformulated into theEuclidean Onofri inequality,namely

log Z

R²

e^udµ₂

− Z

R²

u dµ₂≤ 1

16πk∇uk²_L2(R²,dx) (2)

for anyu∈L¹(R², dµ₂) such that∇u∈L²(R², dx), where

dµ2(x) := dx π(1 +|x|²)² is again a probability measure.

The purpose of this note is to obtain an (optimal) extension of inequality (2) to any space dimension. There is a vast literature on Onofri’s inequality, and we shall only mention a few works relevant to our main result below. Onofri’s inequality with a non-optimal constant was first established by J. Moser in [17], a work prior to

January 7, 2012 c

2012 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

that of E. Onofri, [18]. For this reason, the inequality is sometimes called the Moser-Onofri inequality. We also point out that Onofri’s paper is based on an earlier result of T. Aubin, [2]. We refer the interested reader to [14]

for a recent account on the Moser-Onofri inequality. The inequality has an interesting version in the cylinder R×S¹, see [12], 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

dµd(x) := d

|S^d−1|

dx

1 +|x|^{d−1d d} . Let us denote

Rd(X, Y) :=|X+Y|^d− |X|^d−d|X|^d−2X·Y , (X, Y)∈R^d×R^d, which is a polynomial if dis even. We define

H_d(x, p) :=R_d

−^d|x|

−d−2 d−1

1+|x|^d−1d

x,^d−1_d p

, (x, p)∈R^d×R^d, and

Qd[u] :=

R

R^dHd(x,∇u)dx log R

R^de^udµd

−R

R^du dµd

.

The following is our main result.

Theorem 1.1. With the above notation, for any smooth compactly supported functionu, we have

log Z

R^d

e^udµd

− Z

R^d

u dµd ≤αd

Z

R^d

Hd(x,∇u)dx . (3) The optimal constantαd is explicit and given by

αd = d^1−dΓ(d/2) 2 (d−1)π^d/2 . Small multiples of the function

v(x) = −d x·e

|x|^d−2d−1

1 +|x|^d−1d (4)

for a unit vector eare approximate extremals of (3)in the sense that

ε→0limQd[ε v] = 1 αd

.

A rather unexpected feature of inequality (3) when compared with Onofri’s inequality (2), is that it involves an inhomogeneous Sobolev-Orlicz type norm. As we will see below, as a by-product of the proof we obtain a new Poincar´e inequality in entire space, (7) below, of which the functionv defined by (4) is an extremal.

Example 1.2. Ifd= 2,R

R^dH2(x,∇u)dx= ¹₄R

R²|∇u|²dxand we recover Onofri’s inequality (3) as in [11], with optimal constant 1/α2= 4π. On the other hand, if for instanced= 4, we find thatH4(x,∇u) is a fourth order polynomial in the partial derivatives ofu, sinceR₄(X, Y) = 4 (X·Y)²+|Y|²(|Y|²+ 4X·Y + 2|X|²).

Extensions of inequality (2) to higher dimensions were already obtained long ago. Inequality (1) was generalized to thed-dimensional sphere in [3, 5], where natural conformally invariant, non-local generalizations of the Laplacian were used. Those operators are of different nature than the ones in Theorem 1.1. Indeed, no clear connection through, for instance, stereographic projection is present. See also [16, 15] in which bounded domains are considered.

Inequality (3) determines a natural Sobolev space in which it holds. Indeed, a classical completion argument with respect to a norm corresponding to the integrals defined 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∈L¹(R^d, dµ_d) such that the distribution∇uis a square integrable function. To avoid technicalities, 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 family of optimal interpolation inequalities discovered in [7] and then extended in [8]. These inequalities can be stated as follows.

Theorem 1.3. Letp∈(1, d],a >1such thata≤^p(d−1)_d−p ifp < d, andb=p^a−1_p−1. There exists a positive constant Cp,a such that, for any function f ∈L^a(R^d, dx)with ∇f ∈L^p(R^d, dx), we have

kfk_Lb(R^d)≤Cp,ak∇fk^θ_Lp(R^d)kfk^1−θ_La(

R^d) withθ= (a−1) (d p−(d−p)a)^(a−p)d (5) ifa > p. A similar inequality also holds if a < p, namely

kfk_La(R^d)≤Cp,ak∇fk^θ_Lp(R^d)kfk^1−θ_Lb(

R^d) withθ= _a(d(p−a)+p^(p−a)d_(a−1)) . In both cases, equality holds for any function taking the form

f(x) =A

1 +B|x−x₀|^p−1p −_a−p^p−1 +

∀x∈R^d

for some(A, B, x0)∈R×R×R^d, whereB has the sign ofa−p.

While in [8], only the casep < dwas considered, the proof there actually applies to also cover the casep=d, for anya∈(1,∞).

Fora=p, inequality (5) degenerates into an equality. By substracting it to the inequality, dividing bya−p and taking the limit asa→p+, we obtain an optimal Euclidean L^p-Sobolev logarithmic inequality which goes as follows. Assume that 1< p≤d. Then for any u∈W^1,p(R^d) withR

R^d|u|^pdx= 1 we have Z

R^d

|u|^plog|u|^pdx≤d p log

βp,d

Z

R^d

|∇u|^pdx

, where βp,d:= p d

p−1 e

p−1

1 π^p2





Γ“d 2+1” Γ

„ dp−1

p +1

«





p d

is the optimal constant. Equality holds if and only if for someσ >0 andx0∈R^d

u(x) =

"

1 2π^d2

p p−1

Γ ^d₂ Γ d^p−1_p

p σ

^dp−1_p

#¹_p

e^{−σ1|x−x0|}

p

p−1 ∀x∈R^d.

This inequality has been established in [9] whenp < dand in general in [13]; see also [6, 10].

Whenp < d, the endpointa= ^p(d−1)_d−p corresponds to the usual optimal Sobolev inequality, for which the extremal functions were already known from the celebrated papers by T. Aubin and G. Talenti, [1, 20]. See also [4, 19] 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. In dimensiond= 2, withp= 2,a=q+ 1>2 and b= 2q, it has been recently observed in [11] that

1≤ lim

q→∞C_2,q+1k∇fqk

q−1 2q

L²(R²)kfqk

q+1 2q

L^q+1(R²)

kfqkL^2q(R²)

=e^{161π RR2|∇u|2dx} R

R²e^udµ2

iff_q = (1 +|x|²)^−q−11 (1 + ₂^u_q) andR

R²u dµ₂= 0. In that sense, Onofri’s inequality in dimensiond= 2 replaces Sobolev’s inequality in higher dimensions as an endpoint of the family of Gagliardo-Nirenberg inequalities

kfk_L2q(R^d)≤C_2,q+1k∇fk^θ_L2(R^d)kfk^1−θ_Lq+1(R^d)

withθ=^q−1_{q d+2−q}^d_(d−2). In dimensiond≥3, we will see below that (3) can also be seen as an endpoint of (5).

2 Proof of Theorem 1.1

R^du dµd= 0 and let f_a :=F_a 1 +^d−1_{d a} u

,

whereFa is defined by

F_a(x) =

1 +|x|^d−1d −^d−1_a−d

∀x∈R^d. (6)

From Theorem 1.3, Inequality (5), we know that

1≤ lim

a→+∞Cd,a

k∇fak^θ_Ld(R^d)kfak^1−θ_La(R^d)

kfak_Lb(R^d)

if p=d. Our goal is to identify the right hand side in terms of u. We recall that b= ^d(a−1)_d−1 and θ= _d(a−1)^a−d . Using the fact thatFa is an optimal function, we can then rewrite (5) withf =fa as

R

R^d|fa|^{d(a−1)d−1} dx R

R^d|Fa|^{d(a−1)d−1} dx

≤ R

R^d|∇fa|^ddx R

R^d|∇Fa|^ddx

!_d(d−1)^a−d R

R^d|fa|^adx R

R^d|Fa|^adx and observe that:

(i) lim_a→+∞R

R^d|F_a|^{d(a−1)d−1} dx=R

R^d 1 +|x|^d−1d −d

dx= ¹_d|S^d−1|and

a→+∞lim Z

R^d

|f_a|^{d(a−1)d−1} dx= lim

a→+∞

Z

R^d

F

d(a−1)

a d−1 (1 + ^d−1_{d a} u)^{d(a−1)d−1} dx= Z

R^d

e^u

1 +|x|^d−1d d dx , so that

a→+∞lim R

R^d|fa|^{d(a−1)d−1} dx R

R^d|Fa|^{d(a−1)d−1} dx

= Z

R^d

e^udµ_d.

(ii) Asa→+∞,

Z

R^d

|Fa|^adx≈ 2a π^d/2

d²Γ(d/2) , lim

a→+∞

Z

R^d

|fa|^adx=∞, and

a→+∞lim R

R^d|fa|^adx R

R^d|Fa|^adx = 1. (iii) Finally, as a→+∞, we also find that

R

R^d|∇fa|^ddx R

R^d|∇Fa|^ddx

!_d(d−1)^a−d

≈

1 +^d(d−1)_a αd

Z

R^d

Hd(x,∇u)dx _d^a−d_(d−1)

≈exp

αd

Z

R^d

Hd(x,∇u)dx

.

Here and above`1(a)≈`2(a) means that lim_a→+∞`1(a)/`2(a) = 1. Fact (iii) requires some computations which we make explicit next. First of all, we have

Z

R^d

|∇Fa|^ddx=2d^d−2π^d/2 Γ(d/2) a^1−d. WithXa := 1 +^d−1_{d a} u

∇Fa andYa :=^d−1_{d a} Fa∇u, we can write, using the definition ofRd, that

|∇fa|^d=|∇Fa|^d 1 +^d−1_{d a} u^d

+Fa|∇Fa|^d−2∇Fa· ∇ 1 + ^d−1_{d a} u^d

+Rd(Xa, Ya).

Consider the second term of the right hand side and integrate by parts. A straightforward computation shows that

Z

R^d

Fa|∇Fa|^d−2∇Fa· ∇ 1 +^d−1_{d a} u^d

dx=− Z

R^d

|∇Fa|^d 1 + ^d−1_{d a} u^d dx−

Z

R^d

Fa∆dFa 1 + ^d−1_{d a} u^d dx

where ∆pFa=∇ ·(|∇Fa|^p−2∇Fa) is computed forp=d. Collecting terms, we get Z

R^d

|∇f_a|^ddx=− Z

R^d

F_a∆_dF_a 1 +^d−1_{d a} u^d dx+

Z

R^d

R_d(X_a, Y_a)dx . We may next observe that

a∇F_a(x) =−_a−d^{d a} |x|^{−d−2d−1}x

1 +|x|^d−1d −^a−1_a−d +

→ −d |x|^{−d−2d−1}x

1 +|x|^d−1d a.e. as a→+∞,

whilea∇ 1 +^d−1_{d a} u

=^d−1_d ∇u, so that bothX_a = 1 +^d−1_{d a} u

∇Fa andY_a= ^d−1_{d a} F_a∇uinR_d(X_a, Y_a) are of the order of 1/a. By homogeneity, it follows that

a^dRd(Xa, Ya)→Rd

−^d|x|

−d−2 d−1

1+|x|^d−1d

x,^d−1_d ∇u

=Hd(x,∇u) as a→+∞,

by definition ofHd. Hence we have established the fact that Z

R^d

|∇fa|^ddx=− Z

R^d

Fa∆dFa 1 + ^d−1_{d a} u^d dx+

Z

R^d

Rd(Xa, Ya)dx

=− Z

R^d

F_a∆_dF_a 1 +d^d−1_{d a} u+o(a⁻¹)

dx+a^−d Z

R^d

H_d(x,∇u)dx

Next we can observe that−R

R^dFa∆dFa dx=R

R^d|∇Fa|^ddx, while −lim_a→+∞a^d−1Fa∆dFa =d^d−1|S^d−1|µd, so that

− Z

R^d

Fa∆dFau dx=a^1−dd^d−1|S^d−1| Z

R^d

u dµd+o(a^1−d) =o(a^1−d) as a→+∞

by the assumption thatR

R^du dµ_d= 0. Altogether, this means that R

R^d|∇f_a|^ddx R

R^d|∇Fa|^ddx

!_d^a−d(d−1)

≈

1 + R

R^dH_d(x,∇u)dx a^dR

R^d|∇Fa|^ddx

a−d d(d−1)

≈

1 +^d(d−1)_a α_d Z

R^d

H_d(x,∇u)dx _d(d−1)^a−d

asa→+∞, which concludes the proof of (iii).

Before proving the optimality of the constant αd, let us state an intermediate result which of interest in itself. Let us assume thatd≥2 and defineQd as

Q_d(X, Y) := 2 lim

ε→0ε⁻²R_d(X, ε Y) = d²

dt²|X+t Y|^d

t=0=d|X|^d−4

(d−2) (X·Y)²+|X|²|Y|² .

We also define

G_d(x, p) :=Q_d

−^d|x|

−d−2 d−1

1+|x|^d−1d

x,^d−1_d p

, (x, p)∈R^d×R^d.

Corollary 2.1. With αd as in Theorem 1.1, we have

Z

R^d

|v−v|²dµd≤αd

Z

R^d

Gd(x,∇v)dx with v= Z

R^d

v dµd, (7)

for anyv∈L¹(R^d, dµd)such that ∇v∈L²(R^d, dx).

This inequality is a Poincar´e inequality, which is remarkable. Indeed, if we prove that the optimal constant in (7) is equal toαd, thenαdis also optimal in Theorem 1.1, Inequality (3). We will see below that this is the case.

Proof of Corollary 2.1.Inequality (7) is a straightforward consequence of (3), written withureplaced byε v. In the limitε→0, both sides of the inequality are of orderε². 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). SinceFa is optimal for (3), we can write that

log Z

R^d

|Fa|^{d(a−1)d−1} dx

= logCd,a+_d(d−1)^a−d log Z

R^d

|∇Fa|^ddx

+ log Z

R^d

|Fa|^adx

.

However, equality also holds true if we replaceF_a by F_a,ε with F_a,ε(x) :=F_a(x+εe), for an arbitrary given e∈S^d−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−1

_d(a−1)

d−1 −1R

R^d|Fa|^{d(a−1)d−1} |va|²dx R

R^d|Fa|^{d(a−1)d−1} dx

= _d^a−d_(d−1) R

R^dQd(Xa,^d−1_d Ya)dx R

R^d|∇Fa|^ddx +a(a−1) R

R^d|Fa|^a|va|²dx R

R^d|Fa|^a dx (8)

withX_a=∇Fa,Y_a= _d−1^d F_a∇va andv_a :=e· ∇logF_a, that is

va(x) =− d a−d

x·e

|x|^d−2d−1

1 +|x|^d−1d . Hence, if φis a radial function, we may notice thatR

R^dφ vadx= 0 and

a→+∞lim a² Z

R^d

φ|va|²dx=d² Z

R^d

φ(x)|x|^{d−12 −2}(x·e)²

1 +|x|^d−1d 2 dx=d Z

R^d

φ(x) |x|^d−12

1 +|x|^d−1d 2 dx .

SinceR

R^d|Fa|^{d(a−1)d−1} dx=o R

R^d|Fa|^adx

, the last term in (8) is negligible compared to the other ones. Passing to the limit asa→+∞, withv:= lima→+∞a va, we find thatv is given by (4) and

d d−1

²Z

R^d

|v|²dµd=αd

Z

R^d

Qd

−^d|x|

−d−2 d−1

1+|x|^d−1d

x,^d−1_d Y dx

where Y :=^d−1_d ∇v and where we have used the fact that

d(d−1)αd lim

a→+∞a^d Z

R^d

|∇Fa|^ddx= 1.

Since the function Qd is quadratic, we obtain that

d d−1

2Z

R^d

|v|²dµd=αd

Z

R^d

Gd(x,_d−1^d ∇v)dx=αd d d−1

2Z

R^d

Gd(x,∇v)dx ,

which corresponds precisely to equality in (7) since vgiven by (4) is such that v= 0.

Equality in (3) is achieved by constants. The optimality ofαd amounts to establish that in the inequality Qd[u]≥ 1

αd

,

equality can be achieved along a minimizing sequence. Notice that

Q_d[u] = R

R^dHd(x,∇u)dx log R

R^de^udµ_d if Z

R^d

u dµ_d= 0,

The reader is invited to check that lim_ε→0Qd[ε v] = _α¹

d. In dimensiond= 2,v is an eigenfunction associated to the eigenvalue problem:−∆v=λ1vµ2, corresponding to the lowest positive eigenvalue,λ1. The generalization to higher dimensions is given by (4). Notice that the functionvis an eigenfunction of the linear form associated to Gd, in the spaceL²(R^d, dµd). This concludes the proof of Theorem 1.1.

Whether there are non-trivial optimal functions, that is, whether there exists a non-constant function u such that Qd[u] = _α¹

d, is an open question. At least the proof of Theorem 1.1 shows that there is a loss of compactness in the sense that the limit ofε v,i.e.0, is not an admissible function forQd.

Acknowlegments.J.D. has been supported by the projectsCBDif andEVOLof the French National Research Agency (ANR). M.D. has been supported by grants Fondecyt 1110181 and Fondo Basal CMM. Both authors are participating to theMathAmSud networkNAPDE.

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