Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates

ON THE SPHERE

Jean Dolbeault and Maria J. Esteban

Ceremade (UMR CNRS 7534, Universit´e Paris-Dauphine Place de Lattre de Tassigny, 75775 Paris C´edex 16, France

Micha l Kowalczyk

Departamento de Ingenieria Matem´atica

and Centro de Modelamiento Matem´atico (UMI CNRS 2807) Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

Michael Loss

School of Mathematics, Skiles Building

Georgia Institute of Technology, Atlanta GA 30332-0160, USA

Abstract. This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents.

Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.

1. Introduction. On the d-dimensional sphere, for any real valued functionuin H¹(S^d), let us consider the inequality

de≤i where e:= kuk²_Lp(S^d)− kuk²_L2(S^d)

p−2 and i:=k∇uk²_L2(S^d), (1) where µ is the normalized measure on S^d induced by the Euclidean measure on R^d+1 and p ∈[1,2)∪(2,2^∗) with 2^∗ =∞ if d ≤2 and 2^∗ = _d−2^2d if d≥ 3. The casep= 2^∗ is also covered ifd≥3 and corresponds to Sobolev’s inequality. When p= 1, the inequality is equivalent to the Poincar´e inequality. By taking the limit asp→2, we recover the logarithmic Sobolev inequality

Z

S^d

u²log u² kuk²_L2(S^d)

! dµ≤ 2

dk∇uk²_L2(S^d) ∀u∈H¹(S^d). (2) The constantdin (1) is optimal: see for instance [37]. Whenp= 2, it is consistent to define e as the l.h.s. in (2), that is, e := R

S^du² log u²/kuk²_L2(S^d)

dµ. Hence forp∈(1,2^∗), the functionalseandiare respectively the generalizedentropy and Fisher information functionals.

2010Mathematics Subject Classification. 26D10, 46E35, 58E35.

Key words and phrases. Sobolev inequality, interpolation, Gagliardo-Nirenberg inequalities, logarithmic Sobolev inequality, heat equation, hypercontractivity, spectral decomposition.

In this paper we are interested in improvements of (1) and (2) in the subcritical range, that is, for 1< p <2^∗. Byimproved inequality we mean an inequality of the form

dkuk²_L2(S^d)Φ e kuk²_L2(S^d)

!

≤i ∀u∈H¹(S^d), (3) for some monotone increasing function Φ such that Φ(0) = 0, Φ⁰(0) = 1 and Φ(s)> s for anys. As a straightforward consequence we get a stability result. Indeed, let us set

Ψ(s) :=s−Φ⁻¹(s). (4)

Hence we get i−de=d

"

kuk²_L2(S^d)Φ⁻¹ i dkuk²_L2(S^d)

!

−e

#

+dkuk²_L2(S^d)Ψ i dkuk²_L2(S^d)

! ,

and thus,

i−de≥ dkuk²_L2(S^d)Ψ i dkuk²_L2(

S^d)

! ,

where the inequality is a simple consequence of (3). If, additionally, Ψ is nonde- creasing, then by reapplying (3) we find that

i−de≥dkuk²_L2(S^d)(Ψ◦Φ) e kuk²_L2(S^d)

!

∀u∈H¹(S^d). (5) The function Ψ◦Φ is nondecreasing, positive on (0,∞) and such that (Ψ◦Φ)(0) = (Ψ◦Φ)⁰(0) = 0. Inequality (5) is a stability result sinceecontrols a distance to the optimal functions, which are the constant functions. Our goal is to find the best possible function Φ.

As an important application of the improved version of the inequalities, we can pointstability issues. Some straightforward consequences are:

1. the uniqueness of optimal functions when they exists, and a better character- ization of the equality cases in the inequalities,

2. stability results in spectral theory with application to problems arising from quantum mechanics, like stability of matter,

3. some additional estimates in variational methods (improved convergence of sequences) with applications for instance when one uses Lyapunov-Schmidt reduction methods,

4. improved convergence rates in evolution problems.

The last point is probably the most important in view of applications in physics.

In various cases of interest, it allows to prove that before entering an asymptotic regime of, for instance, exponential decay, a system may have an initial regime with an even faster convergence.

Let us briefly review the literature and give some indications of our motivation and main results. Inequality (1) has been established in [18,19,12]. The limit case p= 2 was known earlier: see for instance [59] in the case of the circle, and [34] for a detailed list of references.

In the case of compact manifolds other than the sphere, the estimates obtained by M.-F. Bidaut-V´eron and L. V´eron in [19] (also see [48, 49, 9, 41, 42]) and by J. Demange in [32, 33] can be improved at the leading order term by considering non-local quantities like in [37].

In the case of the sphere, the leading order term is determined by the constantd in (1). Looking for improvements therefore makes sense. J. Demange observed in [33] that Inequality (1) can be improved whenp∈(2,2^∗). Moreover, in [32] he noticed the existence of a free parameter. The first purpose of this paper is to clarify the range of the free parameter in the method and optimize on it, in order to get the best possible improvement with respect to that parameter. Recent contributions on nonlinear flows and Lyapunov functionals, or entropies, that can be found in [34,37,36], are at the core of our method.

Refined convex Sobolev inequalities have been established in [2] whenp∈(1,2) in the setting of interpolation inequalities involving a probability measure. For a simpler formulation, see [39]. Our second purpose is to adapt the method to interpolation inequalities on the sphere and hence also cover the rangep∈ (1,2).

It is based on refined estimates of entropy decay for a linear heat flow, in the spirit of the Bakry-Emery method.

Our last contribution is inspired by [1]. Under additional orthogonality condi- tions, we show that other (and in some cases better) improved inequalities can be established in the range p ∈ (1,2). The method is based on hypercontractivity estimates for a linear heat flow and a spectral decomposition. It raises an intrigu- ing open question on the possibility of obtaining improvements under orthogonality conditions in the rangep >2.

Now let us explain the strategy of the paper.

1) Standard symmetrization techniques allow to decrease the functioniwhile pre- serving the L² and L^p norms, and the functional e as well. Thus, there are op- timal functions for inequalities (3) and (5) which depend only on the azimuthal angle on the sphere and the interpolation inequalities are therefore equivalent to one-dimensional inequalities for the d-ultraspherical operator, where dcan now be considered as a real parameter. Details are given in Section2. We should point out that this first step simplifies the calculations needed to get an improved inequality, but it is not fundamental for our method. After a change of variables we are led to the following expressions

i= Z 1

−1

|f⁰|²ν dν_d and e= R1

−1|f|^pdνd

_p²

−R1

−1|f|²dνd

p−2 ,

whereνdis a probability measure, andν ≥0 is a smooth function (these functions are explicit but their exact form is irrelevant for the moment). Then we define the self-adjointultraspherical operator through the identity

Z 1

−1

f₁⁰f₂⁰ ν dν_d=− Z 1

−1

Lf₁f₂ν dν_d.

The natural function space for our inequalities is the form domain ofL, that is H:=

f ∈L² (−1,1), dν_d :

Z 1

−1

|f⁰|²ν dν_d<∞

,

and we shall denote bykfk_q the L^q (−1,1), dν_d

norm off.

2) The key to our approach is to combine the ideas of D. Bakry and M. Emery,i.e., take the derivative of i− dealong some flow, with ideas that go back to B. Gidas and J. Spruck in [44] and that were later exploited by M.-F. Bidaut-V´eron and L. V´eron for getting rigidity results in nonlinear elliptic equations. An unessential

but useful trick amounts to write the flow for w withf =w^β for some β ∈R, in the expressions foriande, as we shall see below.

Let us start with the caseβ = 1. We consider the manifold Mp={w∈ H : kwkp = 1}.

The main issue is to choose the right flow for our setting. We observe that if κ=p−1, thenMp is invariant under the action of the flow

wt=Lw+κ|w⁰|²

w . (6)

Notice thatg=w^p evolves according to the equationgt=Lg and we shall there- fore refer to the case associated with this equation as the linear case, or the 1- homogeneous case. At the level of w, the equation is indeed 1-homogeneous since λ wis a solution to (6) for anyλ >0 if wis a solution to (6).

Let

2^]:= 2d²+ 1 (d−1)² and

γ₁:=

d−1 d+ 2

²

(p−1) (2^#−p) if d >1, γ₁:= p−1

3 if d= 1. (7) Ifp∈[1,2)∪(2,2^]] andwis a solution to (6), then

d

dt(i− de)≤ −γ₁ Z 1

−1

|w⁰|⁴

w² dν_d≤ −γ₁ |e⁰|² 1−(p−2)e. Recalling thate⁰=−i, we get a differential inequality

e⁰⁰+de⁰ ≥γ₁ |e⁰|² 1−(p−2)e, which after integration implies an inequality of the form

dΦ(e(0))≤i(0). Details will be given in Sections3.1and5.

3) Now let us consider the case with a general β. The range of p’s for which an improved inequality is valid can be extended to anyp∈(1,2^∗) by considering the nonlinear flow

wt=w^2−2β

Lw+κ|w⁰|² w

(8) for someβ≥1, withκ=β(p−2)+1. ThenMβpis invariant under the flow and our improved functional inequality follows from the computation of _dt^d (i−de) written forf =w^β, with the additional difficulty thate⁰now differs from−i. Details on (8) will be given in Section3and improved inequalities will be established in Section4.

The change of functionf =w^β, for some parameterβ∈R, is convenient to compute

d

dt(i−de) but also sheds light on the strategy used for proving rigidity according to the method of [19]. It moreover shows that the computations are equivalent and explains why a local bifurcation result from constant functions can be extended to a global uniqueness property. This is because the flow relates any initial datum to the constants through the monotonically decreasing quantityi−de.

In his thesis, J. Demange [32] made a computation which is similar to ours.

Compared to his approach, we work in a setting in which the change of function f =w^βclarifies the relation of flow methods with rigidity results in nonlinear elliptic

PDEs. We give an explicit range for the parameterβ and numerically observe that there is no optimal choice ofβ valid for an arbitrary value of the entropy ewhen p >2. We also give a new result whenp∈(1,2) and it turns out that onlyβ = 1 can be used in that range.

4) Our last improvement is of a different nature. If p∈ (1,2) and assuming that the function u is in the orthogonal of the eigenspaces associated with the lowest positive eigenvalues of the Laplace-Beltrami operator ∆_Sd on the sphere, using Nel- son’s hypercontractivity result, it is possible to obtain another improved inequality.

Here we again use the linear flow (6) corresponding to β = 1. Although it is a rather straightforward adaptation of [1], such an approach raises an interesting open question. See Section6.

The nonlinear flow defined by (8) and by (6) whenβ = 1 is the main concep- tual tool for our analysis. It is introduced in Section 3. Explicit stability results rely on generalized Csisz´ar-Kullback-Pinsker type inequalities, which are detailed in Section7. Most of our statements, when written for the ultraspherical operator, are valid for dtaking real values and our proofs can be adapted without changes.

However, for simplicity, we shall assume thatdis an integer throughout this paper, unless explicitly specified.

It is very likely that the improved inequalities presented in this paper are not optimal. What could be an optimal Φ (or even if such a question really makes sense) is an open question, but at least we can construct a whole collection of such functions (depending onpandd), which in most cases improve on previously known results. The strategy that we use to prove the improved inequalities builds up on some previous works, but the way we combine these ideas is new and suggests several directions for future investigations.

After all these preliminaries, we are now in position to state our results. Let us start by giving an expression of the function Φ. For eachp∈(1,2)∪(2,2^]) andγ₁ defined by (7), let

ϕ1(s) :=







2 γ1+2 (p−2)

h

1−(p−2)s−_{2 (p−2)}^γ1

−1 + (p−2)si

if p6= 2−γ1

2 ,

1

2−p 1− (p−2)s

log 1−(p−2)s

if p= 2−γ₁ 2 ,

(9) where we assume thatsisadmissible, that is,s >0 ifp∈(1,2) ands∈(0,_p−2¹ ) if p >2. Next, we define

γ(β) :=− d−1

d+ 2(κ+β−1) ²

+κ(β−1) + d

d+ 2(κ+β−1). (10) We observe thatγ(1) =γ₁, and that there exists someβ ∈Rsuch thatγ(β)>0 only ifp≥1 ifd= 1 or ifp∈[2,2^∗] for anyd >1. Then, we defineB(p, d) by

B(p, d) =n

β∈R : γ(β)>0, β≥1, andβ≤ _4−p² ifp <4o

if p >2, B(p, d) ={1} if 1≤p≤2.

(11) A more explicit description of the regionB(p, d) will be given in the Appendix. For eachβ∈B(p, d) such thatβ >1, let

ϕβ(s) :=

Z s 0

exph _γ(β)

β(β−1)p

(1 −(p−2)z)^1−δ(β)−(1 −(p−2)s)^1−δ(β)i dz

(12)

with

δ(β) :=p− (4−p)β

2β(p−2) . (13)

The reader is invited to check that limβ→1+ϕβ(s) = ϕ1(s) for any admissible s.

Finally we define

Φ(s) := 1 + (p−2)s ϕ

s 1 + (p−2)s

where ϕ(s) := sup

β∈B(p,d)

ϕβ(s). (14) Theorem 1.1. Assume that one of the following conditions is satisfied:

(i) d= 1 andp∈(1,2)∪(2,∞), (ii) d= 2 andp∈(1,2)∪(2,9 + 4√

3), (iii) d≥3 andp∈(1,2)∪(2,2^∗).

For any u∈H¹(S^d) be such thatkuk_L2(S^d)= 1, we have the inequality dΦ

kuk²_Lp(S^d)−1 p−2

!

≤ k∇uk²_L2(S^d),

where Φ is defined by (14). Moreover Φ(0) = 0, Φ⁰(0) = 1 and Φ⁰⁰(s) > 0, for a.e.s >0 if p >2 or for a.e. s∈(0,1/(2−p))if p∈(1,2). Ifp= 2, then for any u∈H¹(S^d)the following improved logarithmic Sobolevinequality

Z

S^d

u² log |u|² kuk²_L2(

S^d)

!

dµ≤ 4

γ₁^∗kuk²_L2(S^d)log 1 +γ₁^∗ 2d

k∇uk²_L2(S^d)

kuk²_L2(

S^d)

!

holds with γ₁^∗= _(d+2)^4d−12.

The reader interested in best constants in logarithmic Sobolev inequalities as in Theorem 1.1 is invited to refer to [22, 23]. In the casep6= 2, a first consequence of the above improved inequality, compared to the standard inequality i≥ de, is that i− de is not only nonnegative, but that it actually measures a distance to the constants in the homogeneous Sobolev norm. With previous notations, one can indeed state that

k∇uk²_L2(S^d)− d p−2

hkuk²_Lp(S^d)−1i

≥dΨ

1

dk∇uk²_L2(S^d)

for any u ∈ H¹(S^d) such that kuk_L2(S^d) = 1, where Ψ is defined by (4). We can rephrase this result without normalization as the following corollary.

Corollary 1. Assume thatp∈(1,2)∪(2,2^∗). With the notations of Theorem1.1, we have

k∇uk²_L2(S^d)− d p−2

hkuk²_Lp(S^d)− kuk²_L2(S^d)

i≥dkuk²_L2(S^d)Ψ k∇uk²_L2(S^d)

dkuk²_L2(

S^d)

! ,

whereΨis defined by (4)in terms of Φ, andΦis given by (14).

Now we can use a generalized Csisz´ar-Kullback-Pinsker inequality to get an es- timate of the distance to the constants using a standard Lebesgue norm. We shall distinguish three cases: (i) p ∈ (1,2), (ii) p ∈ (2,4) , and (iii) p ≥ 4. Let us define q(p) := 2/p, p/2 and p/(p−2) in cases (i), (ii) and (iii) respectively. As- sume also thatr(p) :=p, 2 andp−2 in cases (i), (ii) and (iii) respectively and let s(p) := max{2, p}.

Proposition 1. Assume that p ∈ (1,2)∪(2,∞). With the above notations and (q, r, s) = (q(p), r(p), s(p)), there exists a positive constant C, depending only on p, such that for anyu∈L¹∩L^p(Ω, dµ), we have

1 p−2

hkuk²_Lp(S^d)− kuk²_L2(S^d)

i≥Ckuk^{2 (1−r)}_Ls(

S^d) ku^r−u¯^rk²_Lq(S^d)

whereu¯=kuk_Lr(S^d).

The casep= 2 appears as a limit case and corresponds to the standard Csisz´ar- Kullback-Pinsker inequality. Details on this limit and an explicit estimate of C for all p≥1 will be given in Section7 (see Corollary4). Notice thatC= 1 when p= 1, and the inequality in Proposition1is thus equivalent to a Poincar´e inequality.

A direct consequence of Proposition 1 and Inequality (5) is the followingstability result.

Corollary 2. With the notations of Theorem1.1and Proposition 1, we have

k∇uk²_L2(S^d)− d p−2

hkuk²_Lp(S^d)− kuk²_L2(S^d)

i

≥dkuk²_L2(S^d)(Ψ◦Φ)

C^kuk

2 (1−r) Ls(Sd)

kuk²

L2 (Sd)

ku^r−u¯^rk²_Lq(S^d)

∀u∈H¹(S^d). Now let us turn our attention to the statements corresponding to the last class of improvements studied in this paper. The range of pis now restricted to [1,2) and we consider the linear flow (6).

In [10], W. Beckner gave a method to prove interpolation inequalities between logarithmic Sobolev and Poincar´e inequalities in the case of a Gaussian measure.

The method extends to the case of the sphere as was proved in [34], in the range p∈[1,2), with optimal constants. For further considerations on inequalities that interpolate between Poincar´e and logarithmic Sobolev inequalities, we refer to [2, 1, 7,6,20,21,27,47] and references therein.

Our purpose is to obtain an improved estimate of the optimal constantCp in Z

S^d

|∇u|²dµ≥Cp

"

Z

S^d

|u|²dµ− Z

S^d

|u|^pdµ ^2/p#

∀u∈H¹(S^d, dµ). (15) Without any constraint onu, we have Cp = _2−p^d , and it is natural to expect that this constant will be improved when we impose additional constraints on the set of admissible u’s. It the present case we will assume that u is in the orthogonal complement of the finite dimensional subspace spanned by the spherical harmonics corresponding to the lowest positive eigenvalues of the Laplace-Beltrami operator

∆_Sdon the sphere. Under this additional hypothesis we will obtain an improvement of the estimate (15), similar to what was done [1], in a different setting.

Let us introduce some notations and recall some known results. We consider ∆_Sd

as an operator on L²(S^d, dµ) with domain H²(S^d, dµ), whose eigenvalues are λ_k=k(k+d−1) ∀k∈N,

and denote byEk the corresponding eigenspaces. Recall that dim(E_k) =(k+d−2)!

k! (d−1)! (2k+d−1) ∀k≥0,

according to [16, Corollary C-I-3, page 162]. Notice that E0 is generated by the function 1. Finally, fork= 1,2, . . . let us define constants

αk := 1 dλk+1, and functions

Φk(s) = max

αk

1−(p−1)^αk(1−s), logs log(p−1)

.

With these notations we have:

Theorem 1.2. Assume thatp∈ (1,2). If u∈H¹(S^d) is such that kuk_L2(S^d)= 1 and

Z

S^d

u e dµ= 0 ∀e∈E_j, j= 1,2. . . k , (16) then we have

k∇uk²_L2(S^d)≥dΦ_k

kuk²_Lp(S^d)

.

We may notice thatα7→ _1−(p−1)^α α is an increasing function of α >1 and thus larger that _1−(p−1)^α0 α0 = _2−p¹ sinceα₀= 1. Hence for anyk≥1, we have

d αk

1−(p−1)^αk > d 2−p,

thus showing that we have achieved a strict improvement of the constant compared to the one in (15) (which is optimal without further assumption).

Moreover, lim_s→0+Φk(s) = +∞, thus showing that when the orthogonality con- ditions are satisfied, the estimate of Theorem1.2 is strictly better than the one of Theorem1.1. See Fig.4 for an illustration.

2. Symmetrisation and the ultraspherical operator. As we have pointed out in the introduction, the improved inequality on the d-dimensional sphere can be reduced to the inequalities for functions depending only on the azimuthal angle.

The interested reader can refer to [34]. Alternatively, let us give a sketch of a proof for completeness. More details on the stereographic projection can be found in [35, Appendix B.3]

We denote by ξ = (ξ₀, ξ₁, . . . ξ_d) the coordinates of an arbitrary point in the unit sphereS^d ⊂R^d+1. Consider the stereographic projection Σ :S^d\ {N} →R^d, where N denotes the North Pole, that is the point inS^d corresponding to ξ_d = 1.

Here Σ(ξ) =x means x= (1−ξ_d)⁻¹(ξ₀, ξ₁, . . . ξ_d−1), and to anyu∈ H¹(S^d) we associate a functionσ[u] :=v∈H¹(R^d) such that

u(ξ) = (1−ξd)^−d−22 v(x) ∀ξ∈S^d and x= Σ(ξ).

An elementary computation shows thatk∇uk²_L2(S^d)+^d(d−2)₄ kuk²_L2(S^d)=k∇vk²_L2(R^d), kuk²_L2(S^d)=k4 (1 +|x|²)⁻²vk²_L2(R^d)andkuk²_Lp(S^d)=k(2/(1 +|x|²))^1+dp−d2vk²_Lp(R^d). To v we may apply the standard Schwarz symmetrization and denote the sym- metrized function byv∗. Let us define

u^∗ :=σ⁻¹ (σ[u])_∗

.

Lemma 2.1. Assume thatu∈H¹(S^d). Then we have i[u] =

Z

S^d

|∇u|²dµ≥i[u^∗] and for any q∈[1,2^∗],kuk_Lq(S^d)=ku^∗k_Lq(S^d), so that

e[u] =e[u^∗].

This symmetry result is a kind of folklore in the literature and we can quote [4,50,11] for various related results. Details of the proof are left to the reader. As a straightforward consequence, infi[u]/Φ(e[u]) is achieved by functions depending only onξd, or on the azimuthal angleθ.

Thus, to prove the inequality in Theorem1.1it suffices to prove it for e= 1

p−2

"Z π 0

|v(θ)|^pdσ ²_p

− Z π

0

|v(θ)|²dσ #

, i= Z π

0

|v⁰(θ)|²dσ, and for any functionv∈H¹([0, π], dσ), where

dσ(θ) := (sinθ)^d−1 Zd

dθ with Zd:=√

π Γ(^d₂) Γ(^d+1₂ ).

The change of variablesx= cosθ,v(θ) =f(x) allows to rewrite the expressions for eandi:

e= 1 p−2

"Z 1

−1

|f|^pdνd

²p

− Z 1

−1

|f|²dνd≥

# , i=

Z 1

−1

|f⁰|²ν dνd,

wheredν_d is the probability measure defined by

ν_d(x)dx=dν_d(x) :=Z_d⁻¹ν^d2−1dx with ν(x) := 1−x², Z_d=√

π Γ(^d₂) Γ(^d+1₂ ). We consider the space L²((−1,1), dνd) equipped with the scalar product

hf1, f2i= Z 1

−1

f1f2dνd,

and recall that theultrasphericaloperator is given byhf1,Lf2i=−R1

−1f₁⁰f₂⁰ ν dνd. Explicitly we have:

Lf := (1−x²)f⁰⁰−d x f⁰=ν f⁰⁰+d 2ν⁰f⁰.

With these notations, for any positive smooth functionuon (−1,1), Inequalities (1) and (2) can be rewritten as

− hf,Lfi= Z 1

−1

|f⁰|²ν dν_d≥dkfk²_p− kfk²₂ p−2 ,

− hf,Lfi ≥ d 2

Z 1

−1

|f|²log |f|²

kfk²₂

dνd,

ifp∈[1,2)∪(2,2^∗) andp= 2, respectively. In the framework of the ultraspherical operator, the parameter d can be considered as a positive real parameter, with critical exponent 2^∗ = _d−2^2d if d >2, and 2^∗ =∞if d≤2. We refer to [52, 5, 15, 13, 14,41, 42,34] for more references. The next lemma gives two elementary but very useful identities:

Lemma 2.2. For any positive smooth functionuon(−1,1), we have Z 1

−1

(Lw)²dνd= Z 1

−1

|w⁰⁰|²ν²dνd+d Z 1

−1

|w⁰|²ν dνd,

|w⁰|² w ν,Lw

= d

d+ 2 Z 1

−1

|w⁰|⁴

w² ν²dνd−2d−1 d+ 2

Z 1

−1

|w⁰|²w⁰⁰

w ν²dνd.

3. Flows. As we explained in the introduction, the main step in our methodology is to take the derivative of the expressioni−dealong the manifoldMpof functions whose L^p norm is equal to 1, following the evolution given by well-chosen flows. In this section we will describe a special flow which leaves this manifold invariant and we will carry out the computation of the derivative.

3.1. The 1-homogeneous case (β = 1). On (−1,1), let us consider the flow defined by (6), that is,wt=Lw+κ^|w_w^0|2, and notice that

d dt

Z 1

−1

w^pdν_d= (κ−p+ 1) Z 1

−1

w^p−2|w⁰|²ν dν_d,

so thatR1

−1w^pdν_d is preserved ifκ=p−1. Recall thatg=w^p obeys to the linear equation gt =Lg. A straightforward computation (using the definition of L and Lemma2.2) shows that

1 2

d dt

Z 1

−1

|w⁰|²ν+ d

p−2 |w|²−w²

dνd

=− Z 1

−1

|w⁰⁰|²ν²dν_d+ 2d−1 d+ 2κ

Z 1

−1

w⁰⁰|w⁰|²

w ν²dν_d− d d+ 2κ

Z 1

−1

|w⁰|⁴

w² ν²dν_d,

sincew= R1

−1w^pdν_d1/p

is independent oft. The r.h.s. is negative if

−γ1= d−1

d+ 2κ 2

− d

d+ 2κ≤0,

that is, ifp≤2^]:= _(d−1)^2d2+12 when d >1, orp >1 whend= 1, and this determines the expression (7) ofγ₁. We have proved the following result.

Proposition 2. For all p∈[1,2^]]if d >1,p >1 if d= 1, there exists a constant γ1>0, such that ifw(t)is defined by (6), then

d dt

Z 1

−1

w^pdνd= 0,

−d dt

Z 1

−1

|w⁰|²ν+ d

p−2 w²−w²

dνd ≥2γ1

Z 1

−1

|w⁰|⁴

w² ν²dνd, whereγ1 is given by (7).

3.2. The nonlinear case (β ≥1). On (−1,1), let us consider the flow defined by (8), that is,wt=w^2−2β Lw+κ^|w_w^0|2

, and notice that d

dt Z 1

−1

w^βp dνd=β p(κ−β(p−2)−1) Z 1

−1

w^β(p−2)|w⁰|²ν dνd,

so that w= R1

−1w^βp dνd

1/(βp)

is preserved if κ=β(p−2) + 1. Similarly as in the previous case we calculate:

− 1 2β²

d dt

Z 1

−1

|(w^β)⁰|²ν+ d

p−2 w^2β−w^2β

dνd

= Z 1

−1

|w⁰⁰|²ν²dν_d−2d−1

d+ 2(κ+β−1) Z 1

−1

w⁰⁰|w⁰|² w ν²dν_d

+

κ(β−1) + d

d+ 2(κ+β−1) Z 1

−1

|w⁰|⁴

w² ν²dνd. The r.h.s. is negative if there exists aβ∈Rsuch that

− γ= d−1

d+ 2(κ+β−1) 2

−

κ(β−1) + d

d+ 2(κ+β−1)

= d−1

d+ 2β(p−1) ²

− d

d+ 2β(p−1) + 1 +β(p−2) (β−1)

≤0, i.e.,γgiven by (10) is nonnegative, and in that case we have found that

− 1 2β²

d dt

Z 1

−1

|(w^β)⁰|²ν+ d

p−2 w^2β−w^2β

dν_d≥γ Z 1

−1

|w⁰|⁴

w² ν²dν_d. (17) This definesγas in (10). Since the l.h.s. of the inequality is quadratic inβ and eval- uates to +1 forβ= 0, a necessary and sufficient condition is that the discriminant, which amounts to

4d(d−2)

(d+ 2)² (p−1) (2^∗−p) where 2^∗:= 2d d−2 takes nonnegative values, that is

1≤p≤2^∗ if d≥3.

In dimension d = 2 and 1, the discriminant is respectively 2 (p−1) and ⁴₉(p− 1)(p+ 2) and takes nonnegative values for any p ≥ 1 (we always assume that p≥1). Altogether, we have proved the following result.

Proposition 3. For all p∈ [1,2^∗], there exist two constants, β ∈ R and γ > 0, such that ifw(t) is defined by (8), then

d dt

Z 1

−1

w^βp dν_d= 0,

− 1 2β²

d dt

Z 1

−1

|(w^β)⁰|²ν+ d

p−2 w^2β−w^2β

dν_d≥γ Z 1

−1

|w⁰|⁴

w² ν²dν_d, whereγ is given by (10).

Note that the flow described in Section3.1 is a special case of the flow defined by (8) corresponding toβ = 1. We treat the cases β = 1 and β >1 separately as they cover different ranges ofp, namely (1,2^#) and (2,2^∗) respectively.

4. Improved inequalities in the range p∈(2,2^∗). In this section, we establish the expression ofϕβ that is used for stating Theorem1.1. The computation corre- sponds to the one of J. Demange in [33] whenβ =_6−p⁴ . In [32], J. Demange noticed that there is a free parameter, which is equivalent to the parameterβ in our setting.

Our purpose is to clarify the range of admissibility ofβ and then optimize on it.

From here on, we assume that β > 1. In Appendix A we show that the set B(p, d) of admissible β’s defined by (11) is non-empty if and only if one of the following conditions (Fig.1) holds:

(i)d≥3, p∈(2,2^∗), (ii)d= 2, p∈(2,9 + 4√

3), (iii)d= 1, p >2, and thus from now on we will take for granted that one of these conditions holds. In

0 4 6 8 10

-2 -1 1 2

0 4 6 8 10

-3 -2 -1 1 2 3

0 1 3 4 5 6

-6 -4 -2 2 4 6

1 2 3 4

2 4 6 8 10 12 14

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

1 2 3 4 5 6

1.0 1.5 2.0 2.5

0.5 1.0 1.5 2.0 2.5

Figure 1. Generically γ(β) = 0has two solutions, β±(p, d). Plot of p7→β±(p, d)withd= 1(left) andd= 2(right) on the first row,d= 3 (left) andd= 4(right) on the second row, andd= 5(left) andd= 10 (right) on the third row. The dotted line corresponds top7→_6−p⁴ and we haveβ−(p)≤ _6−p⁴ ≤β+(p) if and only if ₁₇¹⁸_d−2^d ≤p≤2^∗. The dashed lines correspond to β = 1 and β = _4−p² , and the interior of B(p, d) corresponds to the grey area: see definition (11) of B(p, d). Also see Fig.5for more details in the cased= 2.

order to get the improved inequality, we make use of (17) to get a lower bound for

d

dt(−i+de). We note that the factorγis strictly positive forβ∈B(p, d). However, notice that there existβ /∈B(p, d) such that γ(β)>0. Additional restrictions on the set of the admissibleβ’s indeed appear in what follows.

Let us show that we can get a lower bound for R1

−1

|w⁰|⁴

w² ν²dν_d by two different methods, using three points interpolation H¨older’s inequalities.

1. With ¹₂ +^β−1_2β +_2β¹ = 1, H¨older’s inequality shows that Z 1

−1

|w⁰|²ν dν_d= Z 1

−1

|w⁰|²

w ν·1·w dν_d

≤ Z 1

−1

|w⁰|⁴ w² ν²dνd

1

2Z 1

−1

1dνd

β−1

2β Z 1

−1

w^2βdνd

1 2β

,

from which we deduce that Z 1

−1

|w⁰|⁴ w² ν²dν_d

¹2

≥ R1

−1|w⁰|²ν dνd

R1

−1w^2βdνd

_2β¹ ,

becausedν_d is a probability measure.

2. With ¹₂ +_β(p−2)^β−1 +^{2−(4−p)β}_2β(p−2) = 1 and under the condition that 1< β≤ 2

4−p if p <4, (18)

(but no condition ifp≥4, exceptβ >1), H¨older’s inequality shows that 1

β² Z 1

−1

|(w^β)⁰|²ν dν_d

= Z 1

−1

w^2(β−1)|w⁰|²ν dνd= Z 1

−1

|w⁰|² w ν·w

p(β−1)

p−2 ·w^2βδdνd

≤ Z 1

−1

|w⁰|⁴ w² ν²dνd

¹2Z 1

−1

w^βpdνd

β−1 β(p−2)Z 1

−1

w^2βdνd

^δ

withδ=δ(β) := ^p−_2β^(4−p)_(p−2)^β as in (13), from which we deduce that Z 1

−1

|w⁰|⁴ w² ν²dνd

1 2

≥ 1 β²

R1

−1|(w^β)⁰|²ν dνd

R1

−1w^2βdνd

δ ,

because we have chosenR1

−1w^βp dν_d= 1.

By combining the two estimates we have proved the following.

Lemma 4.1. Assume that (18) holds. For all w ∈ H¹ (−1,1), dν_d

, such that R1

−1w^βpdνd= 1, Z 1

−1

|w⁰|⁴

w² ν²dν_d≥ 1 β²

R1

−1|(w^β)⁰|²ν dνdR1

−1|w⁰|²ν dνd

R1

−1w^2βdνd

^δ .

Next, we can use the above estimates to show that there is a differential inequality relatingeandi. From the computations in Section3.2, and Proposition3, we find that

− 1 2β²

d dt

Z 1

−1

|(w^β)⁰|²ν+ d

p−2 w^2β−w^2β

dνd≥γ Z 1

−1

|w⁰|⁴ w² ν²dνd

withw= R1

−1w^pdν_d1/p

= 1, hence

−i⁰+de⁰≥2γ R1

−1|(w^β)⁰|²ν dν_dR1

−1|w⁰|²ν dν_d R1

−1w^2βdνd

δ =− γ 2β²

i e⁰

(1 − (p−2)e)^δ. Let us suppose thatψ_β is a function that satisfies the ODE

ψ⁰⁰_β(e)

ψ⁰_β(e) =− γ

2β² (1 −(p−2)e)^−δ ,

and such that ψ_β⁰ > 0. It is elementary to show that such function exists when β >1. The l.h.s. of the inequality

i⁰− de⁰− γ 2β²

i e⁰

(1 −(p−2)e)^δ ≤0 can then be considered as a total derivative, namely

d

dt iψ⁰_β(e)−d ψ_β(e)

≤0, thus proving after an integration int∈[0,∞) that

dψ_β(e₀) ψ_β⁰(e0) ≤i0. Next, we let

ϕβ(e) := ψ_β(e) ψ_β⁰(e). It is then elementary to check thatϕβ satisfies the ODE

ϕ⁰_β= 1−ϕβ

ψ_β⁰⁰(e)

ψ_β⁰(e) = 1 +ϕβ

γ

2β² (1 −(p−2)e)^−δ

andϕ_β(0) = 0. Solving this linear ODE, we find that ϕ_β is given by (12). Notice thatϕβ is defined for anye∈[0,_p−2¹ ) andϕβ(e)>0, for e>0. From the equation satisfied by ϕβ we get that ϕ⁰_β(e) > 1 and ϕ⁰⁰_β(e) > 0, e > 0, hence ϕβ(e) > e for any admissible β. As a consequence we also get that the functions ϕ⁻¹_β and i7→i−ϕ⁻¹_β (i) are increasing. Let us define (cf. introduction)

ϕ(e) := supn

ϕβ(e) : β∈B(p, d)o .

Numerically we observe thatϕ(e) =ϕ_β(e)(e) for an optimalβ=β(e), which explic- itly depends one. For future reference we observe thatϕ⁰(e)>1 andϕ⁰⁰(e)>0, for a.e.e>0, henceϕ(e)>eand the functionsϕ⁻¹ andi7→i−ϕ⁻¹(i) are increasing.

From the preceding considerations it follows that we have the inequality:

d ϕ(e)≤i.

Now we recall that this inequality is obtained under the assumptionkuk²_Lp(S^d)= 1, namelye= _p−2¹ 1− kuk²_L2(S^d)

. Then, if we do not normalize the L^p norm we get in general:

dkuk²_Lp(S^d)ϕ e kuk²_Lp(

S^d)

!

≤i.

Now, remembering the definition of the improved inequality, we need to find the relation between the l.h.s. of this estimate, which is a function ofe/kuk²_Lp(S^d), and the functiondkuk²_L2(S^d)Φ e/kuk²_L2(S^d)

. This is quite easy since we have kuk²_Lp(S^d)

kuk²_L2(S^d)

= 1 + (p−2)s with s= e kuk²_L2(S^d)

.

By straightforward manipulations we get from this kuk²_Lp(S^d)ϕ e

kuk²_Lp(

S^d)

!

= 1 + (p−2)s ϕ

s 1 + (p−2)s

=: Φ(s) where Φ is the function under consideration in Theorem1.1. Based on the properties ofϕ, it is easy to check that Φ(0) = 0, Φ⁰(0) = 1 and Φ⁰⁰(s)>0. In Fig.2we show the measure of improvement between the improved inequality and the standard inequality.

This ends the proof of Theorem 1.1 in the case p∈ (2,2^∗) if d = 1 or d ≥3, p∈(2,9 + 4√

3), ifd= 2.

0.5 1.0 1.5 2.0

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0 1.2

-0.002 -0.001 0.001

Figure 2. Plot of the function ϕβ corresponding to Theorem 1.1.

Hered= 5,p= 2.5andβ≈2.383,2.267,2.15,2.033,1.917,1.8,1.683, 1.567,1.45. Left: plot ofe7→ϕβ(e)−efore∈(0,1/(p−2)). Right: plot ofe7→ϕβ(e)−ϕβ0(e) whereβ0 = 4/(6−p)corresponds to the choice made in[33].

As a conclusion for this section, let us comment on the literature and emphasize the new results. In [33] J. Demange gives a proof by considering a flow which corresponds to (8), in the special case β = _6−p⁴ . Here we generalize it to a larger but explicit range of β’s, as was done in [32, pages 122–130] (see in particular Proposition 3.12.5). Moreover we explicitly show how to optimize the interpolation inequalities with respect to β and we specify the range of admissible β’s. See AppendixAfor more details.

5. Improved inequalities in the range p∈ (1,2^#): Proof of Theorem 1.1 when p ∈ (1,2) and when p = 2. In this section we adapt the Bakry-Emery approach (which amounts to write a differential inequality fori=−e⁰) and improve it by taking into account the remainder term as in [2]. Here we assume thatβ = 1.

Our result is primarily an improvement of the existing results for p ∈ (1,2), but we work in the larger range p ∈ (1,2^]) (see Section 3.1). We will see that the limitation on the exponent appears naturally since we do not allow any freedom forβ. This special exponent 2^]was already noticed in [8,34]. In the rangep >2, the

limitationp≤2^# is equivalent to require thatβ= 1∈B(p, d). Whenp∈(2,2^#], the computations of this section are a limiting case of those in Section 4. Since estimates have to be adapted and since the range in p is anyway different, we handle this case separately.

Assuming thatkukp= 1 anduis a solution of (6) and following the calculations of Section3.1and Section4, one can check the validity of the differential inequality

e⁰⁰+ de⁰ ≥γ₁ 2

|e⁰|² 1−(p−2)e,

whereγ1>0 has been defined in Section3.1. The estimate is a simple consequence of the Cauchy-Schwarz inequality |e⁰|² = kw⁰k²₂ ≤ kwk2k(w⁰)²/wk2. We observe that we have the boundary conditionse(t= 0) =e₀,i(t= 0) =i₀ and

t→∞lim e(t) = lim

t→∞i(t) = 0.

Proposition 4. With the above notations, and assuming p∈ [1,2)∪(2,2^#] and R1

−1w^pdν_d= 1, we have the following estimate d ϕ1(e)≤i,

where ϕ1 is defined by (9) and such that ϕ1(0) = 0, ϕ⁰₁(0) = 1 andϕ⁰⁰₁(s)>0 for any s >0if p >2, or for any s∈(0,1/(2−p))ifp∈(1,2)

0.0 0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0

2 4 6 8 10

2 4 6 8 10

Figure 3. The plot ofξ 7→ _{2 (2−p)−γ}^{2 (2−p)}

1 ξ²⁺

γ1 p−2−1

ξ²−1 represents the im- provement achieved in Proposition 4 compared to the standard form of the inequality, which is represented by 1. Left: p ∈ (1,2) and ξ = kfk_Lp(S^d)/kfk_L2(S^d) is in the range (0,1). The plot corresponds top= 3/2andd= 2. Right: p∈(2,2^#)andξ=kfk_Lp(S^d)/kfk_L2(S^d) is in the rangeξ >1. The plot corresponds top= 5/2andd= 2,2^#= 9.

Proof. Let us defineh(t) := 1−(p−2)e(t). Whenp∈(2,2^#) we have h⁰ =−(p−2)e⁰>0, h⁰⁰=−(p−2)e⁰⁰<0,

while whenp∈(1,2) these inequalities are changed to their opposities. Alsoh(t)∈ (0,1),h(+∞) = 1, h⁰(+∞) = 0. Our differential inequality takes form:

− 1

p−2(h⁰⁰+ dh⁰)− γ1

2 (p−2)²

|h⁰|² h ≥0, which upon multiplying by ^p−2_h0 and rearranging leads to

d

dtlogh⁰+ γ1

2 (p−2) d dtlogh

(≤ −d if p >2,

≥ −d if p <2,