Jean Dolbeault

Giuseppe Toscani

Improved interpolation inequalities, relative entropy and fast diffusion equations

Keywords: Gagliardo-Nirenberg-Sobolev inequalities, improved inequalities, manifold of optimal functions, entropy -entropy production method, fast diffusion equation, Barenblatt solutions, second moment, intermediate asymptotics, sharp rates, optimal constants AMS subject classification (2010): 26D10, 46E35, 35K55

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Improved interpolation inequalities, relative entropy and fast diffusion equations

Introduction and main results

Consider the following sub-family of the Gagliardo-Nirenberg-Sobolev inequalities [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]

f 2 p ≤ C GN p,d ∇f θ 2 f 1-θ p+1 with θ = θ(p) := p-1 p d d+2-p (d-2) , 1 < p ≤ d d-2 if d ≥ 3 and 1 < p < ∞ if d = 2
. Such an inequality holds for any smooth function f with sufficient decay at infinity and, by density, for any function f ∈ L p+1 (R d ) such that ∇f July 11, 2012 is square integrable. We shall assume that C GN p,d is the best possible constant in [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]. In [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF], it has been established that equality holds in [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]

if f = F p with (2) F p (x) = (1 + |x| 2 ) -1 p-1 ∀ x ∈ R d
and that all extremal functions are equal to F p up to a multiplication by a constant, a translation and a scaling. See Appendix A for an expression of C GN p,d . If d ≥ 3, the limit case p = d/(d -2) corresponds to Sobolev's inequality and one recovers the optimal functions found by T. Aubin and G. Talenti in [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF][START_REF] Talenti | Best constant in Sobolev inequality[END_REF]. When p → 1, the inequality becomes an equality, so that we may differentiate both sides with respect to p and recover the euclidean logarithmic Sobolev inequality in optimal scale invariant form (see [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF][START_REF] Weissler | Logarithmic Sobolev inequalities for the heat-diffusion semigroup[END_REF][START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] for details).

It is rather straightforward to observe that Inequality (1) can be rewritten, in a non-scale invariant form, as a non-homogeneous Gagliardo-Nirenberg-Sobolev inequality:

for any f ∈ L p+1 ∩ D 1,2 (R d ), (3)
R d |∇f | 2 dx + R d |f | p+1 dx ≥ K p,d Å R d |f | 2 p dx ã γ with (4) γ = γ(p, d) := d+2-p (d-2) d-p (d-4) .
The optimal constant K p,d can easily be related with C GN p,d . Indeed, by optimizing the left hand side of (3) written for f λ (x) := λ d/(2 p) f (λ x) for any x ∈ R d , with respect to λ > 0, one recovers that (3) and (1) are equivalent. The detailed relation between K p,d and C GN p,d can be found in Section 7.

Define now

C M := Å M * M ã 2 (p-1) d-p (d-4)
, M * :=

R d Ä 1 + |x| 2 ä -2 p p-1 dx = π d 2 Γ Ä d-p (d-4) 2 (p-1) ä Γ Ä 2 p p-1 ä .
Consider next a generic, non-negative optimal function, f

M,y,σ (x) := σ -d To simplify our statement, we will introduce a normalization constraint and assume that f ∈ L 2 p (R 2 , (1 + |x| 2 ) dx) is such that Such a condition is not restrictive, as it is always possible to cover the general case by rescaling the inequality, but significantly simplifies the expressions. As we shall see in the proof, the only goal is to fix σ = 1.

4 p Å C M + 1 σ |x -y| 2 ã -1 p-1 ∀ x ∈ R d
Our main result goes as follows.

Theorem 1. Let d ≥ 2, p > 1 and assume that p < d/(d -2) if d ≥ 3. For any f ∈ L p+1 ∩ D 1,2 (R d ) such that Condition (5) holds, we have R d |∇f | 2 dx + R d |f | p+1 dx -K p,d Å R d |f | 2 p dx ã γ ≥ C p,d Ä R (p) [f ] ä 2 R d |f | 2 p dx γ
where γ is given by (4).

The constant C p,d is positive and explicit. We do not know its optimal value. See Appendix A for an expression of C p,d , which is such that

lim p→1 + C p,d = 0 and lim p→d/(d-2) - C p,d = 0 .
The space L p+1 ∩ D 1,2 (R d ) is the natural space for Gagliardo-Nirenberg inequalities as it can be characterized as the completion of the space of smooth functions with compact support with respect to the norm • such that

f 2 = ∇f 2 2 + f 2 p+1 .
In this paper, we shall also use the notations f p,q := ( R d |x| p |f | q dx) 1/q , so that f q = f 0,q .

Under Condition (5), we shall deduce from Theorem 4 that ( 6)

R (p) [f ] ≥ C CK f 2 p (γ-2) 2 p inf g∈M (p) d |f | 2 p -g 2 p 2 1 with δ = d + 2 -p (d + 6
) for some constant C CK whose expression is given in Section 3, Eq. [START_REF] Carrillo | Fine asymptotics for fast diffusion equations[END_REF]. Putting this estimate together with the result of Theorem 1, with

C p,d := C d,p C CK 2 ,
we obtain the following estimate.

Corollary 2. Under the same assumptions as in Theorem 1, we have

R d |∇f | 2 dx + R d |f | p+1 dx -K p,d Å R d |f | 2 p dx ã γ ≥ C p,d f 2 p (γ-4) 2 p inf g∈M d (p) |f | 2 p -g 2 p 4 1 .
The critical case p = d/(d-2) corresponding to Sobolev's inequality raises a number of difficulties which are not under control at this stage. However, results which have been obtained in such a critical case, by different methods, are the main motivation for the present paper.

In d is controlled by the difference of the two terms in the critical Sobolev inequality written with an optimal constant. Some partial answers have been provided over the years, of which we can list the following ones. First G. Bianchi and H. Egnell gave in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] a result based on the concentration-compactness method, which determines a non-constructive estimate for a distance to the set of optimal functions. In [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF], A. Cianchi, N. Fusco, F. Maggi and A. Pratelli established an improved inequality using symmetrization methods. Also see [START_REF] Cianchi | Quantitative Sobolev and Hardy inequalities, and related symmetrization principles[END_REF] for an overview of various results based on such methods. Recently another type of improvement, which relates Sobolev's inequality to the Hardy-Littlewood-Sobolev inequalities, has been established in [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF], based on the flow of a nonlinear diffusion equation, in the regime of extinction in finite time. Theorem 1 does not provide an answer in the critical case, but gives an improvement with fully explicit constants in the subcritical regime. Our method of proof enlightens a new aspect of the problem. Indeed, Theorem 1 shows that the difference of the two terms in the critical Sobolev inequality provides a better control under the additional information that f 2,2 p is finite. Such a condition disappears in the setting of Corollary 2.

In this paper, our goal is to establish an improvement of Gagliardo-Nirenberg inequalities based on the flow of the fast diffusion equation in the regime of convergence towards Barenblatt self-similar profiles, with an explicit measure of the distance to the set of optimal functions. Our approach is based on a relative entropy functional. The method relies on a recent paper, [START_REF] Dolbeault | Fast diffusion equations: Matching large time asymptotics by relative entropy methods[END_REF], which is itself based on a long series of studies on intermediate asymptotics of the fast diffusion equation, and on the entropy -entropy production method introduced in [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF][START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF] in the linear case and later extended to nonlinear diffusions: see [START_REF] Newman | A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. I[END_REF][START_REF] Ralston | A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. II[END_REF][START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF][START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity[END_REF][START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF]. In this setting, having a finite second moment is crucial. Let us give some explanations.

Consider the fast diffusion equation with exponent m given in terms of the exponent p of Theorem 1 by

(7) p = 1 2 m -1 ⇐⇒ m = p + 1 2 p .
More specifically, for m ∈ (0, 1), we shall consider the solutions of ( 8)

∂u ∂t + ∇ • î u Ä η ∇u m-1 -2 x äó = 0 t > 0 , x ∈ R d with initial datum u(t = 0, •) = u 0 .
Here η is a positive parameter which does not depend on t. Let u ∞ be the unique stationary solution such that

M = R d u dx = R d u ∞ dx. It is given by u ∞ (x) = Å K + 1 η |x| 2 ã 1 m-1 ∀ x ∈ R d
for some positive constant K which is uniquely determined by M . The following exponents are associated with the fast diffusion equation [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] and will be used all over this paper:

m c := d -2 d , m 1 := d -1 d and ‹ m 1 := d d + 2 .
To the critical exponent 2 p = 2 d/(d -2) for Sobolev's inequality corresponds the critical exponent m 1 for the fast diffusion equation. For d ≥ 3, the con-

dition p ∈ (1, d/(d -2)) in Theorem 1 is equivalent to m ∈ (m 1 , 1) while for d = 2, p ∈ (1, ∞) means m ∈ (1/2, 1).
It has been established in [START_REF] Newman | A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. I[END_REF][START_REF] Ralston | A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. II[END_REF] that the relative entropy (or free energy)

F[u|u ∞ ] := 1 m -1 R d î u m -u m ∞ -m u m-1 ∞ (u -u ∞ ) ó dx decays according to d dt F[u(•, t)|u ∞ ] = -I[u(•, t)|u ∞ ]
if u is a solution of [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], where

I[u(•, t)|u ∞ ] := η m 1 -m R d u ∇u m-1 -∇u m-1 ∞ 2 dx
is the entropy production term or relative Fisher information. If m ∈ [m 1 , 1), according to [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF], these two functionals are related by a Gagliardo-Nirenberg interpolation inequality, namely

(9) F[u|u ∞ ] ≤ 1 4 I[u|u ∞ ] .
We shall give a concise proof of this inequality in the next section (see Remark 1) based on the entropy -entropy production method, which amounts to relate

d dt I[u(•, t)|u ∞ ] and I[u(•, t)|u ∞ ].
We shall later replace the diffusion parameter η in (8) by a time-dependent coefficient σ(t), which is itself computed using the second moment of u, R d |x| 2 u(x, t) dx. By doing so, we will be able to capture the best matching Barenblatt solution and get improved decay rates in the entropy -entropy production inequality. Elementary estimates allow to rephrase these improved rates into improved functional inequalities for f such that |f | 2 p = u, for any p ∈ (1, d/(d -2)), as in Theorem 1.

This paper is organized as follows. In Section 2, we apply the entropyentropy production method to the fast diffusion equation as in [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF]. The key computation, without justifications for the integrations by parts, is reproduced here since we need it later in Section 6, in the case of a time-dependent diffusion coefficient. Next, in Section 3, we establish a new estimate of Csiszár-Kullback type. By requiring a condition on the second moment, we are able to produce a new estimate which was not known before, namely to directly control the difference of the solution with a Barenblatt solution in L 1 (R d ).

Second moment estimates are the key of a recent paper and we shall primarily refer to [START_REF] Dolbeault | Fast diffusion equations: Matching large time asymptotics by relative entropy methods[END_REF] in which the asymptotic behaviour of the solutions of the fast diffusion equation was studied. In Section 4 we recall the main results that were proved in [START_REF] Dolbeault | Fast diffusion equations: Matching large time asymptotics by relative entropy methods[END_REF], and that are also needed in the present paper.

With these preliminaries in hand, an improved entropy -entropy production inequality is established in Section 5, which is at the core of our paper. It is known since [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] that entropy -entropy production inequalities amount to optimal Gagliardo-Nirenberg-Sobolev inequalities. Such a rephrasing of our result in a more standard form of functional inequalities is done in Section 6, which contains the proof of Theorem 1. Further observations have been collected in Section 7. One of the striking results of our approach is that all constants can be explicitly computed. This is somewhat technical although not really difficult. To make the reading easier, explicit computations have been collected in Appendix A.

The entropy -entropy production method

Consider a solution u = u(x, t) of Eq. ( 8) and define z(x, t) := η ∇u m-1 -2 x so that Eq. ( 8) can be rewritten as

∂u ∂t + ∇ • (u z) = 0 .
To keep notations compact, we shall use the following conventions.

If A = (A ij ) d i,j=1 and B = (B ij ) d i,j=1 are two matrices, let A : B = d i,j=1 A ij B ij and |A| 2 = A : A. If a and b take values in R d , we adopt the definitions: a•b = d i=1 a i b i , ∇•a = d i=1 ∂a i ∂x i , a⊗b = (a i b j ) d i,j=1 , ∇⊗a = Å ∂a j ∂x i ã d i,j=1
.

Later we will need a version of the entropy -entropy production method in case of a time-dependent diffusion coefficient. Before doing so, let us recall the key computation of the standard method. With the above notations, it is straightforward to check that

∂z ∂t = η (1 -m)∇ Ä u m-2 ∇ • (u z) ä and ∇ ⊗ z = η ∇ ⊗ ∇u m-1 -2 Id . With these definitions, the time-derivative of 1-m m η I[u|u ∞ ] = R d u |z| 2 dx can be computed as d dt R d u |z| 2 dx = R d ∂u ∂t |z| 2 dx + 2 R d u z • ∂z ∂t dx .
The first term can be evaluated by

R d ∂u ∂t |z| 2 dx = - R d ∇ • (u z) |z| 2 dx = 2 R d u z ⊗ z : ∇ ⊗ z dx = 2 η R d u z ⊗ z : ∇ ⊗ ∇u m-1 dx -4 R d u |z| 2 dx = 2 η (1 -m) R d u m-2 ∇u ⊗ ∇ : (u z ⊗ z) dx -4 R d u |z| 2 dx = 2 η (1 -m) R d u m-2 (∇u • z) 2 dx + 2 η (1 -m) R d u m-1 (∇u • z) (∇ • z) dx + 2 η (1 -m) R d u m-1 (z ⊗ ∇u) : (∇ ⊗ z) dx -4 R d u |z| 2 dx .
The second term can be evaluated by 2

R d u z • ∂z ∂t dx = 2 η (1 -m) R d (u z • ∇) Ä u m-2 ∇ • (u z) ä dx = -2 η (1 -m) R d u m-2 Ä ∇ • (u z) ä 2 dx = -2 η (1 -m) R d î u m (∇ • z) 2 + 2 u m-1 (∇u • z) (∇ • z) + u m-2 (∇u • z) 2 ó dx .
Summarizing, we have found that

R d ∂u ∂t |z| 2 dx + 4 R d u |z| 2 dx = -2 η (1-m) R d u m-2 î u 2 (∇•z) 2 +u (∇u • z) (∇ • z)-u (z⊗∇u) : (∇⊗ z) ó dx .
Using the fact that

∂ 2 z j ∂x i ∂x j = ∂ 2 z i ∂x 2 j , we obtain that R d u m-1 (∇u • z) (∇ • z) dx = - 1 m R d u m (∇ • z) 2 dx - 1 m R d u m d i,j=1 z i ∂ 2 z j ∂x i ∂x j dx and - R d u m-1 (z ⊗ ∇u) : (∇⊗ z) dx = 1 m R d u m |∇z| 2 dx + 1 m R d u m d i,j=1 z i ∂ 2 z i ∂x 2 j dx can be combined to give R d u m-2 î u (∇u • z) (∇ • z) -u ∇u ⊗ z : ∇ ⊗ z ó dx = - 1 m R d u m (∇ • z) 2 dx + 1 m R d u m |∇z| 2 dx .
This shows that (10)

d dt R d u |z| 2 dx + 4 R d u |z| 2 dx = -2 η 1 -m m R d u m Ä |∇z| 2 -(1 -m) (∇ • z) 2 ä dx .
By the arithmetic geometric inequality, we know that

|∇z| 2 -(1 -m) (∇ • z) 2 ≥ 0 if 1 -m ≤ 1/d, that is, if m ≥ m 1 .
Altogether, we have formally established the following result.

Proposition 3. Let d ≥ 1, m ∈ (m 1 , 1)
and assume that u is a nonnegative solution of (8) with initial datum u 0 in L 1 (R d ) such that u m 0 and x → |x| 2 u 0 are both integrable on R d . With the above defined notations, we get that

d dt I[u(•, t)|u ∞ ] ≤ -4 I[u(•, t)|u ∞ ] ∀ t > 0 .
The proof of such a result requires to justify that all integrations by parts make sense. We refer to [START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity[END_REF][START_REF] Carrillo | Fine asymptotics for fast diffusion equations[END_REF] for a proof in the porous medium case (m > 1) and to [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF] for m 1 ≤ m < 1. The case m = 1 was covered long ago in [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF].

Remark 1. Proposition 3 provides a proof of (9). Indeed, with a Gronwall estimate, we first get that

I[u(•, t)|u ∞ ] ≤ I[u 0 |u ∞ ] e -4 t ∀ t ≥ 0 if I[u 0 |u ∞ ] is finite. Since I[u(•, t)|u ∞ ] is non-negative, we know that lim t→∞ I[u(•, t)|u ∞ ] = 0 ,
which proves the convergence of u(•, t) to u ∞ as t → ∞. As a consequence, we also have

lim t→∞ F[u(•, t)|u ∞ ] = 0 and since d dt Ä I[u(•, t)|u ∞ ] -4 F[u(•, t)|u ∞ ] ä = d dt I[u(•, t)|u ∞ ] + 4 I[u(•, t)|u ∞ ] ≤ 0 ,
an integration with respect to t on (0, ∞) shows that

I[u 0 |u ∞ ] -4 F[u 0 |u ∞ ] ≥ 0 ,
which is precisely (9) written for u = u 0 .

A Csiszár-Kullback inequality

Let m ∈ (‹ m 1 , 1) with ‹ m 1 = d d+2 and consider the relative entropy

F σ [u] := 1 m -1 R d î u m -B m σ -m B m-1 σ (u -B σ ) ó dx for some Barenblatt function (11) B σ (x) := σ -d 2 Ä C M + 1 σ |x| 2 ä 1 m-1 ∀ x ∈ R d
where σ is a positive constant and C M is chosen such that B σ 1 = M > 0.

With p and m related by [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF], the definition of C M coincides with the one of Section 1. See details in Appendix A.

Theorem 4. Let d ≥ 1, m ∈ (‹ m 1 , 1) and assume that u is a non-negative function in L 1 (R d ) such that u m and x → |x| 2 u are both integrable on R d . If u 1 = M and R d |x| 2 u dx = R d |x| 2 B σ dx, then F σ [u] σ d 2 (1-m) ≥ m 8 R d B m 1 dx Å C M u -B σ 1 + 1 σ R d |x| 2 |u -B σ | dx ã 2 .
Notice that the condition R d |x| 2 u dx = R d |x| 2 B σ dx is explicit and determines σ uniquely:

σ = 1 K M R d |x| 2 u dx with K M := R d |x| 2 B 1 dx .
For further details, see Lemma 5 and (20) below, and Appendix A for detailed expressions of

K M and R d B m 1 dx. With this choice of σ, since B m-1 σ = σ d 2 (1-m) C M + σ d 2 (mc-m) |x| 2 , we remark that R d B m-1 σ (u -B σ ) dx = 0 so that the relative entropy reduces to F σ [u] := 1 m -1 R d [u m -B m σ ] dx
Proof of Theorem 4. Let v := u/B σ and dµ σ := B m σ dx. With these notations, we observe that

R d (v -1) dµ σ = R d B m-1 σ (u -B σ ) dx = σ d 2 (1-m) C M R d (u -B σ ) dx + σ d 2 (mc-m) R d |x| 2 (u -B σ ) dx = 0 . Thus R d (v -1) dµ σ = v>1 (v -1) dµ σ - v<1 (1 -v) dµ σ = 0 , which, coupled with v>1 (v -1) dµ σ + v<1 (1 -v) dµ σ = R d |v -1| dµ σ , implies R d |u -B σ | B m-1 σ dx = R d |v -1| dµ σ = 2 v<1 |v -1| dµ σ .
On the other hand, a Taylor expansion shows that

F σ [u] = 1 m -1 R d î v m -1 -m (v -1) ó dµ σ = m 2 R d ξ m-2 |v -1| 2 dµ σ
for some function ξ taking values in the interval (min{1, v}, max{1, v}), thus giving the lower bound

F σ [u] ≥ m 2 v<1 ξ m-2 |v -1| 2 dµ σ ≥ m 2 v<1 |v -1| 2 dµ σ .
Using the Cauchy-Schwarz inequality, we get

Å v<1 |v -1| dµ σ ã 2 = Å v<1 |v -1| B m 2 σ B m 2 σ dx ã 2 ≤ v<1 |v -1| 2 dµ σ R d B m σ dx
and finally obtain that

F σ [u] ≥ m 2 Ä v<1 |v -1| dµ σ ä 2 R d B m σ dx = m 8 R d |u -B σ | B m-1 σ dx 2 R d B m σ dx
, which concludes the proof.

Notice that the inequality of Theorem 4 can be rewritten in terms of |f | 2 p = u and g 2 p = B σ with p = 1/(2 m -1). See Appendix A for the computation of R d B m σ dx, σ, C M and K M in terms of R d |x| 2 u dx and M * . In the framework of Corollary 2, we observe that Condition (5) can be rephrased as

(12) σ = 1 K M f 2 p 2,2 p = 1 K 1 f 2 p 2,2 p f 2 p γ 2 p = σ * .
Altogether we find in such a case that

R (p) [f ] = p-1 p+1 F σ * [u] ≥ C CK |f | 2 p -|g| 2 p 2 with (13) C CK = p-1 p+1 d+2-p (d-2) 32 p σ d p-1 4 p * M 1-γ * . Remark 2.
Various other estimates can be derived, based on second order Taylor expansions. For instance, as in [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF], we can write that

F σ [u] = R d î ψ(v m ) -ψ(1) -ψ ′ (1) (v m -1) ó dµ σ
with v := u/B σ and ψ(s) := m 1-m s 1/m , and get

F σ [u] ≥ 1 m 2 -2 m v m -1 2 L 1/m (R d ,dµσ) max ¶ v m L 1/m (R d ,dµσ) , 1 L 1/m (R d ,dµσ) © 2-1 m . Using v m L 1/m (R d ,dµσ) = 1 L 1/m (R d ,dµσ) = B m σ m 1 and R d |u m -B m σ | dx = R d |u m -B m σ | B m (m-1) σ B m (1-m) σ dx ≤ v m -1 L 1/m (R d ,dµσ) B m σ 1-m 1
by the Cauchy-Schwarz inequality, we find

F σ [u] ≥ u m -B m σ 2 1 m 2 2 m B m σ 1
.

With f = u m-1 2 , this also gives another estimate of Csiszár-Kullback type, namely

R (p) [f ] ≥ κ p,d f d 2 (p-1) 2,2 p f 1 2 (d+2-p (d-2)) 2 p inf g∈M (p) d |f | p+1 -g p+1 2 1
, for some positive constant κ p,d , which is valid for any

p ∈ (1, ∞) if d = 2 and any p ∈ (1, d d-2 ] if d ≥ 3.
Also see [START_REF] Unterreiter | On generalized Csiszár-Kullback inequalities[END_REF][START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity[END_REF][START_REF] Cáceres | Nonlinear stability in L p for a confined system of charged particles[END_REF][START_REF] Dolbeault | Large time behaviour of solutions to nonhomogeneous diffusion equations[END_REF] for further results on Csiszár-Kullback type inequalities corresponding to entropies associated with porous media and fast diffusion equations.

Recent results on the optimal matching by Barenblatt solutions

Consider on R d the fast diffusion equation with harmonic confining potential given by ( 14)

∂u ∂t + ∇ • u σ d 2 (m-mc) ∇u m-1 -2 x = 0 t > 0 , x ∈ R d ,
with initial datum u 0 . Here σ is a function of t. Let us summarize some results obtained in [START_REF] Dolbeault | Fast diffusion equations: Matching large time asymptotics by relative entropy methods[END_REF] and the strategy of their proofs.

Result 1. At any time t > 0, we can choose the best matching Barenblatt as follows. Consider a given function u and optimize λ → F λ [u].

Lemma 5. For any given u ∈ L 1 + (R d ) such that u m and |x| 2 u are both integrable, if m ∈ (‹ m 1 , 1), there is a unique λ = λ * > 0 which minimizes λ → F λ [u], and it is explicitly given by

λ * = 1 K M R d |x| 2 u dx where K M = R d |x| 2 B 1 dx. For λ = λ * , the Barenblatt profile B λ satisfies R d |x| 2 B λ dx = R d |x| 2 u dx .
As a consequence, we know that d dλ 14), the value of λ in Lemma 5 may depend on t. Now we choose σ(t) = λ(t), i.e., [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF]

Ä F λ [u] ä λ=λ * = 0 . Of course, if u is a solution of (
σ(t) = 1 K M R d |x| 2 u(x, t) dx ∀ t ≥ 0 .
This makes ( 14) a non-local equation.

Result 2. With the above choice, if we consider a solution of ( 14) and compute the time derivative of the relative entropy, we find that

d dt F σ(t) [u(•, t)] = σ ′ (t) Å d dσ F σ [u] ã |σ=σ(t) + m m -1 R d u m-1 -B m-1 σ(t) ∂u ∂t dx .
However, as a consequence of the choice (15) and of Lemma 5, we know that

Å d dσ F σ [u] ã |σ=σ(t) = 0 ,
and finally obtain

(16) d dt F σ(t) [u(•, t)] = - m σ(t) d 2 (m-mc) 1 -m R d u ∇ u m-1 -B m-1 σ(t) 2 dx .
The computation then goes as in [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] (also see [START_REF] Newman | A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. I[END_REF][START_REF] Ralston | A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. II[END_REF][START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] for details). With our choice of σ, we gain an additional orthogonality condition which is useful for improving the rates of convergence (see [START_REF] Dolbeault | Fast diffusion equations: Matching large time asymptotics by relative entropy methods[END_REF]Theorem 1]) in the asymptotic regime t → ∞, compared to the results of [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] (also see below).

Result 3. Now let us state one more result of [START_REF] Dolbeault | Fast diffusion equations: Matching large time asymptotics by relative entropy methods[END_REF] which is of interest for the present paper. Lemma 6. With the above notations, if u and σ are defined respectively by ( 14) and (15), then the function t → σ(t) is positive, decreasing, with σ ∞ := lim t→∞ σ(t) > 0 and

(17) σ ′ (t) = -2 d (1 -m) 2 m K M σ d 2 (m-mc) F σ(t) [u(•, t)] ≤ 0 .
The main difficulty is to establish that σ ∞ is positive. This can be done with an appropriate change of variables which reduces [START_REF] Cianchi | Quantitative Sobolev and Hardy inequalities, and related symmetrization principles[END_REF] to the case where σ does not depend on t. In [START_REF] Dolbeault | Fast diffusion equations: Matching large time asymptotics by relative entropy methods[END_REF], a proof has been given, based on asymptotic results for the fast diffusion equation that were established in [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF][START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Blanchet | Hardy-Poincaré inequalities and applications to nonlinear diffusions[END_REF][START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF]]. An alternative proof will be given in Remark 3, below.

The scaled entropy -entropy production inequality

Consider the relative Fisher information

I σ [u] := σ d 2 (m-mc) m 1 -m R d u ∇u m-1 -∇B m-1 σ 2 dx .
By applying [START_REF] Brezis | Sobolev inequalities with remainder terms[END_REF] with u ∞ = B 1 and η = 1 to x → σ d/2 u( √ σ x) and using the fact that B 1 (x) = σ d/2 B σ ( √ σ x), we get the inequality

F σ [u] ≤ 1 4 I σ [u] .
Now, if σ is time-dependent as in Section 4, we have the following relations. [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF]. Moreover, for any t ≥ 0, we have

Lemma 7. If u is a solution of (14) with σ(t) = 1 K M R d |x| 2 u(x, t) dx, then σ satisfies
(18) d dt F σ(t) [u(•, t)] = -I σ(t) [u(•, t)] and (19)
d dt I σ(t) [u(•, t)] ≤ - ñ 4 + 1 2 (m -m c ) (m -m 1 ) d 2 |σ ′ (t)| σ(t) ô I σ(t) [u(•, t)] .
Proof. Eq. ( 17) and ( 18) have already been stated respectively in Lemma 6 and in [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF]. They are recalled here only for the convenience of the reader. It remains to prove [START_REF] Dolbeault | Fast diffusion equations: Matching large time asymptotics by relative entropy methods[END_REF].

For any given σ = σ(t), Proposition 3 gives

d dt I σ(t) [u(•, t)] = Å d dt I λ [u(•, t)] ã |λ=σ(t) + σ ′ (t) Å d dλ I λ [u] ã |λ=σ(t) ≤ -4 I σ(t) [u(•, t)] + σ ′ (t) Å d dλ I λ [u] ã |λ=σ(t)
.

Owing to the definition of I λ , we obtain

d dλ I λ [u] = d 2 (m -m c ) 1 λ I λ [u] - m 1 -m λ d 2 (m-mc) R d 2 u Ä ∇u m-1 -∇B m-1 λ ä • d dλ Ä ∇B m-1 λ ä dx .
By definition [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF],

∇B m-1 λ (x) = 2 x λ -d 2 (m-mc) , which implies λ d 2 (m-mc) d dλ Ä ∇B m-1 λ ä = - d λ (m -m c ) x .
Substituting this expression into the above computation and integrating by parts, we conclude with the equality

d dλ I λ [u] = d 2 (m -m c ) 1 λ I λ [u] - 2 d λ (m -m c ) 2 m λ -d 2 (m-mc) 1 -m R d |x| 2 u dx -d R d u m dx .
A simple computation shows that

(20) d R d B m 1 dx = - R d x • ∇B m 1 dx = 2 m 1 -m R d |x| 2 B 1 dx = 2 m 1 -m K M
and, as a consequence, if

λ = σ = 1 K M R d |x| 2 u dx, then 2 m λ -d 2 (m-mc) 1 -m R d |x| 2 u dx = d R d B m λ dx ,
and finally ( 21)

d dλ I λ [u] = d 2 λ (m -m c ) I λ [u] -4 d (1 -m) F λ [u] .
Altogether, we have found that

d dt I σ(t) [u(•, t)] + 4 I σ(t) [u(•, t)] ≤ d 2 (m -m c ) σ ′ (t) σ(t) I σ(t) [u] -4 d (1 -m) F σ(t) [u] .
The last term of the right hand side is non-positive because by [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF] we know that σ ′ (t) ≤ 0 and

I σ(t) [u] -4 d (1 -m) F σ(t) [u] = d (1 -m) I σ(t) [u] -4 F σ(t) [u] + d (m -m 1 ) I σ(t) [u] ≥ d (m -m 1 ) I σ(t) [u] ≥ 0 .
This implies [START_REF] Dolbeault | Fast diffusion equations: Matching large time asymptotics by relative entropy methods[END_REF].

To avoid carrying heavy notations, let us write

f (t) := F σ(t) [u(•, t)] and j(t) := J σ(t) [u(•, t)]
and denote f (0), j(0) and σ(0) respectively by f 0 , j 0 and σ 0 . Estimates ( 17), ( 18) and ( 19) can be rewritten as ( 22)

       f ′ = -j ≤ 0 σ ′ = -κ 1 σ d 2 (m-mc) f ≤ 0 j ′ + 4 j ≤ κ 2 j σ ′ σ
where the constants κ i , i = 1, 2, are given by

κ 1 := 2 d (1 -m) 2 m K M and κ 2 := 1 2 (m -m c ) (m -m 1 ) d 2 .
Using the fact that lim t→∞ f (t) = lim t→∞ f (t) = 0, as in the proof of Proposition 3, we find that j(t) -4 f (t) ≥ 0 and f (t) ≤ f 0 e -4 t for any t ≥ 0.

Remark 3. The decay of σ can be estimated by

- d dt σ d 2 (1-m) = d 2 (1 -m) κ 1 f ≤ d 2 (1 -m) κ 1 f 0 e -4 t , thus showing that σ d 2 (1-m) ∞ ≥ σ d 2 (1-m) 0 -d 2 (1-m) 3 4 m K M f 0 .
Since u 0 and B σ 0 have the same mass and second moment, we know that

f 0 = 1 1 -m R d Ä B m σ 0 -u m 0 ä dx = 2 m K M d (1 -m) 2 σ d 2 (1-m) 0 - 1 (1 -m) R d u m 0 dx .
Hence we end up with the positive lower bound

σ d 2 (1-m) ∞ ≥ d 2 (m -m c ) σ d 2 (1-m) 0 + d 2 (1 -m) 2 4 m K M R d u m 0 dx .
From [START_REF] Ralston | A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. II[END_REF] we get the estimates σ(t) ≤ σ 0 for any t ≥ 0 and

j ′ -4 f ′ = j ′ + 4 j ≤ κ 2 j σ ′ σ = κ 1 κ 2 σ -d 2 (1-m) f f ′ ≤ κ 1 κ 2 σ -d 2 (1-m) 0 f f ′
Integrating from 0 to ∞ with respect to t and taking into account the fact that lim t→∞ f (t) = lim t→∞ f (t) = 0, we get

-j 0 + 4 f 0 ≤ - 1 2 κ 1 κ 2 σ -d 2 (1-m) 0 f 2 0 .
By rewriting this estimate in terms of F σ 0 [u 0 ] = f 0 , I σ 0 [u 0 ] = j 0 and after omitting the index 0, we have achieved our key estimate, which can be written using

C m,d := d 3 2 m K M (m -m c ) (m -m 1 ) (1 -m) 2
as follows.

Theorem 8. Let d ≥ 1, m ∈ (m 1 , 1) and assume that u is a non-negative function in L 1 (R d ) such that u m and x → |x| 2 u are both integrable on R d . Let σ = 1 K M R d |x| 2 u(x) dx where M = R d u(x) dx.
Then the following inequality holds

(23) 4 F σ [u] + C m,d (F σ [u]) 2 σ d 2 (1-m) ≤ I σ [u] .
Recall that

K M = K 1 M γ , with γ = (d+2) m-d d (m-mc)
. See Appendix A for details. Notice that this definition of γ is compatible with the one of Theorem 1 if p = 1/(2 m -1).

Remark 4. If we do not drop any term in the proof of Proposition 3 and Lemma 7, an ODE can be obtained for j, based on (10) and (21) and we can replace (22) by a system of coupled ODEs that reads

       f ′ = -j ≤ 0 σ ′ = -κ 1 σ d 2 (m-mc) f ≤ 0 j ′ + 4 j = d (1 -m) (j -4 f ) σ ′ σ + κ 2 j σ ′ σ -r where r := 2 σ d (m-mc) R d u m Ä |∇z| 2 -(1 -m) (∇ • z) 2 ä dx ≥ 0 and z := σ d 2 (m-mc) ∇u m-1 -2 x.
It is then clear that the estimates σ ≤ σ 0 and j ′ + 4 j ≤ κ 2 j σ ′ σ , which have been used for the proof of Theorem 8, are not optimal.

Proofs of Theorem 1 and Corollary 2

Let us start by rephrasing Theorem 8 in terms of f = u m-1/2 . Assume that

M = R d u dx R d |f | 2 p dx and σ = 1 K M R d |x| 2 u dx = R d |x| 2 |f | 2 p dx
where p = 1/(2 m -1) and using the notation f

(p) M,0,σ ∈ M (p)
d defined in Section 1, consider the functional

R (p) [f ] := -2 p p+1 R d (|f | p+1 - Ä f (p) M,0,σ ä p+1 dx .
In preparation for the proof of Theorem 1, we can state the following result.

Corollary 9. Let d ≥ 2, p > 1 and assume that p < d/(d -2) if d ≥ 3. For any f ∈ L p+1 ∩ D 1,2 (R d ) such that Condition (5) holds, we have R d |∇f | 2 dx + R d |f | p+1 dx -K p,d Å R d |f | 2 p dx ã γ ≥ C p,d Ä R (p) [f ] ä 2 R d |f | 2 p dx γ
where γ is given by (4). This results is slightly more precise than the one given in Theorem 1, as we simply measure the distance to a special function in M (p) d , the one with same mass and second moment, centered at 0. The constant C p,d is the same as in Theorem 1: see Appendix A for its expression.

Proof. By expanding the square in I σ [u] and collecting the terms with the ones of F σ [u], we find that

1 4 I σ [u] -F σ [u] = m (1-m) (2 m-1) 2 σ d 2 (m-mc) R d |∇u m-1 2 | 2 dx + d m-m 1 1-m R d u m dx + 1 1-m Å m K M σ d 2 (1-m) - R d B m σ dx ã .
The last term of the right hand side can be rewritten as

1 1-m Å m K M σ d 2 (1-m) - R d B m σ dx ã = -m 1-m d (m-mc) (d+2) m-d σ d 2 (1-m) C 1 M γ
with γ = (d+2) m-d d (m-mc) (as in the previous Section) and

C 1 = M 1-γ *
(see Appendix A for details). Consequently Inequality ( 23) can be equivalently rewritten as

(24) m (1-m) (2 m-1) 2 σ d 2 (m-mc) R d |∇u m-1 2 | 2 dx + d m-m 1 1-m R d u m dx ≥ m 1-m d (m-mc) (d+2) m-d σ d 2 (1-m) C 1 M γ + d 3 (m-mc) (m-m 1 ) (1-m) 2 8 m K 1 (F σ [u]) 2 M γ σ d 2 (1-m)
.

This inequality is invariant under scaling and homogeneous. As already noticed in [START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity[END_REF], Condition [START_REF] Bianchi | A note on the Sobolev inequality[END_REF]

means σ = σ * , that is m (1-m) (2 m-1) 2 σ d 2 (m-mc) = d m-m 1 1-m .
Using the explicit expressions that can be found in Appendix A and reexpressing all quantities in terms of p = 1 2 m-1 completes the proof of Corollary 9. See Appendix A for an expression of C p,d .

Proof of Theorem 1. It is itself a simple consequences of Corollary 9. Let us consider the relative entropy with respect to a general Barenblatt function, not even necessarilynormalized with respect to its mass. For a given function

u ∈ L 1 + (R d ) with u m ∈ L 1 (R d ) and |x| 2 u ∈ L 1 (R d ), we can consider on (0, ∞) × R d × (0, ∞) the function h defined by h(C, y, σ) = 1 m -1 R d î u m -B m C,y,σ -m B m-1 C,y,σ (u -B C,y,σ ) ó dx
where B C,y,σ is a general Barenblatt function

B C,y,σ (x) := σ -d 2 Ä C + 1 σ |x -y| 2 ä 1 m-1 ∀ x ∈ R d .
An elementary computation shows that

∂h ∂C = m σ d 2 (1-m) 1 -m R d (u -B C,y,σ ) dx , ∇ y h = 2 m σ -d 2 (m-mc) 1 -m R d (x -y) (u -B C,y,σ ) dx , ∂h ∂σ = m d 2 σ -d 2 (m-mc) C R d (u -B C,y,σ ) dx - m -m c 1 -m 1 σ R d |x -y| 2 (u -B C,y,σ ) dx .
Optimizing with respect to Section 1). Notice that optimizing on y amounts to fix the center of mass of the Barenblatt function to be the same as the one of u. This is however required neither in the proof of Corollary 9 nor in the one of Theorem 1.

C fixes C = C M , with M = R d u dx. Once C = C M is
(p) [f ] ≥ R (p) [f ] by def- inition of R (p) (see
Proof of Corollary 2. It is a straightforward consequence of Theorem 1 and of the Csiszár-Kullback inequality (6) when f ∈ D 1,2 (R d ) is such that f 2,2 p is finite. However, f 2,2 p does not enter in the inequality. Since smooth functions with compact support (for which f 2,2 p is obviously finite) are dense D 1,2 (R d ), the inequality therefore holds without restriction, by density.

Concluding remarks

Let us conclude this paper with a few remarks. First of all, notice that Theorem 4 gives a stronger information than Theorem 1, as not only the L 1 (R d , dx) norm is controlled, but also a stronger norm involving the second moment, properly scaled.

No condition is imposed on the location of the center of mass, which simply has to satisfy R d x u dx 2 ≤ R d u dx R d |x| 2 u dx = σ M K M according to the Cauchy-Schwarz inequality. Hence in the definition of R[f ] and R (p) [f ] (in Theorem 1) as well as in Corollary 2, the result holds without optimizing on y ∈ R d . In [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF][START_REF] Dolbeault | Fast diffusion equations: Matching large time asymptotics by relative entropy methods[END_REF], improved asymptotic rates were obtained by fixing the center of mass in order to kill the linear mode associated to the translation invariance of the Barenblatt functions. Here this is not required since, as t → ∞, the squared relative entropy is simply of higher order. Our improvement is better when the relative entropy is large, and is clearly not optimal for large values of t.

Our approach differs from the one of G. Bianchi and H. Egnell in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] and the one of A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF]. It gives fully explicit constants in the subcritical regime. The norms involved in the corrective term are not of the same nature.

Let us list a series of remarks which help for the understanding of our results.

(i) Scaling properties of the Barenblatt profiles. Consider the scaling λ → u λ with u λ (x) := λ d u(λ x) for any x ∈ R d . Then we have

σ λ := 1 K M R d |x| 2 u λ dx = 1 λ 2 1 K M R d |x| 2 u dx = σ λ 2
and may observe that

B σ λ (x) = λ d B σ (λ x) .
As a consequence, we find that As a consequence, we find that

F σ [u λ ] = λ d (m-1) F σ [u]. (
F σ [u λ ] = λ m F σ [u].
(iii) The m = 1 limit. As m → 1, which also corresponds to p → 1, we observe that the constant C p,d in Theorem 1 has a finite limit. Hence we get no improvement by dividing the improved Gagliardo-Nirenberg inequality by (p -1) and passing to the limit p → 1 + , since R (p) [f ] = O(p -1). By doing so, we simply recover the logarithmic Sobolev inequality as in [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF]. This is consistent with the fact that, as m → 1 -, we have C

  and let us define the manifold of the optimal functions asM ,σ : (M, y, σ) ∈ M d .We shall measure the distance to M (p) d with the functionalR (p) [f ] := inf g∈M (p) d R d î g 1-p Ä |f | 2 p -g 2 p ä -2 p p+1 Ä |f | p+1 -g p+1 äó dx .

( 5 )

 5 R d |x| 2 |f | 2 p dx R d |f | 2 p dx γ = d (p-1) σ * M γ-1 * d+2-p (d-2) , σ * (p) := 4 d+2-p (d

[ 9 ,

 9 Question (c), p. 75], H. Brezis and E. Lieb asked the question of what kind of distance to M (p)

d 1 K 1 =

 11 ii) Homogeneity properties of the Barenblatt profiles. Similarly notice that for any m ∈ (m 1 , 1), we haveC M = C 1 M -2 (1-m) d (m-mc ) and K M = K 1 M (m-mc ) .Let u λ := λ u and denote by B σ λ the corresponding best matching Barenblatt function. Using the fact that u λ 1 = λ M if u 1 = M and observing thatK λM = K M λ λM R d |x| 2 u λ dx = λ 2 (1-m) d (m-mc ) σ . Since C λM = λ -2 (1-m) d (m-mc ) C M , we find that B σ λ (x) = λ B σ (x) .

2 exp Ç - d 2 MR d |x| 2 u dx |x| 2 å,by ( 2 )..Ä 2 2 M γ σ d 2 ( 1 2 M γ- 1 *

 2222222121 m,d ∼ (1-m) 2 , σ = O(K -1 M ) = O(1 -m) and, since B σ (x) ∼ B 0 (x) := M Ç d M 2 π R d |x| 2 u dx å d , up to multiplications by a constant, translations and scalings. This allows to compute C GN p,d . All computations done, we findC GN p,d = (p-1) p+1 (p+1) d+1-p(d-1) η = p (d + 2 -p (d -2)).It is easy to relate C GN p,d and K p,d . As in[START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF], apply[START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF] tof λ such that f λ (x) = λ d 2 p f (λ x) for any x ∈ R d . With a := R d |∇f | 2 dx, b := R d |f | p+1 dx, α := d p + 2 -d and β := d p-1 2 p , Inequality (3) amounts to a λ α + b λ -β ≥ K p,d Å R d |f | 2 p dx ã γOptimizing the left hand side with respect to λ > 0 shows thatK p,dLet us consider[START_REF] Unterreiter | On generalized Csiszár-Kullback inequalities[END_REF]. With p = 1 2 m-1 , that is, m = p+1 2 p , andF[u] = m 1-m R (p) [f ] with u = f 2 p , it is straightforward to check that K p,d =since u = B σ always provides the equality case. Hence, using Identity[START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity[END_REF], Inequality[START_REF] Unterreiter | On generalized Csiszár-Kullback inequalities[END_REF] amounts toR d |∇f | 2 dx + R d |f | p+1 dx -K p,d Å R d |f | 2 p dx ã γ ≥ (2 m-1) 2 m (1-m) σ -d 2 (m-mc) * d 3 (m-mc) (m-m 1 ) (1-m) 2 8 m K 1 Ä m 1-m R (p) [f ] ä -m) * . Using K 1 = d (1-m) (d+2) m-d M 1-γ *and expressing everything in terms of p, we finally getC p,d = (2 m-1) 2 8 (1-m) 2 ((d + 2) m -d) d 2 (m -m c ) (m -m 1 )M γ-1 * σ * = (d-p (d-4)) (d-p (d-2)) (d+2-p (d-2))16 p 3 (p-1) σ * (p) .

  assumed, optimizing with respect to σ amounts to choose it such that R d |x| 2 B C,y,σ dx = R d |x -y| 2 u dx as it has been shown inLemma 5. This completes the proof of Theorem 1, since R

Acknowledgments. The authors acknowledge support both by the ANR projects

CBDif-Fr and EVOL (JD) and by MIUR project "Optimal mass transportation, geometrical and functional inequalities with applications" (GT). The warm hospitality of the Department of Mathematics of the University of Pavia, where this work have been partially done, is kindly acknowledged. JD thanks Michael Loss for enlighting discussions on the entropy -entropy production method. Both authors thank D. Matthes

we also get that

Hence, in Theorem 8, the additional term in [START_REF] Talenti | Best constant in Sobolev inequality[END_REF] is of the order of 1 -m and disappears when passing to the limit m → 1 -.

Appendix A. Computation of the constants

Let us recall first some useful formulae. The surface of the d -1 dimensional unit sphere S d-1 is given by

With this formula in hand, various quantities associated with Barenblatt functions can be computed. Applied to the function B(x

Notice that when M = M * , B = B 1 with the notation (11) of Section 3. As a consequence, for

m-1 , a simple change of variables shows that

which determines the value of C M , namely

.

A useful equivalent formula is

.

By recalling [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF] and observing that

, we find that (25)

Consider the sub-family of Gagliardo-Nirenberg-Sobolev inequalities (1). It has been established in [16, Theorem 1] that optimal functions are all given for his careful reading of an earlier version of the manuscript, which has resulted in a significant improvement of our results. c 2012 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.


Anna Gerke
email: anna.gerke@u-psud.fr

Michel Desbordes
email: michel.desbordes@u-psud.fr

Geoff Dickson
email: geoff.dickson@aut.ac.nz

The relationship between inter-organisational citizenship behaviour and innovation within sport clusters -a cross-cultural approach

Keywords: inter-organisational citizenship behaviour, product innovation, sport cluster, crosscultural research, surfing, sailing

The focus of this research is on the relationship between inter-organisational citizenship behaviour (ICB) and innovation within sport clusters. ICB is defined as discretionary and voluntary behaviour of organisations within a cluster that is not formally rewarded but promotes the functioning of the cluster. The innovation of sport through sport equipment, hence product innovation, is subject of this research. Sport clusters are geographic concentrations of interconnected organisations that have an interest in a particular sport as buyer or seller of related services or products. Four clusters are analysed contrasting two different locations, France and Australasia, and two sports of different maturity and level of organisation, sailing and surfing. This research considers the relevance of culture in sport management research by taking a comparative approach across two different cultures. In the first stage qualitative data is collected to map out clusters and their inter-organisational relationships. In the second step multivariate analysis is applied to investigate how much ICB (independent variable) influences product innovation (dependent variable) in those relationships. This research aims at improving the innovativeness of sport clusters and its organisations. Overall, the results are expected to create a better understanding of clusters, their organisations, relationships, and interactions. The objective is to disclose benefits of clusters as industrial structure with regards to innovation. The authors' intention is furthermore to interpret the results in a wider context, such as other sports or consumer goods markets with similar characteristics, and countries and locations with similar conditions.

Introduction

There are a number of studies that examine the innovation phenomenon in sporting equipment industries. [START_REF] Shah | Sources and Patterns of Innovation in a Consumer Products Field: Innovations in Sporting Equipment[END_REF] investigates sources and patterns of innovation in sporting equipment, [START_REF] Tietz | The process of user-innovation: A case study on user innovation in a consumer good setting[END_REF] analyse the process of user-innovations in the kite surf segment, and [START_REF] Hillairet | The dual management of innovation by the Declathon Group. A distinctive strategic system on the sport goods market[END_REF] examine the innovation management of a large French sporting goods company just to name a few.

Other previous research deals with the sporting goods industry, specifically the yachting and the outdoor clothing sector, with regards to internationalisation,. It investigates how New Zealand can serve as a source of country-specific advantages and product innovation as a source of firm-specific advantages with respect to a firm's internationalisation [START_REF] Gerke | The influence of Location and Product Innovation on Internationalisation -Insights from New Zealand's SMEs in the Outdoor Sport Goods Industry[END_REF]. [START_REF] Shilbury | Considering Future Sport Delivery Systems[END_REF] examines sport cluster as potential future sport delivery systems. There are more studies on sport clusters such as the horseracing industry in Southern England [START_REF] Parker | Land-based Economic Clusters and their Sustainability: The Case of the Horseracing Industry[END_REF], the skateboarding cluster in Australia [START_REF] Kellett | A comparison between mainstream and action sport industries in Australia: A case study of the skateboarding cluster[END_REF], the surfing cluster in Torquay, Australia [START_REF] Stewart | Cluster theory and competitive advantage: The Torquay surfing experience[END_REF], the football league in Victoria, Australia [START_REF] Dickson | League Expansion and Interorganisational Power[END_REF] and the motor sport industry in North Carolina, USA (Kimmo, 2007, Connaughton and[START_REF] Connaughton | The Economic Impacts of the North Carolina Motorsports Industry[END_REF]. All these existing studies intrigued us to develop a research project that focuses on sport clusters as unit of analysis.

Research Objectives

The major purpose of this research is to investigate how inter-organisational citizenship behaviour (ICB) influences product innovation in sport clusters. This research seeks to locate ownership and control of innovative knowledge and to explore how value is appropriated from it. Furthermore, it investigates to what extend and how innovative knowledge is dispersed and transferred within sport cluster and what role inter-organisational relationships and interactions play for this phenomenon.

Secondly, this research maps out relationships and interactions between industries and organisations in sport cluster. Hence, a better understanding and use of cluster and their benefits due to interorganisational relationships and behaviour is targeted. Thirdly, this research provides insights with regards to industry restructuring in the context of sport organisations. Overall, this research aims at an increased awareness and understanding of clusters, their organisations, relationships and interactions

Research Question

The core of the research question is how ICB influences product innovation in sport clusters. The consequences of inter-organisational relationships, interactions and behaviour with regards to innovation are focal part of the study [START_REF] Skinner | Some measures of interorganizational citizenship behaviors: Scale development and validation[END_REF]. The linkage between ICB and innovation is constructed based on Porter's (1998) cluster theory and the suggested favourable conditions of a cluster environment for innovation. The discretionary behaviour of individuals and organisations and the geographical proximity facilitate co-operation and knowledge transfer in a cluster (Porter, 2008a).

ICB is also a discretionary behaviour and clusters seem to be a favourable environment for the evolvement of such behaviour between individuals or organisations.

This research exemplifies the cluster as a form of inter-organisational system. Cognitive and physical distance between organisations is reduced in such a system, as well as the cost of knowledge transfer and utilisation. This enables the creation of new knowledge and innovative products and services, while firm's specialisation and individuality is preserved [START_REF] Maskell | Knowledge Creation and Diffusion in Geographic Clusters[END_REF]. This research expands cluster theory by addressing the question how organisations in a cluster are interconnected and how those inter-firm relationships impact on innovation within a cluster [START_REF] Motoyama | What Was New About the Cluster Theory?: What Could It Answer and What Could It Not Answer[END_REF].

Innovation, especially sport product innovation, is an integral part of this research. The factors that facilitate productivity, new business formation and most importantly for this research, innovation, depend on the location of a business (Porter, 2008a). [START_REF] Greve | Bigger and safer: the diffusion of competitive advantage[END_REF] supports this theory in suggesting that a firm's proximity to the location where innovations develop influences the extent to which firms adopt and benefit from those innovations. However, [START_REF] Shilbury | Considering Future Sport Delivery Systems[END_REF] puts forward that the relevance of location and geography is reduced for service industries. He also argues that sport is Prepared for EAMSA by Anna Gerke, 28 July 2011 primarily considered a service sector [START_REF] Shilbury | Considering Future Sport Delivery Systems[END_REF]. On the contrary, sporting goods businesses play an essential role in sport clusters. This applies particularly with regards to the creation and diffusion of product innovation as analysed by [START_REF] Desbordes | Empirical Analysis of the Innovation Phenomena in the Sports Equipment Industry[END_REF]. [START_REF] Desbordes | Innovation management in the sports industry: Lessons from the Salomon case[END_REF] argues that sport equipment firms have developed their individual and sophisticated logic for product innovation and that the sport industry and especially sport equipment firms are an under-researched area.

Literature Review

The focus of this research is on the relationship between ICB and product innovation within sport clusters. It contributes to a number of knowledge bodies. At first it extends inter-organisational research by investigating relationships and interactions between organisations and industries within sport clusters [START_REF] Autry | Interorganizational citizenship behaviors: an empirical study[END_REF]. It contributes to the debate of innovativeness in industrial clusters and addresses the call for more research that investigates drivers of innovation in clusters [START_REF] Caniëls | What drives innovativeness in industrial clusters? Transcending the debate[END_REF]. Furthermore it adds on to research on industry restructuring by investigating and mapping sport clusters as consequence of industrial structure change [START_REF] Shilbury | Considering Future Sport Delivery Systems[END_REF]. This study takes also the under-researched nexus culture and sport management into account by taking a comparative cross-cultural approach [START_REF] Girginov | Culture and the study of sport management[END_REF]. The followings paragraphs give an overview of relevant literature in the knowledge bodies mentioned above: inter-organisational research, innovation, cluster, cross-cultural research.

Inter-organisational research

This research studies inter-organisational relationships and interactions in sport clusters using the concept of ICB. This is because clusters are often an informal and discretionary development resulting from historical events or regional conditions in which relationships and norms of behaviour are not or only little formalised (Porter, 2008a). That is why it is suggested that clusters present a favourable environment for the development of ICB. [START_REF] Autry | Interorganizational citizenship behaviors: an empirical study[END_REF] developed the concept of ICB based on prior research on organisational citizenship behaviour (OCB). OCB was initially coined by Organ [START_REF] Currall | Organizational Citizenship Behavior: The Good Soldier Syndrome[END_REF][START_REF] Organ | Organizational Citizenship Behavior: It's Construct Clean-Up Time[END_REF] by defining it as discretionary behaviour of individuals within an organisation that is not Prepared for EAMSA by Anna Gerke, 28 July 2011 formally rewarded but promotes the functioning of the organisation. [START_REF] Autry | Interorganizational citizenship behaviors: an empirical study[END_REF] apply the OCB concept to study inter-organisational relationships in supply chains. They define ICB as "interfirm behavioural tactics, generally enacted by boundary personnel, that are discretionary, not directly or explicitly included in formal agreements, and that in the aggregate promote the effective functioning of the supply chain" [START_REF] Autry | Interorganizational citizenship behaviors: an empirical study[END_REF]. [START_REF] Shilbury | Considering Future Sport Delivery Systems[END_REF] argues that clusters can be considered as the value chain for all involved organisations. Hence, it is suggested that ICB can be applied in the context of clusters and can be defined as discretionary behaviour of organisations within a cluster that is not formally rewarded or explicitly included in formal agreements, but promotes the effective functioning of the cluster.

The linkage between ICB and innovation in a cluster is suggested based on previous research and existing concepts. [START_REF] Martínez-Sánchez | Interorganizational Cooperation and Environmental Change: Moderating Effects between Flexibility and Innovation Performance[END_REF] investigate the moderator effect of interorganisational cooperation in the relationship between workplace flexibility and innovation performance. Different dimensions are used to operationalise workforce flexibility and one dimension for innovation. The relationship between internal versus external workforce flexibility and innovation performance indicates driver and location of innovativeness in an inter-organisational context. Hence, it can be analysed whether drivers for innovative performance are located inside or outside the firm [START_REF] Martínez-Sánchez | Interorganizational Cooperation and Environmental Change: Moderating Effects between Flexibility and Innovation Performance[END_REF]. This could also indicate ownership and internalisation of innovative knowledge within an inter-organisational context based on [START_REF] Dunning | The Eclectic (OLI) Paradigm of International Production: Past, Present and Future[END_REF] eclectic paradigm. He [START_REF] Dunning | The Eclectic (OLI) Paradigm of International Production: Past, Present and Future[END_REF] argues that the international success of a firm depends on three different sources of advantages: location-based advantages, ownership-based advantages, and internalisation-based advantages. In this research it is investigated whether cluster offer advantages in terms of innovative knowledge that is owned and internalised by certain organisations in the cluster but disseminated and made available to other cluster members due to the unique features of a cluster system.

Innovation

The significance of innovative knowledge for the economic performance of a firm derives from the knowledge-based view of a firm, which is an extension of the resource-based view [START_REF] He | Innovative knowledge assets and economic performance: the asymmetric roles of incentives and monitoring[END_REF]. According to those theories firms naturally differ in their endowment with resources and Prepared for EAMSA by Anna Gerke, 28 July 2011 internal capabilities which serve as a base to gain competitive advantages over competitors [START_REF] Peteraf | The cornerstones of competitive advantage: a resource-based view[END_REF]. Hence, the configuration of a firm with innovative knowledge assets differs as well from firm to firm. Those differences in innovative knowledge offer potential for significant performance advantages for firms with superior knowledge bases or innovative capabilities [START_REF] Peteraf | The cornerstones of competitive advantage: a resource-based view[END_REF].

Furthermore Wang & Chen (2009) argue that firm-specific innovative knowledge leads to improved economic performance because of higher value appropriations. This research also draws on the concept of localised knowledge spillovers and how they foster innovativeness in industrial clusters [START_REF] Caniëls | What drives innovativeness in industrial clusters? Transcending the debate[END_REF]. Based on these concepts this study examines innovative knowledge in the context of particular sport clusters and with respect to the question how ICB influences innovation in sport clusters.

Innovation and especially production innovation is particularly important for sporting goods firms, for both retailers and manufacturers. The reason for that is that technicality and innovativeness of products are important consumption levers [START_REF] Hillairet | The dual management of innovation by the Declathon Group. A distinctive strategic system on the sport goods market[END_REF]. Sporting products are technologically complex products that are often required to fulfil contrary characteristics (such as lightness and resistance for example) [START_REF] Desbordes | Innovation management in the sports industry: Lessons from the Salomon case[END_REF]. This applies primarily to equipment intensive sports and sport industries. Andreff (n.d.) defines 'equipment-intensive' sporting goods as high unit value sporting goods versus 'trite' sporting goods as low unit value sporting goods. He argues that primarily equipment-intensive sporting goods are subject to international trade but that this research area has been neglected by scholars so far (Andreff, n.d.). [START_REF] Desbordes | Innovation management in the sports industry: Lessons from the Salomon case[END_REF] argues further more that the type of innovation in sporting goods depends on the maturity of the sector. Hence, product innovation is more common in young sectors as opposed to process innovation which is more common in mature sectors. Taking these previous studies and findings into account, the choice of the sports sailing/boating and surfing/boarding seem to be appropriate as both sports are considered as equipment-intensive sports (Andreff, n.d.). Furthermore sailing can be considered as a much more mature and organised sport in contrast to surfing as the International Sailing Federation (ISAF) was founded in 1907 in Paris, France (2011b), long before the International Surfing Federation (ISA) was Prepared for EAMSA by Anna Gerke, 28 July 2011 founded in 1964 in Sydney, Australia (2011a). This research can help to clarify the proposition whether the type of innovation depends on maturity of the industry sector.

Cluster

The origin of cluster theory is based on Marshall's (1890) concept of industrial districts. Porter (2008a[START_REF] Porter | Clusters and the new economics of competition[END_REF] related that to competition and developed the cluster theory. He defines cluster as "geographic concentrations of interconnected companies, specialized suppliers, service providers, firms in related industries, and associated institutions" (Porter, 2008a). Clusters are one element of Porter's diamond model that was developed based on his research on the competitive advantage of nations [START_REF] Porter | The Competitive Advantage of Nations[END_REF]. The diamond model illustrates location conditions that influence businesses and their productivity. The element that represents the cluster is called "related and supporting industries". Porter (2008a) argues that location has become increasingly important for the economic success of businesses, regions, and nations. While input factors become abundant in a globalised world independent from the location; productivity, innovation, and new business formation are more likely to be enhanced in clusters. Close co-operation between buyers, suppliers, producers, research institutions, and businesses from related industries is more likely when participants are geographically close. This facilitates any form of formal or informal co-operation and dissemination of knowledge which again might foster productivity, innovation, or new business formation (Porter, 2008a). [START_REF] Shilbury | Considering Future Sport Delivery Systems[END_REF] applies Porter's (2008a) concept of cluster in the sports context and calls for further research of sport clusters. He [START_REF] Shilbury | Considering Future Sport Delivery Systems[END_REF] suggests as method to define sport cluster, to identify the relevant sellers and buyers respective to the sport. Applying this idea, he explores four sport cluster. Another key argument put forward by [START_REF] Shilbury | Considering Future Sport Delivery Systems[END_REF] is that sport develops increasingly towards a series of specialised sport cluster on a sport-by-sport basis as opposed to a single generic sports industry. Sport cluster include a number of interlinked but diverse organisations and industries such as amongst others sporting goods manufacturers, sporting goods retailers, sport media and broadcasting, and sport events [START_REF] Shilbury | Considering Future Sport Delivery Systems[END_REF]. Further research is proposed with regards to the restructuring of a generic single sports industry towards a number of diverse sports industries and sport-by-sport cluster. [START_REF] Shilbury | Considering Future Sport Delivery Systems[END_REF] also identifies a research gap in terms of the investigation Prepared for EAMSA by Anna Gerke, 28 July 2011 of relationships and interactions between industries and organisations within sport clusters. This research addresses that research gap by investigating the relationship between ICB and innovation in sport clusters. Knowledge about the nature, location, and dissemination of innovation is expected to lead to local knowledge spillovers, and hence, enhanced innovativeness within these clusters [START_REF] Caniëls | What drives innovativeness in industrial clusters? Transcending the debate[END_REF]. This is beneficial for the organisations inside the cluster and the region or nation in which the cluster is located.

Cluster development is widely discussed as opportunity to foster economic health and prosperity of related businesses and organisations that are geographically concentrated in one region or nation (European Commission, 2002, Stocker and[START_REF] Stocker | Unite and innovate! European clusters for recovery! Business Europe[END_REF]. Hence, the cluster theory has initiated a number of cluster mapping projects around the world. The scholar Porter initiated the first cluster mapping project in the United States (Harvard Business School, 2011). In 2007 a consortium of six partners launched the European Cluster Observatory as cluster mapping project for Europe (Europe Innova, 2011). Research on cluster and cluster mapping projects are also conducted in Australia [START_REF] Enright | Regional Clustering in Australia[END_REF]Roberts, 2001, Johnston, 2004) and New Zealand [START_REF] Boven | A goal is not a strategy[END_REF], Ministry of Economic Development, 2011).

Cluster emerge with increased sophistication of competition, hence, they are more likely to occur in developed economies (Porter, 2008a). Sport cluster are usually located in developed economies as sporting goods require complex technology and R&D capabilities. Markets for hightechnology sporting equipment are predominantly in developed economies because the population has more disposable income for non-essential products. Sporting products often require varied and complementary competencies. Consumers seek incompatible characteristics combined in their sport equipment. That is why sports equipment design and manufacturing is often highly specialised and complex technology is involved [START_REF] Desbordes | Innovation management in the sports industry: Lessons from the Salomon case[END_REF]. In addition companies are often confronted with peer pressures with regards to product innovation. They have to balance between high initial R&D expenses versus small product volumes in the initial phase of a new product's life cycle [START_REF] Desbordes | Innovation management in the sports industry: Lessons from the Salomon case[END_REF]. Examples for innovation research in sport cluster are [START_REF] Shah | Sources and Patterns of Innovation in a Consumer Products Field: Innovations in Sporting Equipment[END_REF] research on sources and Prepared for EAMSA by Anna Gerke, 28 July 2011 patterns of innovation in sporting equipment and [START_REF] Richard | Application du concept de milieu innovateur dans la filière sports-loisirs: Étude de cas dans l'industrie des articles des port rhônalpine[END_REF] research on the sport articles cluster Rhône-Alpes as innovative milieu.

Cross-cultural context

This paper addresses the under-explored culture-sport management nexus pointed out by [START_REF] Girginov | Culture and the study of sport management[END_REF]. Some sport management scholars (Girginov, 2010, Amis and[START_REF] Amis | Promoting critical and innovative approaches to the study of sport management[END_REF] argue that culture matters for sport management research. Hence, they call for more research that addresses sport management topics from a cultural perspective. This can be done in a number of ways. One suggestion is cross-cultural or comparative studies that take different cultures into account. In contrast to this call for culturally informed sport management research there are scholars that argue for the increasing homogenisation of sport and sport management through internationalisation and globalisation [START_REF] Chadwick | From outside lane to inside track: sport management research in the twenty-first century[END_REF]. This research is embedded in a cross-cultural context, comparing data from two different geographical and cultural locations, France and Australasia. Hence it will contribute to the discussion on a global versus culturally informed research approach in sport management research. In addition to the particular cross-cultural context of this research, it aims at the promotion of greater cultural sensitivity in sport management research. The cross-cultural approach is supported by a multi-cultural research team.

Methodology

The empirical context of this research is sport clusters. In order to define this term precisely a few adjacent concepts are explained briefly. There is a number of concepts that consist of the main idea that a group of different organisations benefit from each other by being part of the group: industrial district (ID) [START_REF] Marshall | Principles of Economics[END_REF][START_REF] Marshall | Industry and Trade[END_REF], innovative milieu (IM) [START_REF] Camagni | Inter-Firm Industrial Networks: --The Costs And Benefits of Cooperative Behaviour[END_REF][START_REF] Camagni | The concept of innovative milieu and its relevance for public policies in European lagging regions[END_REF], and cluster [START_REF] Porter | Clusters and the new economics of competition[END_REF], Porter, 2008a). These concepts share some common characteristics: spatial proximity, high product specialisation, high level of division of labour, positive learning atmosphere; dense input-output relations, high level interaction, strong innovation and entrepreneurship, fast reaction capability in response to external changes; synergies, externalities, and a continuous balance between co-operation and competition. They differ in their comprehensiveness.

Prepared for EAMSA by Anna Gerke, 28 July 2011

While an ID comprises firms that focus on the same or similar product, the IM includes also organisations that are relevant for innovation. The cluster concept covers all organisations that represent one of the five forces that shape competition (Porter, 2008a). The causes of those industrial concentrations vary between a rather natural development due to historical-cultural background and heritage on the one hand, and artificially created or encouraged through economic policies and structural support on the other hand. It can be argues that these different types of firm groups develop in a stage process.

Moreover the concept of networks has been a widely discussed and applied concept and is also relevant for this research [START_REF] Camagni | Inter-Firm Industrial Networks: --The Costs And Benefits of Cooperative Behaviour[END_REF][START_REF] Johanson | Internationalisation in industrial systems -a network approach[END_REF][START_REF] Chetty | Internationalisation of small to medium-sized manufacturing firms: a network approach[END_REF][START_REF] Scott | Social network analysis -a Handbook[END_REF]. The term network refers to relationships between organisations that belong to different groups of organisations as described above (ID, IM, or cluster). Hence, even though a network is built through inter-firm (or even inter-personal) relationships, it represents in fact an intergroup level relationship. This view is also shared by sociologists who use networks as underlying concept in social network analysis. In order to decompose networks into their constituent 'sub-groups' they search for 'clusters' within the network [START_REF] Scott | Social network analysis: a handbook[END_REF].

The cluster concept is chosen as unit of analysis in this research as it is the most comprehensive one. A cluster is defined as "a geographically proximate group of interconnected companies and associated institutions in a particular field, linked by commonalities and complementarities" (Porter, 2008a, p. 215). These concentrations of interdependent organisations consist of different cluster participants. Those cluster organisations can be identified upstream and downstream along the value chain on a vertical level (e.g. specialised suppliers, service providers), and along the chain of related industries and associated institutions on a horizontal level. Further cluster organisations can be governments, regulatory bodies, and non-governmental organisations.

Subject of this research is a special form of cluster, the sport cluster. This term has been defined by [START_REF] Shilbury | Considering Future Sport Delivery Systems[END_REF]: sport cluster include all organisations that have an interest in the sport as buyer or seller.

Prepared for EAMSA by Anna Gerke, 28 July 2011 This research seeks a comparison of sport cluster across the geographical regions France and Australasia, and across sports that differ significantly in their maturity, and level of organisation and institutionalisation, sailing (including boating) and surfing (including boarding). The initially chosen clusters are sailing clusters in Australasia (New Zealand Marine Industry, 2010, New Zealand Trade and Enterprise, 2010, 2011c), surfing clusters in Australasia (Surfing New Zealand, 2011, Surf.co.nz, 2011[START_REF] Stewart | Cluster theory and competitive advantage: The Torquay surfing experience[END_REF], sailing cluster in France (Policy Research Corporation, 2008, Cluster Maritime Francais, 2011) and surfing cluster in France (Fédération Française Surf, 2011[START_REF] Richard | Application du concept de milieu innovateur dans la filière sports-loisirs: Étude de cas dans l'industrie des articles des port rhônalpine[END_REF][START_REF] Eurosima | Le marché des sports de glisse dans le monde: EuroSIMA[END_REF]. All clusters will be mapped out in their respective region outlining relevant organisations and industries in the cluster and their relationships. A cluster is usually regionally concentrated in a location of the country, such as Aquitaine in France as surf and board sport cluster [START_REF] Eurosima | Cluster partner Retrieved 28[END_REF] or the Auckland region in New Zealand as sailing and boating cluster (Auckland Plus, 2011, Farrell).

The first part of the research addresses the question how sport cluster have emerged in different geographical regions and how ICB influences innovation. Cluster organisations and their inter-relationships are mapped out. This part of the research is conducted using social network analysis as a qualitative method to document the research results [START_REF] Scott | Social network analysis -a Handbook[END_REF][START_REF] Scott | Social network analysis: a handbook[END_REF]. Even though social network analysis was originally used to analyse inter-personal relationships, it has also proved to be useful to analyse inter-organisational relationships [START_REF] Scott | Social network analysis: a handbook[END_REF][START_REF] Martin | The role of central brokers and their influence on effectiveness in an intentionally created sports professionalization network[END_REF].

Semi-structured interviews and secondary data are used to construct each cluster as a case study (Eisenhardt, 1989[START_REF] Yin | Case study research: Design and methods[END_REF]. Possible interview partners are boundary representatives of different organisations within the cluster such as professional and leisure sport clubs and schools, sport federations, governmental sport governing bodies, sporting goods manufacturers, sporting goods retailers, sport facility construction firms, sport industry associations, sport adventure businesses, sport education organisations, sport media and broadcasting firms, sport event organisations, sport marketing organisations, sport medicine institutions, and sport research institutions.

The second part of the study is of quantitative nature and targets the question how much ICB (independent variable) influences product innovation (dependent variable). Questionnaires are Prepared for EAMSA by Anna Gerke, 28 July 2011 targeted at boundary personnel that represent cluster organisations and that create the relations and linkages between organisations within the cluster. It is suggested that these people functions as "catalyser" between the organisations in order to facilitate product innovation. Inter-organisationalbehaviour will be operationalized using the dimensions for ICB suggested by [START_REF] Skinner | Some measures of interorganizational citizenship behaviors: Scale development and validation[END_REF]: inter-organisational tolerance, atruism, loyalty, compliance, conscientiousness, constructiveness and advancement. Measurements for the dimensions will be adopted, adjusted to the sport cluster context or newly developed and validated. Product innovation is defined as "a new technology or combination of technologies introduced commercially to meet a user or a market need" (Utterback and Abernathy, 1975, p. 642). Hence, product innovation will be measured through technological and use improvements in the product that are new to the sport. This will be operationalized through different categories depending on whether the innovation improves the use or technique of the product: revolutionary innovations, technical innovations, use innovations, improvements [START_REF] Hillairet | The dual management of innovation by the Declathon Group. A distinctive strategic system on the sport goods market[END_REF]. Sources to track product innovation will be professional journals, manufacturer publications, firm representatives, competitions, trade shows, professional athletes, lead users, schools, clubs, and more. This research uses a mixed-methods research design combining qualitative and quantitative research techniques. These are viewed as complementary rather than rival approaches [START_REF] Jick | Mixing Qualitative and Quantitative Methods: Triangulation in Action[END_REF].

The combination of non-numerical and numerical methods in a two-stages process is chosen because this approach answers the research question best [START_REF] Slack | Understanding sport organizations -The Application of Organization Theory Smaling, A[END_REF]. The research is designed to start with a qualitative element at first in order to explore social structures, relationship and interactions. This is followed by the development of a set of hypotheses that are sought to be confirmed with numerical methods. Depending on the stage of the research process different ontological and epistemological positions are taken. Although the different philosophical paradigms between both approaches are theoretically incompatible, we argue for a pragmatic approach towards social science research (Smaling, 1994). This means that in the first stage of the research process the researcher puts on the interpretivist lenses. The researcher is subjective and co-creator of new knowledge through his function as interpreter of facts. In the second stage the researchers puts on the positivist lenses and Prepared for EAMSA by Anna Gerke, 28 July 2011 becomes an objective observer of facts which are statistically processed. This means that in the first stage reality is seen as socially constructed and interpreted by humans. In an interpretivist perspective reality is subjective and depends on those who live in it (Edwards andSkinner, 2009, Gratton and[START_REF] Gratton | Research methods for sport studies[END_REF]. Knowledge can only be created and understood from social actors who belong to and participate in the researched area [START_REF] Blaikie | Epistemology[END_REF]. In the second stage the epistemological and ontological viewpoints are oppositional. The epistemological position is positivist. Facts are observed and scientifically processed. Reality is objective and the researcher is an observer of phenomena rather than an active participant of reality [START_REF] Slack | Understanding sport organizations -The Application of Organization Theory Smaling, A[END_REF]. The clear distinction between stage one and two makes the conflicting methodological paradigms manageable. Triangulation of the results will respond to potential risks of a mixed-method approach. Overall, the combination of methods aims at a stronger validity of results than a single method approach [START_REF] Edwards | Qualitative Research in Sport Management Eisenhardt[END_REF].

Expected Implications

Implications are expected for researchers in the fields sport management, cluster theory, innovation and inter-organisational research. Practical implications are expected for practitioners in sport organisations including managers of sport businesses, sport federations, sport clubs, and other organisations with an interest in the sport. This research seeks to create knowledge about the impact and role of inter-organisational relationships and interactions with regards to innovation. Insights about the relational mechanism between inter-organisational citizenship behaviour and product innovation should enable firms to take advantage of this mechanism. Knowledge about the creation and dissemination of innovative knowledge within a sport cluster is expected to make organisations within the cluster more innovative, and hence, more successful. It is expected that this improves the overall innovativeness and performance of clusters which positively impact on the region where the cluster is located and the sports organisations around which the cluster is centred. This research seeks to confirm the increasing evolvement of clusters, in particular sport cluster, as a result of industrial restructuring. The cluster as emerging multi-organisational delivery system for sport services and products contains potential benefits for the different organisations in the cluster. This research seeks to increase the understanding and awareness of cluster as economic structure and sets the foundations to explore its potential economic and social benefits by the cluster organisations and by the cluster as a whole.

Prepared for EAMSA by Anna Gerke, 28 July 2011