Matteo Bonforte
email: matteo.bonforte@uam.es

Jean Dolbeault
email: dolbeaul@ceremade.dauphine.fr

Bruno Nazaret
email: bruno.nazaret@univ-paris1.fr

Nikita Simonov
email: simonov.nikita@gmail.com

CONSTRUCTIVE STABILITY RESULTS IN INTERPOLATION INEQUALITIES AND EXPLICIT IMPROVEMENTS OF DECAY RATES OF FAST DIFFUSION EQUATIONS

Keywords: MSC 2020 26D10, 46E35, 35K55; Secondary: 35B40, 49K20 Gagliardo-Nirenberg-Sobolev inequality, Caffarelli-Kohn-Nirenberg inequality, stability, entropy methods, fast diffusion equation, Harnack Principle, selfsimilar solutions, Hardy-Poincaré inequalities, spectral gap, rates of convergence

teaching and research institutions in France or abroad, or from public or private research centers.

Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations

1. Introduction and main results. Let us start with some Gagliardo-Nirenberg-Sobolev inequalities (without weights) and related flow issues in Section 1.1 before extending the results to Caffarelli-Kohn-Nirenberg inequalities (with weights) and related flows in Section 1.2.

1.1. Gagliardo-Nirenberg-Sobolev inequalities and related flows.

The Gagliardo-Nirenberg-Sobolev inequality

∇f θ 2 f 1-θ p+1 ≥ C GNS f 2 p (1)
holds on the space of the functions f ∈ L p+1 (R d ) with ∇f ∈ L 2 (R d ), with exponents given by

θ = d (p-1) (d+2-p (d-2)) p , p ∈ (1, +∞) if d = 1 or 2 , p ∈ (1, p ] if d ≥ 3 , p = d d-2 .
According to [START_REF] Gunson | Inequalities[END_REF][START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF], equality in [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF] is achieved if and only if f is equal to

g(x) = 1 + |x| 2 -1 p-1 ∀ x ∈ R d , (2)
up to a multiplication by a constant, a translation and a scaling. We denote by M the manifold of optimal functions for [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF]. This inequality has a number of interesting limit cases: Sobolev's inequality if d ≥ 3 and p = p , the Euclidean Onofri inequality if d = 2 in the limit as p → +∞, and the scale invariant Euclidean logarithmic Sobolev inequality in the limit as p → 1 + . Let us define the deficit functional and K GNS chosen so that δ[g] = 0. Up to a scaling, the inequality δ[f ] ≥ 0 is equivalent to [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF], as was shown in [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF].

In the critical case p = p , d ≥ 3, optimal functions in (1) are known as the Aubin-Talenti functions and this result goes back to [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF][START_REF] Talenti | Best constant in Sobolev inequality[END_REF][START_REF] Rodemich | The Sobolev inequalities with best possible constants[END_REF]. Later, in [START_REF] Brezis | Sobolev inequalities with remainder terms[END_REF], H. Brezis and E.H. Lieb asked the next natural question: which distance to M is controlled by the deficit δ ? Soon after, an answer was given in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] by G. Bianchi and H. Egnell: there is a positive constant C BE such that

1 (p-1) 2 δ[f ] = ∇f 2 2 -S d f 2 2 * ≥ C BE inf ϕ∈M ∇f -∇ϕ 2 2 ,
where S d is the optimal constant in Sobolev's inequality. This striking result had anyway an important drawback: C BE is obtained by a non-constructive method.

Various extensions and improvements as, e.g., in [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF][START_REF] Figalli | Sharp gradient stability for the Sobolev inequality[END_REF] have been obtained, as well as similar results for (1) in [START_REF] Carlen | Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation[END_REF][START_REF] Seuffert | An extension of the Bianchi-Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions[END_REF], in the subcritical range: we refer to [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF] for a review of the literature. New constructive stability results were recently obtained: we have the estimate

δ[f ] ≥ C inf ϕ∈M R d (p -1) ∇f + f p ∇ϕ 1-p 2 dx ( 3
)
for some explicit C > 0 which depends only on d, p, f 2p and A.

As a function of f or, to be precise, as a function of the mass f 2p and A, C takes positive values on M even if it is not uniformly bounded away from 0 on M. The distance to M is measured by a Fisher information functional and the strategy of the proof involves entropy methods. Inequality (3) is equivalent to an improved entropy -entropy production inequality, which relates the Fisher information with a relative entropy, or free energy, defined for m ∈ [m 1 , 1) with m 1 := 1 -1/d by

F[v] := 1 m -1 R d v m -B m -m B m-1 (v -B) dx (4)
where

B(x) := 1 + |x| 2 1 m-1 ∀ x ∈ R d . (5)
The functional F enters in the study of nonlinear evolution equations as follows.

Let us consider the fast diffusion equation

∂v ∂t + ∇ • v ∇v m-1 = 2 ∇ • (x v) , v(t = 0, •) = v 0 . (6)
By a standard computation, a solution v of ( 6) is such that

d dt F[v(t, •)] = -I[v(t, •)] where I[v] := m 1 -m R d v ∇v m-1 -∇B m-1 2 dx (7)
where I[v] is the relative Fisher information with respect to B. The exponents m in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF] and p in (1) are related by the condition p = 1/(2 m -1). In practice, for m > m 1 (subcritical range), the result of Theorem 1 can be rephrased as a result on decay rates for the solutions of (6).

Corollary 2. [6, Corollary 5.2] Let d ≥ 1 and m ∈ (m 1 , 1). If v is a solution of (6) with nonnegative initial datum v 0 ∈ L 1 (R d ) such that R d v 0 dx = R d B dx, R d x v 0 dx = 0 and A[v 0 ] := sup r>0 r 2 1-m -d |x|>r v 0 dx < ∞ , then we have F[v(t, .)] ≤ F[v 0 ] e -(4+ζ) t ∀ t ≥ 0 (8)
for some positive constant ζ which depends explicitly only on m, d, and

A[v 0 ]. The integral R d x v 0 dx is finite because A[v 0 ] < +∞. Note that if the center of mass of v 0 is finite, then R d x v(t, x) dx = R d x v 0 dx e -2t
for any t ≥ 0. Let us consider the optimized free energy functional

F [v] := inf B∈B 1 m -1 R d v m -B m -m B m-1 (v -B) dx
where B is the set of all Barenblatt functions obtained from B using a multiplication by a constant, translations and scalings.

Corollary 3. Let m ∈ [m 1 , 1) if d ≥ 2, m ∈ (1/2, 1) if d = 1 and consider ζ as in Corollary 2. If v is a solution of (6) with nonnegative initial datum v 0 ∈ L 1 (R d ) such that A[v 0 ] is finite, then F [v(t, .)] ≤ F [v 0 ] e -(4+ζ) t ∀ t ≥ 0 . (9)
In the subcritical range, this result is a straightforward consequence of Corollary 2, but it is new in the critical case m = m 1 corresponding to p = p .

The entropy -entropy production inequality relates rates of convergence for the solutions of ( 6) with [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF], but (1) can also be invoked in the context of the standard fast diffusion equation

∂u ∂t = ∆u m , u(t = 0, •) = u 0 . ( 10
) Assume that m ∈ [m 1 , 1) if d ≥ 2 and m ∈ (1/2, 1) if d = 1.
Using the Rényi entropy powers formalism, we learn from [6, Lemma 2.1] that a solution u of [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] with initial datum

u 0 ∈ L 1 + R d , (1 + |x| 2 ) dx such that u m 0 ∈ L 1 (R d ) satisfies R d u m (t, x) dx ≥ R d u m 0 dx m-mc 1-m + (1-m) C0 m-mc t 1-m m-mc ∀ t ≥ 0 , (11)
for some constant C 0 which explicitly involves C GNS and where m c := (d -2)/d. Equality in [START_REF] Carlen | Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation[END_REF] holds for any t ≥ 0 if and only if u 0 ∈ B. At t = 0, ( 11) is an equality for any u 0 and we can recover the Gagliardo-Nirenberg-Sobolev inequality (1) written with the optimal constant by differentiating with respect to t the growth estimate (11) at t = 0. A more readable estimate is obtained by considering the optimized free energy functional F applied to [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF].

Corollary 4. Assume that d ≥ 1, m ∈ [m 1 , 1)
and consider a solution of (10) with initial datum

u 0 ∈ L 1 + R d , (1 + |x| 2 ) dx such that u m 0 ∈ L 1 (R d ). With κ = ζ/d and ζ as in Corollary 3, we have F [u(t, .)] ≤ F [u 0 ] 1 + d (m -m c ) t -m+κ m-mc ∀ t ≥ 0 .
This new result is remarkable. While the best matching function

B(t, •) ∈ B is such that R d B m (t, x) dx ∼ t (1-m)/(m-mc)
→ +∞ as t → +∞, according to [START_REF] Carlen | Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation[END_REF], it turns out that F[u(t, .)], which involves R d (u m (t, x) -B m (t, x)) dx, decays to 0 at an algebraic rate. As a consequence, we have lim t→+∞ u(t, .) -B(t, .) 1 = 0, with an explicit rate, by the Csiszár-Kullback-Pinsker inequality. The case κ = 0 is a consequence of (1) and the improvement κ > 0 is a consequence of Theorem 1.

1.2. Caffarelli-Kohn-Nirenberg inequalities and related flows. So far the results are simple consequences of the method of [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]. We are now going to extend them to a larger class of inequalities. On R d with d ≥ 1, let us consider the Caffarelli-Kohn-Nirenberg interpolation inequalities

f 2p,γ ≤ C β,γ,p ∇f θ 2,β f 1-θ p+1,γ (12)
with optimal constant C β,γ,p , parameters β, γ and p such that

γ -2 < β < d -2 d γ , γ ∈ (-∞, d) , p ∈ (1, p ] with p := d -γ d -β -2 , (13)
and an exponent

θ = (d -γ) (p -1) p d + β + 2 -2 γ -p (d -β -2)
which is determined by the scaling invariance. This formula for θ extends to [START_REF] Carlson | Une inégalité[END_REF] the expression for [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF], which corresponds to the special case β = γ = 0. Here L q,γ (R d ) and L q (R d ) respectively denote the spaces of all measurable functions f such that

f q,γ := R d |f | q |x| -γ dx 1/q
and f q := f q,0 are finite. Inequality [START_REF] Carlson | Une inégalité[END_REF] holds in the space H p β,γ (R d ) of functions f ∈ L p+1,γ (R d ) such that ∇f ∈ L 2,β (R d ), defined as the completion of the space D(R d \ {0}) of the smooth functions on R d with compact support in R d \ {0}, with respect to the norm given by f → (p -p) f 2 p+1,γ + ∇f 2 2,β . Since the weights are locally integrable, these spaces can also be defined as the completion of the space D(R d ). The limitation p ≤ p in (13) amounts, for a given p > 1 to a restriction to the admissible set of parameters (β, γ), namely β ≥ d -2 + (γ -d)/p. On the other hand, we notice that

β < d -2 d γ ⇐⇒ p < d d -2 .
Inequality [START_REF] Carlson | Une inégalité[END_REF] belongs to a family of inequalities introduced by L. Caffarelli, R. Kohn and L. Nirenberg in [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] and also earlier by V.P. Il'in in [START_REF] Il'in | Some integral inequalities and their applications in the theory of differentiable functions of several variables[END_REF]. The range of admissible parameters (β, γ) is limited by [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF] to a cone in the quadrant β < d -2 and γ < d (see Fig. 1), but the inequality also holds in a cone in the quadrant β > d -2 and γ > d using the property of inversion symmetry: see [4, Section 2.1] for details.

A central issue in Caffarelli-Kohn-Nirenberg inequalities ( 12) is to decide whether the equality case is achieved among radial functions or not when d ≥ 2. We summarize this alternative by symmetry versus symmetry breaking. Symmetry in [START_REF] Carlson | Une inégalité[END_REF] means that the equality case is achieved by the (generalized) Aubin-Talenti type functions

g(x) = 1 + |x| σ -1 p-1 ∀ x ∈ R d , with σ := 2 + β -γ . ( 14
)
This definition of Aubin-Talenti type functions extends the one in (2). In the critical case p = p , θ = 1, V. Felli and M. Schneider proved in [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF] that symmetry breaking holds if

γ < 0 and β FS (γ) < β < d -2 d γ ,
where

β FS (γ) := d -2 -(γ -d) 2 -4 (d -1) . Reciprocally, if γ < d , and γ -2 < β < d -2 d γ and β ≤ β FS (γ) ,
then symmetry holds according to [START_REF] Dolbeault | Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF]. The results are exactly the same in the subcritical case p ∈ (1, p ) as was shown in [4, Theorem 2] and in [START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF]Theorem 1.1]. See Fig. 1. More is known. In the limit case β = (d -2) γ/d corresponding to p = p = d/(d -2), we learn from [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF] that the optimal constant is achieved among radial functions, but that optimal functions exist only for β ≥ 0. The limit case as β → (γ-2) + , that is, p → 1 + , gives rise to a family of logarithmic Hardy inequalities which has been studied in [START_REF] Del Pino | A logarithmic Hardy inequality[END_REF][START_REF] Dolbeault | Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities[END_REF]. We refer to [START_REF] Dolbeault | Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF][START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF] for more details on earlier contributions and to [START_REF] Dolbeault | Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs[END_REF] for a general overview. Stability results à la Bianchi-Egnell, with no constructive estimates, appeared in [START_REF] Wei | Stability of Caffarelli-Kohn-Nirenberg inequality[END_REF] in the critical case of ( 12 as an extension of the results of [START_REF] Bianchi | A note on the Sobolev inequality[END_REF][START_REF] Figalli | Sharp gradient stability for the Sobolev inequality[END_REF]. So far there are no constructive stability results for [START_REF] Carlson | Une inégalité[END_REF].

Exactly as in the case of Inequalities (1), it is interesting to consider on R d the nonlinear flow defined by

∂u ∂t + |x| γ ∇ |x| -β u ∇u m-1 = 0 , u(t = 0, •) = u 0 , (15)
which generalizes [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] to (β, γ) = (0, 0) for m ∈ [m 1 , 1) with a generalized critical exponent defined by

m 1 := 1 -2+β-γ 2 (d-γ) . The computation d dt R d u |x| -γ dx = 0 , d dt R d u m |x| -γ dx = m 2 1 -m R d u ∇u m-1 2 |x| -β dx (16)
enters in the formalism of the generalized Rényi entropy powers. With f = u m-1/2 and p = 1/(2 m -1) so that u = f 2p , we notice that

R d u |x| -γ dx = f 2p 2p,γ , R d u m |x| -γ dx = f p+1 p+1,γ and R d u |∇u m-1/2 | 2 |x| -β dx = 4 (1-m) 2 (2 m-1) 2 ∇f 2 2,β .
As in the non-weighted case, m = m 1 corresponds to p = p . Taking into account ( 12) and ( 16), any solution of (15) satisfies

d dt R d u m |x| -γ dx ≤ C R d u m |x| -γ dx -1-θ θ(p+1)
for some numerical constant C involving C β,γ,p . Altogether, this proves a growth estimate similar to [START_REF] Carlen | Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation[END_REF]. As for (1), this estimate is in fact equivalent to [START_REF] Carlson | Une inégalité[END_REF].

Again, more readable estimates are achieved using self-similar variables and relative entropies. Equation ( 15) can be rewritten in these variables as

|x| -γ ∂v ∂t + ∇ • |x| -β v ∇v m-1 = σ ∇ • x |x| -γ v . ( 17
)
The counterpart of Corollary 2 is an improved decay rate of the free energy F now defined as

F[v] = 2 p 1 -p R d v p+1 2 p -g p+1 - p + 1 2 p g 1-p v -g 2p |x| -γ dx . ( 18
)
where g 2p is now a stationary solution to [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] and m = (p + 1)/(2 p).

Theorem 5. Let d ≥ 1. Assume that (β, γ) = (0, 0) satisfies γ < d , γ -2 < β < d -2 d γ and β < β FS (γ) , (19)
let α = 1+(β-γ)/2 and assume that m ∈ [m 1 , 1). If v solves (17) with a nonnegative initial datum v 0 ∈ L 1,γ (R d ) such that R d v 0 (x) |x| -γ dx = R d g(x) 2p |x| -γ dx and A[v 0 ] := sup R>0 R 2+β-γ 1-m -(d-γ) |x|>R v 0 (x) |x| -γ dx < ∞ ,
then there are some ζ > 0 and some T > 0 which depend explicitly only on m, d, v 0 1,γ and

A[v 0 ] such that F[v(t, .)] ≤ F[v 0 ] e -(4 α 2 +ζ) t ∀ t ≥ 2 T . (20)
Under the restriction that A[v 0 ] is finite (with a definition for A which generalizes the one of Corollary 2), Inequality [START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF] provides us with an improved rate of convergence since the optimal rate of convergence (without the restriction

A[v 0 ] < ∞) for functions in L 1,γ (R d ) is 4 α 2 , see Section 3.2.
It is remarkable that no other condition is needed in the case (β, γ) = (0, 0), which is a major difference with Corollary 2 where the conditions R d v 0 dx = R d B dx and R d x v 0 dx = 0 have to be assumed. Although somewhat hidden, these conditions are also present in Corollaries 3 and 4 as we use the optimized free energy functional F and the optimization with respect to B ∈ B induces a similar normalization condition.

In Theorem 5, T is a threshold time which is similar to the threshold time in the non-weighted case and determines an asymptotic time layer [T, +∞). We actually prove that

F[v(t, .)] ≤ F[v(T, .)] e -(4 α 2 +2 ζ) (t-T ) ∀ t ≥ T . (21)
Compared with Corollary 2, we have no improved decay estimate on the initial time layer [0, T ]. The asymptotic decay rate of F[v(t, .)] as t → +∞ is known from [START_REF] Bonforte | Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods[END_REF][START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF].

Based on a fully quantitative regularity theory, the main progress here is that we give constructive estimates of T and an improved decay rate on [T, +∞).

1.3. Simplifying assumptions and outline of the paper. In this paper, we present some results in the spirit of [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF] and the key ideas of the proofs, with several simplifications: -we do not track the exact dependence of the constants on the parameters, -we do not distinguish in the estimates quantities which depend on the relative entropy and those coming from A[v] defined as in Theorem 5, on the basis of the following result:

Lemma 6. Let d ≥ 1.
Assume that the parameters β, γ and m are as in Theorem 5 and take σ = 2+β -γ. There are two explicit positive numerical constants c 1 and c 2 such that, for any nonnegative function

v ∈ L 1,γ R d , (1 + |x| σ ) dx such that A[v]
is finite, we have

R d |x| σ-γ v dx ≤ R d v |x| -γ dx + c 1 A[v] , R d v m |x| -γ dx 1 m ≤ c 2 R d v |x| -γ dx 1-(d-γ) 1-m σ m R d |x| σ-γ v dx (d-γ) 1-m σ m
.

As a consequence, the relative entropy as defined in ( 18) is controlled as soon as v has finite mass and A[v] is finite. For a proof of Lemma 6, it is easy to adapt the result of [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]Proposition 7.3] for the first inequality and use the Carlson-Levin estimate for the second inequality: see [START_REF] Carlson | Une inégalité[END_REF][START_REF] Levin | Exact constants in inequalities of the Carlson type[END_REF] and [14, Lemma 5] for the proof of a similar result.

This paper is organized as follows. Section 2 is devoted to the regularization properties of the evolution equations ( 10) and [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF]. On the basis of [START_REF] Bonforte | Quantitative a priori estimates for fast diffusion equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Hölder continuity[END_REF], the results on the relative uniform convergence and on threshold time t of [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF] for [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] are extended to [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF]. With less details than in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF], all intermediate estimates are stated but only the differences with [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF] are emphasized: see Theorem 7 for the main result of the section. Entropy methods and improved entropy -entropy production estimates are applied in Section 3 to prove Theorem 5. The fact that no additional constraint has to be imposed to get the improved decay rates for the solutions to [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF], a major difference with the standard fast diffusion equation [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF], is commented there. Section 4 is devoted to a summary of the strategy for proving the stability results for Gagliardo-Nirenberg-Sobolev inequalities, see Theorem 1, whose detailed proof can be found in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]. Our goal here is to explain that the improved decay rates can be extended to the initial time layer using a differential inequality based on the carré du champ method. Such an estimate is missing in the case of the evolution equation ( 15) associated with the Caffarelli-Kohn-Nirenberg inequalities, but a similar property is expected: this motivates the conjecture of Section 5.

2.

A threshold time for the convergence in relative error. Equation ( 15) admits a family of self-similar solutions, that we call Barenblatt solutions as a straightforward generalization of the non-weighted case, as in [START_REF] Bonforte | Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods[END_REF]. These solutions can be written as

B(t, x) := R(t) -d+γ g 2p (x/R(t)) ( 22
)
where g is defined by ( 14), p = 1/(2 m -1), R(t) = c t 1/ξ for some constant c > 0 that depends on m, d, β and γ, and

ξ = 2 + β -γ -(d -γ) (1 -m) . (23)
In order to fix notations, let us define

M := R d g 2p dx . ( 24
)
The purpose of this section is to prove that Barenblatt solutions attract all solutions of [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF]. Notice that the case (β, γ) = (0, 0) is covered.

2.1.

Convergence in relative error. The basin of attraction of the set of Barenblatt solutions in the strong topology of the uniform convergence in relative error has been defined in [START_REF] Vázquez | Asymptotic behaviour for the porous medium equation posed in the whole space[END_REF], and characterized in [START_REF] Bonforte | Quantitative a priori estimates for fast diffusion equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Hölder continuity[END_REF]. It is a key point to estimate the stabilization rates, see [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Bonforte | Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods[END_REF]. The result goes as follows. [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF] and assume that β and γ satisfy [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF]. Let u be a solution to [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF] corresponding to a nonnegative initial datum

Theorem 7. Let d ≥ 2, m ∈ [m 1 ,
u 0 ∈ L 1,γ (R d ) such that R d u 0 |x| -γ dx = M and A[u 0 ] = sup R>0 R 2+β-γ 1-m -(d-γ) |x|>R u 0 (x) |x| -γ dx < ∞ .
Then there exists an explicit ε such that for any ε ∈ (0, ε ) sup

x∈R d u(t, x) B(t, x) -1 ≤ ε ∀ t ≥ t := C ε -a . ( 25
)
Here ε and a > 0 are numerical constants which depend only on d, m, β and γ while C depends also on A[u 0 ].

u(t, x) ≤ B M t + t, x ∀(t, x) ∈ 2 t , +∞] × R d . ( 26
)
where B M (t, x) := (M/M) σ/ξ B t, (M/M) (1-m)/ξ x .

Proposition 9. [8, Theorem 3.1] Under the assumptions of Theorem 7, there exist positive constants t and M such that any solution u to (15) satisfies

u(t, x) ≥ B M t -t, x ∀(t, x) ∈ [2 t , +∞] × R d . ( 27
)
where B M (t, x) is as in Proposition 8.

The proof of Propositions 8 and 9 can be found in [START_REF] Bonforte | Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli-Kohn-Nirenberg weights[END_REF], with another proof in [6, Propositions 4.6 and 4.7] which is better adapted to our purposes. Combining the results of Propositions 8 and 9 we obtain a precise control of the solution u(t, x) which is called in the literature a global Harnack principle, see [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF]. Let us stress that the quantities t, t, M and M can be explicitly computed and their value can be found in [START_REF] Bonforte | Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli-Kohn-Nirenberg weights[END_REF]. We write below their dependencies with respect to the main parameters. In particular, we can chose M = κ 1 M and M = κ 2 M for some positive constants κ 1 and κ 2 which depend only on d, m, γ, and β.

Convergence in relative error and the threshold time. Let us define

ε := M /M σ/ξ -1 , ε := 1 -(M /M) σ/ξ
, and ε m := min 1 2 , ε , ε where M and M are as in Propositions 8 and 9. Integrating inequalities [START_REF] Gunson | Inequalities[END_REF] and ( 26) over the whole space R d , we deduce that M > M and M < M. As a consequence we obtain that ε, ε and ε m are positive, and ε m depends only on d, m, β and γ.

The outer estimate. Here we compare a solution u(t, x) with a Barenblatt profile with same mass M as in [6, Section 4.4.1], outside a large ball in x and for large values of t.

Corollary 10. Under the assumptions of Theorem 7 and for any ε ∈ (0, ε m ) there exist ρ(ε) and T (ε) for which any solution u to (15) satisfies

(1 -ε) B(t, x) ≤ u(t, x) ≤ (1 + ε) B(t, x) if |x| ≥ R(t) ρ(ε) and t ≥ T (ε) . (28)
Furthermore, there exist positive constants C and C such that, for all

x ∈ R d , C B(t, x) ≤ u(t, x) ≤ C B(t, x) ∀ t ≥ 4 T (ε) . ( 29
)
An explicit expression of T (ε) and ρ(ε) can be computed and is not detailed here: see [6, Section 4.4.1] for similar computations. We only remark that ρ(ε

) = O(1/ √ ε ) and T (ε) = (1 + A) 1-m O(1/ε) as ε → 0.
Proof. The proof is based on Propositions 8 and 9 and on the comparison (for t large enough) of B M (t ± c, x) with B(t, x), where c can be either t or t and M either M or M . We observe that the quotient B M (t ± c, x)/B(t, x) can be written as

B M (t ± c, x) B(t, x) = λ(t ± c) λ(t) γ-d 1 + λ(t) σ |x| σ (M/M) σ(1-m) ξ + λ(t ± c) σ |x| σ 1 1-m
, where λ(t) = R(t) -1 . Inequality [START_REF] Il'in | Some integral inequalities and their applications in the theory of differentiable functions of several variables[END_REF] follows from the fact that λ(t ± c) ∼ λ(t) as t → ∞. Inequality (29) follows from a similar analysis performed directly on the same quotient B M (t ± c, x)/B(t, x).

The inner estimate.

We consider what happens inside a ball as in [6, Section 4.4.2].

Corollary 11. Under the assumptions of Theorem 7 and for any ε ∈ (0, ε m ) and for any t ≥ 4 T (ε), there exists a constant K > 0 and an exponent ϑ > 0 such that any solution u to (15) satisfies

u(t, x) B(t, x) -1 ≤ K ε 1 1-m 1 t + F[u 0 ] R(t) ϑ if |x| ≤ 2 ρ(ε) R(t) (30)
The constants K and ϑ are numerical constants, which depend only on d, m, β and γ. Their explicit values follow from the proof as in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF].

Proof. The proof follows the proof of [6, Proposition 4.1] and further properties of parabolic regularity which are detailed in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]. We have to estimate the term |u(t, x)/B(t, x) -1| in the ball of radius 2 ρ(ε) R(t). This can be done by interpolating its L ∞ norm between L p and C µ semi-norms, defined on a bounded open domain Ω as

u C µ (Ω) := sup x,y∈Ω x =y |u(x) -u(y)| |x -y| µ ,
using the interpolation inequality (see [6, Section 3.1.4])

u L ∞ (B R (x)) ≤ C u (d-γ) (d-γ)+p µ C µ (B 2R (x)) u p µ (d-γ)+p µ L p,γ (B 2R (x)) + R -d-γ p u L p,γ (B 2R (x)) . ( 31
)
Here C is a positive constant which depends on d, γ, µ and p and

u L p,γ (B R (x0)) := B R (x0) |u| p |x| -γ dx 1/p .
In order to estimate the C µ semi-norm of u(t, x) and B(t, x) on the domain |x| ≤ 2 ρ(ε)R(t) with t ≥ 4 T (ε), we have to deal with the time dependence of those functions and the domain itself. In order to simplify the analysis, we introduce the scaling ûτ,κ (t, x) := κ σ/(1-m) τ (d-γ)/ξ u τ t, κ τ 1/ξ x for some positive κ and τ > 0. If u is a solution to [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF], then so is ûτ,κ , for any τ, κ > 0. The Barenblatt profile is transformed by the previous scaling as Bτ,κ (t, x) = B κ ξ/(1-m) M t, x where B M (t, x) is as in Proposition 8. The advantage comes from the identity

u(t, x) -B(t, x) B(t, x) = ût,1 1, t -1/ξ x -Bt,1 1, t -1/ξ x Bt,1 1, t -1/ξ x , (32)
as the domain |x| ≤ 2 ρ(ε) R(t) is included in |y| ≤ 2 Z ρ(ε) where y = x t -1/ξ for Z large enough. So, to estimate the C µ semi-norm of the quotient u(t, x)/B(t, x), it is enough to consider the right-hand side of (32) on a domain which is now independent of the time.

The denominator of the right-hand side of ( 32) can be estimated from below by a direct computation, while we use the parabolic regularity theory developed in [START_REF] Bonforte | Quantitative a priori estimates for fast diffusion equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Hölder continuity[END_REF][START_REF] Simonov | Fast diffusion equations with Caffarelli-Kohn-Nirenberg weights: regularity and asymptotics[END_REF] to bound the numerator |û t,1 (1, y) -Bt,1 (1, y)|. We remark that the C µ -norm of Bt,1 can be estimated by a direct computation, while for ût,1 we use the fact that it is a solution to a linear equation

∂u ∂t = |x| γ ∇ • |x| -β A(t, x) ∇u (33)
where the coefficient A(t, x) = m u m-1 (t, x). To obtain an estimate which is independent of ε, we estimate directly the C µ -norm of ût,1 on the whole space R d . We apply a standard trick in regularity theory: we cover R d with subdomains of type B k (0) \ B k/2 (0). In order to estimate the norm of ût,1 , we apply to the rescaled function ûτ,k the identity

ût,k (1, •) C µ (B1(0)\B1/2(0)) = k σ 1-m +µ ût,1 (1, •) C µ (Bk(0)\Bk/2(0)) ∀ t > 0 .
By [START_REF] Levin | Exact constants in inequalities of the Carlson type[END_REF], the function ûτ,k solves [START_REF] Talenti | Best constant in Sobolev inequality[END_REF] with a bounded and bounded away from zero coefficient A(t, x). Therefore, by [START_REF] Bonforte | Quantitative a priori estimates for fast diffusion equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Hölder continuity[END_REF]Proposition 4.2], there exists a constant c 1 , which depends only on d and n, such that

ût,k (1, •) C µ (B1(0)\B1/2(0)) ≤ c 1 ût,k L ∞ (( 1 2 ,4)×B1(0)\B 1/2 (0)) ≤ c 1 k σ 1-m ût,1 L ∞ (( 1 2 ,4)×R d ) . By setting t 0 = 0 and letting R → ∞ in [7, Inequality (2.1)], we obtain the estimate ût,1 L ∞ (( 1 2 ,4)×R d ) ≤ c 2 M
, for some positive constant c 2 which depends only on d, m, β and γ. Combining the above estimates, we find that

ût,1 (1, •) C µ (R d ) ≤ ût,1 (1, •) C µ (B1(0)) + ∞ k=1 ût,1 (1, •) C µ (B 2 k+1 (0)\B 2 k (0)) ≤ c 1 c 2 2 µ 2 µ -1 M . (34)
In order to use Inequality [START_REF] Seuffert | An extension of the Bianchi-Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions[END_REF], we need an estimate of u(t, x) -B(t, x) 1,γ . To do so, we use the Csiszár-Kullback-Pinsker inequality which allows us to control the evolution of u(t, x) -B(t + τ , x) 1,γ where τ > 0 is a time-shift needed for the definition of the relative entropy. Up to a scaling, τ can be defined to be such that B(τ , x) = g 2p (x). It is then convenient to use the triangle inequality as follows

u(t, x) -B(t, x) 1,γ ≤ u(t, x) -B(t + τ , x) 1,γ + B(t + τ , x) -B(t, x) 1,γ .
(35) By a simple although lengthy computation, we have that

B(t + τ , x) -B(t, x) 1,γ ≤ c t ∀ t ≥ T (ε) . ( 36
)
The Csiszár-Kullback-Pinsker inequality as in [6, Lemma 2.12] implies that there exists an explicit constant C, which depends only on d, m, β and γ, such that

u(t, x) -B(t + τ , x) 1,γ ≤ C F[u 0 ] R(t) . ( 37
)
Thanks to [START_REF] Il'in | Some integral inequalities and their applications in the theory of differentiable functions of several variables[END_REF] it is sufficient to estimate the right-hand side of (32) in the domain |y| ≤ 2 Z ρ(ε) for Z large enough. Combining [START_REF] Seuffert | An extension of the Bianchi-Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions[END_REF] with p = 1, applied to the difference ût,1 -Bt,1 together with the estimates ( 35), ( 36) and (37), we obtain for all |y| ≤ 2 Z ρ(ε) and

t ≥ T (ε) ût,1 1, y -Bt,1 1, y ≤ max{c, C} 1 t + F[u 0 ] R(t)
ϑ which allows to estimate the numerator of the right-hand side in [START_REF] Simonov | Fast diffusion equations with Caffarelli-Kohn-Nirenberg weights: regularity and asymptotics[END_REF], and allows to conclude the proof of ( 30) by estimating the denominator of the right-hand side in (32) by a direct computation.

Proof of Theorem 7.

Proof. From Corollary 10 we deduce that

u(t, x) B(t, x) -1 < ε (38) for t ≥ T (ε) and |x| > ρ(ε) R(t), where T (ε) = (1 + A) 1-m O(1/ε). From Corol- lary 11, we deduce that inequality (38) holds if t > 4 T (ε), |x| ≤ 2 ρ(ε) R(t) and t > 0 is such that K ε 1 1-m 1 t + F[u 0 ] R(t) ϑ < ε . Since R(t) ≤ (Ct) 1 ξ , for some positive constant C = C(d, m, γ, β), the last condition is satisfied if t ≥ 1 + F[u 0 ] ξ 2 ε a where a := ϑ ξ 2 -m 1 -m (39)
Combining the above estimate (39) with T (ε) = (1 + A) 1-m O(1/ε), by elementary computations, one finds that there exist a computable constant C which depends on d, m, γ, β, F[u 0 ] and A[u 0 ] for which [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF]

holds for t ≥ C(A[u 0 ], F[u 0 ]) ε -a .
The dependence on F[u 0 ] is eliminated using Lemma 6, although more accurate estimates are obtained if the dependence on F[u 0 ] is kept as in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]. This completes the proof of Theorem 7.

3. Improved entropy -entropy production estimates. We prove Theorem 5 in Section 3.4 using an artificial dimension, entropy methods on the time interval (t , +∞) with t given by Theorem 7 and spectral gap estimates, which are exposed respectively in Sections 3.1, 3.2 and 3.3.

3.1. An artificial dimension. Inequality ( 12) can be recast as an interpolation inequality with same weight in all integrals which, in terms of scaling properties, amounts to introduce an artificial dimension.

To a function f ∈ H p β,γ (R d ), let us associate the function F ∈ H p ν,ν (R d ) with ν := d -n < 0 such that f (x) = F |x| α-1 x ∀ x ∈ R d , (40)
where

α = 1 + β -γ 2 and n = 2 d -γ β + 2 -γ .
Notice that p = n/(n -2). In spherical coordinates, with r = |x| and ω = x/r for any x ∈ R d \ {0}, let us define the derivation operator

D α U := α ∂U ∂r , 1 r ∇ ω U .
With α > 0 and p ∈ (1, p ], we can rewrite [START_REF] Carlson | Une inégalité[END_REF] as

U 2p,ν ≤ K α,n,p D α U θ 2,ν U 1-θ p+1,ν ∀ U ∈ H p ν,ν (R d ) , (41)
for some optimal constant K α,n,p which is explicitly related with the optimal constant in [START_REF] Carlson | Une inégalité[END_REF]: see [4, Proposition 6]. Inequality (41) can be interpreted as a Gagliardo-Nirenberg-Sobolev inequality in the artificial dimension n. As α = 1 unless β = γ, notice that symmetry issues in (41) are in no way simpler than in [START_REF] Carlson | Une inégalité[END_REF]. A remarkable point is that the Aubin-Talenti type function as defined by ( 14) is transformed into the more standard function

x → 1 + |x| 2 1 1-p .
We refer to [4, Section 2.3] and [21, Section 3.1] for further details. Through the transformation u(t, x) = U t, |x| α-1 x , a solution u of ( 15) is transformed into a solution of

∂U ∂t = L α U m (42)
where

D * α denotes the adjoint of D α on L 2 (R d , |x| -ν dx) and L α = -D * α D α is an elliptic self-adjoint differential operator given in spherical coordinates by L α u = α 2 u + n -1 r u + 1 r 2 ∆ ω u .
Functions obtained from B(t, x) by reduction to the artificial dimension n are selfsimilar Barenblatt solutions of (42). If U is a solution of (42) with initial datum B given by ( 5), then

U(t, x) = R(t) -n B x/R(t) if and only if dR dt = α 2 R n (1-m)-1 .
If we additionally assume that R(0) = 1, this can be solved as

R(t) := (1 + α 2 ξ t) 1/ξ
and ξ = n (m-m c ) and m c = (n-2)/n as in [START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF]. Up to a time shift, this definition of R(t) generalizes the definition of Section 2 to the case α = 1. Other self-similar Barenblatt solutions of (42) have same scaling properties and initial data given by B, up to a multiplication by a constant and a scaling.

3.2. Flow and entropies in self-similar variables. Self-similar solutions suggest to rewrite (42) in the corresponding scales using the self-similar change of variables

U (t, x) = λ d R(t) d V 1 2 log R(t), λ x R(t) (43)
where λ n (m-mc) = (1 -m)/(2 m). Hence if U solves (42), then V solves

∂V ∂t = D * α • V D α V m-1 -2 x , V (t = 0, •) = V 0 (44) with nonnegative initial datum V 0 = λ -d U 0 (•/λ) ∈ L 1 (R d ).
Using homogeneity and scaling properties of (44), there is no restriction to fix R d V 0 |x| -ν dx = M, with M defined by [START_REF] Dolbeault | Parabolic methods for ultraspherical interpolation inequalities[END_REF]. As a consequence, we shall assume from now on that

R d V (t, •) |x| -ν dx = M ∀ t ≥ 0
without loss of generality.

The free energy (or relative entropy) and the Fisher information (or relative entropy production) are defined respectively by

F[V ] := 1 m -1 R d V m -B m -m B m-1 (V -B) |x| -ν dx
and, as a generalization of definition [START_REF] Bonforte | Quantitative a priori estimates for fast diffusion equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Hölder continuity[END_REF] of the Fisher information,

I[V ] := m 1 -m R d V D α V m-1 -D α B m-1 2 |x| -ν dx .
With V = |f | 2 p , p = 1/(2 m -1), Inequality (41) is equivalent to the entropyentropy production inequality

I[V ] ≥ 4 α 2 F[V ] ( 45
)
in the symmetry range for [START_REF] Carlson | Une inégalité[END_REF]. If V solves (44), it is a straightforward computation to check that d dt

F[V (t, •)] = -I[V (t, •)] (46)
after one integration by parts (which has to be justified: see [START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF]), and as a consequence, we obtain that

F[V (t, •)] ≤ F[V 0 ] e -4 α 2 t ∀ t ≥ 0 . (47)
3.

3. An improved spectral gap. We consider the linearized free energy and the linearized Fisher information given respectively by

F[h] := m 2 R d |h| 2 B 2-m |x| -ν dx and I[h] := m (1 -m) R d |D α h| 2 B |x| -ν dx .
These quadratic forms are obtained as

F[h] = lim ε→0 ε -2 F B + ε B 2-m h and I[h] = lim ε→0 ε -2 I B + ε B 2-m h .
The following result is taken from [4, Proposition 4].

Proposition 12. Let d ≥ 2, α ∈ (0, +∞), ν = d -n < 0 and δ = 1/(1 -m) ≥ n.
Then the Hardy-Poincaré inequality

R d |D α h| 2 B |x| -ν dx ≥ Λ R d |h| 2 B 2-m |x| -ν dx holds for any h ∈ L 2 (R d , B 2-m |x| -ν dx) such that R d h B 2-m dx = 0, with an optimal constant Λ given by Λ =    2 α 2 (2 δ -n) if 0 < α 2 ≤ (d-1) δ 2 n (2 δ-n) (δ-1) , 2 α 2 δ η if α 2 > (d-1) δ 2 n (2 δ-n) (δ-1)
, where η is given by

η = d-1 α 2 + n-2 2 2 -n-2 2 = 2 2+β-γ d -1 + d-2-β 2 2 -d-2-β 2+β-γ .
3.4. Proof of Theorem 5. The constant in the entropy -entropy production inequality (45) can be improved for a solution to (44) if time is large enough and the initial datum v 0 is such that

A[v 0 ] < ∞.
Lemma 13. Assume that (β, γ) = (0, 0) satisfies [START_REF] Dolbeault | Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities[END_REF]. Let α and ν be as in Section 3.1 and define ζ := 2 (1 -m) Λ -4 α 2 /4 > 0 with Λ as in Proposition 12.

If V is a solution to (44) with an initial datum V 0 such that

sup R>0 R n(m-mc ) 1-m |x|>R V 0 (x) |x| -ν dx < ∞
where m c = (n -2)/n as in [START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF], then there exists T > 0 such that

4 α 2 + 2 ζ F[V (t, •)] ≤ I[V (t, •)] ∀ t ≥ T . (48)
Proof. By applying the change of variables (40) and (43), solutions to [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] are tranformed in solutions to (44) and in particular the shifted time-dependent Barenblatt profile B(t + τ , x) is transformed into the stationary solution B defined in [START_REF] Bonforte | Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods[END_REF]. As in Section 2, τ > 0 is such that B(τ , x) = g 2p (x) and B is as in [START_REF] Dolbeault | Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs[END_REF]. Since

B(t, x) B(t + τ , x) L ∞ (R d ) ≤ c 1 and B(t + τ , x) -B(t, x) B(t + τ , x) L ∞ (R d ) ≤ c 2 t
for t > 0 large enough as a consequence of Theorem 7, we find that for any ε > 0 small enough there exists an explicit T = T (ε) > 0 such that sup

x∈R d V (t, x) B(x) -1 ≤ ε ∀ t ≥ T .
According to [5, Lemma 18], one can deduce that

I[h(t, •)] ≤ (1 + ε) 3-2m (1 -ε) I[V (t, •)] + ε s ε F[h(t, •)] ,
where h(t, •) = V (t, •) B m-2 -B m-1 and s ε is a positive function of ε such that lim ε→0 s ε > 0. As in the proof of [START_REF] Bonforte | Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods[END_REF]Lemma 19], using Proposition 12 and the fact that

(1 + ε) m-2 ≤ F[V (t, •)]/F[h(t, •)] ≤ (1 -ε) m-2 by [5, Lemma 15], we obtain 2 (1 -m) Λ -ρ ε ε F[V (t, •)] ≤ I[V (t, •)] ,
where ρ ε is bounded and stays bounded as ε → 0. A discussion has to be made depending on the cases in Proposition 12. When 2

(1-m) Λ = 4 α 2 (1-m) (2 δ -n),
satisfies |(v(t, •) -g 2p )/g 2p | < ε for any t > t where t is as in [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF]. So, we are in the position of using Lemma 13: under the center of mass condition, the improved entropy -entropy production inequality Step 2: improved entropy -entropy production inequality in the initial time layer. By integrating the differential inequality of Lemma 14 backwards on [0, t ], under the assumption that Q(v(t )) ≥ 4 + ζ, one finds

I[v(t, •)] ≥ (4 + ζ) F[v(t, •)] ∀ t ≥ t (
I[v(t, •)] ≥ (4 + µ) F[v(t, •)] ∀ t ∈ [0, t ] where µ = 4 ζ e -4 t 4 + ζ -ζ e -4 t .
As a consequence, the improvement obtained in the asymptotic time layer [t , +∞) can be transferred to the initial time layer [0, t ] and up to the initial datum, with a smaller improvement of the constant. By multiplying the improved inequality by 4/(4 + µ), we obtain

I[v] -4 F[v] ≥ µ 4 + µ I[v]
with v = v(t, •) for any t ≥ 0 and, as a special case, for v = v 0 . Notice that the constant µ/(4 + µ) can be estimated explicitly since the dependence of t is given by [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF].

Step 3: rescaling and proof of Inequality (3). So far we have proven, with the above notation, that

p + 1 p -1 δ[f ] ≥ C I[v] ,
where C = 4/(4 + µ). To obtain inequality (3) we simply observe that

I[v] ≥ inf ϕ∈M R d (p -1) ∇f + f p ∇ϕ 1-p 2 dx
where the infimum is taken on the manifold of all optimal functions for (1). To obtain the result in its general form it is enough to rescale and go back to the original variables.

Step 4: the critical case p = p . Due to the lack of an improved spectral gap in

Step 1 under the previous scheme, with mass and center of mass constraints, a second moment constraint is also needed. This amounts to change the evolution equation to a new one which allows to control the evolution of the second moment as well: in practice we need to rescale Equation (6) as done in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]Chapter 5]. Entropy and entropy production should be optimized, i.e., considered with respect to the best matching Barenblatt profile which is g 2p up to a time-dependent rescaling.This rescaling amounts to an additional time shift. Explicit estimates of the time shift requires some more work based on a system of ODEs and a phase portrait analysis. For more details see [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]Chapter 6]. Proof of Corollary 2. The entropy of the initial datum F[v 0 ] is finite under the current assumptions as a consequence of Lemma 6. The proof of inequality [START_REF] Bonforte | Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli-Kohn-Nirenberg weights[END_REF] follows from identity (46) by combining a Grönwall argument with inequality (49).

In this way we obtain

F[v(t, •)] ≤ F[v 0 ] e -(4+ζ) t ∀ t ≥ 0 . ( 50
)
Proof of Corollary 3. Let us consider the sub-critical case m 1 < m < 1. We can obtain inequality (47) from (50) as follows. First, we can rescale the initial datum in such a way that F[v 0 ] = F [v 0 ]. Then it is enough to take the infimum over all the Barenblatt profiles in the left-hand side of (50).

In the critical case m = m 1 , we have to deal with an additional time-dependent scaling in order to control the evolution of the second moment. For more details see [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]Chapter 5].

5.

A conjecture on stability for Caffarelli-Kohn-Nirenberg inequalities. In Gagliardo-Nirenberg-Sobolev inequalities (1) and in Caffarelli-Kohn-Nirenberg inequalities [START_REF] Carlson | Une inégalité[END_REF], the invariance under scalings plays an important role, as well as in the corresponding fast diffusion equations [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] and [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF]. This explains why entropy methods are so efficient to get sharp results for the best constants as discussed in [START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF]. A key feature is the carré du champ method, which has been rigorously implemented for [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] in [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF], in the context of parabolic equations. The regularity and decay estimates needed to justify the carré du champ method in the context of the fast diffusion flow associated with [START_REF] Carlson | Une inégalité[END_REF] are so far missing, although some partial estimates are known from [START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF][START_REF] Dolbeault | Parabolic methods for ultraspherical interpolation inequalities[END_REF]. This is why the symmetry results in [START_REF] Dolbeault | Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF] were based on an elliptic version of the method, which formally also applies to parabolic equation [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF]. Proving that all necessary integrations by parts can be justified would establish the following conjecture:

For some ζ > 0, Inequality (20) holds for any t ≥ 0. In other words, this means that in Theorem 5, one can take T = 0. Indeed, the carré du champ estimate would allow us to extend the estimate on the asymptotic time layer (T, +∞) to the initial time layer (0, T ) and find a smaller but still constructive value for ζ depending only on the initial datum, by the same scheme as in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]. A straightforward consequence would be an improved entropy -entropy production inequality that would provide us with a stability result with a constructive estimate for Caffarelli-Kohn-Nirenberg inequalities [START_REF] Carlson | Une inégalité[END_REF] similar to the result of Theorem 1. With this method, the stability would be measured by a relative Fisher information as for [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF]. Such a result is to be expected as, in the critical case of ( 12), a stability result without constructive estimate has already been established by F. Seuffert in [START_REF] Seuffert | An extension of the Bianchi-Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions[END_REF].

to the project MTM2014-52240-P (Spain) and by the E.U. H2020 MSCA programme, grant agreement 777822, by the Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR), by the DIM Math-Innov of the Region Île-de-France and by the "Fondation Mathématique Jacques Hadamard". The authors thank an anonymous referee for his detailed reading, comments and suggestions. c 2022 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

2 p

 2 with χ = d+2-p (d-2) d-p (d-4)

Theorem 1 . [ 6 ,

 16 Corollary 5.4 and Theorem 6.1] Let d ≥ 3 and p ∈ (1, p ], or d = 1, 2 and p ∈ (1, +∞). For any f ∈ L 2p (R d ) with ∇f ∈ L 2 (R d ) and A dx < ∞ ,

Figure 1 .

 1 Figure 1. In dimension d = 4, the critical exponent p = p = d/(d -2) = 2 corresponds to the left figure, while p = 6/5 is subcritical and corresponds to the right figure. The half cone of admissible regions of the parameters (β, γ) appear in grey, with symmetry breaking in dark grey and symmetry in light grey (the symmetry region is bounded if and only if p < p ).

  49) holds with ζ = 2 d (m -m 1 ). The computation of the precise value of ζ can be found in [6, Proposition 2.10]. The additional constraint R d x v 0 dx = 0 is needed to have ζ > 0.

4. 3 .

 3 Proof of Corollaries 2 and 3.

Acknowledgments. M.B. has been supported by the Projects MTM2017-85757-P and PID2020-113596GB-I00 (Ministry of Science and Innovation, Spain) and the Spanish Ministry of Science and Innovation, through the "Severo Ochoa Programme for Centres of Excellence in R&D" (CEX2019-000904-S) and by the E.U. H2020 MSCA programme, grant agreement 777822. J.D. has been supported by the Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR). N.S. has been supported by the Spanish Ministry of Science and Innovation, through the FPI-grant BES-2015-072962, associated

it follows from m > m 1 that (1 -m) (2 δ -n) > 1 and we find 2 (1 -m) Λ -ρ ε ε > 4 α 2 + ζ for ε small enough. When 2 (1 -m) Λ = 4 α 2 η, the result follows because the condition η > 1 is equivalent to [START_REF] Dolbeault | Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities[END_REF].

Proof of Theorem 5. As a consequence of (48) and (46), we have

Inequalities [START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF] and ( 21) are consequences of Grönwall's lemma.

4.

A stability result for GNS. In this section, we deal with the non-weighted case (β, γ) = (0, 0). Detailed proof are given in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]. Here we simply outline the main steps of the proofs and emphasize the differences with the weighted case studied in Section 3. [START_REF] Dolbeault | Parabolic methods for ultraspherical interpolation inequalities[END_REF], which is smooth enough and sufficiently decaying at infinity let us consider the quotient

A quotient estimate. For any function

where F[v] and I[v] are defined respectively by ( 4) and [START_REF] Bonforte | Quantitative a priori estimates for fast diffusion equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Hölder continuity[END_REF]. Lemma 14. Assume that v solves (6). Then we have

Equation ( 6) corresponds to the fast diffusion equation without weights, i.e., β = γ = 0 and the result follows from

which itself arises from the carré du champ method adapted to nonlinear flows. See for instance [START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF] and references therein. With (β, γ) = (0, 0), such an estimate is so far formal.

4.2. Scheme of the proof of Theorem 1. In the absence of weights the result follows by considering the improved spectral gap obtained for the flow [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]. We shall only sketch the main steps of the proof and the interested reader may find the whole proof in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]Chp. 5]. We first consider the subcritical case 1 < p < p .

Step 0: normalization. We notice that the deficit functional δ[•] defined in (1.1) is invariant by translations so that, without loss of generality, we assume that |f | 2p has zero center of mass. As well, there is no harm to assume R d |f | 2p dx = R d g 2p dx where g is as in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF], since the general case can be recover by scalings, see [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]Chapter 5]. We learn from [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF]Lemma 1.12]

with the notation v = |f | 2p and p = 1/(2 m -1). Notice that translations are not allowed if (β, γ) = (0, 0), but that R d x |f | 2p dx = 0 is not required in that case.

Step 1: improved entropy -entropy production inequality in the asymptotic time layer. Let v(t) be the solution the Cauchy problem [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method[END_REF] with initial datum v(0) = |f | 2p . By Theorem 7 we know that, for ε > 0 small enough, the relative error