Dominique Bakry
email: bakry@math.univ-toulouse.fr

Ivan Gentil
email: gentil@math.univ-lyon1.fr

Grégory Scheffer
email: gregory.scheffer@gmail.com

Sharp Beckner-type inequalities for Cauchy and spherical distributions

Keywords: Cauchy distribution, Beckner inequality, curvature-dimension condition, Poincaré inequality, Spherical analysis, Bessel processes, Stochastic calculus Mathematics Subject Classification (2010): 60G10, 60-XX, 58-XX

Using some harmonic extensions on the upper-half plane, and probabilistic representations, and curvature-dimension inequalities with some negative dimensions, we obtain some new optimal functional inequalities of the Beckner type for the Cauchy type distributions on the Euclidean space. These optimal inequalities appear to be equivalent to some non tight optimal Beckner inequalities on the sphere, and the family appears to be a new form of the Sobolev inequality.

Introduction

The so-called Γ 2 criterium is a way to prove functional inequalities such as Poincaré or logarithmic Sobolev inequalities. For instance, let Ψ : R d → R be a smooth function such that ∇ 2 ψ ρ I with ρ > 0, then the probability measure

dµ ψ = e -ψ Z dx,
where Z is the normalization constant which turns µ ψ into a probability, satisfies both the Poincaré inequality,

f 2 dµ ψ - f dµ ψ 2 ≤ 1 ρ |∇f | 2 dµ ψ ,
and the logarithmic Sobolev inequality

f 2 log f 2 dµ ψ -f 2 dµ ψ log f 2 dµ ψ ≤ 2 ρ |∇f | 2 dµ ψ ,
for any smooth function f . More generally, the same hypothesis leads to a family of interpolation inequalities between Poincaré and logarithmic Sobolev, namely, for any p ∈ (1, 2],

p p -1 f 2 dµ ψ - |f | 2/p dµ ψ p ≤ 2 ρ |∇f | 2 dµ ψ ,
where the case p = 2 corresponds to the Poincaré inequality and the limit p → 1 to the logarithmic Sobolev inequality. One of the technics used to prove such inequalities is based on the Γ 2 calculus (the CD(ρ, +∞) condition) introduced by the first author with M. Émery in [B É85] (see also [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]). This family is the so-called Beckner inequalities, proved for the usual Gaussian measure, that is when ψ = x 2 2 , by W. Beckner in [START_REF] Beckner | A generalized Poincaré inequality for Gaussian measures[END_REF] (in that case ρ = 1). Some improvements of such inequalities can be found in [AD05, [START_REF] Dolbeault | On the Bakry-Emery criterion for linear diffusions and weighted porous media equations[END_REF][START_REF] Bolley | Phi-entropy inequalities for diffusion semigroups[END_REF]. Let us note also that recent developments for the Beckner inequalities applied to the Gaussian distribution in an optimal way have been proved in [START_REF] Ivanisvili | Improving Beckner's bound via Hermite functions[END_REF], also for uniformly strictly log-concave measures. Let us finish to quote the recent article of Nguyen [START_REF] Nguyen | Phi-entropy inequalities and asymmetric covariance estimates for convex measures[END_REF] where the author generalizes some results of this paper with a completely different method, the method is extended later in [START_REF] Dupaigne | A family of Beckner inequalities under different curvature-dimension conditions[END_REF].

On the other hand, on the sphere S d ⊂ R d , with similar arguments, the following Beckner inequalities can be stated, p ∈ (1, 2],

p p -1 f 2 dµ S - |f | 2/p dµ S p ≤ 2 d |∇f | 2 dµ S , (1)
see [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF], where in this situation, ρ = d -1.

It is important to notice that all Beckner inequalities (for the spherical and the Gaussian models) are optimal in the sense that constants in front of the right hand side are optimal. However, only the constant functions saturate these inequalities when p = 2. The aim of this article is two-fold. First, we explain and improve in a general context the method introduced by the last author [START_REF] Scheffer | Local Poincaré inequalities in non-negative curvature and finite dimension[END_REF] to obtain some new optimal inequalities. This method is mainly probabilistic, and makes strong use of various processes associated with the underlying structure behind the inequalities. He used this method to obtain an optimal Poincaré inequality under a CD(ρ, n) condition, which is a stronger form of the condition ∇ 2 ψ ρId, and without integration by parts. More precisely, the method is enough robust to be applied to a nonsymmetric operator even if our example is the usual Laplacian which is symmetric. This method is rather technical and is based on the CD(ρ, n) condition with a negative dimension n, together with a construction of some sub-harmonic functional. Let us note that negative dimension in the curvature-dimension CD(ρ, n) condition has been introduced and used in [START_REF] Scheffer | Local Poincaré inequalities in non-negative curvature and finite dimension[END_REF][START_REF] Ohta | K, N )-convexity and the curvature-dimension condition for negative N[END_REF][START_REF] Milman | Beyond traditional curvature-dimension. I: New model spaces for isoperimetric and concentration inequalities in negative dimension[END_REF]. Stochastic calculus is also a main tool of our approach.

Secondly, improving unpublished results of the last author, we apply this method to obtain Beckner inequalities for the generalized Cauchy type distributions, that is the probability measure on R d with density

dν b (x) = 1 Z(1 + |x| 2 ) b dx
where b > d/2 and Z is the normalization constant. This measure does not satisfy any CD(ρ, +∞) condition with ρ 0 but some CD(0, n) condition with n < 0, as will be explained in detail in the paper. We prove the following new optimal family of inequalities,

p p -1 f 2 dν b - |f | 2/p dν b p ≤ 1 (b -1) Γ(f )(1 + |y| 2 )dν b . where b d + 1 and p ∈ [1 + 1/(b -d), 2].
Of course, when p = 2 we recover a weighted Poincaré inequality for the Cauchy type distributions proved in many papers, see [Sch01, BBD + 07, [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF][START_REF] Nguyen | Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Prékopa's theorem[END_REF].

But more surprisingly, this family is equivalent to the following one on the sphere S d ,

h 2 dµ S ≤ A |h| 2/p dµ S p + 16 (m + 2 -d)(3d -2 + m) Γ S (h)dµ S , (2)
where

m d + 2, p = 1 + 2 m -d ∈ [1, 2]
and A 1 is some explicit constant to be described below, converging to 1 as p converges to 1, or equivalently when m converges to infinity. Inequality (1) is a tight inequality, that is constant functions are optimal, on the other hand inequality (2) is a non-tight inequality. And this inequality (2) is optimal with explicit extremal functions. Moreover this new family of inequalities, or more precisely its behavior when p converges to 1, appears as a new form of a Sobolev inequality on the sphere, as many other ones described for example in [START_REF] Bakry | Sobolev inequalities in disguise[END_REF].

The paper is organized as follows. In the next section, we describe the curvature-dimension condition CD(ρ, n), even for negative n, and derive a general form of associated sub-harmonic functionals. In Section 3, we define an operator related to the generalized Cauchy type distributions, and show that it satisfies the generalized curvature-dimension condition. In Section 4, we prove the new Beckner inequalities for these generalized Cauchy distributions. Finally, in Section 5, we prove the non-tight Becker inequalities on the d-dimensional sphere and show how they relate to the Sobolev inequality.

Notations: In all this paper, d will be the dimension of the main space, and satisfies d 1, and d 2 in sections 4.2 and 5. For every

X, Y ∈ R d , X • Y = d i=1 X i Y i is the usual scalar product and |X| 2 is the Euclidean norm in R d .

General properties on curvature-dimension condition

Let (M d , g) be a smooth connected d-dimensional Riemannian manifold and ∆ g its Laplace-Beltrami operator. The associated Ricci tensor is denoted Ric g . For any diffusion operator L on M (that is a second order semi-elliptic differential operator with no zero-order term, see [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]) we define the so-called carré du champ operator

Γ L (f, g) = 1 2 [L(f g) -f Lg -gLf ] with Γ L (f, f ) = Γ L (f ) and its iterated operator Γ L 2 (f, f ) = Γ L 2 (f ) = 1 2 L(Γ L (f )) -Γ L (f, Lf ),
for any smooth functions f and g. We write Γ (resp. Γ 2 ) instead Γ L (resp. Γ L 2 ) when there is no possible confusion.

Definitions

Definition 1 (CD(ρ, n) condition) An operator L = ∆ g + X with X a smooth vector field satisfies a CD(ρ, n) condition with ρ ∈ R and n

∈ R \ [0, d) if Γ 2 (f ) ρΓ(f ) + 1 n (Lf ) 2 , (3)
for any smooth function f .

As we shall see in the proof in Lemma 2, this condition will never hold when n ∈ [0, d) but we will extend it for n = 0.

Lemma 2 ([Bak94]) For any ρ ∈ R and n / ∈ [0, d] the operator L = ∆ g + X satisfies a CD(ρ, n) if and only if n -d n (Ric(L) -ρg) X ⊗ X n , (4)
and when n = d it reduces only to X = 0 and the condition becomes Ric g ρg. In the definition, Ric(L) = Ric g -∇ S X where ∇ S X is the symmetrized tensor of ∇X.

Proof. -For completeness, we give a sketch of proof of this result which can be helpful for the rest of the paper. For any smooth function f , the Bochner-Lichnerowicz-Weitzenbock formula states that (cf. [BGL14, Sec C.5]),

Γ 2 (f ) = ||∇ 2 f || 2 H.S. + Ric g (∇f, ∇f ) -∇X S (∇f, ∇f ) = ||∇ 2 f || 2 H.S. + Ric(L)(∇f, ∇f ), (5)
where ||∇ 2 f || H.S. is the Hilbert-Schmidt norm of f . On a fixed point x ∈ M , we choose a local chart such that the metric g = (g i,j ) is the identity matrix at the point x. In that case,

∇ 2 f = (∇ i,j f ) is a d-dimensional symmetric matrix denoted Y , Γ(f ) = d i=1 (∂ i f ) 2 and let denote Z = ∇f . Since d i,j=1 Y 2 i,j = tr(Y 2 ), the CD(ρ, n) condition (3) becomes tr(Y 2 ) + Z • RZ ρZ • Z + 1 n (tr(Y ) + X • Z) 2 ,
where R is a matrix representing the coordinates of the tensor Ric(L) at x.

Y 2 ii + Z • RZ ρZ • Z + 1 n ( d i=1 Y ii + X • Z) 2 . By taking Y ii all equals, from d i=1 Y 2 ii 1 d ( d i=1 Y ii ) 2 , the previous inequality is equivalent to 1 d ( d i=1 Y ii ) 2 + Z • RZ ρZ • Z + 1 n ( d i=1 Y ii + X • Z) 2 .
Let y = d i=1 Y ii , then the inequality becomes

y 2 d + Z • RZ ρZ • Z + 1 n (y + X • Z) 2 .
In other words, the CD(ρ, n) condition is satisfied if and only if, for every y ∈ R and,

Z ∈ R d y 2 1 d - 1 n -y 2 n X • Z + Z • (R -ρId)Z - 1 n (X • Z) 2 0.
Then, either n = d, then we need to assume that X = 0 and Z • (R -ρId)Z 0, either 1 d -1 n > 0 (with translates to n / ∈ [0, d]) and for any vector Z,

(X • Z) 2 n ≤ n -d n Z • (R -ρId)Z,
that is condition (4).

Next we define the tensor curvature-dimension condition, T CD(ρ, n).

Definition 3 (T CD(ρ, n) condition and quasi-models

QM (ρ, n)) Let ρ ∈ R, n ∈ R \ {d}
and an operator L = ∆ g + X with X a smooth vector field.

• L satisfies a T CD(ρ, n) condition if n -d n Ric(L) -ρg X ⊗ X n , (6)
when n = 0, and

-d(Ric(L) -ρg) ≤ X ⊗ X, when n = 0. • The operator L is a quasi model QM (ρ, n) if (n -d)(Ric(L) -ρg) = X ⊗ X. (7)
Indeed, this new definition allows us to extend (3) when n ∈ [0, d), and we shall see later crucial examples of such operators, which are even quasi-models.

Sub-harmonic functionals

In this section, we construct sub-harmonic functionals from harmonic functions under T CD(ρ, n).

We systematically explore the most general ones, in view of further use. The next result is rather technical, and we shall see in the next sections how it leads to Poincaré inequalities or Beckner inequalities.

Theorem 2.1 (Sub-harmonic functionals under T CD(ρ, n) or QM (ρ, n))

Let Φ : I × [0, ∞) → R be a smooth function (I open interval of R), and denote Φ i and Φ ij its first and second derivatives, i, j ∈ {1, 2}. Let L = ∆ g + X, where X is a smooth vector field.

Let ρ ∈ R and n ∈ R \ {d}. We assume that L satisfies the T CD(ρ, n) condition with n ∈ R \ (0, d] (or the QM (ρ, n) condition). Then, if F : M → I is smooth and satisfies L(F ) = 0, then L[Φ(F, Γ(F ))] 0 as soon as Φ satisfies at any point (y, z) ∈ I × [0, ∞), one of the three conditions below,

             Φ 2 (y, z) > 0; Φ 2 (y, z) n + 1 -d n -d + 2zΦ 22 (y, z) > 0;
(n -d) nΦ 2 (y, z) + 2(n -1)zΦ 22 (y, z) > 0;

2ρΦ 2 (y, z) + Φ 11 (y, z) -2(n -1)

zΦ 2 12 (y,z) nΦ 2 (y,z)+2(n-1)zΦ 22 (y,z) 0, (8) or            Φ 2 (y, z) = 0; zΦ 22 (y, z) = 0; zΦ 12 (y, z) = 0; zΦ 11 (y, z) 0, (9) or      Φ 2 (y, z) = 0; zΦ 22 (y, z) > 0; Φ 11 (y, z)Φ 22 (y, z) -Φ 2 12 (y, z) 0. ( 10
)
Remark 4

• For simplicity, we omit the variables y and z inside the function

Φ. When n / ∈ [0, d], the condition Φ 2 n+1-d n-d + 2Φ 22 z > 0 is contained into Φ 2 + 2 n-1 n zΦ 22 > 0. Hence, provided that n / ∈ [0, d], then (8) boils down to            Φ 2 > 0; Φ 2 + 2 n -1 n zΦ 22 > 0; 2ρΦ 2 + Φ 11 -2(n -1) Φ 2 12 z nΦ 2 + 2(n -1)Φ 22 z 0. ( 11
)
It is interesting to notice that when n / ∈ [0, d] the dimension d does not appear in the previous conditions (11).

• As pointed out by a referee, condition (8) has a nice formulation by using a new variable coming from [START_REF] Ivanisvili | Isoperimetric functional inequalities via the maximum principle: the exterior differential systems approach[END_REF]. The authors set

M (y, u) = Φ(x, √ u), then condition (8) is equivalent to, for any (u, y) ∈ R × (0, ∞), (n -d) M 11 (y, u) + ρ u M 12 (y, u) M 12 (y, u) M 12 (y, u) M 22 (y, u) + 1 (n-1)u M 2 (y, u) 0,
together with the conditions M 2 (y, u) > 0 and

M 22 (y, u) + 1 (n -1)u M 2 (y, u) > 0
where we use again the notation

M ij = ∂ 2 ij M .
At the moment, we are not aware of any implications of this new formulation of our condition (8).

Proof. -The proof is an improvement of the proof of Lemma 2.

If LF = 0, then from the diffusion property of L (cf. [BGL14, Sec 1.14]),

L(Φ(F, Γ(F ))) = Φ 2 LΓ(F ) + Φ 11 Γ(F ) + 2Φ 12 Γ(F, Γ(F )) + Φ 22 Γ(Γ(F ), Γ(F )) = 2Φ 2 Γ 2 (F ) + Φ 11 Γ(F ) + 2Φ 12 Γ(F, Γ(F )) + Φ 22 Γ(Γ(F ), Γ(F )), ( 12
)
since LΓ(F ) = 2Γ 2 (F ) when LF = 0 and for simplicity we note Φ ij = Φ ij (F, Γ(F )). At some point x ∈ M , we write all the expressions appearing in L(Φ(F, Γ(F ))) in terms of local charts. In view of (5), the resulting expressions involve only ∇∇F and ∇F . We chose a system of coordinates such that at this point the metric g is identity and ∇∇F is diagonal, with eigenvalues (λ i ) 1≤i≤d . The components of ∇F are denoted Z i , and the components of X are denoted -X i . Since L(F ) = 0, the constraint can be written,

d i=1 λ i = d i=1 X i Z i . and Γ(F, Γ(F )) = 2 d i=1 λ i Z 2 i , Γ(Γ(F ), Γ(F )) = 4 d i=1 λ 2 i Z 2 i , Γ(F ) = d i=1 Z 2 i .
From the T CD(ρ, n) conditions and (5), the Γ 2 operator has a lower bound,

Γ 2 (F ) = d i=1 λ 2 i + Ric(L)(Z, Z) d i=1 λ 2 i + ρ d i=1 Z 2 i + 1 n -d d i=1 X i Z i 2 .
Under the T CD(ρ, n) this is an inequality and we need to assume at this stage that n / ∈ (0, d]. When the operator is a quasi-model QM (ρ, n) we only need to assume that n = d.

Since Φ 2 0, from (12), we just have to minimize the following expression,

L(Φ(F, Γ(F ))) 2 2Φ 2 d i=1 λ 2 i + ρ d i=1 Z 2 i + 1 n -d d i=1 X i Z i 2 + Φ 11 d i=1 Z 2 i + 4Φ 12 d i=1 λ i Z 2 i + 4Φ 22 d i=1 λ 2 i Z 2 i ,
for all (λ i ) and Z i , under the constraint

d i=1 λ i = d i=1 X i Z i . Let z = d i=1 Z 2 i = Γ(F )
and Z 2 i = z i , then the expression becomes by using the constraint (after dividing by 2)

Φ 2 d i=1 λ 2 i + ρz + 1 n -d d i=1 λ i 2 + Φ 11 2 z+2Φ 12 d i=1 λ i z i +2Φ 22 d i=1 λ 2 i z i .
We first minimize on the simplex {z i 0, d i=1 z i = z} where z > 0 is fixed. We forget the constraint { d i=1 ±X i √ z i = d i=1 λ i } since it is enough to minimize along a bigger set. We first observe that the expression is affine in z i . The vector (z i ) 1≤i≤d belongs to the simplex {z i 0, d i=1 z i = z}, so that the extremals of this expression ((λ i ) 1≤i≤d and z > 0 are fixed) is obtained at the one of the extreme points of the simplex. That is one of the z i is z and all the others are 0. Assuming that z 1 = z = 0, one is now led to minimize

λ 2 1 (Φ 2 + 2Φ 22 z) + 2Φ 12 λ 1 z + Φ 2 d i=2 λ 2 i + Φ 2 n -d d i=1 λ i 2 + z Φ 2 ρ + Φ 11 2 = λ 2 1 (Φ 2 n + 1 -d n -d + 2Φ 22 z) + λ 1 2Φ 12 z + 2 Φ 2 n -d d i=2 λ i + Φ 2 d i=2 λ 2 i + 1 n -d d i=2 λ i 2 + z Φ 2 ρ + Φ 11 2
Then, we minimize in λ 1 . This imposes the second condition of (8), that is

Φ 2 n + 1 -d n -d + 2Φ 22 z > 0,
recall that z = Γ(F ). And the minimizer is given by

L(Φ(F, Γ(F ))) 2 Φ 2 d i=2 λ 2 i + 1 n -d d i=2 λ i 2 + z Φ 2 ρ + Φ 11 2 - (Φ 12 z + Φ 2 n-d d i=2 λ i ) 2 Φ 2 n+1-d n-d + 2Φ 22 z Let assume that Φ 2 > 0. Decompose the vector Λ = (λ i ) 2≤i≤d into a vector parallel to (1, • • • , 1) and a vector u = (u i ) 2≤i≤d orthogonal to it, that is Λ = λ(1, • • • , 1) + u with d i=2 u i = 0. From this decomposition, we have d i=2 λ i = (d -1)λ and d i=2 λ 2 i = d i=2 (λ + u i ) 2 (d -1)λ 2 .
We see that we need to find the minimum in λ of

Φ 2 (d -1)λ 2 + (d -1) 2 n -d λ 2 + z Φ 2 ρ + Φ 11 2 - (Φ 12 z + (d-1)Φ 2 n-d λ) 2 Φ 2 n+1-d n-d + 2Φ 22 z = λ 2 Φ 2 (d -1) nΦ 2 + 2(n -1)zΦ 22 (n -d)(Φ 2 n+1-d n-d + 2Φ 22 z) -λ 2(d -1)Φ 2 Φ 12 z (n -d)(Φ 2 n+1-d n-d + 2Φ 22 z) - Φ 2 12 z 2 Φ 2 n+1-d n-d + 2Φ 22 z + z Φ 2 ρ + Φ 11 2 .
Then, once again, to get a minimizer in λ, we also need

(n -d)(nΦ 2 + 2(n -1)Φ 22 z) > 0,
which is the third assumption in (8). Then, the minimizer is

L(Φ(F, Γ(F ))) 2 - (d -1)Φ 2 Φ 2 12 z 2 (n -d)(nΦ 2 + 2(n -1)zΦ 22 )(Φ 2 n+1-d n-d + 2Φ 22 z) - Φ 2 12 z 2 Φ 2 n+1-d n-d + 2Φ 22 z + z Φ 2 ρ + Φ 11 2 = -z 2 (n -1)Φ 2 12 nΦ 2 + 2z(n -1)Φ 22 + z Φ 2 ρ + Φ 11 2 0,
which is the fourth assumption in (8) since z 0. When Φ 2 = 0 at some point (y, z), we follow the inequality to check the assumptions needed.

Corollary 5 (Sub-harmonic functionals under T CD(0, n)) Let assume that L = ∆ g + X satisfies a T CD(0, n) with n < 0. Let θ be a positive and smooth function on an interval

I ⊂ R such that 2 n -1 n (θ ) 2 ≤ θθ . ( 13
)
Then for any function F : M → I such that LF = 0,

L(θ(F )Γ(F )) 0. ( 14
)
In particular, for any β ∈ [n/(2 -n), 0] and nonnegative harmonic function F ,

L(F β Γ(F )) 0.
When n goes to -∞, the condition (13) on θ degenerates into 2θ ≤ θθ . This means that the function Φ such that Φ = θ is admissible : Φ is convex and 1/Φ is concave.

Moreover, when n ∈ (-∞, 0), the extremal case is given by θ(x) = x n/(2-n) , where ( 13) is an equality.

Although we shall mainly use Corollary 5 instead of general Theorem 2.1, the proof in this last case is not really simpler than the general one.

The operators

Q (m) t Definition 6 (The operators Q (m) t ) For any t 0, m > 0, x ∈ R d , and any bounded func- tion f : R d → R, let Q (m) t (f )(x) = 1 c(m, d) f (yt + x) (1 + |y| 2 ) m+d 2 dy, (15)
where

c(m, d) = 1 (1 + |y| 2 ) m+d 2 dy = Γ( m 2 ) Γ( m+d 2 ) π d/2 (16) is the normalization constant, such that Q (m)
t (1) = 1. In other words, for any t > 0,

Q (m) t (f )(x) = f (y)q t (x, dy),
where the kernel q t is given by

q t (x, dy) = 1 c(m, d) t m (t 2 + |x -y| 2 ) m+d 2 dy. (17)
The probability measure 1 c(m,d)

(1 + |y| 2 ) -m+d 2 dy
in R d , with m > 0, is known in statistics as the multivariate t-distribution with m degrees of freedom. There is a huge literature on the subject, see for instance [START_REF] Kotz | Multivariate t distributions and their applications[END_REF]. Through an integration by parts (or from the computation of normalization constants) we may notice the useful formula,

c(m, d) c(m -2, d) = m -2 m -2 + d , (18)
for any m > 2.

Proposition 7 (Harmonicity of

Q (m) t f ) Let m > 0.
For any smooth and compactly supported function f , the map (0, +∞)

× R d (t, x) → Q (m) t (f )(x) is solution of the elliptic equation ∆ (m) Q (m) t (f ) = 0 where ∆ (m) = ∆ R d + ∂ 2 tt + 1 -m t ∂ t . (19)
Proof. -This can be proved directly using the expression of the kernel (17).

From Proposition 7 (or from a direct computation) the operators (Q

(m) t ) t 0 admit another formulation. Let (Z s ) s 0 = (X s , Y s ) s 0 ∈ R d × R be a diffusion process with generator ∆ (m) starting from (x, t) ∈ R d × (0, ∞). That is (X s ) s 0 is a Brownian motion up to a factor √ 2 and (Y s ) s 0 is independent of (X s ) s 0 with generator ∂ 2 t + 1-m t ∂ t .
The process (Y s ) s 0 is a Bessel process with parameter m starting from t > 0. When m > 0, the stopping time S = inf{s > 0, Y s = 0} is finite a.s., and since Q

Q (m) t (f )(x) = E x,t (f (X S )), (20)
for every smooth and bounded function f . In other words, since S and (X s ) s 0 are independent, if σ m (s, t)ds denotes the law of the hitting time S starting from t > 0,

Q (m) t (f )(x) = ∞ 0 P s f (x)σ m (s, t)ds, (21)
where (P s ) s 0 is the heat semi-group in R d associated to the Euclidean Laplacian ∆. From the definition of the operators

Q (m) t
and the definition of the equation (P s ) s 0 ,

P s f (x) = f (y) exp - |x -y| 2 4s dy (4πs) d/2 ,
and this gives a way to compute the law σ m of the hitting time. One obtains, for any t > 0, the probability measure

σ m (s, t)ds = 1 2 m Γ(m/2) t m exp(-t 2 /4s) s m/2+1 ds. (22)
This formula can also be found for instance in [START_REF] Borodin | Handbook of Brownian motion -facts and formulae[END_REF].

The map (t, s) → σ m (t, s) satisfies

∂ s = ∂ 2 tt + 1 -m t ∂ t ,
with Dirichlet boundary conditions, σ m (t, 0) = 0 for every t > 0, and σ m (0, s)ds = δ 0 (s). Moreover the RHS of the previous formula is the generator of the Bessel process.

Let us note that the links between the law of the stopping time and the generator explained above is quite general and has nothing to do in particular with Bessel processes, see for instance [Pav14, Sec 7.2].

From this observation and an integration by parts, one recovers that the map (t,

x) → Q (m) t f (x) satisfies the elliptic equation ∆ (m) Q (m) t (f ) = 0.
Lemma 8 Let m > 0 and p ∈ (0, m/2), then for any smooth and bounded function g,

E x,t (S p g(X S )) = t 2p Γ( m 2 -p) 4 p Γ( m 2 ) Q (m-2p) t (g)(x).
In particular, when p = 1 and m > 2,

E x,t (Sg(X S )) = t 2 2(m -2) Q (m-2) t (g)(x). ( 23
)
Proof.

-The formula can be obtained from a direct computation since we have

E x,t (S p g(X S )) = 1 2 m Γ(m/2)(4π) d/2 g(y) ∞ 0 s p-m+d 2 -1 exp - |x -y| 2 4s - t 2 4s ds dy = t 2p Γ( m 2 -p) 2 m Γ(m/2)(4π) d/2 P s g(y)σ m-2p (s)ds.
Lemma 9 (QM condition for ∆ (m) ) For any m = 1, the operator

∆ (m) on R d × (0, ∞) is a quasi-model QM (0, d -m + 2).
Proof. -For that model,

∆ (m) = ∆ R d+1 + 1 -m t ∂ t , that is X = 1-m t ∂ t . Then Ric(L) = 1-m t 2 (∂ t ) 2 and X ⊗ X = (1-m) 2 t 2
(∂ t ) 2 and the result follows from Definition (7).

Beckner and Φ-entropy inequalities for Cauchy type distributions 4.1 Beckner inequalities for Cauchy type distributions

Let f : R → [0, ∞) be a smooth and compactly supported function, and let

F (t, x) = Q (m) t (f )(x) with m > 0.
Let β ∈ R. From proposition (7), F is ∆ (m) -harmonic, and

∆ (m) (F β+2 ) = (β + 2)(β + 1)F β Γ ∆ (m) (F ).
As in Section 3, we can write,

E x,t (F (Z S ) β+2 ) = E x,t (F (Z 0 ) β+2 ) + (β + 2)(β + 1) ∞ 0 E x,t (F β (Z s )Γ ∆ (m) (F )(Z s )1l s≤S )ds,
where (Z s ) s 0 is the Markov process in R d+1 with generator ∆ (m) . Since F (Z S ) = f (X S ), from (20) we have the general equation,

Q (m) t (f β+2 ) = Q (m) t (f ) β+2 + (β + 2)(β + 1) ∞ 0 E x,t (F β (Z s )Γ ∆ (m) (F )(Z s )1l s≤S )ds. (24)
Again, from Dynkin's formula applied to F β Γ ∆ (m) (F ), we have

E x,t (F β (Z s )Γ ∆ (m) (F )(Z s )1l s≤S ) = E x,t (F β (Z S )Γ ∆ (m) (F )(Z S )1l s≤S ) -E x,t S s ∆ (m) (F β Γ ∆ (m) (F ))(Z u )du1l s≤S .
In other words,

Q (m) t (f β+2 ) -Q (m) t (f ) β+2 = (β + 2)(β + 1)E x,t SF β (Z S )Γ ∆ (m) (F )(Z S ) -(β + 2)(β + 1) ∞ 0 E x,t S s ∆ (m) (F β Γ ∆ (m) (F ))(Z u )du1l s≤S ds. ( 25
)
Let now assume that m d + 2, and denote

n = d -m + 2 (n ≤ 0). Let also assume that β ∈ [ n 2-n , 0] (that is β ∈ [-1 + 2 m-d , 0], 2 m-d > 0). Since n ≤ 0, ∆ (m) satisfies T CD(0, n) (Lemma 9) and from Corollary 5, ∆ (m) (F β Γ ∆ (m) (F )) 0. Equation (25) becomes Q (m) t (f β+2 ) -Q (m) t (f ) β+2 ≤ (β + 2)(β + 1) ∞ 0 E x,t (F β (Z S )Γ ∆ (m) (F )(Z S )1l s≤S )ds, which is Q (m) t (f β+2 ) -Q (m) t (f ) β+2 ≤ (β + 2)(β + 1)E x,t (SF β (Z S )Γ ∆ (m) (F )(Z S )). Starting form Z 0 = (x, t), F β (Z S )Γ ∆ (m) (F )(Z S ) = f β (X S )Γ ∆ (m) (F )(X S , 0) = f β (X S ) Γ(f )(X S ) + (∂ t F ) 2 (X S , 0) , since Γ ∆ (m) (F ) = Γ(F ) + (∂ t F ) 2 where Γ(f ) = |∇f | 2 , (f : R d → R) is the carré du champ operator associated to the Euclidean Laplacian in R d . From Proposition 7, since m > 1, ∂ t F (x, 0) = 0 then F β (Z S )Γ(F )(Z S ) = f β (X S )Γ(f )(X S ).
We can now apply Lemma 8, to get

Q (m) t (f β+2 ) -Q (m) t (f ) β+2 ≤ (β + 2)(β + 1) t 2 2(m -2) Q (m-2) t (f β Γ(f )). ( 26
)
Replacing f β+2 by f 2 and setting p = β + 2 in (26) we have obtained,

p p -1 Q (m) t (f 2 ) -Q (m) t (f 2/p ) p ≤ 2t 2 m -2 Q (m-2) t (Γ(f )). ( 27
)
When p = 2 (m d + 2), we obtain the Poincaré inequality for every smooth function f ,

Q (m) t (f 2 ) -Q (m) t (f ) 2 ≤ t 2 (m -2) Q (m-2) t (Γ(f )). ( 28
)
For t = 1 and x = 0 in the previous inequality (recall that the inequality depends also on the space variable x ∈ R d ), we have obtained the Beckner inequality for the Cauchy distribution

1 c(m, d) 1 (1 + |y 2 |) m+d 2
.

So, for any m d + 2 and p

∈ [1 + 2 m-d , 2] one has, p p -1 f 2 (1 + |y| 2 ) m+d 2 dy c(m, d) - f 2/p (1 + |y| 2 ) m+d 2 dy c(m, d) p ≤ 2 m -2 Γ(f ) (1 + |y| 2 ) m+d-2 2 dy c(m, d) . (29)
The inequality can be written as a weighted Poincaré-type inequality for a generalized Cauchy

∈ [1 + 1 b-d , 2], p p -1 f 2 dν b - f 2/p dν b p ≤ 1 (b -1) Γ(f )(1 + |y| 2 )dν b , (30)
for any smooth and nonnegative function f . Moreover, the inequality is optimal, that is, fixing p ∈ [1 + 1 b-d , 2], the constant 1 (b-1) is the best possible in the inequality (30).

The case p = 2 can be extended to any smooth function f (that may not be nonnegative). Then we obtain the following inequality, proved in [Sch01, BBD + 07, BDGV10, Ngu14].

Corollary 10 (Poincaré inequality for Cauchy type distributions) For any b d + 1, the measure ν b satisfies a Poincaré type inequality,

f 2 dν b - f dν b 2 ≤ 1 2(b -1) Γ(f )(1 + |y| 2 )dν b , (31)
for any smooth function f . Moreover, functions y → y i with 1 ≤ i ≤ d are extremal functions.

Proof. -We only have to prove optimality. Apply the Poincaré inequality (31) to the function y → y i for some 1

≤ i ≤ d, ν b (y 2 i ) ≤ 1 2(b -1) ν b (1 + |y| 2 ) = 1 2(b -1) (1 + ν b (|y| 2 )), or equivalently 1 d ν b (|y| 2 ) ≤ 1 2(b -1) (1 + ν b (|y| 2 )).
But the last inequality is an equality since, from (18),

ν b (|y| 2 ) = d 2b-2-d .
Remark 11

• Inequality (30) is equivalent to (27); indeed we only have to replace the map y → f (y) by y → f (ty + x).

• We cannot reach the logarithmic Sobolev inequality, since in our computation p 1 + 1/b and the logarithmic Sobolev inequality requires that p goes to 1.

• Setting f = 1 + εg in inequality (30) and using a Taylor expansion for small ε we get back the optimal Poincaré inequality (31), since

p p -1 f 2 dν b - f 2/p dν b p = 2ε 2 g 2 dν b - gdν b 2 + o(ε 2 ).
We do not know if there are extremal functions apart of constant functions.

• Corollary 10 has also been proved by Nguyen [Ngu14, Cor. 14], using the Brunn-Minkowski theory (see also the approach in [START_REF] Bobkov | Weighted Poincaré-type inequalities for Cauchy and other convex measures[END_REF][START_REF] Arnaudon | Intertwinings and generalized Brascamp-Lieb inequalities[END_REF]).

In the one dimensional case, the exact value has been computed by Bonnefont, Joulin and Ma in [BJM16, Thm 3.1]. Their result is more general, since they compute the optimal constant for every b > 1/2. The constant of (31) with d = 1 is

     1 2(b -1) if b 3/2; 4 (2b -1) 2 if 1/2 < b ≤ 3/2.
It is interesting to notice that there are two regimes, depending on the range of the parameter b. Actually, we are not able with our method to reach this range, even in dimension 1.

• By applying inequality (30) to the function x → f ( √ 2bx) and letting b → +∞, the inequality becomes the optimal Beckner inequality for the Gaussian measure

γ in R d , p p -1 f 2 dγ - f 2/p dγ p ≤ 2 Γ(f )dγ,
for any smooth and positive function f . This inequality is proved by W. Beckner [START_REF] Beckner | A generalized Poincaré inequality for Gaussian measures[END_REF].

In that case the inequality holds for any p ∈ (1, 2], and the limit case (when p → 1) leads to the optimal Logarithmic Sobolev inequality for the Gaussian measure.

• At the same time, Nguyen proves in [START_REF] Nguyen | Phi-entropy inequalities and asymmetric covariance estimates for convex measures[END_REF] the Beckner inequality (30) for a range of parameter p strictly contained in our interval

[1 + 1/(b -d), 2].
Even if, with his method, the parameter p can not reach the full interval, Nguyen is able to extend the result to a general class of probability measures with a convexity assumption. The method is actually improved in a more general context in [START_REF] Dupaigne | A family of Beckner inequalities under different curvature-dimension conditions[END_REF].

Tightness of the inequalities

We are interested in the tightness of inequality ( 27), in the sense of the curvature-dimension condition. In this whole of this section we assume that d 2.

Lemma 12 (Taylor expansion of (Q

(m)
t ) For any m > 4, we have

Q (m) t = Id + ∆ 2(m -2) t 2 + ∆ 2 8(m -2)(m -4) t 4 + o(t 4 ). ( 32
)
The formula has to be understood as follows, for any smooth and compactly supported function f , the remainder term o(t 4 ) is uniformly bounded with respect to the variable x.

Proof. -The formula can be checked directly. On the other hand, one can deduce the formula with functional analysis, at least formally. Since Q

(m) t is ∆ (m) -harmonic from Proposition 7 then, Q (m) t
is a function G m (t, ∆), from the functional analysis point of view. By the definition (19) of ∆ (m) , the function G m satisfies for t 0, x < 0, the differential equation

xG m (t, x) + 1 -m t ∂ t G m (t, x) + ∂ 2 tt G m (t, x) = 0, with the boundary condition G m (0, x) = 1. Solution of such differential equation is given by, G m (t, x) = H m (-t 2 x)
where H m satisfies for any λ 0,

4λH m (λ) -2(m -2)H m (λ) -H m (λ) = 0.
An asymptotic development H m (λ) = p 0 c p λ p gives c 0 = 1 (since G m (0, x) = 1) and

c p+1 = 1 2(p + 1)(2p -m + 2) c p .
Considering the fourth order term in Taylor's expansion of Q

t , we get (32), that is

c 1 = 1 2(2-m) and c 2 = 1 8(m-4)(m-2) .
This Lemma is fundamental. The Taylor expansion in (27), or equivalently in (28), gives in return the CD(0, d) conditions. We say then that the inequalities are Γ 2 -tight and nothing has been lost in terms of curvature-dimension condition (3), in other words the Taylor expansion of the inequality implies the CD(0, d) condition.

To see that, it is enough to compute the Taylor expansion of the inequality (27), when t goes to 0. It is important to fix β = 2-m+d m-d = -1 + 2 m-d the extremal bound of the admissible interval of β.

It is clear that the zero order terms vanish, it is also not difficult to observe that the second order terms also vanish. Therefore we only have to compute the fourth order terms and we obtain the following expression,

1 8(m -4)(m -2) ∆ 2 (f β+2 ) - (β + 2) 8(m -4)(m -2) f β-1 ∆ 2 (f ) - (β + 2)(β + 1) 8(m -2) 2 f β-1 ∆(f ) 2 ≤ β(β + 1)(β + 2) 2(m -4)(m -2) Γ(f, Γ(f ))f β-1 + β(β + 1)(β + 2) 4(m -4)(m -2) f β-1 ∆(f )Γ(f ) + (β -1)β(β + 1)(β + 2) 4(m -4)(m -2) f β-2 Γ(f ) 2 + (β + 1)(β + 2) 4(m -4)(m -2) f β ∆Γ(f ).
Expanding the term ∆ 2 (f β+2 ) (recall that β = 2-m+d m-d ), after a bit of algebra, we finally get,

Γ 2 (f ) β + 1 d(β + 1) -2β (∆f ) 2 -β Γ(f, Γ(f )) f - β(β -1) 2 Γ(f ) 2 f 2 . (33)
First, observe that if β = 0 that is m = d + 2 then the previous inequality becomes

Γ 2 (f ) 1 d (∆f ) 2 ,
in others words we recover the CD(0, d) condition on the Laplacian ∆.

It is more subtle to notice that, for any β ∈ (-1, 0) fixed, inequality (33) implies the same condition CD(0, d). It is more tricky since, setting f = 1 + εg with ε → 0, one infers that (33) already implies Γ 2 (g)

β+1 d(β+1)-2β (∆g) 2 which is the CD(0, d(β+1)-2β β+1
) condition, weaker than the CD(0, d) condition, since d(β+1)-2β β+1 d. The way to recover the CD(0, d) condition is to replace f by Φ(f ) in (33), where Φ is a smooth function.

By using the diffusion properties in (33) one has,

Φ 2 (Γ 2 (f ) -a(∆f ) 2 ) + Φ Γ(f ) 2 (1 -a) -c Φ 4 Φ 2 Γ(f ) 2 + Φ Φ (Γ(f, Γ(f )) -2a∆f Γ(f )) -2b Φ 2 Φ Φ Γ(f ) 2 -b Φ 3 Φ Γ(f, Γ(f )) 0,
where a = β+1 d(β+1)-2β , b = -2β and c = -β(β -1) and for simplicity we omit the variable f in the function Φ. It implies that the quadratic form with respect to the variables Φ , Φ 2 /Φ and Φ is positive, then the determinant of its matrix is positive. We get

Γ 2 (f ) 1 d (∆f ) 2 + -ac -a 2 c(a -1) -b 2 - 1 d (∆f ) 2 + -b 2 -c -b 2 a 4(c(a -1) -b 2 ) Γ(f, Γ(f )) Γ(f ) 2 + b 2 + ac c(a -1) -b 2 Γ(f, Γ(f )) Γ(f ) ∆f.
After an unpleasant computation, the inequality is independent of β, and may be written as follows

Γ 2 (f ) 1 d (∆f ) 2 + d d -1 Γ(f, Γ(f )) 2Γ(f ) - 1 d ∆f 2 ,
which is a reinforced CD(0, d) condition. Indeed, in all these computations, we could have replaced the Laplace operator ∆ in R d by any operator satisfying the CD(0, d) condition. Then we could have constructed the associated Q (m) t operator through (21), and could have obtained also a similar family of Beckner inequalities for this Q (m) t , still equivalent to the starting CD(0, d) condition. The main difference in the Euclidean case is that then the family of inequalities for Q (m) t (x) reduces to the same family for Q (m) 1 (0), through dilations and translations.

Φ-entropy inequalities for Cauchy type distributions

We can extend Beckner's inequalities to a general Φ-entropy inequality. For any convex function Φ on an interval I ⊂ R, let us define the Φ-entropy Ent Φ µ for a measure µ (or a kernel) and a function f on I,

Ent Φ µ (f ) = Φ(f )dµ -Φ f dµ .
Results of Section 4.1 can be put in the setting of the Φ-entropy easily.

Definition 13 (n-Admissible functions) Let Φ : I → R be a smooth function on an interval I ⊂ R and let n < 0. We say that Φ is n-admissible if Φ > 0 on I and (Φ )

2-n n is concave. In other words, Φ is n-admissible if and only if Φ > 0 and moreover

2 n -1 n (Φ (3) ) 2 ≤ Φ Φ (4) ,
that is θ = Φ satisfies condition (13).

When n goes to -∞, an (-∞)-admissible function is just a function Φ such that, Φ > 0 on I and (1/Φ ) is concave. Many Φ-entropy inequalities have been proved for such a function Φ under the curvature-dimension condition CD(ρ, ∞) (see [START_REF] Chafaï | Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities[END_REF][START_REF] Bolley | Phi-entropy inequalities for diffusion semigroups[END_REF]). In our computations, the case n = -∞ is then similar to the case n = +∞.

We can state the following result.

Ent Φ Q (m) t (f ) ≤ t 2 2(m -2) Q (m-2) t (Φ (f )Γ(f )), (34)
for any smooth function f on I.

The inequality is optimal in the sense that if inequality (34) holds for a some function Φ, the constant t 2 2(m-2) is the optimal one.

Corollary 14 (Φ-entropy inequality for Cauchy type distributions) Let b d + 1 and Φ an (2d -2b + 2)-admissible function on an interval I, then,

Ent Φ ν b (f ) ≤ 1 4(b -1) Φ (f )Γ(f )(1 + |x| 2 )dν b ,
for any smooth function f on I. The inequality is optimal.

The proof is the same as the one of Theorem 4.1, using the sub-harmonicity inequality (14), so we skip it, the optimality being proved in the same way.

A Beckner-type inequality on the sphere

In this section we are looking for the link between the family of Beckner inequalities (4.2) for ν b (or Q (m) t ) and a family of Beckner inequalities on the sphere. In the whole of this section we assume that d 2.

The unit sphere S d ⊂ R d+1 can be seen in R d through the stereographic projection, with the carré du champ operator on the sphere

Γ S (f ) = ρ 4 4 Γ(f ),
where ρ(x) = 1 + |x| 2 and the spherical measure

µ S (dx) = 1 c(d, d) 1 ρ 2d dx,
where the normalization constant c(d, d) has been defined in (16) (see [BGL14, Sec 2.2]). The Laplace-Beltrami operator takes the following representation after the stereographic projection

∆ S = (1 + |x| 2 ) 2 4 ∆ - d -2 2 (1 + |x| 2 ) d i=1 x i ∂ i ,
where ∆ is the classical Laplacian in R d . The operator is symmetric in L 2 (µ S ).

It is interesting to recall that the map S d x → x d+1 is an eigenvector associated to the eigenvalue -d. This function, from the stereographic projection, takes the form

u(x) = 1 -|x| 2 1 + |x| 2 , x ∈ R d . (35)
Then, u satisfies ∆ S u = -du and

Γ S d (u) = 1 -u 2 since Γ S d (u) = (1 + |x| 2 ) 2 4 Γ 1 -|x| 2 1 + |x| 2 = 1 (1 + |x| 2 ) 2 Γ(|x| 2 ) = 4|x| 2 (1 + |x| 2 ) 2 = 1-u 2 .
Indeed this computation is much easier (and indeed completely elementary) using the standard representation on the sphere, see [BGL14, Sec 2.2].

We can now transform the Beckner inequality for Q (m) t into a new Beckner inequality on the sphere. With these new notations, inequality (26) for t = 1 and x = 0 becomes for a smooth and nonnegative function f ,

c(d, d) c(m, d) f β+2 ρ d-m dµ S - c(d, d) c(m, d) f ρ d-m dµ S β+2 ≤ 2(β + 1)(β + 2) m -2 c(d, d) c(m -2, d) f β Γ S (f )ρ d-m-2 dµ S . (36)
Let us focus on the case β = d+2-m m-d , where β is on the limit of the admissible interval. Let f = ρ m-d g and let us compute the various terms in (36). First, by using again the diffusion properties of the carré du champ operator Γ S ,

f β Γ S (f )ρ d-m-2 = g β ρ 2(d-m) Γ S (ρ m-d g) = g β Γ S (g) + 2(m -d)g β+1 Γ S (log ρ, g) + (m -d) 2 g β+2 Γ S (log ρ),
and then

f β Γ S (f )ρ d-m-2 dµ S = g β Γ S (g)dµ S + (m -d) 2 β + 2 Γ S (log ρ, g β+2 )dµ S + (m -d) 2 g β+2 Γ S (log ρ)dµ S .
With an integration by parts, we get Γ S (log ρ, g β+2 )dµ S = -g β+2 ∆ S log ρdµ S , and then

f β Γ S (f )ρ d-m-2 dµ S = g β Γ S (g)dµ S + g β+2 Kdµ S ,
where

K = (m -d) 2 2 ∆ S log ρ d -m -2 + Γ S (log ρ) , since β + 2 = (d -m -2)/(d -m). Secondly we have f β+2 ρ d-m dµ S = g β+2 ρ 2 dµ S and f ρ d-m dµ S = gdµ S .
Then, inequality (36) can be written as

c(d, d) c(m, d) g β+2 ρ 2 dµ S - c(d, d) c(m, d) gdµ S β+2 ≤ 2(β + 1)(β + 2) m -2 c(d, d) c(m -2, d) g β Γ S (g)dµ S + g β+2 Kdµ S , or
g β+2 Rdµ S - c(d, d) c(m, d) gdµ S β+2 ≤ 2(β + 1)(β + 2) m -2 c(d, d) c(m -2, d) g β Γ S (g)dµ S , with R = c(d, d) c(m, d) ρ 2 - 2(β + 1)(β + 2) m -2 c(d, d) c(m -2, d) K.
From (18), the definition of β and the fact that ρ 2 = 2/(1 + u) where u has been defined in (35), we can write

R = c(d, d) c(m, d) 2 1 + u -2 m -d + 2 (m -d) 2 K m -2 + d ,
and the inequality

g β+2 Rdµ S - c(d, d) c(m, d) gdµ S β+2 ≤ 4(m -d + 2) (m -d) 2 (m -2 + d) c(d, d) c(m, d) g β Γ S (g)dµ S .
We need now to compute ∆ S log ρ and Γ S (log ρ). We have

∆ S (log ρ) = 1 2 ∆ S log ρ 2 = - 1 2 ∆ S log(1 + u) = d 2(1 + u) u + 1 2(1 + u) 2 Γ S d (u) = d 2(1 + u) u + 1 2(1 + u) 2 (1 -u 2 ). Then, finally ∆ S (log ρ) = 1 + u(d -1) 2(1 + u) , and
Γ S (log ρ) = 1 -u 4(1 + u) .
If we plug those values in the definition of K, then a miracle occurs. The function R is constant,

R = c(d, d) c(m, d) 3d + m -2 d + m -2 . Setting now f 2 = g β+2 = g d-m-2
d-m , we obtain Theorem 5.1 (Beckner-type inequalities on the sphere) For any smooth nonnegative function f on the sphere S d , and m d + 2, we have

f 2 dµ S ≤ A f 2/p dµ S p + 16 (m + 2 -d)(3d -2 + m) Γ S (f )dµ S , (37)
where

p = 1 + 2 m -d ∈ (1, 2] and A = c(d, d) c(m, d) 2 m-d m + d -2 m + 3d -2 .
Remark 15

• Function R is constant only for the parameter p = 1 + 2 m-d . • When m > d + 2, A > 1.
In that case inequality (37) is not tight, in the sense that when g = 1, we do not have the equality. But the inequality is still optimal since it is saturated for f = ρ d-m-2 2 .

• When m = d + 2, then A = 1, p = 2 and inequality (37) is nothing else than the optimal Poincaré inequality on the sphere,

f 2 dµ S - f 2 dµ S 2 ≤ 1 d Γ S (f )dµ S .
• The classical Beckner inequalities hold for the spherical model (see e.g. [BGL14, Remark 6.8.4]): for any p ∈ (1, 2]

f 2 dµ S ≤ |f | 2/p dµ S p + p -1 pd Γ S (f )dµ S . (38)
Inequality (38) is optimal in the sense that p-1 pd is the best constant. Only constant functions saturate inequality (38) for p ∈ (1, 2). And when p goes to 1, inequality (38) provides the Logarithmic Sobolev inequality on the sphere

f 2 log f 2 f 2 dµ S dµ S ≤ 2 d Γ S (f )dµ S ,
proved in [START_REF] Mueller | Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n-sphere[END_REF] (see also [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]Thm. 5.7.4]). On the other hand, this last logarithmic Sobolev inequality implies back the full family (38), p ∈ (1, 2].

Let us mention that inequality (38) has been generalized in [START_REF] Dolbeault | Improved interpolation inequalities on the sphere[END_REF] in the following way,

Ψ 1 -f 2/p dµ S p -1 ≤ Γ S (f )dµ S ,
for every nonnegative function f such that f 2 dµ S = 1 and for some explicit function Ψ.

• So (37) appears as a new and optimal inequality on the sphere. It is probably worth to compare it to the del Pino-Dolbeault family of optimal Gagliardo-Nirenberg inequalities,

f 2 a-1 a-2 ≤ C ∇f θ 2 f 1-θ 2a a-2 , a d,
where θ is fixed by scaling properties, see [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF].

• The Taylor expansion (when m → ∞) of the constant A is given by

A = 1 + d log(m) m + C m + o(1/m),
where C is a constant depending on the dimension d. This will be used in the rest of the section.

Even if we are not able from Theorem 5.1 to reach directly the Logarithmic Sobolev inequality, inequality (37) contains enough information to obtain a Sobolev inequality for the spherical model, however with a non optimal constant.

For this, we are going to obtain from (37) a Nash inequality on the sphere. First, it is not difficult to see that there exits two constants α, β > 0 depending only on d, such that, for any m d + 2, A ≤ α Let us prove that for any m d + 2,

x m ≤ α m d/2 1 + βE 2αm 1+d/2 = a m .

To prove such an inequality, it is enough to prove that, for any m d + 2, ϕ m (a m ) ≤ a m . We have

ϕ m (a m ) = α m d/2 1 + βE 2αm 1+d/2 1-2 m-d + β m 2 E ≤ α m d/2 1 + βE 2αm 1+d/2 1-2 m + β m 2+ d 2 E ≤ α m d/2 1 + βE 2αm 1+d/2 = a m ,
where we used the inequality (1 + x) u ≤ 1 + ux, for x > 0 and u ∈ (0, 1). Modifying the numerical constants (depending only on d), we see that for any m d + 2,

f 2 dµ S ≤ Cm d/2 + C m d/2 Γ S (f )dµ S .
Optimizing with respect to the parameter m d + 2 and using Jensen inequality, we obtain, for some other constant C > 0, . This last inequality is a so-called Nash inequality, and is known to be equivalent to the Sobolev inequality (up to the optimal constant) for the spherical model, see [BGL14, Prop. 6. (39)

Conclusion:

The family of optimal inequalities (37) appears as a new form of optimal inequalities on the sphere, implying a Sobolev inequality.

t

  (f ) is a solution of the elliptic equation (15), from Itô's Lemma, or Dynkin's formula[START_REF]Stochastic differential equations. An introduction with applications[END_REF] Sec 7.3]

Theorem 4. 1 (

 1 Beckner inequalities for Q (m) t ) For any m d + 2 and p ∈ [1 + 2 m-d , 2], and for any smooth function f 0,

  distribution. Let us define for b > d 2 , the probability measure dν b (y) = 1 c(2b -d, d) 1 (1 + |y| 2 ) b dy. Setting b = (m + d)/2 in inequality (29), we obtain, using (18), Theorem 4.2 (Beckner inequalities for Cauchy type distributions) For any b d + 1 and p

  Theorem 4.3 (Φ-entropy inequality for Q (m) t ) Let m d + 2 and Φ an (d -m + 2)admissible function on an interval I Then,

  which is just a precise form of the previous Taylor expansion, and16 (m + 2 -d)(3d -2 + m) ≤ β m 2 . From Hölder's inequality, if p ∈ [1, 2], |f | 2/p dµ S with p = 1 + 2/(m -d) ∈ [1, 2]. Inequality (37) becomes, for any nonnegative function f such that f dµ S = 1, and any m m + 2,f 2 dµ S ≤ α S (f )dµ S .Let us define for x 0, the mapϕ m (x) = αwhere E = Γ S (f )dµ S . The function ϕ m has a unique fixed point x m > 0, and the previous inequality implies that f 2 dµ S ≤ x m .

f 2 dµ S ≤ C |f |dµ S 2 +

 2

≤≤

  C f 2 dµ S + C Γ S (f )dµ S .By means of Poincaré inequality (m = d + 2) and [BGL14, Prop. 6.2.2], this inequality implies a tight Sobolev inequality. We have obtained Theorem 5.2 (From Beckner inequalities to Sobolev inequality on the sphere) The family of Beckner inequalities (37) implies that there exists a constant C > 0 depending only on d such that, for any smooth functions f on the sphere S d , f 2 dµ S + C Γ S (f )dµ S .

  One can choose a new base in R d such that Y is diagonal. If we denote again Z in the new base in R d , we have to prove the following inequality, for every vectors (Y ii ) 1≤i≤d and (Z i ) 1≤i≤d ,

	d
	i=1

Acknowledgements

This work was partly written while the two first authors were visiting Institut Mittag-Leffler in Stockholm; it is a pleasure for them to thank this institution for its kind hospitality. We would like also to warmly thank the referee who gave us useful comments and pointed out references.

This research was supported by the French ANR-17-CE40-0030 EFI project.


Cintia Okamura
email: cintiaokamura@hotmail.com

Jacques Lolive

Patrick Romieu

Jean-Paul Thibaud
email: paul.thibaud@grenoble.archi.fr

Nicolas Tixier

Towards a sensory risk culture Risk communication about contamination at Barão de Mauá (Brazil)

Keywords: Participative risk communication, risk ambiances, analysis of controversies, public policies, contaminated areas

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

Towards a sensory risk culture

This research 1 is based on an interdisciplinary collaboration between the CRESSON team (UMR AAU n°1563, France), which specializes in in the themes of urban ambiances, affective atmospheres and sensory environments [START_REF] Augoyard | Éléments pour une théorie des ambiances architecturales et urbaines[END_REF][2] [START_REF]Ambiances tomorrow[END_REF], the CETESB (Environmental Agency of the State of São Paulo, Brazil), which is in charge of managing at-risk territories and examines the social stakes of such inhabited spaces, and the Territorial team (UMR 5194 PACTE), which develops socio-political approaches to at-risk territories. The research recommends tackling an atrisk territory by combining a sensory approach of residing on it with an approach in terms of public policies. The goal is to give an account of the sensory experience of an at-risk territory (approach via ambiances) and to relate it to its social, environmental and political history (approach via its related controversies [START_REF] Latour | Esquisse d'un parlement des choses[END_REF]).

The chosen field of study is the Condominío Barão de Mauá in the city of Mauá, located 30 km from São Paulo, Brazil. Comprising a total of 54 buildings in which there are programme entitled "Da Cominicação de Riscos at Cultura de Risco" (CETESB, Brazil). [START_REF] Papermann | Les émotions dans l'espace public[END_REF],000 residents, it was built in 1996 on land owned by COFAP, a shock absorber company, which had buried solid industrial waste, mainly foundry residues, on the premises. As there was no land control by the owners, other toxic substances of unknown origin were also illegally deposited there. In other words, the buildings had been constructed on a ground contaminated by organic and inorganic compounds (some of which are volatile), including methane, benzene, chlorobenzene, trimethylbenzene and decane. People became aware of the contamination in April 2000 after an explosion that had likely been caused by methane. This explosion happened during the maintenance of a pump in one of the underground water tanks that had been installed in the condominium. It killed one worker, and another suffered serious burns. Since then, legal proceedings have dragged on and are still ongoing, while these condominiums' residents continue to be exposed to a double risk: contamination by carcinogens and methane explosions.

To study the complexity of the situation and the problems such a territory creates, our hypothesis is that a risk environment is embodied in specific ambiances and lends itself to sensory manifestations that can be documented by methodologically appropriate means. Therefore, it was necessary to examine the modalities of the emergence of inhabitants' sensitivity to the risks, to question what might constitute a sensory risk culture and to update the stakes and socio-political consequences associated with the daily vulnerability of being in such a living environment. The approach adopted was based on the idea of a risk environment and on the commitment shown to the inhabitants with regard to this public problem.

Controversies regarding the in situ investigation

An investigation carried out in Barão de Mauá in June 2015 took both a controversial and an environmental (or ambience-based) approach. These two approaches have proved to be quite complementary and intertwined. The first approach made it possible to understand the stakes of this contaminated territory, trace its history and its public life, update all the actors involved in the controversy and contextualize the in situ investigation [START_REF] Grosjean | L'espace urbain en méthodes[END_REF]. The second approach gave access to residents' everyday sensory experiences, to the way in which a risky territory can be lived from within and can mobilize inhabitants and, lastly, to the way in which the atmosphere creates a particular feeling in such an at-risk territory.

The controversy is a problematic situation where there is a combination of scientific uncertainty, divergent strategies of actors and strong social mobilization. It is characterized by the absence of a consensus on the definition of reality. Controversy analysis can be used as a methodological tool to understand how new public issues are formed, and contaminated areas are a good example in this regard. It is not a question of taking sides in the controversy in question but of describing the full panorama (actors, positions, arguments, strategies, instruments, evolution), identifying the salient aspects and analyzing the points of blockage.

In addition, the ambiance approach took the form of a fiveday survey involving a group of 13 investigators. The aim was to document as accurately as possible the scattered contamination risks in Barão de Mauá. An ethnography of the senses was developed to be as close as possible to the situations faced by the inhabitants. The investigation [START_REF] Soubeyran | Les enjeux du virage aménagiste[END_REF] was simultaneously immersed, plural, collective and interactive: immersed, because it was an in situ investigation within the Barão de Mauá residential complex; plural, because it is based on a variety of light approaches (ethnographic observations, explorations in progress, sensory reactivations, group meetings, inhabitants' stories, floating listening and sound recordings, photographs, shared logbook, reporting on the survey with the residents); collective, because of the role played by the group of investigators' internal dynamics as well as that of the inhabitants; interactive, because of the round trips between the data collected, the analyses in progress and the presentations made to the inhabitants and other city actors.

Sharing first impressions

The power of the site's ambiance was both rich and complex. Upon arrival at the condominium, not all the researchers shared the same experience regarding Brazil, São Paulo or the Portuguese language. However, it is striking that the tone of the researchers' initial notes, and even of the final syntheses, was so similar. The floating attention that emerged thanks to the predominant use of the descriptive form, as well as the pedestrian drift and the priority given to a first sensorytype knowledge, has certainly contributed to a strong coherence between the theoretical affect resulting from the immersion in the condominium. The strong contact with the site seems to have made a mark on all the investigators. According to the logic of the condominium, the threshold effect is both practical (barrier and control of individuals) and symbolic. For example, the tension that can be felt by a European in a Latin American metropolis decreases immediately, as it gradually becomes possible to have more relaxed interaction with the inhabitants, even if they remain unfamiliar. At the same time, there is an extreme ambivalence regarding the site, as well as the compartmentalization where this ambivalence originated.

The spatial experience seems to have been based from the outset on the perception of both visual and auditory discontinuity on the disparity of the grounds distributed between ground rather welcoming to being tread on and the many technical stigmas related to the primordial accident. It is also noteworthy that the sound space reveals repetitions via dry echo much more easily than through an enveloping reverberation. In the same way, the fortress or castle aspect orients the representations on the side of a strong personality of the building or a certain resistance of the place with respect to its surroundings, while the sensory experience attests to the site's strong dependence on the perceptible horizons (visual, auditory and olfactory), which evoke intermediate spaces that are highly undifferentiated and ultimately not very elastic.

The site is wrapped up inside the sound ring of peripheral sounds, to such an extent that the residents have long perceived auditory threats that can express a new disaster in relation to underground gas. This border around the condominium is noted by everyone and quite logically colors the visual barrier of industrial chimneys, whose chemical smoke smells can easily be transposed into site-specific miasmas. In the same way, all the researchers notice the "weight of the sky" that the birds cannot make lighter. Perceptible thanks to their singing, the oral investigation reveals, however, that the birds feed on fruits that have almost certainly been contaminated. The sky has many aircraft passing through at low altitude, which contributes to the shared sense of an atmosphere of heaviness. Finally, beyond the details that provide consensus for a final report, we must note a fine sharing of all the rhythms of discovery and crystallization of theoretical affect.

Controversies in Barão de Mauá

The controversial analysis allowed us to reconstruct the complex history of the Condomínio Barão de Mauá, its public life with a few highlights of what has transformed the inhabitants' perception and increased the controversy.

In 1995, the sale of COFAP's contaminated land was approved by the City Council. However, according to one resident, "everyone knew there was waste -the hospital, COFAP, the other companies… everyone was dumping waste here". This event would later provoke the indignation of the inhabitants concerning the lack of respect shown to them by institutions, especially the Town Hall.

20 April 2000: There is an explosion in the condominium. It kills one worker. This dramatic event leads to the condominium put on hold (see below). It causes panic among the inhabitants. This is a traumatic event, especially for those who were on the scene. The complex's inhabitants have been living in fear ever since of another explosion happening.

August 2001: The Secretary of the Environment of the state of São Paulo invites the press, as well as nine sindicos from the condominium, to the CETESB to share the results of the first study into the causes of the explosion. The meeting with the press is held at 10 a.m., the one with the sindicos at 11 a.m. This first public meeting provokes a feeling of injustice among the inhabitants. "We were informed after the journalists, while we were the main ones concerned." Before the meeting, the inhabitants are informed by the press in a sensationalist way: "52 carcinogenic substances: the Chernobyl of Brazil". This event reinforces the anxiety of the inhabitants who are already frightened of another explosion.

On 26 September 2006, the judge of the 3rd civil court of Mauá ruled on the inhabitants' compensation and the demolition of the condominio's 72 buildings. In September 2007, Mauá's "Environmental Prosecutor" decides to suspend legal action and sign a Conduct Adjustment Compromise with COFAP to present an environmental recovery plan. This evolution of the judiciary's positions and a procedure that drags on cause disappointment and anger among the inhabitants.

The controversial analysis also focused on describing the position of each of the social actors (administrations, companies, politicians, mobilized communities, etc.). Each group offers its own solution in accordance with its vision of the problem without engaging in dialogue with the others. For example, for the mobilized inhabitants, the solution is compensation in order to repair the lack of respect that they faced. For our interlocutors in the Public Ministry, only a final decision of the justice will be able to put an end to the conflict that feeds off the population's anger. For the CETESB Contaminated Area Service, the solution is a good risk communication plan that will help re-establish a connection with the residents. For our interlocutors in the Health Administration, the use of the ATSDR epidemiological methodology from the United States can contribute to solving the problem of the population's exposure to benzene. If each institution defends a different position, it is undoubtedly because each of them defines its activities and priorities in its own field of action and because they define their priorities based on the imperfections of their own actions.

This erratic risk management can be understood as a nondirected collective learning process [START_REF] Papermann | Les émotions dans l'espace public[END_REF] because no one really knew how to deal with this unprecedented situation; no one had a real strategy. This learning on the part of the institutions had a positive result, namely the realization that it was necessary to radically change practices -in particular, risk communication design -and that the institutions would not be able to improve the situation by settling for the remediation plan that the Geoklock company has been implementing since 2014 to decontaminate the site. From the point of view of the people who have been at risk for 20 years, the process is much too protracted, considering that this situation had been known to those in charge for 15 years by the time we started our field investigation work.

Pervasiveness of the senses: vision, smell, silence

Following the investigation of the controversies, the in situ investigation focused on the ambiances of Barão de Mauá and the pervasiveness of the senses. In this respect, three main aspects emerged: in visual, olfactory and auditory terms.

On the visual level, scattered throughout various parts of the condominium, there are unfinished buildings, ghost buildings without doors or windows, showing rough, grey, tagged and neglected pieces of rubble. This vivid and disturbing image, which calls to mind a territory at war or a post-disaster zone, sets the tone for the entire condominium, harnesses the imagination and constantly brings to mind the perception of risk with which the residents are living. On the olfactory level, there is a smell, more prominent in some places than others, even if it is difficult to identify and delimit the specific areas. According to the inhabitants, the pervasiveness of this smell varies significantly according to the weather. The smell is stronger when it rains; when it is hot, only smoke rises from the ground, and the smells spread more evenly. While the ghost buildings provide an immediate shock, this indistinct, omnipresent and indefinite smell spreads slowly, gradually. It keeps the residents in a more or less worried state of hypervigilance: Is it the smell of methane, the gas that caused the explosion in 2000? Or is it the smell of benzene, an index of contamination? Or both? Or a combination of all of this and the odors of the nearby petrochemical complex? On the auditory level, many people remember the explosion of 2000, so when they hear a sudden noise, an impact sound, they cannot help but think of the possibility of another explosion. A state of auditory vigilance persists as if the inhabitants remain on the lookout for an accident to come. Moreover, there is an enigmatic silence all over the neighborhood. In particular, the sound of cars being parked -and sometimes, of Geoklock's work -is very noticeable. Following the investigation and attentive listening, it is necessary to note the little sound presence of the residents, the noise of (human) life. Everything happens as if the sound space of the condominium was inconsistent, punctuated from the outside (there is a very lively district just opposite) by the diffuse sounds of the city in the distance (noise of the main road and the petrochemical complex) and the very low sound ceiling, as if the sounds from the inside could not really be heard or could not create their own atmosphere.

A neighborhood on hold

The methane explosion that revealed -or at least made public -the problem of soil pollution was a cataclysmic event shared by the whole neighborhood, the starting point for putting its entire development on hold. If the past is shared among the inhabitants (completely new neighborhood, dates of moves of people over the same period of ten years), the present and the future are also just as much. A shared present: Few people left, the neighborhood is no longer developing, and the problem is always there. But also a shared future or, more precisely, the absence of a possible future: between personal situations that are stuck and the need to wait for potential compensation and/or strong remedial action.

A place that is on hold. The condominium is located on a hillock, cut off from other areas by fairly steep slopes, bordered by small streams (córrego) and set off by an abandoned green buffer and an oil pipeline. Although it is true that the primary function of the condominium is housing, there are activities such as traffic and commercial life, school, sports, craft, industrial on the outside. The visual horizon, as well as the luminous horizon in the evening and the sound horizon, bears witness to a more intense life all around. The forming of methane, which continually has to be extracted from buried waste, puts people and buildings on ground that is not firm and homogeneous but literally degassing. There is something below that might be dangerous. One is reminded of this by children's games that jut out more than 1.5 mets above the ground thanks to artificial mounds and inside free space, and one grasps that the land is not used (no vegetable garden or outdoor sports activity).

A becoming that is on hold. The future seems to be on hold. On hold first because of an explosion that may well be unlikely but is still possible. Mostly, however, it is on hold because of the results of surveys and expertise, the revelation of a level of toxicity that makes it necessary for the residents to leave or the possible result of epidemiological analyses showing infection through diffuse contamination. The inhabitants' own future is tied up with the future of this place, which makes it difficult to make normal decisions in all areas of life (to leave, to rent, to sell, to buy, to renovate) -almost all of these decisions depend on the results of studies that are underway, of legal decisions, possible compensation and more satisfactory risk treatments. In the meantime, there is no possibility for the neighborhood to develop: Although sought, no common room was built, and there is no opportunity to install rooftops in the outdoor parking area because any new foundation would be risky. It is just as difficult to start renovating the buildings and even to demolish or transform what has been stopped and long abandoned.

A temporality that is on hold. There are buildings with faded facades, the one next to the other, others made of concrete and exposed blocks, their construction stopped, and others, more infrequent, have been repainted, giving a semblance of something new. In the same space, there are inhabited buildings and the ruins of these same buildings. Here the difference is not in the typology of the buildings (they are almost identical, apart from the color and the orientation) but in the state of the building where all the temporalities come together to produce a strange effect of a temporal mirror.

Since the construction was halted, the district has not seen any renewal, neither in its buildings nor in its inhabitants. Few things have evolved, except one, which is very much alive: children, the only ones who really make use of time every day to enjoy the outdoor spaces and the only ones to show how long the time has passed: They are born, they grow up and leave... At the bottom of the condominium buildings, green spaces have been planted and can be seen but not really used, with very pruned, season-less plants, some potted next to plastic animals and decorative white wrought-iron benches. Activity in the area is static, a point reflected by these immobile elements. Traces of Home Sweet Home, the lawn is well maintained and the seasons are represented in the halls of buildings that often have artificial potted flowers, painted flowers and colorful paintings. The whole contrasts sharply with the offbeat nature on display around the condominium.

Although the past, the present and the future are now shared among the inhabitants of the neighborhood, life is nonetheless still there. A solidarity, a mixture of attachment to a collective home, public actions and expertise, has developed between the inhabitants. Although the condominium has no public space, no collective space, there are places to meet and talk. An equipment that we learn to decipher... "Here is our playground, where the children play. At the beginning, it was at the ground level. However, due to the contamination, it was raised about 2 meters. So that the children wouldn't have direct contact with the soil. This is where the kids play. That is the only space we have for them."

Collective expression of affects

On the whole, the investigation took place within an emotionally charged context. The inhabitants' attachment to their initial housing project, the explosion in 2000 that revealed the extent of the contamination, the protracted legal battle and the sense of profound injustice all contribute to the pervasiveness of a formidable, emotion-filled climate. The photographic reactivation workshop of Thursday, 4 June 2015 bears witness to this. It presented a collection of photographs taken at the site (26 photographs chosen by the research team) to the inhabitants to serve as a basis for and stimulate discussion. The meeting turned into a public meeting, and it was very challenging to discuss the photos with the inhabitants. This meeting was a highlight of the survey, as the anger and distress of the people facing this painful and blocked situation were expressed forcefully. It looked like the remediation work had a powerful impact on their memory: "Now that they have started digging, our memories are starting to surface ... When we see people making holes in the ground, when they come here, then everything rushes back to our memory, and in particular a reminder of the fear that we had."

Complaints include, first and foremost, the public expression of vulnerability when the participants mention their suffering: the trauma of "those who saw the burned, naked, headless guy"; "the dream (of living here) that turns into a nightmare", the terror of living here "with these monsters that look like they are ready to explode at any moment"; the stigma they face "But we are becoming victims today; former dreamers, we have become victims... now we have become something even worse, and then when (they) see us, 'Look, look there: There's a ghost'." Then they also remember the diseases related to contamination, the children who may have cancer, the sick neighbors who leave, the epidemiological tests and this diffuse nature of contamination.

Other emotions have a more political tone. The expression of emotions in public is sometimes linked to a moral evaluation [START_REF] Grosjean | L'espace urbain en méthodes[END_REF], and emotions can then be the cement for mobilization. This is the case when the inhabitants mention the lack of respect (desrespeito) from the institutions: "the public authorities have only one solution: to give five cents to everyone, so that they go to the pharmacy to buy five cents' worth of drugs, shame in the face. But the residents here are not rubbish, I am a human being, and here the others treat you like s***." When they express the indignation, it provokes at home. "But my indignation is about the lack of respect for people, you know? ... Everyone knew that there was a garbage dump; the city let it happen. The city authorized the houses to be built, and the city gave them the right to do it. You know what? Everyone is implicated, and they got City Hall loose from the situation." During the workshop's two sessions, emotions backed up the very strong criticism of the institutions, particularly the CETESB, which has not lived up to residents' expectations. We ourselves were challenged and criticized, sometimes vehemently, because some of the team members are from the CETESB.

Thus, the photographic reactivation workshop became a means for people to complain and to ask questions, a kind of obligatory passage so that the inhabitants, once placated, could participate in the two sound workshops that took place in the neighboring rooms. The sound investigation and the proposals for collective listening revealed the ambivalence of the inhabitants between anger and a deep-seated desire for re-enchantment. Thus, the proposal of a collective outcry presented as a mode of salvation shared between investigators and users was initially received with a curiosity made stronger by the fear of making noise -as if the risks of a noisy and magical contamination, acting as a reminder of the original catastrophic noise, became possible -a fear that very quickly gave way to the prospect of a liberating and productive gesture. Even though there were not many participants, the production of a sound space through a scream and weakly projected onto the walls because of the distance represented, for a brief moment, the intense desire for a possible form of becoming.

Becoming of a contaminated space

The two approaches -controversy and ambiancesproved to be very complementary from different points of view: in the implementation of various temporalities involved in the risky situation (long temporality when it falls under the socio-political history of the neighborhood and short when it relates to the inhabitants' day-to-day experiences) and in the articulation of the various levels of analysis involved (analysis of actors' logics and residents' experiences). This complementarity is reflected in the results of the analysis: The two approaches reveal the existence of two incommensurable realities: the world of public policies that is relatively closed in on itself, where the sectoral actions of institutions collide with each another without any real exchanges or willingness to collaborate, and the lived world of the inhabitants who are torn between the restrictive nature of considering all the risks and the aspiration for their lives to return to normal, between the affliction caused by a painful and blocked situation and the hope for better living conditions.

This telescoping effect of the two very different realities makes it possible for us to better understand the difficulties of life inside the condominium. For example, why the inhabitants do not spontaneously adhere to the remediation project that seemed to be the solution for some of the institutions. To begin to improve the situation, we propose connecting the two worlds to implement a participative and partnership risk management in Barão de Mauá. The two sessions to present the results that we organized separately with the inhabitants and with the institutions made it possible to legitimize the perception of the risk by the inhabitants and to reinforce awareness of institutional compartmentalization without making the actors feel guilty. This first awareness based on the results of our research has served as a lever to build a new participation system composed of two complementary forums that are called on to work together: one composed of the condominium's inhabitants and the other of the various institutions concerned.

Thus, the combination of these two approaches has made it possible to broaden the notion of risk communication by giving full scope to the qualitative perception of risk, the complexity of vulnerability situations and leading to a proposal for participatory risk communication, which "includes the public as a collaborating agent", as Covello and Sandman [START_REF] Covello | Risk communication: Evolution and Revolution[END_REF] so beautifully describe it.

Figure 1 .

 1 Figure 1. 3D perspective of the Condominío Barão de Mauá

Figure 2 .

 2 Figure 2. Barão de Mauá residential condominium. Halted constructions next to the inhabited buildings.

Figure 3 .

 3 Figure 3. Controversy Time-Line

Figure 4 .

 4 Figure 4. Barão de Mauá at dusk.

Figure 5 .

 5 Figure5. An equipment that we learn to decipher... "Here is our playground, where the children play. At the beginning, it was at the ground level. However, due to the contamination, it was raised about 2 meters. So that the children wouldn't have direct contact with the soil. This is where the kids play. That is the only space we have for them."

Figure 6 .

 6 Figure 6. We have seen the above condo that feels like a fort-castle built on a hill. It appears as a fortress, banished by a vast area of impassable vegetation... and block the horizon. The condo is isolated from the other inhabited neighborhood, like a quarantine.

Acknowledgements

This text was supported by the SFR Territoire (Univ. Grenoble Alpes) for proofreading English. A shorter version of this text has been published in Portuguese in ICUR2016, International Conference on Urban Risks, Lisbon, June 2016.