Sergey G Bobkov

Cyril Roberto

ENTROPIC ISOPERIMETRIC INEQUALITIES

Keywords: 2010 Mathematics Subject Classification. Primary 60E, 60F Rényi entropy, Fisher information, isoperimetric inequality

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Introduction

The entropic isoperimetric inequality asserts that N (X) I(X) ≥ 2πe n (1.1)

for any random vector X in R n with a smooth density. Here N (X) = exp -2 n p(x) log p(x) dx and I(X) = |∇p(x)| 2 p(x) dx denote the Shannon entropy power and respectively the Fisher information of X with density p (with integration with respect to Lebesgue measure dx on R n which may be restricted to the supporting set supp(p) = {x : p(x) > 0}). This inequality was discovered by Stam [START_REF] Stam | Some inequalities satisfied by the quantities of information of Fisher and Shannon[END_REF] where it was treated in dimension one. It is known to hold in any dimension, and the standard normal distribution on R n plays an extremal role in it. Later on, Costa and Cover [START_REF] Costa | On the similarity of the entropy power inequality and the Brunn-Minkowski inequality[END_REF] pointed out a remarkable analogy between (1.1) and the classical isoperimetric inequality relating the surface of an arbitrary body A in R n to its volume vol n (A). The terminology "isoperimetric inequality for entropies" goes back to Dembo, Costa and Thomas [START_REF] Dembo | Information Theoretic Inequalities[END_REF].

As Rényi entropies have become a focus of numerous inverstigations in the recent time, it is natural to explore more general relations of the form

N α (X) I(X) ≥ c α,n (1.2)
for the functional 1) .

N α (X) = p(x) α dx -2 n(α-
(1.

3)

It is desirable to derive (1.2) with optimal constants c α,n independent of the density p, where α ∈ [0, ∞] is a parameter called the order of the Rényi entropy power N α (X). Another representation N α (X)

-n 2 = p L α-1 (p(x) dx)
shows that N α is non-increasing in α. This allows one to define the Rényi entropy power for the two extreme values by the monotonicity to be

N ∞ (X) = lim α→∞ N α (X) = p -2 n ∞ , (1.4)
N 0 (X) = lim α→0 N α (X) = vol n (supp(p))

2 n ,
where p ∞ = ess sup p(x). As a standard approach, one may also put N 1 (X) = lim α↓1 N α (X) which returns us to the usual definition of the Shannon entropy power N 1 (X) = N (X) under mild moment assumptions (such as N α (X) > 0 for some α > 1).

Returning to (1.1)-(1.2), the following two natural questions arise.

Question 1. Given n, for which range A n of the values of α does (1.2) hold with some positive constant? Question 2. What is the value of the optimal constant c α,n and can the extremizers in (1.2) be described?

The entropic isoperimetric inequality (1.1) answers both questions for the order α = 1 with an optimal constant c 1,n = 2πe n. As for the general order, let us first stress that, by the monotonicty of N α with respect to α, the function α → c α,n is also non-increasing. Hence, the range in Question 1 takes necessarily the form A n = [0, α n ) or A n = [0, α n ] for some critical value α n ∈ [0, ∞]. The next assertion specifies these values.

Theorem 1.1. We have

A n =      [0, ∞] for n = 1, [0, ∞)
for n = 2, [0, n n-2 ] for n ≥ 3.

Thus, in the one dimensional case there is no restriction on α (the range is full). In fact, this already follows from the elementary sub-optimal inequality N ∞ (X)I(X) ≥ 1, (1.5) implying that c α,1 ≥ 1 for all α. To see this, assume that I(X) is finite, so that X has a (locally) absolutely continuous density p, thus differentiable almost everywhere. Since p is non-negative, any point y ∈ R such that p(y) = 0 is a local minimum, and necessarily p (y) = 0 (as long as p is differentiable at y). Hence, applying the Cauchy inequality, we have It follows that p has a bounded total variation not exceeding I(X), so p(x) ≤ I(X) for every x ∈ R. This amounts to (1.5) according to (1.4) for n = 1.

Turning to Question 2, we will see that the optimal constants c α,1 together with the extremizers in (1.2) may be explicitly described in the one dimensional case for every α using the results due to Nagy [START_REF] Nagy | Über integralungleichungen zwischen einer Funktion und ihrer Ableitung[END_REF]. Since the transformation of these results in the informationtheoretic language is somewhat technical, we discuss this case in detail in the next three sections (Sections 2-4). Let us only mention here that

4 ≤ c α,1 ≤ 4π 2 ,
where the inequalities are sharp for α = ∞ and α = 0 respectively, with extremizers

p(x) = 1 2 e -|x| and p(x) = 2 π cos 2 (x) 1 {|x|≤ π 2 } .
The situation in higher dimensions is more complicated, and only partial answers to Question 2 will be given here. Anyway, in order to explore the behavior of the constants c α,n , one should distinguish between the dimensions n = 2 and n ≥ 3 (which is also suggested by Theorem 1.1). In the latter case, these constants can be shown to satisfy

4πn(n -2) Γ( n 2 ) Γ(n) 2 n ≤ c α,n ≤ 4π 2 n, 0 ≤ α ≤ n n -2 ,
where the left inequality is sharp and corresponds to the critical order α = n n-2 . With respect to the growing dimension, these constants are asymptotically 2πen + O(1), which exhibits nearly the same behaviour as for the order α = 1. However (which is rather surprising), the extremizers for the critical order exist for n ≥ 5 only and are described as densities of the (generalized) Cauchy distributions on R n . We discuss these issues in Section 7, while Section 6 deals with dimension n = 2, where some description of the constants c α,2 will be given for the range α ∈ [ 1 2 , ∞). We end this introduction by giving an equivalent formulation of the isoperimetric inequalities (1.2) in terms of functional inequalities of Sobolev type. As was noticed by Carlen [START_REF] Carlen | Superadditivity of Fisher's information and logarithmic Sobolev inequalities[END_REF], in the classical case α = 1, (1.1) is equivalent to the logarithmic Sobolev inequality of Gross [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF], cf. also [START_REF] Bobkov | Bounds on the deficit in the logarithmic Sobolev inequality[END_REF]. However, when α = 1, a different class of inequalities should be involved. Namely, using the substitution p = f 2 / f 2 (here and in the sequel integrals are understood with respect to the Lebesgue measure on R n ), we have

N α (X) = f 2α -2 n(α-1) f 2 2α n(α-1)
and

I(X) = 4 |∇f | 2 / f 2 .
Therefore (provided that f is square integrable), (1.2) can be equivalently reformulated as a homogeneous analytic inequality

|f | 2α 2 n(α-1) ≤ 4 c α,n |∇f | 2 f 2 α(2-n)+n n(α-1) , (1.6)
where we can assume that f is smooth and has gradient ∇f (however, when speaking about extremizers, the function f should be allowed to belong to the Sobolev class W 2 1 (R n )). Such inequalities were introduced by Moser [START_REF] Moser | On Harnack's theorem for elliptic differential equations[END_REF][START_REF] Moser | A Harnack inequality for parabolic differential equations[END_REF] in the following form

|f | 2+ 4 n ≤ B n |∇f | 2 f 2 2 n . (1.7)
More precisely, (1.7) corresponds to (1.6) for the specific choice α = 1 + 2 n . Here, the one dimensional case is covered by Nagy's paper with the optimal factor B 1 = 4 π 2 . This corresponds to α = 3 and n = 1, and therefore c 3,1 = π 2 which complements the picture depicted above. To the best of our knowledge, the best constants B n for n ≥ 2 are not known. However, using the Euclidean log-Sobolev inequality and the optimal Sobolev inequality, Beckner [START_REF] Beckner | Asymptotic estimates for Gagliardo-Nirenberg embedding constants[END_REF] proved that asymptotically B n ∼ 2 πen . Both Moser's inequality (1.7) and (1.6) with a certain range of α enter the general framework of Gagliardo-Nirenberg's inequalities

|f | r 1 r ≤ κ n (q, r, s) |∇f | q θ q |f | s 1-θ s (1.8) with 1 ≤ q, r, s ≤ ∞, 0 ≤ θ ≤ 1, and 1 r = θ ( 1 q -1 n ) + (1 -θ) 1
s . We will make use of the knowledge on Gagliardo-Nirenberg's inequalities to derive information on (1.2).

In the sequel, we denote by f r = ( |f | r ) 1 r the L r -norm of f with respect to the Lebesgue measure on R n (and use this functional also in the case 0 < r < 1).

Nagy's theorem

In the next three sections we focus on dimension n = 1, in which case the entropic isoperimetric inequality (1.2) takes the form

N α (X) I(X) ≥ c α,1 (2.1)
for the Rényi entropy

N α (X) = p(x) α dx -2 α-1
and the Fisher information

I(X) = p (x) 2 p(x) dx = 4 d dx p(x) 2
dx.

In dimension one, our basic functional space is the collection of all (locally) absolutely continuous functions on the real line whose derivatives are understood in the Radon-Nikodym sense. We already know that (2.1) holds for all α ∈ [0, ∞].

According to (1.6), the family (2.1) takes now the form

|f | 2α ≤ 4 c α,1 α-1 2 f 2 α-1 2 f 2 α+1 2 (2.2)
when α > 1, and

f 2 ≤ 4 c α,1 1-α 1+α f 2 1-α 1+α |f | 2α 2 1+α
(2.3) when α ∈ (0, 1).

In fact, these two families of inequalities can be seen as sub-families of the following one, studied by Nagy [START_REF] Nagy | Über integralungleichungen zwischen einer Funktion und ihrer Ableitung[END_REF],

|f | γ+β ≤ D |f | p β pq |f | γ 1+ β(p-1) pq with p > 1, β, γ > 0, q = 1 + γ(p -1) p . (2.4)
and some constants D = D γ,β,p depending on γ, β and p, only. For such parameters, introduce the functions y p,γ = y p,γ (t) defined for t ≥ 0 by

y p,γ (t) =      (1 + t) p p-γ if p < γ, e -t if p = γ, (1 -t) p p-γ 1 [0,1] (t) if p > γ.
To involve the parameter β, define additionally y p,γ,β implicitly as follows. Put y p,γ,

β (t) = u, 0 ≤ u ≤ 1, with t = 1 u s γ (1 -s β ) -1 p ds if p ≤ γ. If p > γ, then y p,γ,β (t) = u, 0 ≤ u ≤ 1
, is the solution of the above equation for

t ≤ t 0 = 1 0 s γ (1 -s β ) -1
p ds and y p,γ,β (t) = 0 for all t > t 0 . With these notations, Nagy established the following result.

Theorem 2.1 [START_REF] Nagy | Über integralungleichungen zwischen einer Funktion und ihrer Ableitung[END_REF]. Under the constraint (2.4), for any (locally) absolutely continuous function f : R → R, (i)

f ∞ ≤ q 2 1 q |f | p 1 pq |f | γ p-1 pq . (2.5)
Moreover, the extremizers take the form f (x) = ay p,γ (|bx + c|) with a, b, c constants (b = 0). (ii)

|f | β+γ ≤ q 2 H q β , p -1 p β q |f | p β pq |f | γ 1+ β(p-1) pq (2.6)
where

H(u, v) = Γ(1 + u + v) Γ(1 + u) Γ(1 + v) u u + v u v u + v v , u, v ≥ 0.
Moreover, the extremizers take the form f (x) = ay p,γ,β (|bx+c|) with a, b, c constants (b = 0).

Here, Γ denotes the classical Gamma function, and we use the convention that H(u, 0) = H(0, v) = 1 for u, v ≥ 0. It was mentioned by Nagy that H is monotone in each variable. Moreover, since H(u,

1) = (1 + 1 u ) -u is between 1 and 1 e , one has 1 > H(u, v) > (1 + 1 u ) -u > 1 e
for all 0 < v < 1. This gives a two-sided bound

1 ≥ H q β , p -1 p > 1 + β q -q β > 1 e .

One dimensional isoperimetric inequalities for entropies

The inequalities (2.2) and (2.3) correspond to (2.6) with parameters

p = γ = q = 2, β = 2(α -1) in the case α > 1 and p = 2, β = 2(1 -α), γ = 2α, q = 1 + α in the case α ∈ (0, 1),
respectively. Hence, as a corollary from Theorem 2.1, we get the following stetement which solves Question 2 when n = 1. Note that, by Theorem 2.1, the extremal distributions (their densities p) in (2.1) are determined in a unique way up to non-degenerate affine transformations of the real line. So, it is sufficient to indicate just one specific extremizer for each admissible collection of the parameters. Recall the definition of the optimal constants c α,1 from (2.1).

Theorem 3.1. (i) In the case α = ∞, we have c ∞,1 = 4.
Moreover, the density p(x) = 1 2 e -|x| (x ∈ R) of the two-sided exponential distribution represents an extremizer in (2.1).

(ii) In the case 1 < α < ∞, we have

c α,1 = 2π α -1 2 α + 1 α-3 α-1 Γ( 1 α-1 ) Γ( α+1 2(α-1) ) 2 .
Moreover, the density p(x) = a cosh(x)

-2 α-1 with a normalization constant a = 1 √ π Γ( α+1 2(α-1) ) Γ( 1 α-1 )
represents an extremizer in (2.1).

(iii) In the case 0 < α < 1,

c α,1 = 2π 1 -α 2 1 + α 1+α 1-α Γ( 1+α 2(1-α) ) Γ( 1 1-α ) 2 .
Moreover, the density p(x) = a cos(x)

2 1-α 1 [-π 2 , π 2 ] (x) with constant a = 1 √ π Γ( 3-α 1-α ) Γ( 3-α 2(3-α) ) represents an extremizer in (2.1).
To prove the theorem, we need a simple technical lemma. Lemma 3.2. (i) Given a > 0 and t ≥ 0, the (unique) solution y ∈ (0, 1] to the equation

1 y ds s √ 1-s a = t is given by y = cosh at 2 -2 a . (ii) Given a, b > 0 and c ∈ R, we have ∞ -∞ cosh(|bx + c|) -a dx = √ π b Γ( a 2 ) Γ( a+1 2 )
.

(iii) Given a ∈ (0, 1) and u ∈ [0, 1], we have

1 u ds s a 1 -s 2(1-a) = 1 1 -a arccos(u 1-a ). Remark 3.3. Since Γ( a+1 2 ) = Γ(m + 1 2 ) = (2m)! 4 m m!
√ π for a = 2m with an integer m ≥ 1, for such particular values of a, we have

∞ -∞ cosh(|bx + c|) -a dx = 1 b • 4 m m! (m -1)! (2m)! .
Proof of Lemma 3.2. Changing the variable u = √ 1 -s a , we have

1 y ds s √ 1 -s a = 2 a √ 1-y a 0 du 1 -u 2 = 1 a log 1 + √ 1 -y a 1 - √ 1 -y a .
Inverting this equality leads to the desired result of item (i).

For item (ii) we use the symmetry of the cosh-function together with the change of variables u = bx + c and then t = sinh(u)

2 to get ∞ -∞ cosh(|bx + c|) -a dx = 1 b ∞ -∞ cosh(|u|) -a du = 2 b ∞ 0 cosh(u) -a du = 1 b ∞ 0 t -1 2 (1 + t) -a+1 2 dt.
To obtain the result, we need to perform a final change of variables v = 1 1+t . This turns the last integral into

1 0 (1 -v) -1 2 v a 2 -1 dv = B 1 2 , a 2 = √ π Γ( a 2 ) Γ( a+1 2 )
, where we used the beta function B(x, y)

= 1 0 (1 -v) x-1 v y-1 dv = Γ(x)Γ(y) Γ(x+y) ,
x, y > 0. Finally, in item (iii), a change of variables leads to

1 u ds s a 1 -s 2(1-a) = 1 1 -a 1 u ds 1-a 1 -s 2(1-a) = 1 1 -a 1 u 1-a dv √ 1 -v 2 = 1 1 -a arccos(u 1-a ). Proof of Theorem 3.1. When α = ∞ as in the case (i), (2.2) with f 2 = 1 becomes f ∞ ≤ 4 c ∞,1 f 2 1 4 .
This corresponds to (2.5) with parameters p = q = γ = 2. Therefore, item (i) of Theorem 2.1 applies and leads to

f ∞ ≤ f 2 1 4 ,
that is, c ∞,1 = 4. Moreover, the extremizers in (2.5) are given by

f (x) = ay 2,2 (|bx + c|) = a e -|bx+c| , b = 0, a, c ∈ R.
But, the extremizers in (2.1) are of the form p = f 2 / f 2 with f an extremizer in (2.5). The desired result then follows after a change of variables.

Next, let us turn to the case (ii), where 1 < α < ∞. Here (2.1) is equivalent to (2.2) and corresponds to (2.6) with p = γ = q = 2 and β = 2(α -1). Therefore, by Theorem 2.1,

( 4 c α,1 ) α-1 2 = H( 1 α-1 , 1 2 ) α-1 , so that c α,1 = 4 H( 1 α-1 , 1 2 ) 2 = 4 Γ(1 + 1 α-1 ) 2 Γ( 3 2 ) 2 Γ( 3 2 + 1 α-1 ) 2 1 α-1 + 1 2 1 α-1 2 α-1 1 α-1 + 1 2 1 2 = π 1 α-1 α+1 2(α-1) 2 Γ( 1 α-1 ) 2 Γ( α+1 2(α-1) ) 2 α + 1 2 2 α-1 α + 1 α -1
where we used the identities Γ(3/2) = √ π/2 and Γ(1 + z) = zΓ(z). This leads to the desired expression for c α,1 .

As for extremizers, item (ii) of Theorem 2.1 applies and asserts that the equality cases in (2.2) are reached, up to numerical factors, for functions

f (x) = y(|bx + c|), with b = 0, c ∈ R, and y : [0, ∞) → R defined implicitly for t ∈ [0, ∞) by y(t) = u, 0 ≤ u ≤ 1, with t = 1 u s 2 (1 -s 2(α-1) ) -1 2 ds = 1 u 1 s 1 -s 2(α-1)
ds. Now, Lemma 3.2 provides the solution y(t) = (cosh((α -1) t)) -1 α-1 . Therefore, the extremizers in (2.2) are reached, up to numerical factors, for functions of the form

f (x) = (cosh(|bx + c|)) -1 α-1 , b = 0, c ∈ R.
Similarly to the case (i), the extremizers in (2.1) are of the form p = f 2 / f 2 with f an extremizer in (2.2). Therefore, by Lemma 2.3, with some b > 0 and c ∈ R,

p(x) = cosh(|bx + c|) -2 α-1 cosh(|bx + c|) -2 α-1 dx = b √ π Γ( α+1 2(α-1) ) Γ( 1 α-1 ) cosh(bx + c) -2 α-1
as announced.

Finally, let us turn to item (iii), when α ∈ (0, 1). As already mentioned, (2.1) is equivalent to (2.3) and therefore corresponds to (2.6) with p = 2, β = 2(1 -α), γ = 2α and q = 1 + α. An application of Theorem (2.1) leads to the desired conclusion after some algebra (which we leave to the reader) concerning the explicit value of c α,1 . In addition, the extremizers are of the form p(x) = ay 2 (|bx + c|), with a a normalization constant, b = 0 and c ∈ R. Here y = y(t) is defined implicitly by the equation

t = 1 y 1 s α 1 -s 2(1-α) ds for t ≤ t 0 = 1 0 1 s α √ 1-s 2(1-α)
ds and y(t) = 0 for t > t 0 . Item (iii) of Lemma 3.2 asserts that

t 0 = π 2(1 -α)
and y(t) = cos((1 -α) t)

1 1-α 1 [0, π 2(1-α) ] (t).
This leads to the desired conclusion.

Special orders

As an illustration, here we briefly mention some explicit values of c α,1 and extremizers for specific values of the parameter α in the one dimensional entropic isoperimetric inequality

N α (X) I(X) ≥ c α,1 (4.1)
The order α = 0. The limit in item (iii) of Theorem 3.1 leads to the optimal constant

c 0,1 = lim α→0 c α,1 = 4π 2 .
Since all explicit expressions are continuous with respect to α, the limits of the extremizers in (2.1) for α → 0 represent extremizers in (2.1) for α = 0. Therefore, the densities

p(x) = 2b π cos 2 (bx + c) 1 [-π 2 ; π 2 ] (bx + c), b > 0, c ∈ R, are extremizers in (2.1) with α = 0.
The order α = 1 2 . Direct computation leads to c 1 2 ,1 = (4/3) 3 π 2 . Moreover, the extremizers in (2.1) are of the form

p(x) = 8b 3π cos 4 (bx + c) 1 [-π 2 ; π 1 ] (bx + c), b > 0, c ∈ R.
The order α = 1. This case corresponds to Stam's isoperimetric inequality for entropies. Here c 1,1 = 2πe, and, using the Stirling formula, one may notice that indeed

c 1,1 = lim α→1 c α,1 = 2πe.
Moreover, Gaussian densities can be obtained from the extremizers p(x) = cosh(bx + c)

-2 α-1 with b = b √ α -1, c = c
√ α -1 in the limit as α ↓ 1. (Note that the limit α ↑ 1 would lead to the same conclusion.)

The order α = 2. A direct computation leads to c 2,1 = 12 with extremizers of the form

p(x) = b 2 cosh 2 (bx + c) , b > 0, c ∈ R.
In this case, the entropic isoperimetric inequality may equivalently be stated in terms of the Fourier transform p(t) = e itx p(x), t ∈ R, of the density p. Indeed, thanks to Plancherel's identity, we have

N 2 (X) -1/2 = p 2 = 1 2π |p| 2 .
Therefore, the (optimal) isoperimetric inequality for entropies yields the relation

|p| 2 ≤ π I(X) 3
which is a global estimate on the L 2 -norm of p. In [START_REF] Zhang | Inequalities for characteristic functions involving Fisher information[END_REF], Zhang derived the following pointwise estimate: If the random variable X with density p has finite Fisher information I(X), then (see also [START_REF] Bobkov | Fisher information and the central limit theorem[END_REF] for an alternative proof)

|p(t)| ≤ I(X) I(X) + t 2 , t ∈ R.
The latter leads to some bounds on c 2,1 , namely

N 2 (X) -1/2 = 1 2π |p| 2 ≤ 1 2π I(X) 2 (I(X) + t 2 ) 2 dt = 1 2 I(X).
Hence N 2 (X)I(X) ≥ 4 that should be compared to N 2 (X)I(X) ≥ 12.

The order α = 3. Then c 3,1 = π 2 , and the extremizers are of the form

p(x) = b π cosh(bx + c) , b > 0, c ∈ R.
The order α = ∞. From Theorem 3.1, c ∞,1 = 4, and the extremizers are of the form

p(x) = b e -|bx+c| , b > 0, c ∈ R.

Fisher information in higher dimensions

In order to perform the transition from the entropic isoperimetric inequality (1.2) to the form of the Gagliardo-Nirenberg inequality such as (1.8) via the change of functions p = f 2 / f 2 and back, and to justify the correspondence of the constants in the two types of inequalities, let us briefly fix some definitions and recall some approximation properties of the Fisher information. This is dictated by the observation that in general f in (1.8) does not need be square integrable, and then p will not be defined as a probability density. The Fisher information of a random vector X in R n with density p may be defined by means of the formula

I(X) = I(p) = 4 |∇ √ p| 2 . (5.1)
This functional is well-defined and finite if and only if f = √ p belongs to the Sobolev space

W 2 1 (R n ).
There is the following characterization: A function f belongs to

W 2 1 (R n ), if and only if it belongs to L 2 (R n ) and sup h =0 1 |h| f (x + h) -f (x) 2 < ∞.
In this case, there is a unique vector-function g = (g 1 , . . . , g n ) on R n with components in L 2 (R n ), called a weak gradient of f and denoted g = ∇f , with the property that

gv = -f ∇v for all v ∈ C ∞ 0 (R n ). (5.2)
As usual, C ∞ 0 (R n ) denotes the class of all C ∞ -smooth, compactly supported functions on R n . Still equivalently, there is a representative f of f which is absolutely continuous on almost all lines parallel to the coordinate axes and whose partial derivatives

∂ x k f belong to L 2 (R n ).
In particular, g k (x) = ∂ x k f (x) for almost all x ∈ R n (cf. [START_REF] Ziemer | Weakly differentiable functions. Sobolev spaces and functions of bounded variation[END_REF], Theorems 2.1.6 and 2.1.4).

Applied to f =

√ p with a probability density p on R n , the property that f ∈ W 2 1 (R n ) ensures that p has a representative p which is absolutely continuous on almost all lines parallel to the coordinate axes and such that the functions ∂ x k p/ √ p belong to L 2 (R n ). Moreover,

I(p) = n k=1 ∂ x k p √ p 2 2
.

Note that W 2 1 (R n ) is a Banach space for the norm defined by

f 2 W 2 1 = f 2 2 + ∇f 2 2 = f 2 2 + g 1 2 2 + • • • + g n 2 2
(g = ∇f ).

We use the notation N α (X) = N α (p) when a random vector X has density p.

Proposition 5.1. Given a (probability) density p on R n such that I(p) is finite, there exists a sequence of densities

p k ∈ C ∞ 0 (R n ) satisfying as k → ∞ a) I(p k ) → I(p), and b) N α (p k ) → N α (p) for any α ∈ (0, ∞), α = 1.
Proof. Let us recall two standard approximation arguments. Fix a non-negative function ω ∈ C ∞ 0 (R n ) supported in the closed unit ball Bn (0, 1) = {x ∈ R n : |x| ≤ 1} and such that ω = 1, and put ω ε (x) = ε -n ω(x/ε) for ε > 0. Given a locally integrable function f on R n , one defines its regularization (mollification) as the convolution

f ε (x) = (f * ω ε )(x) = ω ε (x -y)f (y) dy = f (x -εy)ω(y) dy, x ∈ R n . (5.
3)

It belongs to C ∞ (R n ), has gradient ∇f ε = f * ∇ω ε , and is non-negative, when f is nonnegative. From the definiton it follows that, if f ∈ L 2 (R n ), then

f ε 2 ≤ f 2 , lim ε→0 f ε -f 2 = 0. Moreover, if f ∈ W 2 1 (R n ), then, by (5.2)-(5.3), we have ∇f ε = ∇f * ω ε . Hence ∇f ε 2 ≤ ∇f 2 , lim ε→0 ∇f ε -∇f 2 = 0, so that f ε W 2 1 ≤ f W 2 1 , lim ε→0 f ε -f W 2 1 = 0. (5.4) Thus, C ∞ (R n ) ∩ W 2 1 (R n ) is dense in W 2 1 (R n ). To obtain a), define f = √ p. Given δ ∈ (0, 1 2 ), choose ε > 0 such that f ε -f W 2 1 < δ. Let us take a non-negative function w ∈ C ∞ 0 (R n ) with w(0) = 1 and consider a sequence u l (x) = f ε (x)w(x/l).
These functions belong to C ∞ 0 (R n ), and by the Lebesgue dominated convergence theorem,

u l → f ε in W 2 1 (R n ) as l → ∞. Hence u -f W 2 1 < δ for some u = u l , which implies | u 2 -1| = | u 2 -f 2 | ≤ u -f 2 < δ and thus u 2 > 1 2 . As a result, the normalized function f = u/ u 2 satisfies f -f W 2 1 = u -u 2 f W 2 1 u 2 ≤ δ + δ f W 2 1 u 2 < 4δ f W 2 1 ,
where we used

f W 2 1 ≥ f 2 = 1. This gives | ∇ f 2 -∇f 2 | < 4δ f W 2 1 ≤ 2 f W 2 1
and hence

∇ f 2 2 -∇f 2 2 ≤ 4δ f W 2 1 ∇ f 2 + ∇f 2 ≤ 4δ f W 2 1 2 f W 2 1 + 2 ∇f 2 = 8δ f 2 W 2 1 + f W 2 1 ∇f 2 .
Here f 2

W 2 1 = 1 + I(p) and f W 2 1 ∇f 2 ≤ 1 2 f 2 W 2 1 + 1 2 ∇f 2 2 ≤ 1 2 + I(p).
Eventually, the probability density p = f 2 satisfies

|I(p) -I(p)| ≤ 4δ (3 + 4I(p)). (5.5)
With δ = δ k → 0, we therefore obtain a sequence p k = p such that I(p k ) → I(p) as k → ∞, thus proving a).

Let us see that similar functions p k may be used in b) when

p(x) α dx = f (x) 2α dx = ∞
which corresponds to the case where N α (p) = 0 for α > 1 and N α (p) = ∞ for 0 < α < 1.

Returning to the previously defined functions u l , we observe that u l 2α → f ε 2α as l → ∞. Hence, it is sufficient to check that f ε 2α → f 2α = ∞ for some sequence ε = ε k → 0. Indeed, since f 2 = 1, the function f is locally integrable, implying that f ε (x) → f (x) as ε → 0 for almost all points x ∈ R. This follows from (5.3) and the Lebesgue differentiation theorem which yields

|f ε (x) -f (x)| ≤ ω ε (x -y) |f (y) -f (x)| dy ≤ ω ∞ ε -n |y-x|<ε |f (y) -f (x)| dy → 0 a.e.
Hence, by Fatou's lemma, f 2α ≤ lim inf ε→0 f ε 2α , and we are done. Now, let us turn to the basic case where p(x) α dx < ∞, α ∈ (0, ∞). To prove b), we borrow arguments from the proof of Theorem 2.3.2 in [START_REF] Ziemer | Weakly differentiable functions. Sobolev spaces and functions of bounded variation[END_REF]. Consider a partition {w i } ∞ i=0 of unity of R n subordinate to the covering G i = B n (0, i + 1) \ Bn (0, i -1), in which B n (0, -1) = B n (0, 0) = ∅. Every function w i is supposed to be in C ∞ 0 (R n ) with a support lying in G i , to be non-negative, and all of them satisfy

∞ i=0 w i (x) = 1, x ∈ R n .
(5.6)

As before, let f = √ p. Given 0 < δ < 1 2 , for each i ≥ 0 choose ε i > 0 small enough such that (w i f

) ε i is still supported in G i and (w i f ) ε i -w i f W 2 1 < 2 -i-1 δ. (5.7)
The latter is possible due to the property (5.4) applied to w i f . By the integrability assumption on p, we have

w i f 2α < ∞, implying (w i f ) ε -w i f 2α → 0 as ε → 0 (5.8)
as long as 2α ≥ 1. Since f ∈ L 2 (R n ), we similarly have (w i f ) ε -w i f 2 → 0. The latter implies that (5.8) holds in the case 2α < 1 as well, since w i f is supported on a bounded set. Therefore, in addition to (5.7), we may require that 1) . (

|(w i f ) ε i -w i f | 2α dx < (2 -i-1 δ) max(2α,
, f (x) = ∞ i=0 w i (x)f (x) 5.9) Now, by (5.6)
, where the series contains only finitely many nonzero terms. More precisely,

f (x) = m i=0 w i (x)f (x), |x| < m + 1.
Similarly, for the function u

(x) = ∞ i=0 (w i (x)f (x)) ε i , we have u(x) = m i=0 (w i (x)f (x)) ε i , |x| < m + 1.
This equality shows that u is non-negative and belongs to the class C ∞ 0 (R n ). In addition, by (5.7),

u -f W 2 1 ≤ ∞ i=0 (w i f ) ε i -w i f W 2 1 < δ. Hence u -f 2 < δ, (5.10)
and repeating the arguments from the previous step, we arrive at the bound (5.5) for the density p = f 2 with f = u/ u 2 . Next, if α ≥ 1 2 , by the triangle inequality in L 2α , from (5.9) we also get u -

f 2α < δ, so | u 2α -f 2α | < δ. (5.11)
If α < 1 2 , then, applying the inequality (a

1 + • • • + a N ) 2α ≤ a 2α 1 + • • • + a 2α N (a k ≥ 0), from (5.9) we deduce that |u -f | 2α dx ≤ ∞ i=1 |u -w i f | 2α dx < δ.
This yields u 2α dx -f 2α dx < δ and therefore, by Jensen's inequality,

| u 2α -f 2α | < (2δ) 1/(2α) .
(5.12)

In view of (5.10), inequalities similar to (5.11)-(5.12) hold also true for the function f = u/ u 2 in place of u. Applying this with δ = δ k → 0, we obtain a sequence fk such that the probability densities p = f 2 satisfy a) -b) for any α = 1.

Corollary 5.2. For any α > 0, α = 1, the infimum inf

I(p)<∞ N α (p)I(p)
may be restricted to the class of compactly supported, C ∞ -smooth densities p on R n with finite Fisher information.

Two dimensional isoperimetric inequalities for entropies

In this section we deal with dimension n = 2. As will be clarified, the entropic isoperimetric inequality

N α (X)I(X) ≥ c α,2 (6.1)
holds true for any α ∈ [0, ∞) with a positive constant c α,2 and does not hold for α = ∞ which answers Question 1 in the introduction. In addition, we will give a certain description of the optimal constants c α,2 in (6.1) for the range α ∈ [ 1 2 , ∞), thus answering partially Question 2. When n = 2, the family of inequalities (1.6) takes now the form

|f | 2α 1 2α ≤ 4 c α,2 α-1 2α |∇f | 2 θ 2 f 2 1-θ 2 (6.2)
with θ = α-1 α when α > 1, and

f 2 1 2 ≤ 4 c α,2 1-α 2 |∇f | 2 θ 2 |f | 2α 1-θ 2α (6.3)
with θ = 1 -α when α ∈ (0, 1). Both inequalities enter the framework of Gagliardo-Nirenberg's inequality (1.8). The best constants and extremizers in (1.8) are not known for all admissible parameters. The most recent paper on this topic is due to Liu and Wang [START_REF] Liu | On the best constant for Gagliardo-Nirenberg interpolation inequalities[END_REF] (see references therein and historical comments). The case q = s = 2 in (1.8) that corresponds to (6.2) with r = 2α goes back to Weinstein [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF] who related the best constants to the solutions of non-linear Schrödinger equations.

We present now part of the results of [START_REF] Liu | On the best constant for Gagliardo-Nirenberg interpolation inequalities[END_REF] that are useful for us. Since we will use them for any dimension n ≥ 2, the next statement does not deal only with the case n = 2. Also, since all the inequalities of interest for us deal with the L 2 -norm of the gradient only, we may restrict ourselves to q = 2 for simplicity, when (1.8) becomes

|f | r 1 r ≤ κ n (2, r, s) |∇f | 2 θ 2 |f | s 1-θ s (6.4)
with parameters satisfying 1 ≤ r, s ≤ ∞, 0 ≤ θ ≤ 1, and

1 r = θ( 1 2 -1 n ) + (1 -θ) 1
s . This inequality may be restricted to the class of all smooth, compactly supported functions f ≥ 0 on R n . Once (6.4) holds in C ∞ 0 (R n ), this inequality is extended by a regularization and density arguments to the Sobolev space of functions f ∈ L s (R n ) such that |∇f | ∈ L 2 (R n ) (the gradients in this space are understood in a weak sense).

The next statement relates the optimal constant in (6.4) to the solutions of the ordinary non-linear equation

u (t) + n -1 t u (t) + u(t) r-1 = u(t) s-1 (6.5)
on the positive half-axis. Put

σ = n+2 n-2 if n ≥ 3, ∞ if n = 2.
We denote by |x| the Euclidean norm of a vector x ∈ R n .

Theorem 6.1 [START_REF] Liu | On the best constant for Gagliardo-Nirenberg interpolation inequalities[END_REF]. In the range 1 ≤ s < σ, s < r < σ + 1,

κ n (2, r, s) = θ -θ 2 (1 -θ) θ 2 -1 r M -θ n s , M s = R n u s r,s (|x|) dx,
where the functions u r,s = u r,s (t) are defined for t ≥ 0 as follows.

(i) If s < 2, then u r,s is the unique positive decreasing solution to the equation (6.5) in 0 < t < T (for some T ), satisfying u (0) = 0, u(T ) = u (T ) = 0, and u(t) = 0 for all t ≥ T .

(ii) If s ≥ 2, then u r,s is the unique positive decreasing solution to (6.5) in t > 0, satisfying u (0) = 0 and lim t→∞ u(t) = 0.

Moreover, the extremizers in (6.4) exist and have the form

f (x) = au r,s (|bx + c|) with a ∈ R, b = 0, c ∈ R n .
Note that (6.2) corresponds to Gagliardo-Nirenberg's inequality (6.4) with s = 2, r = 2α and θ = α-1 α for α > 1, while (6.3) with α ∈ [ 1 2 , 1) corresponds to (6.4) with r = 2, s = 2α and θ = 1 -α. Applying Corollary 5.2, we therefore conclude that

κ 2 (2, r, s) = (4/c α,2 ) α-1 2α when α > 1, κ 2 (2, r, s) = (4/c α,2 ) 1-α 2 when α ∈ [1/2, 1).
Together with Liu-Wang's theorem, we immediately get the following corollary, where we put as before

M s = R 2 u s (|x|) dx = 2π ∞ 0 u s (t) tdt.
Corollary 6.2. (i) For any α > 1, we have

c α,2 = 4(α -1) α -1 α-1 M 2 ,
where M 2 is defined for the unique positive decreasing solution u(t) on (0, ∞) to the equation u (t) + u (t) t + u(t) 2α-1 = u(t) with u (0) = 0 and lim t→∞ u(t) = 0. (ii) For any α ∈ [ 1 2 , 1), we have

c α,2 = 4(1 -α) α α 1-α M 2α ,
where M 2α is defined for the unique positive decreasing solution u(t) to u (t)+ 1 t u (t)+u(t) = u(t) 2α-1 in 0 < t < T with u (0) = 0, u(T ) = u (T ) = 0, and u(t) = 0 for all t ≥ T .

In both cases the extremizers in (6.1) represent densities of the form p

(x) = b M u 2 (|bx + c|), x ∈ R 2 , with b > 0 and c ∈ R 2 .
So far, we have seen that (6.1) holds for any α ∈ [1/2, ∞). Since, as observed in the introduction, α → c α,n is non-increasing, (6.1) holds also for α < 1/2 and therefore for any α ∈ [0, ∞). Note that the case α = 1, which is formally not contained in the results above, is the classical isoperimetry inequality for entropies (1.1). Let us now explain why (6.1) cannot hold for α = ∞. The functional form for (6.1) should be the limit case of (6.2) as α → ∞, when it becomes

f 2 ∞ ≤ D |∇f | 2 dx (6.6) with D = 4/c ∞,2 .
To see that (6.6) may not hold with any constant D, we reproduce Example

1.1.1 in [14]. Let, for x ∈ R 2 , f (x) = log | log |x| | if |x| ≤ 1/e, 0 otherwise.
Then, passing to radial coordinates, we have

|∇f | 2 = 2π
1/e 0 dr r| log r| 2 = 2π, while f is not bounded. In fact, (6.6) is also violated for a sequence of smooth bounded approximations of f . 7. Isoperimetric inequalities for entropies in dimension n = 3 and higher One may exhibit two different behaviors between n = 3, 4, and n ≥ 5 in the entropic isoperimetric inequality N α (X)I(X) ≥ c α,n .

Let us rewrite the inequality (1.6) separately for the three natural regions, namely as

|f | 2α 1 2α ≤ 4 c α,n n(α-1) 4α |∇f | 2 θ 2 f 2 1-θ 2 (7.2) with θ = n(α-1) 2α when 1 < α ≤ n n-2 , |f | 2α θ 2α f 2 1-θ 2 ≤ 2 √ c α,n |∇f | 2 1 2 (7.3)
with θ = 2α n(α-1) when α > n n-2 (observe that θ ∈ (0, 1) in this case), and finally

f 2 1 2 ≤ 4 c α,n n(1-α) 2[α(2-n)+n] |∇f | 2 θ 2 |f | 2α 1-θ 2α (7.4) with θ = n(1-α)
α(2-n)+n when α ∈ (0, 1). Both (7.2) and (7.4) enter the framework of Gagliardo-Nirenberg's inequality (1.8). As for (7.3), we will show that such an inequality cannot hold. To that aim, we need to introduce the limiting case θ = 1 in (7.2), which corresponds to α = n n-2 . It amounts to the classical Sobolev inequality

|f | 2n n-2 n-2 2n ≤ S n |∇f | 2 1 2 (7.5)
which is known to hold true with best constant

S n = 1 πn(n -2) Γ(n) Γ( n 2 ) 1 n .
Moreover, the only extremizers in (7.5) have the form

f (x) = a (1 + b |x -x 0 | 2 ) n-2 2 , a ∈ R, b > 0, x 0 ∈ R n (7.6)
(sometimes called the Barrenblat profile), see [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF][START_REF] Talenti | Best constant in Sobolev inequality[END_REF][START_REF] Del Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF].

If f ∈ L 2 (R n ) and |∇f | ∈ L 2 (R n ), then , by (7.
3), we would have that f ∈ L p (R n ) with p = 2α > 2n n-2 which contradicts the Sobolev embeddings. Therefore (7.3) cannot be true, so that (7.1) holds only for α ∈ [0, n n-2 ]. As for the value of the best constant c α,n in (7.1) and the form of the extremizers, we need to use again Theorem 6.1 which can however be applied only for n ≤ 5. As in Corollary 6.2, we adopt the notation

M s = R n u s (|x|) dx
for a function u satisfying the non-linear ordinary differential equation

u (t) + n -1 t u (t) + u(t) 2α-1 = u(t), 0 < t < ∞, (7.7)
or (in a different scenario)

u (t) + n -1 t u (t) + u(t) = u(t) 2α-1 , 0 < t < T. (7.8) Corollary 7.2. Let 3 ≤ n ≤ 5. (i) For any 1 < α < n n-2 , we have c α,n = 2n(α -1) α 2α α(2 -n) + n n(α-1)-2 n(α-1) M 2 n 2
where M 2 is defined for the unique positive decreasing solution u(t) to (7.7) on (0, ∞) with u (0) = 0 and lim t→∞ u(t) = 0.

(ii) For any α ∈ [ 1 2 , 1),

c α,n = 4 n(1 -α) α(2 -n) + n 2α α(2 -n) + n 2α n(1-α) M 2 n 2α
where M 2α is defined for the unique positive decreasing solution u(t) to (7.8) with u (0) = 0, u(T ) = u (T ) = 0 and u(t) = 0 for all t ≥ T .

In both cases, the extremizers in (7.1) are densities of the form p(x) = b M u 2 (|bx + c|), x ∈ R n , with b > 0 and c ∈ R n .

For the critical value of α, the picture is more complete, but is different. (i) For n = 3 and n = 4, (7.1) has no extremizers, i.e. there does not exist any density p for which equality holds in (7.1) with the optimal constant.

(ii) For n ≥ 5, the extremizers in (7.1) exist and have the form In particular, c α,n ≥ 2πen -c 0 for all 0 ≤ α ≤ n n-2 with some absolute constant c 0 > 0. To get a similar upper bound, it is sufficient to test (7.1) with α = 0 on some specific probability distributions. In this case, this inequality becomes vol n (supp(p))

p(x) = a (1 + b|x -x 0 | 2 ) n-2 , a, b > 0, x 0 ∈ R n . ( 7
2 n I(X) ≥ c 0,n .

(7.10) Suppose that the random vector X = (X 1 , . . . , X n ) in R n has independent components such that every X k has a common density w(s) = 2 π cos 2 (s), |s| ≤ π 2 . As we already mentioned in Section 4, this one dimensional probability distribution appears as an extremal one in the entropic isoperimetric inequality (1.2) for the parameter α = 0. The random vector X has density p(x) = w(x 1 ) . . . w(x n ), x = (x 1 , . . . , x n ) ∈ R n , so that N 0 (X) = N 0 (X 1 ) = π 2 , I(X) = nI(X 1 ) = 4n. Therefore, from (7.10) we may conclude that c 0,n ≤ 4π 2 n. (7.11) Therefore 4 cα,n = S 2 n from which the explicit value of c α,n follows (recalling Corollary 5.2). Now, in order to analyze the question about the extremizers in (7.1), suppose that we have an equality in it for a fixed (probability) density p on R n . In particular, we should assume that the function f = √ p belongs to W 2 1 (R n ). Rewriting (7.1) in terms of f , we then obtain an equality in (7.11), which is the same as (7.5). As mentioned earlier, this implies that f must be of the form (7.6), thus leading to (7.9). However, whether or not this function p is integrable depends on the dimension. Using polar coordinates, one immediately realizes that

dx (1 + b|x -x 0 | 2 ) n-2
has the same behavior as

Corollary 7 . 3 .

 73 Let n ≥ 3 and α = n n-2 . Then c α,n = 4πn(n -

. 9 ) 7 . 4 .

 974 Remark Recall that c 1,n = 2πen. Using the Stirling formula, it is easy to see that,for α = n n-2 , c α,n ∼ 2πen -2πe (2 + log 2) + O 1 n as n → ∞.

Proof of Corollaries 7 . 2 - 7 . 3 .

 7273 The first corollary is obtained by a straight forward application of Theorem 6.1 withs = 2, r = 2α, θ = n(α -1) 2α , κ n (2, r, s) = (4/c α,n ) n(α-1) 4αwhen 1 < α < n n-2 , and withq = r = 2, s = 2α, θ = n(1 -α) α(2 -n) + n , κ n (2, r, s) = (4/c α,n ) n(1-α) 2[α(2-n)+n]when α ∈ (0, 1). Details are left to the reader.For the second corollary, we first observe that (7.2) can be recast for n ≥ 3 and α = n

∞ 1 1 r

 1 n-3 dr. But, the latter integral converges only if n ≥ 5.

Acknowledgement. Research of the first author was partially supported by the NSF grant DMS-1855575. Research of the second author was partially supported by the grants ANR-15-CE40-0020-03 -LSD -Large Stochastic Dynamics, ANR 11-LBX-0023-01 -Labex MME-DII and Fondation Simone et Cino del Luca in France.


Abdelkader Aguir

The Impact of Central Bank Independence on The Performance of Inflation Targeting Regimes : Emerging Economies

Keywords: Inflation Targeting, Monetary Policy, Central Bank Independence, Credibility, Transparency, Performance, Efficiency and Stability

This paper examines the mac ro economic effects of inflation targeting on inflation in emerging economies. It focuses on both theoretical and empirical analysis of the inflation targeting regime. However, when we differentiate the impact of inflation targeting based on the degree of central bank independence, we find large effects in emerging economies wit h low central bank independence. Our results therefore suggest that central bank independence is not a prerequisite for emerging economies to experience significant declines in inflation following the adoption of inflation targeting. Furthermore, we provide evidence that one channel through which inflation targeting lowers inflation and lowers volatility of inflation more in countries with low central bank independence is the reduction of budget deficits following the adoption of an inflation target.

Introduction

Proceedings of 3rd European Business Research Conference 4 -5 September 2014, Sheraton Roma, Rome, Italy, ISBN: 978-1-922069-59-7 3 -Table 1: Summary of some definitions. [START_REF] Mishkin | Inflation targeting in emerging market countries[END_REF] "The inflation targeting regime has two characteristics: an explicit numerical inflation target by specifying the index, the target level, the tolerance interval, the horizon and t he definition of possible situations which the monetary authorities will change the target. . . Method for Inflation Forecasting is to use all information that could provide an indication of future inflation and implement a procedure prospective (forward looking) in order to manage the driving instrument which will depend on the early assessment compared to the predefined target rate. "

The inflation targeting policyas a new framework for monetary policy analysis which consists of an official announcement from an interval target for one or more horizons. They evok e the uniqueness of the objective: that of price stability. They suggest the explicit announcement of this strategy. In addition, they consider that this policy generates a growth of the degree of communication with the public around the plans and objectives to be implemented.

Inflation targeting is a monetary policy strategy that encompasses five essential elements: (i) A n announcement of a numerical inflation target over the medium term; (ii) an institutional commitment to consider the stability of prices as the overriding objective of monetary policy, which are subordinated the other objectives. (iii) A n information strategy in which several variables are used (not not only monetary aggregates, the exchange rate) determining the implementation of the policy instrument. (iv) The increas e of the degree of transparency via the communication with the public and the market on plans, objectives and decisions of the monetary authorities. (v) The increase in the responsibility of the Central Bank in order to achieve the inflation targets." A structure of monetary policy designed to redress inflation... The Proceedings of 3rd European Business Research Conference 4 -5 September 2014, Sheraton Roma, Rome, Italy, ISBN: 978-1-922069-59-7 4 [START_REF] Capistrán | Does inflation targeting affect the dispersion of inflation expectations?[END_REF] countries pursuing inflation targeting undertake to consider the price stability as their primary objective. They consider inflation as the single nominal anchor on the medium-term."

Defines the inflation targeting policy as a monetary policy strategy aimed at maintaining price stability using all the information available to the Central Bank mainly the prices of financial assets. To examine the effect of monetary instability (via the volatility of inflation on economic growth).

---

Studies

the average growth of output as a function of aggregate inflation volatility.

The results were in favor of a negative effect.

-------

2 -13

 2 Faculty of Economic Sciences and Management, University of Sousse, Research Unit: MO2FID, Tunisia / Institute of Political Studies (IEP) Research Laboratory Public Policy, Political Action, Territories (PACTE) Grenoble, France. 2 Fourteen of the 30 OE CD countries have explicit inflation ta rgets. However, twelve of thes e countries are in the EMU and operate under a single central banks in the OECD target inflation.Proceedings of 3rd European Business Research Conference 4 -5 September 2014, Sheraton Roma, Rome, Italy, ISBN: 978-1-922069-59[START_REF] Martin | Term structure of interest rates, monetary rule and identification of shocks activity[END_REF], " Term structure of interest rates, monetary rule and identificat ion of shocks activity " CREREG, University Rennes 1

  [And] the absence of an explicit intermediate target such as monetary aggregate target or exchange rate targeting." "Inflation targeting is based on the definition of an explicit inflation target,..." [the definition] clear and unambiguous indications which constitute the overarching objectives leading to the stability of inflation [...].

Table 3

 3 Descriptive Statistics for Macroeconomic VariablesMean and (standard deviation) for all variables Proceedings of 3rd European Business Research Conference 4 -5 September 2014, Sheraton Roma, Rome, Italy, ISBN: 978-1-922069-59-7

	Variable	IT Sample	Non-IT Sample
		(456 obs.)	(577 obs.)
	GDP growth (%)	1.11	1.00
		(5.93)	(7.84)
	GDP gap (%)	-0.11	0.29
		(3.86)	(4.62)
	Inflation (%)	5.10	9.80
		(4.21)	(9.15)
	Interest rate (%)	7.92	12.60
		(6.09)	(10.25)
	Real exchange rate change	2.50	-0.49
	(%)	(5.76)	(13.27)
	Foreign reserve change (%)	3.25	4.66
		(7.89)	(22.82)

4 Level, volatility persistence

5 Aguir Abdelkader (2014) « Inflation Targeting: An Alternative to Monetary Policy » International Journal of Economics and Finance Vol

6, No 7 Proceedings of 3rd European Business Research Conference 4 -5 September 2014, Sheraton Roma, Rome, Italy, ISBN: 978-1-922069-59-7 6


José - María

Valid Andrés Castán

Mwatati Mwalukuku

Antonio J Riquelme

Johan Liotier

Quentin Huaulmé

Juan A Anta

Pascale Maldivi

Renaud Demadrille

Photochromic spiro-indoline naphthoxazines and naphthopyrans in dye-sensitized solar cells

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Photochromic compounds are a specific class of molecules that can undergo photo-induced transformations between isomers with different optical properties. 1 During the last decades, many families of organic molecules with photochromic properties have been identified and studied in detail. 2,3 Most of them involve monomolecular reaction mechanisms such as photocyclizations, isomerization reactions and intramolecular proton transfer to carry out their photoswitch. Among photochromic molecules, diphenyl-naphthopyrans, 4 spiroindoline naphthopyrans, 5 spiro-indoline naphthooxazines, 6 diarylethenes 7 or azobenzenes 8 stand out. Due to their appealing properties, they can be employed in a wide variety of applications, 9 ranging from data storage, 10 actuators, 11 sensing 12 and ion detection, 13 optical lenses, 14 holographic recording devices 15,16 or bioimaging, 17,18 to name a few. Despite decades of research on photochromes, their use in photovoltaics is very recent. 19 However, these compounds could be particularly interesting for the design of semi-transparent devices because their reversible transformation between the two forms with different absorption would allow obtaining solar cells whose transparency could be modulated according to the light intensity. 20 Such a property would be particularly appreciable in Building Integrated Photovoltaics (BIPV). 21,22 Nevertheless, very few photochromes have been tested in photovoltaic devices. In 2015, a bis-thienylethene derivative was first used in dyesensitized solar cells (DSSCs). 23 For this type of molecules, the interconversion between the different photo-isomers is only possible by alternating ultraviolet and visible light irradiations. In addition, the colored isomer led to a poorer power conversion efficiency (PCE). Photochromic dyes from the spirobenzopyran and spirobenzo and naphthoxazine families were also tested in DSSCs. 24 Here again the isomerization process required intense ultraviolet irradiation and the molecules did not lead to the reversible optical properties expected within the device. Besides the fact that they were not stable, their photovoltaic performances were extremely low with PCEs below 0.1%. The low performance of these molecules can be explained by an unsuitable molecular design for this application. Indeed, in most of these systems, the anchor function was not conjugated with the photochromic core, which resulted in poor charge injection into the photoelectrode.

We have recently demonstrated that naphthopyran-based derivatives, when properly designed, show a high potential in semitransparent DSSCs. 20 Push-pull photochromic naphthopyrans can sensitize efficiently TiO2-based mesoporous electrodes while keeping their reversible photochromic behavior after integration in complete devices. The resulting smart solar cells can generate electricity and self-adapt themselves depending on the light intensity in order to be more transparent under low light intensity. In this work, we propose a new strategy for designing spiroindoline naphthoxazine (SINO) and spiro-indoline naphthopyran (NIPS) dyes with the goal to explore their potential for solar cells. These two families of photochromic dyes are well known for their excellent switching kinetics and good resistance to photodegradation. 25 As with diphenylnaphthopyrans, the mechanism of the photochromic reaction of SINO and NIPS involves the cleavage and reformation of the carbon-oxygen single bond of the oxazine or pyran rings, respectively, as shown in Scheme 1. In the previous work on this type of dyes, the spiro-indoline benzopyran (BIPS) sensitizer carried the carboxylic anchor away from the conjugated system and its SINO counterpart did not even have a suitable anchoring function, leading to extremely poor performances in solar cells. 24 For this reason, we designed two SINO and two NIPS photosensitizers following a donor--acceptor (D--A) structure (Scheme 1). Then, we investigated their optical and photochromic properties with the goal to establish new structure-properties relationships for this class of molecules. Then we used these molecules as photosensitizers in DSSCs to compare their photovoltaic performances. Scheme 2. Synthetic pathway followed to obtain precursors 1, 5 and 6. DIPEA: N,N-diisopropylethylamine; Tf: trifluoromethylsulfonyl, dba: benzylideneacetone. Scheme 3. Synthetic pathway used to obtain the new dyes SINO-1, SINO-2, NIPS-1 and NIPS-2. dppf: (diphenylphosphino)ferrocene.

Design and synthesis

Upon irradiation, SINO and NIPS are known to lead to the formation of merocyanine forms (MC) with a cation localized on the indoline nitrogen after opening the oxazine or pyran ring of their closed forms (CF). 26 Therefore, it seems reasonable that the introduction of the anchoring function must be on the naphthalene part of the molecule to favor the injection into the oxide. To verify this hypothesis and to identify the position on which this function must be introduced, density-functional theory (DFT) calculations have been performed on the SINO reference (Figure S24). 27 Three positions easily accessible by synthesis have been selected for this modeling study, i.e. the 8', 5, and 6' positions for SINO8', SINO5 and SINO6' respectively. From the calculated energies, we noticed that all the molecules have highest/lowest occupied molecular orbitals (HOMO and LUMO respectively) energy levels correctly positioned with respect to the conduction band of the oxide and with respect to the potential of the redox mediator. The HOMO levels are found between -5.3 and -5.7 eV and the LUMO are lying between -2.9 and -3.2 eV with respect to the vacuum level. In other words, the three molecules can sensitize the oxide and be regenerated by the iodine-based electrolyte, both in their initial and isomerized state. The determination of the HOMO and LUMO orbital distributions is more useful. Indeed, the LUMO of the open forms of SINO8' and SINO5 are delocalized on the whole molecule so they will not be suitable for a good injection, while the LUMO of SINO6' is a little more localized on the acceptor part. Therefore, it could potentially lead to a more efficient injection of electrons into the oxide. For this reason, the anchoring functions, either a p-phenyl-carboxylic acid or a pphenyl-cyanoacrylic acid, were introduced in the 6' position. In addition, to reinforce the electron-donating character of the indoline unit, a diphenylamine unit substituted by two octyl chains was introduced in the 5 position. Therefore, we carried out the synthesis of four derivatives: SINO-1, SINO-2, NIPS-1 and NIPS-2, whose structures and synthesis are depicted in Schemes 2-3. In the case of the SINO dyes, the 4-bromo-1-nitrosonaphthalen-2-ol precursor 1 was obtained from 4-bromonaphthalen-2-ol via direct nitrosilation in acid medium 28 and it was directly used in the subsequent step without further purification. In the case of 4-formyl-3hydroxynaphthalen-1-yl trifluoromethanesulfonate required for the synthesis of the NIPS derivatives, a less straightforward pathway was followed. First, naphthalene-1,3-diol was methylated in position 3 prior to introduce the triflate group in position 1. 29 The carbonyl group was then introduced in position 4 through a Rieche formylation and eventually the alcohol was deprotected in the presence of boron tribromide to obtain the naphthol 5. On the other hand, to prepare the donating block 6, the Fischer's base was first brominated according to previously reported conditions 30 and then the diphenylamine moiety was introduced through a Buchwald-Hartwig amination. The condensations between the Fischer's bases and the naphthol derivatives were carried out in ethanol or toluene for the NIPS and SINO derivatives respectively to obtain intermediates 7-10 (Scheme 3). Finally, and in view of the lack of stability of the photochromes in the conditions used in the Knoevenagel condensations, the anchoring functions were directly introduced in the Suzuki coupling together with the phenyl spacer, for either the benzoic or the 2-cyano-3phenylacrylic acceptors. This procedure, which is a deviation from the conventional synthetic way of introducing the anchoring functions, allowed obtaining the target molecules in a straightforward manner with overall yields between 5% and 18%.

Optical properties under different stimuli

Photochromic properties

Despite the large number of studies devoted to SINO and NIPS molecules, structure-photochromic property relationships are only available for a restricted number of compounds and only few reports are dealing with push-pull effects within these families. 31 Since toluene, tetrahydrofuran (THF) and dimethylformamide (DMF) are commonly used solvents to characterize the photochromic behavior of photochromic dyes, we decided to use them in this study. (Figures S25-S27, Table 1). As an example, we show the spectra obtained in DMF in Figure 1. In every solvent, the absorption patterns of the colorless forms exhibit localized -* transitions in the UV range, corresponding to the two orthogonal halves of each molecule. 32 In good agreement with different examples in the literature, both SINO derivatives exhibit a photochromic behavior upon irradiation. In these conditions, the ICT band of the open-form isomers emerges above 600 nm, slightly redshifted in the case of SINO-2. This effect is consistent with the higher push-pull effect induced by the two substituents. Compared to SINO compounds, their NIPS counterparts exhibited a limited photochromic behavior, a phenomenon that has been already reported for this type of derivatives. 33 However, a slight band at ca. 570 nm could be observed under illumination in the case of NIPS-1, in both toluene and DMF, but surprisingly this compound was not photochromic when studied in THF. To the contrary, NIPS-2 does not show photochromic properties in any of the solvents. For SINO-1, SINO-2 and NIPS-1, it is noteworthy that the maxima of the absorption bands do not change much from one solvent to the other for either the CF or MC forms. It is also important to note that the three photochromic dyes can be colored with exposure to polychromatic light, an important aspect for the targeted application. Then, we investigated the thermal discoloration kinetics of the new compounds. Once the photostationary state (PSS) was reached under illumination (after circa 60 s), the irradiation source was turned off and the obtained discoloration curves were then modelled through the following equations of first and second order respectively depending on the obtained experimental results:

𝐴(𝑡) = 𝑎 1 𝑒 -𝑘 1 𝑡 + 𝐴 ∞ ; 𝐴(𝑡) = 𝑎 1 𝑒 -𝑘 1 𝑡 + 𝑎 2 𝑒 -𝑘 2 𝑡 + 𝐴 ∞

Wavelength (nm)

In dark Under light where, A(t) is the absorbance of the solution, kn is the thermal discoloration kinetic constant (in s -1 ), an is the amplitude of the kinetics of this process, and A∞ is the residual absorbance. The photochromic SINO derivatives showed extremely fast bleaching kinetics and the discoloration was even faster for NIPS-1. In all cases, no persistent colored isomers are formed after exposure to light (Table 2, Figure 2 and Figure S28) and the solutions are fully decolored in less than 10 s for the SINO dyes and less than 1s for the photochromic NIPS dye. However, they are more than 100 times faster compared to those of the best performing naphthopyran photochromic dyes in DSSCs, which make these molecules particularly appealing for application in semitransparent solar cells with dynamic optical properties. These values are in good accordance with the ones reported for unsubstituted SINO and NIPS. 25 It is also noticeable that in every case a mono-exponential decay was obtained except for SINO-2, which exhibits the slowest kinetics and a bi-exponential decay in toluene likely due to the formation of different isomers.

Acidochromic properties in the dark

The compounds synthesized in this study are designed to be integrated in a DSSC device configuration with a liquid electrolyte, which usually contains many additives, I3 -/I -couple as redox mediator and electrolytes often show a pH below 7. However, it is well known that SINO and NIPS are acid-sensitive and they usually possess acidochromic properties. [34][35][36] According to the literature, this type of derivatives are rapidly converted into their respective protonated open forms (MCH) in the presence of strong acids in an E conformation (Scheme 4). This is why it seemed important to address this aspect. We found that the four new derivatives revealed acidochromic properties in absence of light. In order to study the interaction between protons and our new SINO and NIPS molecules, a titration with HCl was carried out in THF in the dark (Figure 3). It can be observed that the increase in the absorption of the band related to the MCH form when HCl is added. In addition, a sole isosbestic point was monitored in every case, suggesting that the protonation gives rise to a single compound, namely the MCH form. 37,38 It appears from the spectra presented in Figure 3 that the SINO compounds are less sensitive to acidic conditions compared to their NIPS analogs, as only a small fraction of the molecules undergoes the protonation reaction.

For SINO-1, SINO-2 and NIPS-1, the absorption band of the MCH form in the visible is blue-shifted compared to the MC. The shifts varied between 15 and 70 nm for the SINO derivatives and from 70 to 90 nm for NIPS-1 depending on the solvent. In contrast to the CF and MC forms, the maxima of the bands associated with the MCH form are shifted depending on the solvent used due to its charged nature, 39 although it does not follow a specific trend. This indicates that only MCH are subjected to solvatochromism. In addition, the acidochromism was found to greatly depend on the strength of the acid used. This observation confirms the results reported for the BIPS family. 40 The acidochromic effect is observed when strong acids that can protonate the MC form are used. For instance, in the case of NIPS-1, the colored form was observed when acids with pKa below 0.5 were employed (i.e. nitric, sulfuric, hydrochloric and perchloric acids) but not in the case of "weaker" acids (i.e. trifluoroacetic, phosphoric or acetic acids) (Figure S29). Our experiments show that the nature of the acid plays a critical role on the protonation and the formation of the MCH. Further experiments are required to elucidate the exact role of the counter ions on the acidochromic properties of these dyes.

Acidochromic properties under illumination

Under the conditions of use that we envisaged for these compounds in solar cells, the dyes will not only be exposed to a low pH environment but also to light. This is why it seemed interesting to investigate their photochromic and acidochromic behaviors by combining the two stimuli. Therefore, the compounds in their protonated form in solution were therefore exposed to light. First, it should be noted that the SINO derivatives still exhibited a positive photochromism, i.e. a coloration upon irradiation, but the discoloration kinetic constants were found to be lower in the presence of acid (Figure S30). This could be explained by the fact that a large proportion of these compounds still exists in closed form even in the presence of acid as suggested by the weak intensity of the band corresponding to MCH in Figure 3. More interestingly, in stark contrast to the SINO compounds, the MCH forms of the NIPS exhibit a negative photochromic behavior. 41 According to the literature of spiro-indoline benzopyrans, this process implies an E-Z isomerization followed by the deprotonation and the consequent ring closure. 42 Although the effect is not intense in the case of NIPS-2, NIPS-1 reverts to its initial state under illumination and it is able to reopen quite rapidly in the dark (with a k of circa 2.6•10-2 s-1) (Figure 4). The high reversibility of this process in THF has to be highlighted, as NIPS-1 bleaches almost quantitatively under illumination even reaching the same level of coloration in the dark showing no sign of photodegradation. Though this effect has been observed in the case of some spiro-indoline benzopyran derivatives, 43 to the best of our knowledge, it has not been reported for NIPS compounds. It should also be noted that this negative photochromism observed in solution when acidic conditions and light are combined could be modulated by changing the solvent and the nature of the acid (see Figure 4). To conclude this section, it seems that the molecules once incorporated in a device in the presence of a liquid electrolyte, will be able to exist in multiple forms: either in a closed form, an open form of MC type but also in an open MCH form after protonation. For this reason, it is important to verify if the different possible forms of these molecules present energy levels appropriately positioned for an efficient sensitization of the oxide and an efficient regeneration by the redox mediator that we have chosen, namely the I3 -/I -couple. 44 The energy levels of the dyes are discussed in the next section.

Energy levels and theoretical calculations

With the goal to unravel how the optoelectronic properties of the dyes change by swapping from the closed to the opened forms, and how the substitution and the nature of the photochromic unit influence the energy levels of the frontier orbitals, we performed cyclic voltammetry experiments in dichloromethane in the dark (Figure S31). Contrarily to diphenyl-naphthopyran dyes, 20 the open-form isomers of SINO and NIPS compounds cannot be accumulated in sufficient amount in solution to allow the determination of the HOMO and LUMO energy levels for these isomers, so only the HOMO of the CF could be experimentally characterized. Therefore, the energy levels of the frontier orbitals of the opened forms were evaluated by ground-state density-functional theory calculations at the B3LYP/TZ2P level of theory (Figure 5 and supporting information for details). However, it should be taken into account that the electronic landscape of the dyes in real device configuration may vary. First, the HOMO energies obtained for the CF by means of theoretical calculations are in good accordance with the one obtained by cyclic voltammetry (all comprised between -4.9 eV and -5.6 eV), thus validating the theoretical approach. Then, when we compare equivalent forms for each dye, the theoretical trend is the same within the Apart from the energy levels, the spatial distribution of the frontier orbitals is also a critical parameter because the efficiency of the injection and regeneration processes will strongly depend on it. Even though in all the CF the HOMO is localized on the donor and the LUMO on the acceptor showing a suitable electron distribution along the molecules, it is not the case for all the open forms. In the case of NIPS-1 and SINO-1, the HOMO and LUMO distributions of either the MC or MCH lie basically in the same region, i.e. the photochromic core, thus probably hampering the injection of electrons into the conduction band of the TiO2 electrode. However, though for the MC form of NIPS-2 and SINO-2 there is clearly a suitable spatial distribution between the donor and the acceptor for a charge transfer effect, we observe again the LUMO contribution in the central part of the molecule for their MCH forms, which may hinder the electron injection as well.

Photovoltaic properties

The photovoltaic performances of the SINO and NIPS dyes were evaluated in opaque DSSC configurations. We recorded currentvoltage characteristics in the dark and at different time intervals of exposure to light, and the short-circuit current density (Jsc), open-circuit voltage (Voc), fill factor (FF) and PCE were determined. Two types of electrolyte were employed for the fabrication of the solar cells, a homemade electrolyte 20 that was specifically designed for photochromic naphthopyran dyes, and a commercial one known as Iodolyte (Solaronix). The main difference of our electrolyte in comparison to the latter is a 3fold increase in I2 and the lack of some additives, such as guanidinium thiocyanate and tert-butylpyridine, thus modifying the final pH. The results are reported in Figure 6 and Table 3. The TiO2 mesoporous electrodes were sensitized with the dyes in the presence of chenodeoxycholic acid (CDCA) to avoid aggregation (see ESI for details). It should be noted that during this step a very light coloration of the dyes occurs on the surface of the electrode (Figures S32-S33). This faint coloration is attributed to the formation of MCH due to residual traces of acid in the pores of TiO2 after the TiCl4 treatment. The huge different proton affinity between NIPS and SINO observed in solution is less appreciable once the molecules are grafted onto the semiconductor. Then, after sealing the electrodes, the electrolyte was introduced in the devices. Note that when our homemade electrolyte is introduced and reaches the electrode, all four dyes switch instantaneously to the open MCH form and appear to be locked in this conformation (Figure S33). This coloration is a clear manifestation of the acidochromic behavior of the dyes and the stabilization of the MCH form due to the acidic pH of this electrolyte. Indeed, we could determine a pH between 1 and 2 for this electrolyte. This acidochromic phenomenon, which consequently induces the loss of photochromic behavior, has not been reported in previous reports on spiropyran and spiroxazine-based DSSCs, even though a comparable iodine-based redox mediator was used. 24 On the contrary to what was observed in solution, the negative photochromism of NIPS derivatives was not observed once the dyes were grafted on the electrodes and embedded in the devices. This is confirmed by the absence of decrease in Jsc after several successive J-V measurements. Even though, the current density for all dyes appears limited with this electrolyte (< 3.5 mA cm-2). Referring to theoretical calculations in previous section, in the SINO series the formation of the MCH strongly decreases the ability to inject electrons into the conduction band of TiO2. Indeed, the LUMO level is lowered, thus reducing the driving force, and its spatial localization is quite far from the anchor group, not allowing a correct sensitization of the TiO2. This explains the poorer performance of SINO compared to their NIPS counterparts. The same problems appear with the NIPS compounds, but in that case, the LUMO energy level of the MCH is higher giving rise to a slightly higher driving force towards the injection. In contrast, when the commercial Iodolyte was used, the level of staining was considerably lower (Figure 6). This is explained by its pH around 7, which allows the closed form to be present in the absence of illumination. In this case, the photochromic behavior is preserved for the molecules that were already demonstrated photochromic in solution, i.e. the two SINO derivatives and NIPS-1. In addition, Iodolyte contains additives such as tert-butylpyridine that is known to shift the TiO2 conduction band upwards thus increasing the Voc. However, in spite of all that, the overall performances of the three photochromes remain in the same range. According to theoretical calculations, the spatial distribution of the LUMO that delocalizes to the core of the molecule on the MC forms may explain this result for SINO-1 and NIPS-1. Surprisingly, in the case of NIPS-2, despite its lack of photochromic behavior and thus an absence of open MC form absorbing in the visible, a better current density was recorded contrarily to the other sensitizers. NIPS-2 demonstrates a Jsc higher than 5 mA• cm -2 and a Voc over 0.6 V, thus leading to the highest PCE ever reported for this family of molecules: 2.73%. In addition, these devices are relatively stable as they are able to maintain their Voc after ca. 5000 h under ISOS-D1 ageing test 45 though the Jsc is substantially reduced, resulting in an efficiency loss of 47% (Figure S34). The lack of coloration of this device may indicate that most of NIPS-2 stays in its CF. To confirm the contribution of each form in NIPS-2/Iodolyte-based devices, the absorption patterns of 2 m transparent devices made of both electrolytes were recorded (Figure 7). It is noteworthy that when the homebased electrolyte is used, an intense absorption band appears in the visible corresponding to the formation of MCH, whereas with Iodolyte the device shows the typical yellow of the Iodine based electrolyte and the absorption spectrum remains flat in the visible. In addition, as it can be observed in the incident photon to current efficiency (IPCE) spectrum of the 13 m device composed of the homemade electrolyte (Figure S35), in spite of the great coloration, the injection of electrons related to the MCH form is negligible, in good agreement with the results provided by the theoretical calculations. This also demonstrates that the main contributor to the photogeneration is indeed the CF, favored by the good spatial distribution of its LUMO orbital. Then, to understand the electrolyte-driven behavior of the devices presented in Figure 6 and Table 3, an electrochemical impedance investigation was carried out on the DSSCs made of the best performing dye, NIPS-2, and both electrolyte compositions.

The main arc of the impedance spectra (see Figure S36 a-b) was fit to a single -RC-equivalent circuit element where the resistance is considered the charge transfer resistance (dark) or recombination resistance (RCT) and the capacitance is the chemical capacitance (Cµ). Both parameters are known to show an exponential voltage dependence following equations S1-S2 (Figure S36 c-f). 46,47 The extracted α and β for the NIPS-2 DSSC using Iodolyte and the homebased electrolyte at dark conditions and under illumination are collected in Table S1. Typical values for α and β values normally reported for DSSCs are in the range of 0.15 -0.35 for α and 0.5 -0.8 for β. [48][49][50][51] While the α parameter, which shows the average depth of the traps, remained within the range typical for a DSSC configuration irrespective of the redox couple used,  varied depending on the electrolyte. When Iodolyte was employed, this parameter changed from 0.56 to 0.73 between the irradiation conditions but still within the DSSC typical values. However, when our homebased electrolyte was used in turn,  changed from 0.75 in the dark to 1.04 under operating conditions, which suggests a possible change in the dominant recombination pathway, which normally in DSSCs correspond to triiodide reduction. 52,53 Whatever the redox couple, no TiO2 conducting band shift was induced at irradiation conditions (Figure S36). However, as it can be observed in Figure S37, the change of the electrolyte induces a bandshift of +226 mV and +220 mV with Iodolyte under dark and white light irradiations, respectively, explaining the higher photovoltage observed in the reported JV curves.The Nyquists plots under irradiation, in which the size of the real part of the recombination arc is approximately equal to the recombination resistance, were compared applying a voltage correction to account for the bandshift, ensuring that both measurements were done at a similar electron concentration at the quasi-Fermi level (Figure S38). The diameter is larger in homebased electrolyte-based cells despite the high I2 concentration and the lower Voc therein. Certainly, the acidic homemade electrolyte gives rise to the protonated merocyanine form facilitating the interaction of the positively charged center with I3 -hence lowering its local concentration at the photoanode compared to Iodolyte. This can also explain the change observed in the β values compared to the range usually reported for DSSCs. Eventually, in an attempt to obtain the most performing semitransparent device with the couple NIPS-2/Iodolyte, DSSCs were fabricated using a transparent 13 m TiO2 anode (Figure 9). On such thick electrodes, the coloration is more pronounced due to the formation of the MCH form with the remaining traces of acid, thus leading to slightly greenish electrodes. Despite of that, a PCE of 2.18% was then recorded while keeping the AVT at 52%, mostly absorbing below 500 nm, demonstrating once again that the majority of the molecules is in its CF as the transmittance at longer wavelengths is always kept above 50%. This corresponds to a light utilization efficiency (LUE) of 1.13.

Then, to determine the aesthetic quality of the semitransparent devices, the Commission Internationale de l'Eclairage (CIE) color coordinates were determined, as well as the color rendering index (CRI), which revealed an acceptable value of 80.4% (Figure S39). 54,55 Therefore, despite its lack of photochromic behavior, this result shows that this type of dyes can be used in semi-transparent DSSCs. Eventually, though modest, they remain as the best results for SINO and NIPS derivatives in the literature.

Conclusion

In this study, we investigated the potential of two families of photochromic dyes, i.e. spiro-indoline naphthopyran and spiroindoline naphthoxazine for their use in dye-sensitized solar cells. We proposed an original molecular design to obtain photochromic molecules that can sensitize TiO2 and be regenerated by an iodine-based electrolyte. We studied the photochromic and acidochromic behavior of these compounds. We highlight a positive photochromism in solution for only three of the compounds, photochromism which is characterized by extremely fast thermal discoloration kinetics. On the other hand, all the compounds have an acidochromic behavior and lead to the formation of protonated merocyanines absorbing in the visible range under acidic conditions. When the compounds are studied in acidic conditions under illumination, they show different behaviors. The spiro-indoline naphthoxazines keep a positive photochromism while the spiro-indoline naphthopyrans show a negative photochromism. These studies allow us to rationalize the behavior of these molecules when they are incorporated in solar cells. We demonstrate that the choice of the electrolyte is crucial as it drives the optical behavior of these compounds. The use of an acidic electrolyte suppresses the photochromic behavior in device whereas with a neutral pH electrolyte, the photochromic behavior is preserved. Although the photochromic behavior is not preserved in all devices with these dyes, these results show that they can be used in DSSCs. The results remain modest with a maximum PCE of 2.7%, but they are significantly improved compared to previous reports on spiropyran and spiroxazine derivatives in the literature. Eventually, the best performing dye, NIPS-2, despite its lack of photochromism, allowed obtaining a PCE of 2.7 % in opaque solar cells and 2.2% in semitransparent ones while showing an AVT at 52%. This study is the first to investigate in details the complex relationships between photochromic, acidochromic and photovoltaic properties for these two families of dyes. The structure-property relationships that we establish in this study will undoubtedly be useful for the development of new photochromic compounds with optimized optoelectronic properties for applications in various fields, including photovoltaics.

Scheme 1 .

 1 Scheme 1. General structure of the SINO (X=N) spiro[indoline-2,3'-[3H]naphth[2,1b][l,4]oxazine and NIPS (X=CH) spiro[indoline-2,3'-[3H]naphtho[2,1-b]pyran derivatives synthesized in the present work as well as the interconversion reaction between their close (CF) and open quinoidal and merocyanine forms (MC).

Figure 2 .

 2 Figure 2. Normalized discoloration curves of SINO-1, SINO-2 and NIPS-1, example given in DMF in 2•10 -5 M solutions at 25 °C in the dark after irradiation 60 s with a 200-600 nm/200 W xenon lamp.

Figure 1 .

 1 Figure 1. UV-vis absorption spectra of a) SINO-1, b) SINO-2, c) NIPS-1 and d) NIPS-2 in 2•10 -5 M DMF solutions at 25 °C in the dark and under light using a 200-600 nm/200 W xenon lamp.

Scheme 4 .

 4 Scheme 4. Interconversion between the close (CF) and open protonated forms (MCH) of SINO (X = N) and NIPS (X = CH) derivatives in the presence of strong acids.

Figure 3 .

 3 Figure 3. Absorption titration of a) SINO-1, c) NIPS-1 (5•10 -5 M), b) SINO-2 and d) NIPS-2 (2.5•10 -5 M) with increments of 70 and 140 eq. of HCl in THF solutions in the dark at 25 °C.

Figure 4 .

 4 Figure 4. Coloration kinetics of NIPS-1 (2•10 -5 M in the dark at 25 °C after illuminating with a 200-600 nm/200 W xenon lamp) a) after the addition of 1 drop of different acids in THF and b) in THF, DMF and toluene after the addition of 1 drop of HCl

NIPS- 1 /

 1 SINO-1 and NIPS-2/SINO-2 pairs. The results indicate that the delocalisation of the orbitals on the backbone of the molecule and the energy levels positions are mostly influenced by the modification of the electron-donor and electronacceptor groups, and not by the nature of the photochromic unit. First, we observe that for all the compounds the band gap is reduced going from the close to the open forms, being the levels of the MCH forms substantially downshifted. As for diphenyl naphthopyran dyes, the LUMO energy levels of the open-form isomers are shifted towards more negative values. This leads, in the case of SINO derivatives, to LUMO energy levels of MCH close to the one of the TiO2 conducting band, which could end up in a reduced driving force for the photoinduced injection of electrons from the excited state of the dyes. For the NIPS family, the shift of the LUMO energy levels for the MCH forms is less problematic since a driving force of circa 0.3 eV is conserved. Comparing SINO-1 to SINO-2 and NIPS-1 to NIPS-2, it appears that the stronger electron withdrawing character of the cyano-acrylic acid function shifts the LUMO energy levels downward except for the MCH forms. It is also clear that the introduction of a diphenylamine donor group on the indoline part has a strong impact on the HOMO level of the molecules that are shifted towards more positive values.

Figure 5 .

 5 Figure 5. DFT-calculated energy levels (versus the vacuum level) of the frontier orbitals of the four new dyes and their spatial localizati ons in closed (CF), merocyanine (MC) and protonated merocyanine forms (MCH) (full lines). Experimental values of the CF obtained by CV are shown in dashed li nes. The potential of the I3 -/I -pair (ca. -4.95 eV) and the TiO2 conduction band (ca. -4.10 eV) are shown in yellow and grey dashed lines respectively.

Figure 6 .

 6 Figure 6. J-V curves of the 13 m DSSCs using a) SINO-1, b) SINO-2, c) NIPS-1, and d) NIPS-2 as sensitizers under standard irradiation conditions AM 1.5 G, 1000 W m -2 ; 25 °C (active area= 0.36 cm 2 ) using our homemade electrolyte (black) and Iodolyte (red) and the pictures of the respective devices in the absence of illu mination.

Table 3 .

 3 Photovoltaic parameters of the DSSCs made of SINO-1, NIPS-1, SINO-2 and NIPS-2 under standard irradiation conditions AM 1.5 G, 1000 W m -2 , 25 °C (active area: 0.36 cm 2 ). Average values in brackets over minimum two cells.

Figure 7 .

 7 Figure 7. UV-Vis spectra and pictures of the respective devices of NIPS-2-based DSSCs on 2 m transparent TiO2 electrodes using Iodolyte and our homebased electrolyte in the absence of illumination.

Figure 8 .

 8 Figure 8. Charge transfer resistances comparisons at the same quasi-Fermi level of 13 m DSSCs made of NIPS-2 and both electrolytes a) in the dark and b) under operating conditions.

Figure 9 .

 9 Figure 9. a) J-V curve of the 13 m NIPS-2/Iodolyte-based DSSCs under standard irradiation conditions AM 1.5 G, 1000 W m -2 ; 25 °C (active area= 0.36 cm 2 ) and b) respective transmittance spectrum together with a picture of the actual device(right).

Table 1 .

 1 Optical properties of the four new compounds in 2•10 -5 M solutions in toluene, THF and DMF at 25 °C. The MC forms were generated by irradiation with a 200-600 nm/200 W xenon lamp and the MCH forms by addition of HCl.

	Dye	k (s -1 ) Toluene	k (s -1 ) THF	k (s -1 ) DMF
	SINO-1	0.59	1.2	0.77
	SINO-2	0.58; 3.8•10 -2 *	0.83	0.51
	NIPS-1	28	-	8.1

Table 2 .

 2 . Kinetic discoloration constants of SINO-1, SINO-2 and NIPS-1 in 2•10 -5 M solutions in toluene, THF and DMF at 25 °C from their MC form. *k2 constant of the bi-exponential decay.

Acknowledgements

R. D. acknowledges ANR for funding through the ODYCE project (grant agreement number ANR-14-OHRI-0003-01). R. D., V.M.M., and J.M.A.C. acknowledge the European Research Council (ERC) for funding. This work was funded under the European Union's Horizon 2020 research and innovation program (grant agreement number 832606; project PISCO). J. L. acknowledges CEA for funding through a CFR PhD grant. J.A.A acknowledges the Ministerio de Ciencia e Innovación of Spain, Agencia Estatal de Investigación (AEI) and EU (FEDER) under grants PID2019-110430GB-C22 and PCI2019-111839-2 (SCALEUP) and Junta de Andalucia under grant SOLARFORCE (UPO-1259175). A.J.R. thanks the Spanish Ministry of Education, Culture and Sports for its supports via a PhD grant (FPU2017-03684). Dr. D. Joly is acknowledged for preliminary work on the synthesis of the compounds.

Author Contributions

Q.H., J.L. and JM.AC. synthesized and characterized the dyes. P.M. performed the DFT calculations. V.M.M., and JM.AC. fabricated and characterized the solar cells and V.M.M performed the EIS measurements . A.J.R. and J.A.A. performed the IPCE measurements. R.D. designed the materials and experiments. R.D. and JM.AC. treated the data and wrote the manuscript, with contributions from all authors. All authors have given approval to the final version of the manuscript.

Conflicts of interest

There are no conflicts to declare.