Pascal Lefèvre

Daniel Li

Hervé Queffélec

Luis Rodríguez-Piazza

Compact composition operators on the Dirichlet space and capacity of sets of contact points

Keywords: MSC 2010. Primary: 47B33 -Secondary: 28A12, 30C85, 31A15, 46E20, 46E22, 47B10 Bergman space -Bergman-Orlicz space -composition operator -Dirichlet space -Hardy space -Hardy-Orlicz space -logarithmic capacity -Schatten classes

We prove that for every compact set K ⊆ ∂D of logarithmic capacity Cap K = 0, there exists a Schur function ϕ both in the disk algebra A(D) and in the Dirichlet space D * such that the composition operator C ϕ is in all Schatten classes S p (D * ), p > 0, and for which K = {e it ; |ϕ(e it )| = 1} = {e it ; ϕ(e it ) = 1}. We show that for every bounded composition operator C ϕ on D * and every ξ ∈ ∂D, the logarithmic capacity of {e it ; ϕ(e it ) = ξ} is 0. We show that every compact composition operator C ϕ on D * is compact on the Bergman-Orlicz space B Ψ2 and on the Hardy-Orlicz space H Ψ2 ; in particular, C ϕ is in every Schatten class S p , p > 0, both on the Hardy space H 2 and on the Bergman space B 2 . On the other hand, there exists a Schur function ϕ such that C ϕ is compact on H Ψ2 , but which is not even bounded on D * . We prove that for every p > 0, there exists a symbol ϕ such that C ϕ ∈ S p (D * ), but C ϕ / ∈ S q (D * ) for any q < p, that there exists another symbol ϕ such that C ϕ ∈ S q (D * ) for every q < p, but C ϕ / ∈ S p (D * ). Also, there exists a Schur function ϕ such that C ϕ is compact on D * , but in no Schatten class S p (D * ).

1 Introduction, notation and background

Introduction

Recall that a Schur function is an analytic self-map of the open unit disk D. Every Schur function ϕ generates a bounded composition operator C ϕ on the Hardy space H 2 , given by C ϕ (f ) = f • ϕ. Let us also introduce the set E ϕ of contact points of the symbol with the unit circle (equipped with its normalized Haar measure m), namely:

(1.1)

E ϕ = {e it ; |ϕ * (e it )| = 1}.

In terms of E ϕ , a well-known necessary condition for compactness of C ϕ on H 2 is that m(E ϕ ) = 0. This set E ϕ is otherwise more or less arbitrary. Indeed, it was proved in [START_REF] Gallardo-Gutiérrez | Hausdorff measures, capacities and compact composition operators[END_REF] that there exist compact composition operators C ϕ on H 2 such that the Hausdorff dimension of E ϕ is 1. This was generalized in [START_REF] El-Fallah | Level sets and composition operators on the Dirichlet space[END_REF]: for every Lebesgue-negligible compact set K of the unit circle T, there is a Hilbert-Schmidt composition operator C ϕ on H 2 such that E ϕ = K, and in [START_REF] Li | Estimates for approximation numbers of some classes of composition operators[END_REF]: Theorem 1.1 ( [START_REF] Li | Estimates for approximation numbers of some classes of composition operators[END_REF]) For every Lebesgue-negligible compact set K of the unitcircle T and every vanishing sequence (ε n ) of positive numbers, there is a composition operator C ϕ on H 2 such that E ϕ = K and such that its approximation numbers satisfy a n (C ϕ ) ≤ C e -n εn .

We are interested here in a different Hilbert space of analytic functions, on which not every Schur function defines a bounded composition operator, namely the Dirichlet space D. Recall its definition: the Dirichlet space D is the space of analytic functions f : D → C such that:

(1.2) f 2 D := |f (0)| 2 + D |f ′ (z)| 2 dA(z) < +∞ .
If f (z) = ∞ n=0 c n z n , one has:

(1.3)

f 2 D = |c 0 | 2 + ∞ n=1 n |c n | 2 .
Then D is a norm on D, making D a Hilbert space. Whereas every Schur function ϕ generates a bounded composition operator C ϕ on the Hardy space H 2 , it is no longer the case for the Dirichlet space (see [START_REF] Mccluer | Angular derivatives and compact composition operators on the Hardy and Bergman spaces[END_REF], Proposition 3.12, for instance).

In [START_REF] Gallardo-Gutiérrez | Exceptional sets and Hilbert-Schmidt composition operators[END_REF], the study of compact composition operators on the Dirichlet space D associated with a Schur function ϕ in connection with the set E ϕ was initiated. In particular, it is proved there that if the composition operator C ϕ is Hilbert-Schmidt on D, then the logarithmic capacity Cap E ϕ of E ϕ is 0, but, on the other hand, there are compact composition operators on D for which this capacity is positive. The optimality of this theorem was later proved in [START_REF] El-Fallah | Level sets and composition operators on the Dirichlet space[END_REF] under the following form: Theorem 1.2 (O. El-Fallah, K. Kellay, M. Shabankhah, H. Youssfi) For every compact set K of the unit circle T with logarithmic capacity Cap K equal to 0, there exits a Hilbert-Schmidt composition operator C ϕ on D such that E ϕ = K.

In this paper, we shall improve on this last result. We prove in Section 4 (Theorem 4.1) that for every compact set K ⊆ ∂D of logarithmic capacity Cap K = 0, there exists a Schur function ϕ ∈ A(D) ∩ D * such that the composition operator C ϕ is in all Schatten classes S p (D * ), p > 0, and for which E ϕ = K (and moreover E ϕ = {e it ; ϕ(e it ) = 1}). On the other hand, in Section 2, we show (Theorem 2.1) that for every bounded composition operator C ϕ on D * and every ξ ∈ ∂D, the logarithmic capacity of E ϕ (ξ) = {e it ; ϕ(e it ) = ξ} is 0.

In link with Hardy and Bergman spaces, we prove, in Section 2 yet, that every compact composition operator C ϕ on D * is compact on the Bergman-Orlicz space B Ψ2 and on the Hardy-Orlicz space H Ψ2 . In particular, C ϕ is in every Schatten class S p , p > 0, both on the Hardy space H 2 and on the Bergman space B 2 (Theorem 2.5). However, there exists a Schur function ϕ such that C ϕ is compact on H Ψ2 , but which is not even bounded on D * (Theorem 2.6).

In Section 3, we give a characterization of the membership of composition operators in the Schatten classes S p (D * ), p > 0 (actually in S p (D α, * ), where D α, * is the weighted Dirichlet space). We deduce that for every p > 0, there exists a symbol ϕ such that C ϕ ∈ S p (D * ), but C ϕ / ∈ S q (D * ) for any q < p, and that there exists another symbol ϕ such that C ϕ ∈ S q (D * ) for every q < p, but C ϕ / ∈ S p (D * ) (Theorem 3.3). We also show that there exists a Schur function ϕ such that C ϕ is compact on D * , but in no Schatten class S p (D * ) (Theorem 3.4).

Notation and background.

We denote by D the unit open disk of the complex plane and by T = ∂D the unit circle. A is the normalized area measure dx dy/π of D and m the normalized Lebesgue measure dt/2π on T.

As said before, a Schur function is an analytic self-map of D and the associated composition operator is defined, formally, by

C ϕ (f ) = f • ϕ. The function ϕ is called the symbol of C ϕ .
The Dirichlet space D is defined above. We shall actually work, for convenience, with its subspace D * of functions f ∈ D such that f (0) = 0. In this paper, we call D * the Dirichlet space.

An orthonormal basis of D * is formed by e n (z) = z n / √ n, n ≥ 1. The reproducing kernel on D * , defined by f (a) = f, K a for every f ∈ D * , is given by K a (z) = ∞ n=1 e n (a) e n (z), so that:

(1.4) K a (z) = log 1 1 -az •
Compactness of composition operators on D was characterized in terms of Carleson measure by D. Stegenga ([24]) and by B. McCluer and J. Shapiro in terms of angular derivative ( [START_REF] Mccluer | Angular derivatives and compact composition operators on the Hardy and Bergman spaces[END_REF]). Another characterization, more useful for us here, was given by N. Zorboska ([29], page 2020): for ϕ ∈ D, C ϕ is bounded on D if and only:

(1.5) sup h∈(0,2) sup |ξ|=1 1 A[W (ξ, h)] W (ξ,h) n ϕ (w) dA(w) < ∞ ,
where W (ξ, h) = {w ∈ D ; 1 -|w| ≤ h and | arg(w ξ)| ≤ πh} is the Carleson window of size h ∈ (0, 2) center at ξ ∈ T and n ϕ is the counting function of ϕ:

(1.6) n ϕ (w) = ϕ(z)=w 1 , w ∈ ϕ(D) ,
(we set n ϕ (w) = 0 for w ∈ D \ ϕ(D)). In particular, every Schur function with bounded valence defines a bounded composition operator on D. Moreover, C ϕ is compact if and only if:

(1.7) sup |ξ|=1 1 A[W (ξ, h)] W (ξ,h) n ϕ (w) dA(w) -→ h→0 0 .
For further informations on the Dirichlet space, one may consult the two surveys [START_REF] Arcozzi | The Dirichlet space: a survey[END_REF] and [START_REF] Ross | The classical Dirichlet space, Recent advances in operatorrelated function theory[END_REF], for example.

Logarithmic capacity

The notion of logarithmic capacity is tied to the study of the Dirichlet space by the following seminal and sharp result of Beurling ( [START_REF] Beurling | Sur les ensembles exceptionnels[END_REF]; see also [START_REF] Kahane | Ensembles parfaits et séries trigonométriques[END_REF]).

Theorem 1.3 (Beurling) For every function f (z) = ∞ n=0 c n z n ∈ D,
there exists a set E ⊆ ∂D, with logarithmic capacity 0, such that, if t ∈ T \ E, then the radial limit f * (e it ) := lim r→1 -f (re it ) exists (in C). Moreover, the result is optimal: if a compact set E ⊆ T has zero logarithmic capacity, there exists

f (z) = ∞ n=0 c n z n ∈ D such that f * (e it ) does not exist on E.
Let us recall some definitions (see [START_REF] Kahane | Ensembles parfaits et séries trigonométriques[END_REF], Chapitre III, [START_REF] Conway | Functions of One Complex Variable II[END_REF], Chapter 21, § 7, or [START_REF] Ross | The classical Dirichlet space, Recent advances in operatorrelated function theory[END_REF], Section 4, for example).

Let µ be a probability measure supported by a compact subset K of T. The potential U µ of µ is defined, for every z ∈ C, by:

U µ (z) = K log e |z -w| dµ(w) .
The energy I µ of µ is defined by:

I µ = K U µ (z) dµ(z) = K×K log e |z -w| dµ(w) dµ(z) .
The logarithmic capacity of a Borel set E ⊆ T is: If µ is the equilibrium measure of the compact set K, we have Frostman's Theorem ( [START_REF] Conway | Functions of One Complex Variable II[END_REF], Theorem 21.7.12, or [START_REF] Kahane | Ensembles parfaits et séries trigonométriques[END_REF], Chapitre III, Proposition 5 and Proposition 6): U µ (z) ≤ I µ for every z ∈ C and

Cap E = sup µ e -Iµ ,
(1.8) U µ (z) = I µ for almost all z ∈ K .
Suppose that the compact set K has zero logarithmic capacity. For ε > 0, let K ε = {z ∈ T ; dist (z, K) ≤ ε}, µ ε its equilibrium measure, and I µε its energy. Then ( [START_REF] Conway | Functions of One Complex Variable II[END_REF], Proposition 21.7.15):

(1.9)

lim ε→0 I µε = ∞ .

Bounded and compact composition operators

In [START_REF] Gallardo-Gutiérrez | Exceptional sets and Hilbert-Schmidt composition operators[END_REF], E. A. Gallardo-Gutiérrez and M. J. González showed that for every Hilbert-Schmidt composition operator C ϕ on D * , the logarithmic capacity of the set E ϕ = {e iθ ∈ ∂D ; |ϕ(e iθ )| = 1} is zero. On the other hand, they showed that there are compact composition operators on D * for which E ϕ has positive logarithmic capacity. We shall see that if we replace |ϕ| by ϕ in the definition of E ϕ , the result is very different.

Theorem 2.1 For every bounded composition operator C ϕ on D * and every ξ ∈ ∂D, the logarithmic capacity of E ϕ (ξ) = {e it ; ϕ(e it ) = ξ} is 0.

We first state the following characterization of Hilbert-Schmidt composition operators on D * . This result is stated in [START_REF] Gallardo-Gutiérrez | Exceptional sets and Hilbert-Schmidt composition operators[END_REF], but not entirely proved.

(2.1) D |ϕ ′ (z)| 2 (1 -|ϕ(z)| 2 ) 2 dA(z) < ∞ . Proof. Let e n (z) = z n / √ n; then (e n ) n≥1 is an orthonormal basis of D * and ∞ n=1 C ϕ (e n ) 2 = ∞ n=1 ϕ n 2 n = D |ϕ ′ (z)| 2 (1 -|ϕ(z)| 2 ) 2 dA(z) . Hence (2.1) is satisfied if C ϕ is Hilbert-Schmidt. To get the converse, we need to show that (2.1) implies that C ϕ is bounded on D * . Let f ∈ D * and write f (z) = ∞ n=1 c n z n . Then C ϕ f = ∞ n=1 c n ϕ n and C ϕ f ≤ ∞ n=1 |c n | ϕ n ≤ ∞ n=1 n |c n | 2 1/2 ∞ n=1 ϕ n 2 n 1/2 = D |ϕ ′ (z)| 2 (1 -|ϕ(z)| 2 ) 2 dA(z) 1/2 f .
Then (2.1) implies that C ϕ is Hilbert-Schmidt. Now Theorem 2.1 will follow from the next proposition. Taking this proposition for granted for a while, we can prove the theorem.

Proof of Theorem 2.1. Making a rotation, we may, and do, assume that ξ = 1.

Then, if σ is the map of Proposition 2.3, C ϕ C σ = C σ•ϕ is Hilbert-Schmidt. By [6], the set E σ•ϕ has zero logarithmic capacity. But σ has modulus 1 only at 1; hence e iθ ∈ E σ•ϕ if and only if e iθ ∈ E ϕ (1).
To prove Proposition 2.3, it will be convenient to use the following criteria, where

ϕ a (z) = z-a 1-āz • Lemma 2.4 Let f ∈ D such that Re f ≥ 1. Then if σ = ϕ a • e -1/f
, where a = e -1/f (0) , the composition operator C σ is Hilbert-Schmidt on D * .

Proof. Let σ 0 = e -1/f . If u = Re f and v = Im f , one has:

|σ 0 | 2 = exp - 2u u 2 + v 2 and |σ ′ 0 | 2 = u ′ 2 + v ′ 2 (u 2 + v 2 ) 2 exp - 2u u 2 + v 2 • Then |σ 0 | < 1 and so σ 0 is a self-map of D. Since u ≥ 1 > 0, one has |σ ′ 0 | 2 ≤ (u ′ 2 + v ′ 2 )/(u 2 + v 2 ) 2 ≤ u ′ 2 + v ′ 2 = |f ′ | 2 ; hence σ 0 ∈ D. For 0 ≤ x ≤ 2, one has 1 -e -x ≥ x/4. Therefore, since u ≥ 1 implies 2u/(u 2 + v 2 ) ≤ 2/u ≤ 2, one has: 1 -|σ 0 | 2 ≥ u 2(u 2 + v 2 ) •
It follows that:

|σ ′ 0 | 2 (1 -|σ 0 | 2 ) 2 ≤ u ′ 2 + v ′ 2 (u 2 + v 2 ) 2 4(u 2 + v 2 ) 2 u 2 ≤ 4(u ′ 2 + v ′ 2 ) = 4|f ′ | 2 . Since f ∈ D, |f ′ | 2 has a finite integral and therefore (2.1) is satisfied. It follows that C σ0 is Hilbert-Schmidt on D and hence C σ = C σ0 • C ϕa is Hilbert-Schmidt on D * , since σ(0) = 0.
Proof of Proposition 2.3. Let Ω be the domain defined by:

Ω = {z ∈ C ; Re z > 1 and |Im z| < 1/(Re z) 2 } . Let f be a conformal map from D onto Ω such that f (1) = ∞. Since A(Ω) < ∞,
we have f ∈ D. By Lemma 2.4, the function σ = e -1/f has the required properties.

For the next result, recall that an Orlicz function Ψ is a nondecreasing convex function such that Ψ(0) = 0 and Ψ(x)/x → ∞ as x goes to infinity. We refer to [START_REF] Lefèvre | Composition operators on Hardy-Orlicz spaces[END_REF] for the definition of Hardy-Orlicz and Bergman-Orlicz spaces. In the following result, one set Ψ 2 (x) = exp(x 2 ) -1.

Theorem 2.5 Every compact composition operator C ϕ on D * is compact on the Bergman-Orlicz space B Ψ2 and on the Hardy-Orlicz space H Ψ2 . In particular, C ϕ is in every Schatten class S p , p > 0, both on the Hardy space H 2 and on the Bergman space B 2 .

Proof. Consider the normalized reproducing kernels Ka = K a / K a , a ∈ D. When |a| goes to 1, they tends to 0 uniformly on compact sets of D; hence C * ϕ ( Ka ) tends to 0, by compactness of the adjoint operator

C * ϕ . But C * ϕ (K a ) = K ϕ(a) and K a 2 = K a , K a = log 1 1-|a| 2 , so we get: (2.2) lim |a|→1 log 1 1-|ϕ(a)| 2 log 1 1-|a| 2 = 0 .
This condition means that C ϕ is compact on the Bergman-Orlicz space B Ψ2 ( [START_REF] Lefèvre | Composition operators on Hardy-Orlicz spaces[END_REF], page 69) and implies that C ϕ is in all Schatten classes S p (B 2 ), p > 0 ( [START_REF] Lefèvre | Compact composition operators on Bergman-Orlicz spaces[END_REF]).

In the same way, it suffices to show that C ϕ is compact on H Ψ2 , because that implies that C ϕ is in all Schatten classes S p (H 2 ) ([11], Theorem 5.2).

Compactness of C ϕ on H Ψ is equivalent to say ( [START_REF] Lefèvre | Composition operators on Hardy-Orlicz spaces[END_REF], Theorem 4.18) that:

ρ ϕ (h) := sup |ξ|=1 m {e it ; ϕ(e it ) ∈ W (ξ, h)} = o h→0 1 Ψ AΨ -1 (1/h)
for every A > 0 .

When Ψ = Ψ 2 , this means that ρ ϕ (h) = o (h A ) for every A > 0. Now, by [START_REF] Lefèvre | Nevanlinna counting function and Carleson function of analytic maps[END_REF], Theorem 4.2, this is also equivalent to say that:

(2.3) sup |ξ|=1 1 A[W (ξ, h)] W (ξ,h) N ϕ (w) dA(w) = o (h A ) for every A > 0 ,
where N ϕ is the Nevanlinna counting function of ϕ:

(2.4) N ϕ (w) = ϕ(z)=w (1 -|z| 2 ) , w ∈ ϕ(D) ,
and N ϕ (w) = 0 otherwise. But (2.2) is equivalent to the fact that for every ε > 0 there exists δ ε > 0 such that:

(2.5) 1 -|ϕ(z)| ≥ δ ε (1 -|z|) ε , ∀z ∈ D .
Since ϕ(0) = 0, we have |ϕ(z)| ≤ |z|, by Schwarz's lemma; hence one has

N ϕ (w) ≤ 2δ -1 ε (1 -|w|) 1/ε n ϕ (w). It follows that (since 1 -|w| ≤ h for w ∈ W (ξ, h)): 1 A[W (ξ, h)] W (ξ,h) N ϕ (w) dA(w) ≤ 2δ -1 ε h 1/ε 1 A[W (ξ, h)] W (ξ,h) n ϕ (w) dA(w) ,
which is o (h 1/ε ), uniformly for |ξ| = 1, by (1.7).

Remarks. 1) One may argue that compactness of C ϕ on H Ψ2 implies its compactness on B Ψ2 ([15], Proposition 4.1, or [START_REF] Li | Compact composition operators on Hardy-Orlicz and Bergman-Orlicz spaces[END_REF], Theorem 9). One may also use the forthcoming Corollary 3.2 saying that

C ϕ ∈ S p (H 2 ) implies that C ϕ ∈ S p (B 2 ).
2) To show the compactness of C ϕ on H Ψ2 , we used its compactness on D * twice. However, due to the fact that ε > 0 is arbitrary, we may replace o (h 1/ε ) by O (h 1/ε ); hence to end the proof, we only have to use (1.5), i.e. the boundedness of C ϕ on D * , instead of (1.7).

Note that (2.2) does not suffice to have compactness on H Ψ2 (in [START_REF] Lefèvre | Composition operators on Hardy-Orlicz spaces[END_REF], Proposition 5.5, we construct a Blaschke product satisfying (2.2)).

In the opposite direction, we have the following result.

Theorem 2.6 There exists a Schur function ϕ such that C ϕ is compact on H Ψ2 , but which is not even bounded on D * .

To prove this theorem, we first begin with the following key lemma. Lemma 2.7 There exists a constant κ 1 > 0 such that for any f ∈ H(D) having radial limits f * a.e. and which satisfies, for some α ∈ R:

(2.6) Im f (0) < α and f (D) ⊆ {z ∈ C ; 0 < Re z < π} ∪ {z ∈ C ; Im z < α},
we have, for all y ≥ α:

m {z ∈ T ; Im [f * (z)] ≥ y} ≤ κ 1 e α-y .
Proof. Suppose that f satisfies (2.6), and define

f 1 (z) = -if (z) + π 2 i -α. Then either Re [f 1 (z)] < 0, or -π 2 < Im [f 1 (z)] < π 2 for every z ∈ D. Therefore, defining h(z) = 1 + exp[f 1 (z)], we have h : D → H, that is Re [h(z)] > 0 for every z ∈ D. Finally define h 1 (z) = h(z) -i Im [h(0)].
Then h 1 : D → H and h 1 (0) ∈ R (and so h 1 (0) > 0). Kolmogorov's inequality yields that, for some absolute constant C 1 , one has, for every λ > 0:

(2.7) m {z ∈ T ; |h * 1 (z)| ≥ λ} ≤ C 1 h 1 (0) λ •
Observe that, since Im [f (0)] < α, we have Re [f 1 (0)] < 0, and then:

(2.8) |Im [h(0)]| < 1 and h 1 (0) = Re [h(0)] < 2.
Suppose now that, for y > α and z ∈ D, we have

Im [f (z)] > y; then exp[f 1 (z)] ∈ H, and |h(z)| ≥ | exp[f 1 (z)]| > e y-α
. Taking radial limits we get, up to a set of null Lebesgue-measure:

{z ∈ T ; Im [f * (z)] ≥ y} ⊆ {z ∈ T ; |h * (z)| ≥ e y-α }.
We consider two cases: e y-α ≥ 2 and e y-α < 2. When e y-α ≥ 2, then |h * (z)| ≥ e y-α yields:

|h * 1 (z)| ≥ e y-α -|Im [h(0)]| > e y-α -1 ≥ 1 2 e y-α ,
by the first part of (2.8). Then, using (2.7) and the second part of (2.8), we have:

m {z ∈ T ; Im [f * (z)] ≥ y} ≤ m {z ∈ T ; |h * 1 (z)| > (1/2) e y-α } ≤ 2 C 1 h 1 (0) e y-α ≤ 4C 1 e y-α ,
and, in this case, the lemma is proved, if one takes κ 1 ≥ 4C 1 . When e y-α < 2, then e α-y > 1/2, and, because:

m {z ∈ T ; Im [f * (z)] ≥ y} ≤ 1 < κ 1 e α-y ,
since κ 1 > 2, the lemma is proved. Now, we give a general construction of Schur functions with suitable properties.

Proposition 2.8 Let g : (0, ∞) → (0, ∞) be a continuous non-increasing function such that:

lim t→0 + g(t) = +∞,
and lim t→+∞ g(t) = 0.

Let h : (0, ∞) → (0, ∞] be a lower semicontinuous function such that M := sup{h(t) ; t ≥ π} < +∞ and consider the simply connected domain:

Ω = {x + iy ; x ∈ (0, ∞) and g(x) < y < g(x) + h(x)} . Let f : D → Ω ∪ {∞} be a conformal mapping from D onto Ω such that f(0) = π + i(g(π) + h(π)/2).
Then the symbol ϕ : D → D defined by ϕ(z) = exp[-f(z)], for every z ∈ D, satisfies, for some ε 0 ,k 0 > 0:

1) For all h ∈ (0, ε 0 ):

(2.9) m({z ∈ T ; |ϕ * (z)| > 1 -h}) ≤ k 0 exp -g(2h)
.

2) Assume that, for some r ∈ (0, ∞] and integers 0 ≤ n < N ≤ ∞, one has {h(t) ; t ≤ r} ⊆ (2 nπ, 2N π]. Then, for all z ∈ D, such that |z| > e -r , we have n ≤ n ϕ (z) ≤ N .

In particular, {z ∈ D ; |z| > e -r } ⊆ ϕ(D) ⊆ D \ {0}, when n ≥ 1.

Remarks.

1. When N = 1, the map ϕ is univalent. 2. When r = ∞ and n ≥ 1, we have ϕ(D) = D \ {0}.
3. With g(t) = 1/t, the operator C ϕ is compact on H Ψ2 , therefore belongs to all Schatten classes S p (H 2 ), p > 0.

4. When N < ∞, the operator C ϕ is bounded on the Dirichlet space. 5. When n ≥ 1, the operator C ϕ is not compact on the Dirichlet space (since the averages on the windows of the function n ϕ cannot uniformly vanish).

Proof of Proposition 2.8. We shall apply Lemma 2.7 with α = M + g(π).

Suppose that, for z ∈ T and 0 < h < 1, we have |ϕ

* (z)| > 1 -h. Then, if h is small enough, e -2h < 1 -h < |ϕ * (z)| = exp -Re [f * (z)] ,
and therefore 2h > Re [f * (z)]. But observe that f * (z) ∈ Ω ∪ {∞}, and so, if 2h > Re [f * (z)], we necessarily have Im [f * (z)] ≥ g(2h). Again, if h is small enough, we have y = g(2h) > α, and may apply the lemma to obtain:

m {z ∈ T ; |ϕ * (z)| > 1 -h} ≤ m {z ∈ T ; Im [f * (z)] ≥ g(2h)} ≤ κ 1 e α-g(2h) .
We get (2.9).

On the other hand, let Z ∈ D such that |Z| > e -r , we can write Z = e -x e iθ with x < r. We can find θ ′ i s such that g(x) < θ 1 < . . . < θ s < g(x) + h(x) and θ j ≡ θ[2π] with n ≤ s ≤ N . For each j, there exists a unique z j ∈ D, such that Re f(z j ) = x and Im f(z j ) = θ j ; hence ϕ(z j ) = Z. Moreover no other z ∈ D can satisfy ϕ(z) = Z. Hence n ϕ (Z) = s.

Proof of Theorem 2.6. As said before, if one takes g(t) = 1/t in Proposition 2.8, then C ϕ is compact on H Ψ2 and hence is in all Schatten classes S p (H 2 ), p > 0. On the other hand, if one choose also h(t) = 1/t, then, for every r > 0, {h(t) ; t ≤ r} = [1/r, ∞) and for |z| > e -r , we get that n ϕ (z) ≥ [1/(2πr)] (the integer part of 1/(2πr)). It follows that, for some constant c > 0, one has, with e -r = 1h:

1 A[W (ξ, h)] W (ξ,h) n ϕ (z) dA(z) ≥ c 1 log[1/(1 -h)] -→ h→0 ∞ .
Therefore, C ϕ is not bounded on D * , by (1.5).

Remarks. 1. Actually, as we may take g growing as we wish, the proof shows, using [START_REF] Lefèvre | Composition operators on Hardy-Orlicz spaces[END_REF], Theorem 4.18, that for every Orlicz function Ψ, one can find a Schur function ϕ such that C ϕ is not bounded on D * , though compact on the Hardy-Orlicz space H Ψ . 2. This construction also allows to produce a univalent map ϕ, with an arbitrary small Carleson function ρ ϕ (h) = sup |ξ|=1 m {e it ; ϕ * (e it ) ∈ W (ξ, h)} , and such that C ϕ is not compact on the Dirichlet space (note we cannot replace "compact" by "bounded" since any Schur function with a bounded valence is bounded on the Dirichlet space).

Indeed, take h(t) = 2π and g be C 1 : g(t) = 1/t for instance. We have N = 1 and so ϕ is univalent. Now it suffices to notice that the range of the curve Γ = e -x-ig(x) ; x ∈ (0, ∞) = t cos(1/ ln(t)), t sin(1/ ln(t)) ; t ∈ (0, 1) ⊆ D has a null area measure. The range of ϕ is D \ (Γ ∪ {0}) and for each w / ∈ Γ, we have n ϕ (w) = 1 Then, for h ∈ (0, 1), we have:

1 h 2 W (1,h) n ϕ (w) dA(w) = 1 h 2 W (1,h)\Γ dA(w) = 1 h 2 A[W (1, h) \ Γ] = 1 h 2 A[W (1, h)] ≈ 1 ,
and so C ϕ in not compact on D * , by (1.7).

Composition operators in Schatten classes

Characterization

In this section, we give a characterization of the membership in the Schatten classes of composition operators on D * . This characterization will be deduced from Luecking's one for composition operators on the Bergman space. Actually, we shall give it for weighted Dirichlet spaces D α, * . Boundedness and compactness has been characterized by B. McCluer and J. Shapiro in [START_REF] Mccluer | Angular derivatives and compact composition operators on the Hardy and Bergman spaces[END_REF] and, in other terms, by N. Zorboska in [START_REF] Zorboska | Composition operators on weighted Dirichlet spaces[END_REF].

Recall that for α > -1, the weighted Dirichlet space D α is the space of analytic functions f : D → C such that (3.1)

D |f ′ (z)| 2 (1 -|z| 2 ) α dA(z) < ∞ .
This is a Hilbert space for the norm given by:

(3.2) f 2 α = |f (0)| 2 + (α + 1) D |f ′ (z)| 2 (1 -|z| 2 ) α dA(z) < ∞ .
The standard Dirichlet space D corresponds to α = 0; the Hardy space H 2 to α = 1 and the standard Bergman space to α = 2. For more general weights, see [START_REF] Kellay | Compact composition operators on weighted Hilbert spaces of analytic functions[END_REF]. We denote by D α, * the subspace of the f ∈ D α such that f (0) = 0.

If ϕ is a Schur function, one defines its weighted Nevanlinna counting function N ϕ,α at w ∈ Ω := ϕ(D) as the number of pre-images of w with the weight (1 -|z|) α :

(3.3) N ϕ,α (w) = ϕ(z)=w (1 -|z| 2 ) α . For w ∈ D \ ϕ(D), we set N ϕ,α (w) = 0. One has N ϕ,1 = N ϕ and N ϕ,0 = n ϕ .
With this notation, recall the change of variable formula:

(3.4) D F [ϕ(z)] |ϕ ′ (z)| 2 (1 -|z| 2 ) α dA(z) = Ω F (w) N ϕ,α (w) dA(w) .
Denote by R n,j , n ≥ 0, 0 ≤ j ≤ 2 n -1, the Hastings-Luecking windows:

R n,j = z ∈ D ; 1 -2 -n ≤ |z| < 1 -2 -n-1 and 2jπ 2 n ≤ arg z < 2(j + 1)π 2 n .
We can now state.

Theorem 3.1 Let α > -1. Let ϕ be a Schur function and p > 0. Then C ϕ ∈ S p (D α, * ) if and only if:

(3.5) ∞ n=0 2 n -1 j=0 2 n(α+2) Rn,j N ϕ,α (w) dA(w) p/2 < ∞ .
If ϕ is univalent, (3.5) can be replaced by the purely geometric condition:

(3.6) ∞ n=0 2 n -1 j=0 2 n(α+2) A α (R n,j ∩ Ω) p/2 < ∞ ,
where A α is the weighted measure dA α (w) = (α + 1) (1 -|w| 2 ) α dA(w).

Remark. Of course, every operator in a Schatten class is compact, but we may note that condition (3.5) implies the compactness of C ϕ , by [START_REF] Zorboska | Composition operators on weighted Dirichlet spaces[END_REF], Theorem 1 (and [START_REF] Lefèvre | Some examples of compact composition operators on H 2[END_REF], Proposition 3.3).

Proof of Theorem 3.1. First, we compute C * ϕ C ϕ . Let us fix f and g in the Dirichlet space D α, * . We have:

(α + 1) D (C * ϕ C ϕ )(f ) ′ (z) g ′ (z) (1 -|z| 2 ) α dA(z) = f • ϕ, g • ϕ Dα, * = (α + 1) D (f ′ • ϕ)(z)(g ′ • ϕ)(z) |ϕ ′ (z)| 2 (1 -|z| 2 ) α dA(z).
By the change of variable formula, we get:

D (C * ϕ C ϕ )(f ) ′ (z)g ′ (z) (1 -|z| 2 ) α dA = D f ′ (w) g ′ w) N ϕ,α (w) dA(w) ,
which is equivalent to:

D (C * ϕ C ϕ )(f ) ′ (z) G(z) (1 -|z| 2 ) α dA(z) = D f ′ (w) G(w) N ϕ,α (w) dA(w)
for every function G belonging to the weighted Bergman space [START_REF] Zhu | Operator Theory in Function Spaces[END_REF], § 6.4.1), we obtain that for every z ∈ D:

B 2 α . That means that (C * ϕ C ϕ )(f ) ′ -f ′ .N ϕ,α /(1 -|w| 2 ) α is orthogonal to the weighted Bergman space B 2 α . But (C * ϕ C ϕ )(f ) ′ ∈ B 2 α . Hence (C * ϕ C ϕ )(f ) ′ is the orthogonal projection onto B 2 α of the function f ′ .N ϕ,α /(1 -|w| 2 ) α . Thus (see
(C * ϕ C ϕ )(f ) ′ (z) = (α + 1) D f ′ (w) (1 -wz) α+2 N ϕ,α (w) (1 -|w| 2 ) α (1 -|w| 2 ) α dA(w) = (α + 1) D f ′ (w) (1 -wz) α+2 dµ(w) = (α + 1) T µ (f ′ )(z) ,
where µ is the positive measure A with weight N ϕ,α and T µ is the Toeplitz operator on B 2 α is introduced in [START_REF] Luecking | Trace ideal criteria for Toeplitz operators[END_REF] (let us point out that α in [START_REF] Luecking | Trace ideal criteria for Toeplitz operators[END_REF] corresponds to -(α + 1) in our work).

In other words, introducing the map

∆(h) = h ′ , which is an isometry from D α, * onto B 2 α , we have ∆ • (C * ϕ C ϕ ) = T µ • ∆.
We have the following diagram:

D α, * ∆ C * ϕ Cϕ / / D α, * ∆ B 2 α Tµ / / B 2 α
Hence the approximation numbers of T µ (viewed as an operator on B 2 α ) and the ones of C * ϕ C ϕ (viewed as an operator on D α, * ) are the same. In particular, the membership in the Schatten classes are the same and the final result follows from the main theorem in [START_REF] Luecking | Trace ideal criteria for Toeplitz operators[END_REF]:

C ϕ ∈ S p (D α, * ) if and only if C * ϕ C ϕ ∈ S p/2 (D α, *
) and that holds if and only if:

∞ n=0 2 n -1 j=0 2 n(α+2) µ(R n,j ) p/2 < ∞ .
Hence C ϕ ∈ S p (D α, * ) if and only if:

∞ n=0 2 n -1 j=0 2 n(α+2) Rn,j N ϕ,α (w) dA(w) p/2 < ∞ ,
and that ends the proof of Theorem 3.1.

Remark. In the same way, we can obtain other characterizations for D α, * by using the ones for B 2 α given in [START_REF] Luecking | Composition operators belonging to the Schatten ideals[END_REF] and [START_REF] Zhu | Schatten class composition operators on weighted Bergman spaces of the disk[END_REF]:

C ϕ ∈ S p (B 2 α ) if and only if N ϕ,α+2 (z)/ log(1/|z|) α+2 ∈ L p/2 (λ), where dλ(z) = (1 -|z| 2 ) -2 dA(z) is the
Möbius invariant measure on D, and, when ϕ has bounded valence and p ≥ 2, if and only if

(1 -|z| 2 )/ 1 -|ϕ(z)| 2 ∈ L p(α+2)/2 (λ)
. Such a result can be found in [START_REF] Xu | Schatten-class composition operators on weighted Dirichlet spaces[END_REF].

Applications

We give several applications of the previous theorem.

Corollary 3.2 Let -1 < α ≤ β, p > 0, and ϕ be a Schur function. Then

C ϕ ∈ S p (D α, * ) implies that C ϕ ∈ S p (D β, * ). In particular, C ϕ ∈ S p (D * ) implies that C ϕ ∈ S p (H 2 ), which in turn implies that C ϕ ∈ S p (B 2 ). Proof. Assume that C ϕ ∈ S p (D α, * ). Then ∞ n=0 2 n -1 j=0 2 n(α+2) Rn,j N ϕ,α (w) dA(w) p/2 < ∞. Since, thanks to Schwarz's lemma, N ϕ,β (w) ≤ N ϕ,α (w)(1 -|w| 2 ) β-α , we have N ϕ,β (w) ≤ (2.2 -n ) β-α N ϕ,α (w) for w ∈ R n,j . It follows that ∞ n=0 2 n -1 j=0 2 n(β+2) Rn,j N ϕ,β (w) dA(w) p/2 < ∞ ,
and that proves Corollary 3.2.

It is known ( [START_REF] Lefèvre | Some examples of compact composition operators on H 2[END_REF]) that composition operators on H 2 separate Schatten classes, but the difficulty is that we must not only control the shape of ϕ(∂D), but also the parametrization t → ϕ(e it ), even if ϕ is univalent. In the case of the Dirichlet space, this difficulty disappears, because only the areas come into play, and we can easily prove the following result.

Theorem 3.3 The composition operators on D * separate Schatten classes, in the following sense. Let 0 < p 1 < ∞. Then, there exists a symbol ϕ such that:

C ϕ ∈ p>p1 S p (D * ) \ S p1 (D * ) .
Similarly, there exists a symbol ϕ such that:

C ϕ ∈ S p1 (D * ) \ p<p1 S p (D * ) .
In particular, for every 0 < p 1 < p 2 < ∞, there exists ϕ such that

C ϕ ∈ S p2 (D * ) \ S p1 (D * ).
Proof. Let (h n ) n≥1 , with 0 < h n < 1, be a sequence of real numbers with limit 0 to be adjusted, and J the Jordan curve formed by the segment [0, 1] and the north and (truncated) north-east sides of the curvilinear rectangles

{1 -2 -n ≤ |z| < 1 -2 -n-1 } × {0 ≤ arg z < 2 -n h n }.
Let Ω 0 be the interior of J and Ω = Ω 0 ∪ D(0, 1/8). Let ϕ : D → Ω be a Riemann map such that ϕ(0) = 0. Since ϕ is univalent and bounded, it defines a symbol on D * , and the necessary and sufficient condition (3.6) for membership in S p (D * ) reads:

(3.7) ∞ n=0 [4 n 4 -n h n ] p/2 = ∞ n=0 h n p/2 < ∞.
Indeed, it is clear that, for fixed n, the Hastings-Luecking windows R n,j satisfy:

R n,0 ∩ Ω = ∅; R n,j ∩ Ω = ∅ for 1 ≤ j < 2 n .
Therefore, only the Hastings-Luecking windows R n,0 matter. Since:

A(R n,0 ∩ Ω) = 1-2 -n ≤r<1-2 -n-1 , 0≤θ<2 -n hn r dr dθ ≈ 4 -n h n ,
we can test the criterion (3.7). Now, it is enough to take h n = (n + 1) -2/p1 to get:

C ϕ ∈ p>p1 S p (D * ) \ S p1 (D * ) .
Similarly, the choice h n = (n + 1) -2/p1 [log(n + 2)] -4/p1 , gives a symbol ϕ such that:

C ϕ ∈ S p1 (D * ) \ p<p1 S p (D * ) .
This ends the proof.

T. Carroll and C. Cowen ([3] proved, but only for α > 0, that there exist compact composition operators on D α which are in no Schatten class (see also [START_REF] Jones | Compact composition operators not in the Schatten classes[END_REF]). In the next result, we shall see that this still true for α = 0. Proof. It suffices to use the proof of Theorem 3.3 and to take, instead of the above h n , h n = 1/ ln(n + 2).

For the next application, which will be used in Section 4, we need to recall the definition of the cusp map χ, introduced in [START_REF] Lefèvre | Compact composition operators on Bergman-Orlicz spaces[END_REF], and later used, with a slightly different definition in [START_REF] Li | Estimates for approximation numbers of some classes of composition operators[END_REF]. Actually, we have to modify it slightly again in order to have χ(0) = 0. We first define:

χ 0 (z) = z -i iz -1 1/2 -i -i z -i iz -1 1/2 + 1
, then:

χ 1 (z) = log χ 0 (z), χ 2 (z) = - 2 π χ 1 (z) + 1, χ 3 (z) = a χ 2 (z)
, and finally:

χ(z) = 1 -χ 3 (z) ,
where a = 1 -2 π log( √ 2 -1) ∈ (1, 2) is chosen in order that χ(0) = 0. The image Ω of the (univalent) cusp map is formed by the intersection of the inside of the disk D a 2 , a 2 and the outside of the two disks D ia 2 , a 2 and D -ia 2 , a 2 .

Corollary 3.5 If χ is the cusp map, then C χ belongs to all Schatten classes S p (D * ), p > 0.

Proof. Since χ is univalent, χ(0) = 0, and Ω = χ(D) has finite area, we have χ ∈ D * . A little elementary geometry shows that, for some constant C, we have:

(3.8) w ∈ Ω, 0 < h < 1 and |w| ≥ 1 -h =⇒ |Im w| ≤ Ch 2 .
It follows (changing C if necessary) that R n,j ∩ Ω is contained in a rectangle of sizes 2 -n and C 4 -n and with area C 8 -n . Hence, for a given n, at most C of the Hastings-Luecking windows R n,j can intersect Ω. Therefore, the series in Theorem 3.1 reduces, up to constants, to the series:

∞ n=0 (4 n 8 -n ) p/2 = ∞ n=0 2 -np ,
which converges for every p > 0.

Logarithmic capacity and set of contact points

In view of the result of [START_REF] Gallardo-Gutiérrez | Exceptional sets and Hilbert-Schmidt composition operators[END_REF] mentioned in the introduction, if Cap K > 0, there is no hope to find a symbol ϕ such that E ϕ = K and C ϕ is Hilbert-Schmidt on D * . But as was later proved in [START_REF] El-Fallah | Level sets and composition operators on the Dirichlet space[END_REF], Cap K > 0 is the only obstruction. We can improve on the results from [START_REF] El-Fallah | Level sets and composition operators on the Dirichlet space[END_REF] as follows: our composition operator is not only Hilbert-Schmidt, but in any Schatten class; moreover, we can replace E ϕ = K by E ϕ = E ϕ (1) = K. Theorem 4.1 For every compact set K of the unit circle T with logarithmic capacity Cap K = 0, there exists a Schur function ϕ with the following properties:

1) ϕ ∈ A(D) ∩ D * := A, the "Dirichlet algebra";

2)

E ϕ = E ϕ (1) = K; 3) C ϕ ∈ p>0 S p (D * ).
In fact, the approximation numbers of

C ϕ satisfy a n (C ϕ ) ≤ a exp(-b √ n).
This theorem actually results of the particular following case and the properties of the cusp map seen in Section 3.2. Theorem 4.2 For every compact set K ⊆ ∂D of logarithmic capacity Cap K = 0, there exists a Schur function q ∈ A(D) ∩ D * which peaks on K and such that the composition operator C q : D * → D * is bounded (and even Hilbert-Schmidt).

Recall that a function q ∈ A(D), the disk algebra, is said to peak on a compact subset K ⊆ ∂D (and is called a peaking function ) if:

q(z) = 1 if z ∈ K ; |q(z)| < 1 if z ∈ D \ K .
Proof of Theorem 4.1. We simply take for ϕ the composed map ϕ = χ • q, where χ is the cusp map and q our peaking function. Recall that χ ∈ A(D) and that χ peaks on {1}. We take advantage of this fact by composing with q, for which C q : D * → D * is bounded as well as C χ (since χ is univalent). We clearly have ϕ ∈ A(D), ϕ(z) = χ(1) = 1 for z ∈ K, and |ϕ(z)| < 1 for z / ∈ K, since then |q(z)| < 1. Therefore E ϕ (1) = K. Moreover, C ϕ being bounded on D * , we have in particular ϕ = C ϕ (z) ∈ D * . Since C ϕ = C q • C χ , we get 3), by Corollary 3.5.

In [START_REF] Lefèvre | Approximation numbers of composition operators on the Dirichlet space[END_REF], we prove that a n (C χ ) ≤ a exp(-b √ n). Since a n (C ϕ ) ≤ C q a n (C χ ), by the ideal property of approximation numbers, this ends the proof of Theorem 4.1.

In turn, the proof of Theorem 4.2 relies on the following crucial lemma. Lemma 4.3 Let K ⊆ ∂D be a compact set such that Cap K = 0. Then, there exists a function U : D → R + ∪ {∞}, such that:

1) U (z) = ∞ if and only if z ∈ K; 2) U ≥ 1 on D; 3) U is continuous on D \ K, harmonic in D and D |∇U | 2 dA < ∞; 4) lim z→K, z∈D U (z) = ∞; 5) the conjugate function V = Ũ is continuous on D \ K.
Proof of Theorem 4.2. Taking this lemma for granted, let us end the proof of the theorem. We set f = U + iV , a = e -1/f (0) and q = ϕ a • e -1/f , where ϕ a (z) = z-a 1-āz . In view of the third and fourth items of the lemma, we have q ∈ A(D). Since U ≥ 1, Lemma 2.4 shows that C q is Hilbert-Schmidt on D * . Moreover, for z ∈ K, one has f (z) = ∞ and hence q(z) = 1 since ϕ a (1) = 1 because a ∈ R (since f (0) = U (0)). On the other hand, when z / ∈ K, one has |f (z)| < ∞ and hence |q(z)| < 1. Therefore q peaks on K.

Proof of Lemma 4.3. This proof is strongly influenced by that of Theorem III, page 47, in [START_REF] Kahane | Ensembles parfaits et séries trigonométriques[END_REF]. Let: For 0 < ε < 1/2, let K ε = {z ∈ T ; dist (z, K) ≤ ε}, µ ε its equilibrium measure, and U ε the logarithmic potential of µ ε , that is:

(4.1) L(z) = log e 1 -z = P (z) + i Q(z), with
P (z) = log e |1 -z| and Q(z) = -arg(1 -z), |Q(z)| ≤ π 2 , z ∈ D \ {1} ,
U ε (z) = Kε log e |z -w| dµ ε (w) ,
that we could as well write (since K ε ⊆ T):

U ε (z) = Kε P (z w) dµ ε (w) .
Let us set:

(4.2) f ε (z) = Kε L(z w) dµ ε (w) = U ε (z) + iV ε (z) , with V ε (z) = Kε Q(z w) dµ ε (w) .
Then, if I ε is the energy of µ ε , one has (see [START_REF] Ross | The classical Dirichlet space, Recent advances in operatorrelated function theory[END_REF], Section 4)

I ε = 1+ ∞ n=1 | µε(n)| 2 n
, where µ ε (n) = T w n dµ ε (w) is the n-th Fourier coefficient of µ ε , and:

(4.3) f ε ∈ D and f ε 2 D = I ε . Note that f ε D ≥ 1.
We claim that there exist δ > 0 and 0 < r < 1 such that:

(4.4) z ∈ D and dist (z, K) ≤ δ =⇒ U ε (rz) ≥ I ε /2
Indeed, let P a (t) =

(4.5) U ε (z) = π -π U ε (e it ) P z (t) dt 2π .
Let now δ ≤ ε/4, to be adjusted later, and take 1δ ≤ r < 1. Suppose that dist(z, K) ≤ δ, with z ∈ D, and let u ∈ K such that |z -u| ≤ ε/4. Note that then |rz -u| ≤ (1r) + |z -u| ≤ ε/2. It follows from (4.5) that:

I ε -U ε (r z) = π -π [I ε -U ε (e it )] P rz (t) dt 2π
(it is useful to recall that U ε (z) ≤ I ε for every z ∈ C). Set:

J 1 =
|e it -rz|≤ε/2 For the integral J 1 , we have:

|e it -u| ≤ |e it -rz| + |rz -u| ≤ ε ;
therefore e it ∈ K ε . Since U ε = I ε Lebesgue-almost everywhere on K ε , by Frostman's Theorem, we get J 1 = 0.

For the integral J 2 , we have:

P rz (t) ≤ 2(1 -r |z|) (ε/2) 2 ≤ 2 (1 -r) + r(1 -|z|) (ε/2) 2 ≤ 4δ (ε/2) 2 = 16δ ε 2 ;
hence (since U ε (e it ) ≥ 0):

J 2 ≤ 16δ ε 2 I ε .
Therefore, if we choose 0 < δ ≤ ε 2 /32, we get:

0 ≤ I ε -U ε (r z) ≤ I ε /2 ,
which gives (4.4). Now, as Cap K = 0, we know from (1.9) that lim ε→0 + I ε = ∞, and we can adjust a sequence ε j → 0 + so that: (4.6)

I εj ≥ 4 j 6 .

Using (4.4), we find two sequences (δ j ) j and (r j ) j , with 0 < δ j → 0 and 1 > r j → 1, such that, for every j ≥ 1, (4.7)

z ∈ D and dist (z, K) ≤ δ j =⇒ U εj (r j z) ≥ I εj /2.

Finally, let us set:

(4.8) f j (z) = f εj (r j z) and (4.9)

f = U + iV = 1 + ∞ j=1 j -2 f j f j D •
The series defining f is absolutely convergent in D. Note that f (0) is real. We now have: 1) f is continuous on D \ K. Indeed, let z ∈ D\K. Then, dist (z, K) > 0 and there exists a neighbourhood ω of z in D, an integer j 0 = j 0 (z) and a positive number δ > 0 such that: w ∈ ω and j ≥ j 0 =⇒ dist (r j w, K εj ) ≥ δ.

We then have, for w ∈ ω and j ≥ j 0 : 3) lim z→K,z∈D U (z) = ∞. Indeed, let A > 0. Take an integer j ≥ A and suppose that dist (z, K) ≤ δ j . Then, using the positivity of the U ε k 's as well as (4.3), (4.6) and (4.7), we have:

U (z) ≥ j -2 U εj (r j z) f εj D ≥ j -2 I εj /2 I εj ≥ j ≥ A.
This ends the proof of our claims, and of Lemma 4.3.

To end this paper, let us mention the following version of the classical Rudin-Carleson Theorem. Though it is not the main subject of this paper, it has the same flavor as Theorem 4.2. We do not give a proof, but only mention that it can be obtained by mixing the proofs of Theorems III.E.2 and III.E.6 in [START_REF] Wojtaszczyk | Banach spaces for analysts[END_REF] (see pages 181-187).

Theorem 4.4 Let K be a compact subset of T with Cap K = 0. Given any continuous strictly positive function s ∈ C(T) equal to 1 on K, we can find, for every h ∈ C(K) and every ε > 0, a function f ∈ A(D) ∩ D such that f |K = h and:

|f (θ)| ≤ (1 + ε) h ∞ s(θ) , ∀θ ∈ T ; f D ≤ (1 + ε) h ∞ .

Lemma 2 . 2

 22 Let ϕ ∈ D * be an analytic self-map of D. Then C ϕ is Hilbert-Schmidt on D * if and only if

Proposition 2 . 3

 23 There exists an analytic self-map σ of D, belonging to D * and to the disk algebra A(D), such that σ(1) = 1 and |σ(ξ)| < 1 for ξ ∈ ∂D \ {1} and such that the associated composition operator C σ is Hilbert-Schmidt on D * .

Theorem 3 . 4

 34 There exists a Schur function ϕ such that C ϕ is compact on D * , but in no Schatten class S p (D * ).

γ n z n , with γ n = 1 /

 1 (2 |n|) if n = 0 , and γ 0 = 1 .

  |f εj (w)| = Kε j log e r j wu dµ εj (u)since µ εj is a probability measure supported by K εj . Therefore, the series defining f is normally convergent on ω since its general term is dominated by j -2 C on ω. Since the functions f j are continuous on D, this shows that f is continuous at z.2) U (z) := Re f (z) ≥ 1.This is obvious since, for every z ∈ D,U ε (z) := Re f ε (z) =Kε log e |z -u| dµ ε (u) ≥ 0 .

  1-|a| 2 |e it -a| 2 be the Poisson kernel at a ∈ D. Since U ε is harmonic in D and integrable on T ([4], Proposition 19.5.2), one has, for every z ∈ D:

* Supported by a Spanish research project MTM 2009-08934.


Dr Thibault Mura
email: t-mura@chu-montpellier.fr

Cécile Proust-Lima

Tasnime N Akbaraly

Hélène Amieva

Christophe Tzourio

Hugues Chevassus

Marie-Christine Picot

Hélène Jacqumin-Gadda

Claudine Berr

Mura Thibault

PROUST-LIMA Cécile

Ph.D Akbaraly Tasnime

Picot Marie-Christine

Ph.D Jacqumin-Gadda Hélène

Chronic use of benzodiazepines and latent cognitive decline in the elderly: Results from the Three-city study

Keywords: cognitive decline, cognitive impairment, benzodiazepines, latent process, longitudinal study

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

INTRODUCTION

In Europe, benzodiazepines are among the most widely prescribed psychotropic drugs, especially in the elderly population in which benzodiazepines are used for extended periods of time compared to younger age ranges [START_REF] Morgan | Prevalence, frequency, and duration of hypnotic drug use among the elderly living at home[END_REF][START_REF] Ohayon | Use of psychotropic medication in the general population of France, Germany, Italy, and the United Kingdom[END_REF]. While the European authorities and Food and Drugs Administration have recommended limiting the duration of treatment to 4 months (Food and Drug Administration, 1980;Haute Autorité de Santé, 2007;National Health Service Grampian, 2008), some European studies have shown that almost 25% of elderly subjects reported daily use of benzodiazepines for more than 6 months [START_REF] Lechevallier | Benzodiazepine use in the elderly: the EVA Study][END_REF]. Alterations in the pharmacokinetics and pharmacodynamics of benzodiazepines, combined with age-related decrease in the reserve of the central nervous system are liable to lead the elderly to be particularly sensitive to the cognitive side effects of benzodiazepines.

Although acute effects of occasional use of benzodiazepines on psycho-motor memory and attention performances have been demonstrated [START_REF] Buffett-Jerrott | Cognitive and sedative effects of benzodiazepine use[END_REF][START_REF] Curran | Benzodiazepines, memory and mood: a review[END_REF][START_REF] Stewart | The effects of benzodiazepines on cognition[END_REF], less is known regarding the effect of the long-term impact on cognitive outcome. Some studies have suggested a cross-sectional deleterious impact of longterm benzodiazepine use on memory performance [START_REF] Bierman | The effect of chronic benzodiazepine use on cognitive functioning in older persons: good, bad or indifferent?[END_REF][START_REF] Hanlon | Benzodiazepine use and cognitive function among community-dwelling elderly[END_REF], fluid intelligence [START_REF] Bierman | The effect of chronic benzodiazepine use on cognitive functioning in older persons: good, bad or indifferent?[END_REF] and global cognition [START_REF] Paterniti | Long-term benzodiazepine use and cognitive decline in the elderly: the Epidemiology of Vascular Aging Study[END_REF], but the longitudinal effects of benzodiazepines on cognitive change with age (cognitive decline) are more difficult to assess and have led to conflicting results [START_REF] Verdoux | Is benzodiazepine use a risk factor for cognitive decline and dementia? A literature review of epidemiological studies[END_REF]. Whereas one study highlighted accelerated cognitive decline [START_REF] Paterniti | Long-term benzodiazepine use and cognitive decline in the elderly: the Epidemiology of Vascular Aging Study[END_REF], two others failed to find any association [START_REF] Dealberto | Psychotropic drug use and cognitive decline among older men and women[END_REF][START_REF] Lagnaoui | The risk of cognitive impairment in older community-dwelling women after benzodiazepine use[END_REF]. The inconsistency of these findings may be due to difficulties in assessing adequately age-related cognitive change.

Indeed, in these studies the use of arbitrary cut-offs to define subjects with or without cognitive decline may constitute a major limitation. Furthermore, statistical approaches using differences between two points when studying cognitive change may lead to biased results due to the metrological properties of neuropsychological tests not taken into account [START_REF] Galasko | Measuring cognitive change in a cohort of patients with Alzheimer's disease[END_REF][START_REF] Proust-Lima | Misuse of the Linear Mixed Model When Evaluating Risk Factors of Cognitive Decline[END_REF] (i.e. ceiling effects, varying sensitivity to change, also called curvilinearity). The classical statistical alternative which consists in using a linear mixed model does not correct this bias.

In the present paper, we aimed to analyze the association between chronic benzodiazepine use and cognitive outcomes in subjects over 65, by distinguishing the crosssectional effects of benzodiazepines on cognitive level, and the longitudinal effects on cognitive decline. To do this, we applied a nonlinear multivariate mixed model involving a latent cognitive process underlying several neuropsychological tests considered simultaneously and measured repeatedly over time. This model makes it possible to account for the metrological properties of tests (curvilinearity and ceiling/floor effects) when estimating the effects of a covariate, and distinguishes the global effect on cognition and the specific effects on the cognitive domains covered by neuropsychological tests [START_REF] Proust-Lima | Gender and education impact on brain aging: a general cognitive factor approach[END_REF][START_REF] Proust | A nonlinear model with latent process for cognitive evolution using multivariate longitudinal data[END_REF].

EXPERIMENTAL PROCEDURES

Study Design & Participants:

Data came from the multi-site prospective "Three City Study" (E3C), a cohort study of 9,294 community-dwelling persons aged 65 years and over, recruited from the electoral rolls of three French cities between 1999and 2001(The 3C Study Group, 2003). The study protocol was approved by the Ethical Committee of the University-Hospital of Bicêtre (France) and written informed consent was obtained from each participant. The cohort was followed up at 2, 4 and 7 years with face-to-face interviews and medical examinations in a medical center or at home. Because the definition of chronic use of benzodiazepines required data on drug consumption from the baseline and 2-year follow up, we excluded subjects who (Figure 1): 1) (N=970) had reported a change in benzodiazepine use between the baseline and the 2-year follow up; 2) were dead (N=274) or missing (N=918) at the 2-year follow-up; or 3) (N=327) were diagnosed with prevalent or 2-year incident dementia cases. Subjects were not excluded when these events occurred at 4 or 7 years of follow-up. Among these 6,805 subjects, 212 who were not subjected to any measurement for each neuropsychological test during their follow-up, and 1,398 subjects with missing data for baseline adjustment covariates (listed in Table 1) were excluded from the analysis. The present report was finally based on 5,195 participants.

Compared to those excluded, the participants included in the present analysis were younger, had higher occupational grades, practiced more frequently regular physical activity, were less likely to report depressive or anxious symptoms and performed better in cognitive tasks at baseline. (Data not shown)

Data Collection

Neuropsychological evaluation: The cognitive tests were administered by trained psychologists at baseline and repeated at the 2, 4 and 7-year follow-up visits.

-The Mini Mental State Examination (MMSE) [START_REF] Folstein | Mini-mental state". A practical method for grading the cognitive state of patients for the clinician[END_REF] was used as an index of global cognitive performance. The scores range from 0 to 30.

-The Benton Visual Retention Test (BVRT) [START_REF] Benton | Manuel pour l'application du test de rétention visuelle[END_REF], which evaluates immediate visual memory, consists in the presentation of a stimulus card displaying a geometric figure for 10 seconds after which individuals are asked to identify the initial figure from 4 possibilities. Fifteen figures are successively presented and scores range from 0 to 15.

-The Isaacs Set Test (IST) [START_REF] Isaacs | The Set test as an aid to the detection of dementia in old people[END_REF] consisting in generating words belonging to 4 semantic categories (cities, fruits, animals and colors) in 15 seconds, measured mostly semantic verbal fluency abilities but also the speed of verbal production. The scores range from 0 to 66.

-The Trail Making Test, forms A and B (TMT-A and TMT-B) [START_REF] Reitan | Validity of the Trail Making Test as an indicator of organic brain damage[END_REF], evaluates attention, visuomotor processing speed and mental flexibility capacities. TMT-A assesses attention and visuomotor processing speed by requiring patients to connect numbered circles in sequence. TMT-B assesses attention and visuomotor processing speed, but also demands mental flexibility as it requires patients to connect circles by alternating numerical and alphabetical sequences. The average number of correct displacements in 10 seconds can be calculated for each subject ((number of correct items/total time)*10). The range observed varied from 0 to 13.4 for TMT-A, and from 0 to 7.8 for TMT-B.

Assessment of benzodiazepine use: At each phase of examination, an inventory of all drugs (prescription and over-the-counter drugs) taken more than once a week during the preceding month was included in a standardized interview. Medical prescriptions and, where feasible, the medications themselves were checked by the interviewer. No data on dose, duration of treatment or reason for prescribing the drug were collected. Drugs were recorded and coded according to the WHO Anatomical Therapeutic Chemical (ATC) classification system (WHO Collaborating Centre for Drug Statistics Methodology, 2000).

All classes of benzodiazepines were considered: anxiolytic (ATC code: N05BA), hypnotic and sedative (N05CD), antiepileptic (N03AE), and myorelaxing (M03BX07).

Hypnotic drugs derived or connected with benzodiazepines (N05CF) were also considered.

Participants who did not report taking benzodiazepines both at baseline and 2-year follow-up were classified as "non-users". Those who used benzodiazepines both at inclusion and at 2year follow-up were defined as "chronic users". As previously described (figure 1), to avoid classification bias linked to punctual use, subjects who reported a change in benzodiazepine use between the baseline and the 2-year follow up (N=970) were excluded from this analysis.

In order to minimize prescription bias and to limit the interference of acute effect in the longitudinal part of the analysis, cognitive data were censored at 4 and/or 7-year follow-up visits when a subject was diagnosed as demented, or when they reported a change in benzodiazepine use (new prescription or discontinuation).

Covariates: Socio-demographic and health behavior consisted of sex, educational level, socio-professional status, alcohol consumption, tobacco use and physical activity. Health status was ascertained using a number of measurements: glycaemia (normal: fasting glucose<6.1mmol/l ; high 6.1 to 7mmol/l ; diabetes >7mmol/l or treated), cholesterol levels (Hypercholesterolemia: total cholesterol level ≥7.2 mmol/l or treated by lipid lowering agents) and ApoE genotyping, as described previously [START_REF] Ritchie | The neuroprotective effects of caffeine: a prospective population study (the Three City Study)[END_REF], hypertension (defined as having systolic blood pressure >160 mm Hg or diastolic blood pressure > 95 mm Hg or use of antihypertensive drugs), cardiovascular disease history (self-reported angina pectoris, myocardial infarction or cardio-vascular surgery), insomnia (at least two positive responses in the following issues: poor sleep; recurrent difficulty to fall asleep; frequent waking at night or in the morning; taking medication for this indication). Finally, depressive symptoms, assessed by the Center for Epidemiological Studies-Depression Scale (CES-D, cut-off point 17 for men, 23 for women) [START_REF] Führer | La version française de l'échelle CES-D (Center for Epidemiologic Studies Depression Scale)[END_REF] or antidepressant use, and anxiety symptoms, assessed by the Spielberger inventory trait (we used the 75 th percentile as cut-off to define high anxiety: >43 for men and >48 for women), were also considered.

Statistical analysis

Baseline characteristics were compared using the Khi2 test for qualitative variables and Wilcoxon's non-parametric test for quantitative variables. To analyze the relationship between benzodiazepine use and cognitive trajectory, nonlinear mixed models with latent process for multivariate longitudinal data were used [START_REF] Proust | A nonlinear model with latent process for cognitive evolution using multivariate longitudinal data[END_REF]

Effect of the covariates on the latent cognitive process (referred to as (a)

). The evolution of the common latent cognitive process was modeled using a linear mixed model [START_REF] Laird | Random-effects models for longitudinal data[END_REF], which evaluates changes of a repeated outcome over time (i.e. the latent cognitive process) and accounts for correlation between the repeated measurements of each subject. The linear mixed model included a subject-specific random intercept and random coefficients for time after inclusion (in years of follow-up). A Brownian motion was added to account for individual deviations from this quadratic trend, thus relaxing the parametric form of the model. The time of first assessment of cognitive level for each patient was identified by a covariate and included in the model to adjust for learning effects [START_REF] Jacqmin-Gadda | A 5-year longitudinal study of the Mini-Mental State Examination in normal aging[END_REF].

Test-specific effects of the covariates (Contrasts) referred to as (b):

The test-specific models defined flexible links between the neuropsychological tests and the latent cognitive process. The model assumed that a parameterized test-specific nonlinear transformation of each test was a noisy measurement of the latent cognitive process adjusted for covariates, time, and test-and subject-specific variability. This part of the model also evaluated the specific effect of the covariate on each test after adjustment for the latent cognitive process and the common effect of the covariate on it. Thus, these test-specific effects, called contrasts, were used to evaluate whether the covariate had a different specific impact on each neuropsychological test in addition to its global effect on the latent cognitive process, and quantify this differential impact.

Overall effect on each specific test (referred to as (c)): the (cross-sectional or longitudinal) overall effect of a covariate on each specific test was calculated by adding together the effect of the covariate on the latent cognitive process (a) and the test-specific effect (contrast) (b).

The nonlinear test-specific transformations that link each neuropsychological test with the latent cognitive process were used to account for the global metrological properties of the tests [START_REF] Proust-Lima | Sensitivity of four psychometric tests to measure cognitive changes in brain aging-population-based studies[END_REF]. These transformations are covariate-and-time-invariant parametric functions depending on parameters that are estimated simultaneously with the other parameters of the model. Beta cumulative distribution functions were chosen as flexible transformations. These functions offered a large variety of shapes (concave, convex, sigmoid, or simply linear) and thus accounted for the curvilinearity of the tests. The latent cognitive process was defined in [0, 100] where 0 and 100 corresponded respectively to the minimal and maximal latent cognitive level. The complete methodology was detailed elsewhere [START_REF] Proust-Lima | Gender and education impact on brain aging: a general cognitive factor approach[END_REF][START_REF] Proust | A nonlinear model with latent process for cognitive evolution using multivariate longitudinal data[END_REF].

All models were adjusted for the age of the participants at inclusion in the study, and its interaction with time after inclusion. Covariates marginally associated (p < 0.05) with chronic use of benzodiazepines in a univariate analysis were considered to be potential confounders of the benzodiazepine-cognition relationship and were first included in the multivariate analyses. Only covariates associated with latent cognitive level or latent cognitive change were finally retained in the model. Further analyses were conducted to assess the effects of benzodiazepines according to their different categories according to ATC classification. Similar analyses were repeated by distinguishing anxiolytic benzodiazepines (ATC Code: N05BA) from sedative or hypnotic benzodiazepines (N05CD or N05CF).

Due to missing data (1,034 subjects without anxiety score), the main analyses were performed without adjustment for anxiety. However, sensitivity analyses were performed, with this adjustment, in order to exclude any confounding bias due to this covariate. Statistical tests were performed at the conventional two-tailed α level of 0.05. Data were analyzed using SAS (Cary, North Carolina) statistical software and a FORTRAN90 executable for the nonlinear mixed model with latent process (program NLMULTIMIX available at http://biostat.isped.u-bordeaux2.fr/).

RESULTS

Characteristics of participants:

969 of the 5,195 participants included in the present analyses were chronic benzodiazepine users. The mean age was 73.47 years old (SD = 5.1) with women making up 59.9% of the sample. At baseline, 69.9% (n=677) of chronic users were taking benzodiazepines classified as anxiolytic, 7.4% (n=72) as sedatives, 27.0% (n=262) as hypnotic drugs derived or connected with benzodiazepines, 4.1% (n=40) as antiepileptic, and 2.0% (n=19) as myorelaxing.

The characteristics of the participants as a function of chronic benzodiazepine are detailed in Table 1. Compared to non users, chronic users were more likely to be women, older, and less likely to have higher level of education or occupational grade, smoke, or practice regular physical activity. Chronic users were more likely to report cardiovascular antecedents, depressive symptoms, and have hypertension or hypercholesterolemia. Chronic users also presented significantly lower cognitive performances at baseline compared to nonusers whatever the cognitive test considered.

Cognitive latent process modeling:

The estimated transformation between each test and the latent cognitive process are displayed in figure3. Except for IST, transformations indicate nonlinear relationships between neuropsychological tests and the latent process. For MMSE, Figure 3 shows that the maximum value of the test was reached for a value of the latent process (approximately 85) lower than its maximum (100), indicating a strong ceiling effect. Furthermore, the concave shape of the MMSE transformation shows that a decline of MMSE does not correspond to the same intensity of decline of the latent process over the whole range of the test. Indeed, a loss of 10 points in the latent cognitive process scale between 70 and 60 represents a loss of 1 MMSE point, whereas the same loss between 40 and 30 corresponds to a loss of 3 MMSE points, suggesting that: 1) changes in MMSE cannot be interpreted without accounting for the initial value, and 2) MMSE is more appropriate for detecting decline in individuals with low or moderate cognitive levels rather than in individuals with a high cognitive level.

The BVRT shape is also concave with a smaller ceiling effect. The shape of IST is quite linear, revealing that a decline in this test represents roughly the same intensity of decline for the latent process whatever the initial cognitive levels. TMTA and TMTB present convex shapes, indicating that these tests are more sensitive to cognitive changes in high levels of cognition than in low ones. A considerable floor effect is observed for TMTB.

Association between chronic benzodiazepine use and cognitive trajectories:

The results from the nonlinear mixed model with a latent cognitive process (Table 2) showed that chronic benzodiazepine use was significantly associated with cross-sectional lowering of the latent cognitive level both in crude analyses (benzodiazepines=-2.51 p-value<0.001) and in adjusted analyses (benzodiazepines=-1.79 p-value<0.001), (Part (a)-Table 2). As shown by test-specific contrasts (Part (b)-Table 2), this cross-sectional association between benzodiazepines and lower cognitive performance was significantly more pronounced for immediate visual memory (BVRT) and for attention and visuomotor processing speed (TMT-A), but significantly less so on the global cognitive functioning measured by MMSE. The overall cross-sectional effect of benzodiazepines (c) on each specific test, estimated by adding together the global effect (a) and the test-specific effects (contrast) (b), are presented in Figure 4 and show that chronic benzodiazepine use was significantly associated with lower performance in each test, except on MMSE.

Regarding cognitive change over time, we did not find any longitudinal association between chronic benzodiazepine use and the acceleration of latent cognitive decline (a) in both the crude and adjusted models (respectively benzodiazepines*time=-0.06 p-value=0.11 and benzodiazepines*time=0.010 p-value=0.812). Consistent with this result, we did not find any longitudinal test-specific effect (part (b) of table 2), or overall longitudinal effect (c) for each test (figure 4) regarding the link between chronic use and cognitive decline. Sensitivity analyses adjusted for the anxiety score led to the same conclusions (data not shown).

When investigating the effect of benzodiazepines by separating sedatives from anxiolytics using ATC codes (table 3), after adjustment we found a significant lowering of the latent cognitive level for chronic users of both classes of benzodiazepines, with a greater effect observed in chronic users of anxiolytics. However, we did not find any association with accelerated latent cognitive decline for these two classes.

DISCUSSION

The present study, carried out on a large, general elderly population followed-up over 7 years, focused on investigating the impact of chronic benzodiazepine use on cognitive trajectories, by measuring cognition with a wide range of neuropsychological tests, and by modeling a common latent cognitive process. This study emphasized predominant impairment of cognitive level in chronic benzodiazepine users regarding their immediate visual memory and visuomotor processing speed, after adjustment for a wide range of socio-demographic, health behavior and health status factors. However, our results did not highlight an impact of chronic use on accelerated cognitive decline in this elderly population.

Our results differ from those reported in the EVA Study (Paterniti et al, 2002) (n=1,176 subject aged from 60 to 70). In this study, subjects who reported benzodiazepines use in three consecutive examinations over the 4-year follow up presented an accelerated decline of cognitive performance in MMSE, and Digit Symbol Substitution Tests. In the EVA study, subjects were younger than in E3C, fewer (only 80 chronic users), and cognitive decline was analyzed using a dichotomous statistical approach (the outcome considered was the percentage of subjects who declined between the baseline and 4-year follow-up). Our study showed that the sensitivity of some tests is different when detecting cognitive change according to initial cognitive level. Thus, a test like MMSE could have a greater propensity to detect a decline in subjects with a low baseline cognitive level, like chronic users, than in subjects with a higher baseline cognitive level, like non-users. The two other studies carried out on this topic [START_REF] Dealberto | Psychotropic drug use and cognitive decline among older men and women[END_REF][START_REF] Lagnaoui | The risk of cognitive impairment in older community-dwelling women after benzodiazepine use[END_REF] and using the same dichotomous statistical approach, did not find any association with cognitive decline. However, these studies produced little evidence and were subject to methodological limitations (few subjects, only one follow-up point, and no account taken of interference with acute effects).

One major issue when studying drug effects in observational studies is indication bias.

Benzodiazepines are frequently prescribed for anxiety and depression, which are both known to impair cognitive function. Concerning the cross-sectional association that we found, these and other potential confounders -like age, level of education, etc... -have been accounted for in the multivariate analysis. Moreover, our results were coherent with effects observed in occasional use of benzodiazepine [START_REF] Buffett-Jerrott | Cognitive and sedative effects of benzodiazepine use[END_REF][START_REF] Curran | Benzodiazepines, memory and mood: a review[END_REF][START_REF] Stewart | The effects of benzodiazepines on cognition[END_REF]. However it is always difficult to be sure that a residual confounding effect does not affect our results.

Concerning the absence of association with the cognitive decline, the direction of the bias could have led to a false association, but not to the attenuation of a true association with cognitive decline. This bias may be responsible for the statistical trend observed in our crude analysis (beta=-0.06 pvalue=0.10), a trend which is not confirmed in the multivariate analyze, (after adjustment beta=0.01 pvalue=0.81) suggesting its correction by the inclusion in the model of confounders such as anxiety or depression.

Another type of indication bias is the "depletion of susceptible effect" [START_REF] Lagnaoui | Benzodiazepine use and risk of dementia: a nested case-control study[END_REF][START_REF] Moride | Evidence of the depletion of susceptibles effect in nonexperimental pharmacoepidemiologic research[END_REF]. This bias is the consequence of ceasing benzodiazepine prescriptions for subjects with prodromal or incident dementia and may explain the protective effect of benzodiazepines on dementia reported by some studies [START_REF] Fastbom | Benzodiazepines may have protective effects against Alzheimer disease[END_REF]. It can be illustrated by a recent publication on the discontinuation of psychotropic drugs after initiation of memantine in dementia [START_REF] Vidal | Evaluation of the impact of memantine treatment initiation on psychotropics use: a study from the French national health care database[END_REF]. In the present analysis, we tried to limit this bias by keeping in the analysis subjects who reported a change (cessation or new prescription) in benzodiazepine use with 4 or 7-year follow up, and by considering in the analysis only their cognitive data before changing. We also kept incident cases of dementia with 4 or 7-year follow up, and in the same way, we censored their cognitive data at the time of diagnosis. This method of selecting subjects also allowed dealing with another type of bias, which was the interference of acute cognitive effect of benzodiazepines in the longitudinal study of cognitive change.

Indeed, the real impact on cognitive decline due to chronic benzodiazepine use should be distinguished from the immediate change of cognitive level induced by the initiation or the discontinuation of benzodiazepines and thus by the appearance or disappearance of their acute cognitive effects [START_REF] Dealberto | Psychotropic drug use and cognitive decline among older men and women[END_REF]. This interference is also the reason why we had to exclude subjects who reported a change in their benzodiazepine use between baseline and 2 year follow-up (cognitive changes could not be interpreted as a chronic effect of benzodiazepines, or as the initiation or discontinuation of their acute effects).

A potential limitation of our study is the exclusion of some subjects: those who had only one point of follow-up (with no evaluable cognitive decline) and those with missing data. Some cognitive data were also censored to avoid indication bias or interference with acute effect of benzodiazepine (cf. supra). It is always difficult to know the exact influence of these potential selection biases, and such potential diluting factors can complicate the generalization of a lack of difference between groups. However, it seems likely that, if any exists, a major effect of benzodiazepines on cognitive decline would have been detected in spite of these factors.

In our study subjects were categorized on the basis of current use of drugs. Information on the dose and on the treatment history between successive follow-up examinations was not available. A French study carried on elderly [START_REF] Lechevallier | Benzodiazepine use in the elderly: the EVA Study][END_REF] showed that users who reported taking benzodiazepines at one visit, took it actually every day or almost every day (>70% of users) for at least two years (>75% of users), and in a normal range of recommended doses. Our study defined chronic users from the consumption of benzodiazepines reported at two consecutive visits. Thus it seems reasonable to assume that proportions of daily and/or long-term consumption are higher in our study and that subjects defined as chronic users have a high probability to have been treated between visits. Furthermore, in order to investigate, at least partially, the possibility of a different effect between a daily single low dose to aid sleep and repeated higher doses to manage chronic anxiety, we performed a complementary analysis by distinguishing anxiolytic benzodiazepines, from sedative benzodiazepines. In this analysis, we found a more pronounced cross-sectional lowering of the cognitive performance for subjects taking anxiolytics than for those taking sedatives. However, no such effect was evidenced or suspected on the longitudinal part.

Our study is the first to propose a quantitative approach for studying the association between benzodiazepine use and cognitive decline. The statistical model handles unbalanced repeated measurements of several bounded quantitative outcomes, and takes into account their metrological properties [START_REF] Proust | A nonlinear model with latent process for cognitive evolution using multivariate longitudinal data[END_REF]. This gave us the opportunity to show that the five tests of the neuropsychological battery had different sensitivities to cognitive changes within the entire range of cognition, and thus that most of these tests (MMSE, TMT, BVRT) should not be interpreted without accounting for their initial value. We were able to take these properties into account to construct a latent cognitive process covering the entire range of cognition to evaluate the effect of benzodiazepines. It is also noteworthy that this latent cognitive process was defined using a set of standardized neuropsychological tests used to assess different domains of cognition (visual memory, verbal fluency, attention, visuomotor processing speed and mental flexibility capacities), and could be interpreted as a global cognitive factor. One of the strengths of this model is that it maximizes the power of the analysis by analyzing simultaneously several neuropsychological tests with different metrological properties. Moreover, it avoids the misspecification of the linear mixed model when evaluating predictors of cognitive decline [START_REF] Proust-Lima | Misuse of the Linear Mixed Model When Evaluating Risk Factors of Cognitive Decline[END_REF]. In consequence, in view of the methodology and of the number of subjects included in our analyses, our study is, to our knowledge, the most powerful analysis conducted on this topic.. The cognitive decline observed in the seven years of our study was highly significant (beta=-0,125 p<0.001) with a major interaction with age (Interaction= -0,395 p= <0.001). The rate of decline in our study was of 0.9% of decline per year at 75 years old and 1.7% per year at 85 years old. These rates were very similar to those observed in other studies in general population (for review see [START_REF] Park | A systematic review of cognitive decline in the general elderly population[END_REF]. The improvement of cognitive functioning observed in both groups at the two-year follow up (i.e. variable "first passing effect") was probably due to practice and/or learning effects (Collie et al, 2003). Similarly to some others studies (Collie et al, 2003;[START_REF] Jacqmin-Gadda | A 5-year longitudinal study of the Mini-Mental State Examination in normal aging[END_REF] the observed improvement was limited to the second testing.

Although the present paper suggests that long-term benzodiazepine use did not have a significant longitudinal effect on cognitive decline, cross-sectional impairment of cognitive performance is an adverse effect which should not be minimized. Indeed, this impairment could result in the depletion of cognitive reserves and precipitate the onset of dementia, thereby explaining its increased incidence observed in chronic users in some studies [START_REF] Lagnaoui | Benzodiazepine use and risk of dementia: a nested case-control study[END_REF][START_REF] Wu | The association between dementia and long-term use of benzodiazepine in the elderly: nested case-control study using claims data[END_REF]. Furthermore, despite the possible recovery of cognitive functions after withdrawal from long-term benzodiazepine use, this recovery after withdrawal could be incomplete with some remaining impairment in some areas of cognition [START_REF] Barker | Persistence of cognitive effects after withdrawal from long-term benzodiazepine use: a meta-analysis[END_REF].

Finally other problems associated with prolonged benzodiazepine therapy, such as withdrawal difficulty, dependence or falls and fractures must also be considered in the prescription renewal of benzodiazepines in elderly people [START_REF] Cumming | Epidemiology of medication-related falls and fractures in the elderly[END_REF][START_REF] Madhusoodanan | Safety of benzodiazepines in the geriatric population[END_REF].

The present study did not provide proof of an absence of effect of benzodiazepines chronic use on cognitive decline. Indeed, in observational studies, the possibility remains that unmeasured confounders or uncontrolled selection biases -due to missing data, death, lost of follow-up-may partly explain the absence of association observed, and this precludes any conclusion of absence of causality. However, the sample size and the duration of the study, the extensive adjustment we made, the robustness of the results after taking into account the potential confounding factors, and the use of an elaborated model constitute strong arguments that increase the likelihood of our observations. These observations suggest that chronic use of benzodiazepines is associated with poorer cognitive performance but not with an accelerated cognitive decline with age. The potential absence of a longitudinal effect of benzodiazepines should not hide the fact that a cross-sectional cognitive effect could exists and persists with chronic use, and that other adverse effects have to be considered in the long-term prescription of benzodiazepines, especially in the elderly.

Bibliography

* assessed by the Spielberger inventory trait (we used the 75 th percentile as cut off) on 3426 non users and 735 chronics users of Bzd. **TMTA and TMTB: average number of good displacements in 10 seconds.

Cross-sectional

Sedative

-1,007 0,470 0,031

Longitudinal

Sedative *Time * -0,028 0,100 0,779

Cross-sectional Anxiolytic

-1,854 0,298 <0,001

Longitudinal

Anxiolytic *Time * 0,069 0,055 0,209

Table 3: Impact of chronic benzodiazepines use on the latent cognitive process and its change with time, when distinguishing sedative from anxiolytic classes of BZD.

Results from a nonlinear mixed model with a latent process. * Time variable represents the number of years after inclusion. † Age at inclusion was included in the model as decades after 65 years old: (age-65)/10; ‡ Estimates adjusted for center, gender, education, socio-professional status, depression, insomnia, high blood pressure, hypercholesterolemia, tobacco, alcohol and sports activity. .

  (illustrated in figure 2). The statistical model assumes that the correlation between the five neuropsychological tests is induced by a latent cognitive process representing the common factor of the neuropsychological tests. The model is divided into two parts estimated simultaneously: (a) a linear mixed model describes change over time in the latent cognitive process and evaluates the cross-sectional and longitudinal common effects of covariates on this latent cognitive trajectory, and (b) test-specific measurement models link each administration of the neuropsychological tests with the latent cognitive process, by accounting for cross-sectional and longitudinal test-specific associations with covariates and metrological properties of the tests.

Figure 1 :

 1 Figure 1: Flow chart diagram mapping the selection of the 5195 3C Study participants included in the present analyses

  (a) a linear mixed model describes change over time in the latent cognitive process and evaluates the global effect of covariates on this latent cognitive trajectory (b) test-specific measurement nonlinear models relates each administration of the neuropsychological tests with the latent cognitive process, and evaluates specific effects of the covariates on each test in addition to their global effect on the latent cognitive process. (c) the overall effect of a covariate on each specific test is calculated by adding together the global (a) and the test-specific effect (b) of the covariate.

Figure 2 :

 2 Figure 2: Conceptualisation of the Nonlinear mixed model with a latent process modelling global cognition from 5 neuropsychological tests.

Figure 3 :

 3 Figure 3: Estimated beta transformations between each neuropsychological test and the latent cognitive process.

Non users N=4226 Chronic users N=969 p-value

	Sex, Women, %	56.0	76.9	<0.001
	Age, Mean(sd)	73.2 (5.0)	74.6 (5.4)	<0.001
	Education
	9 to 12 years, %	29.6	33.8	<0.001
	12 years or more, %	40.9	32.1
	Center
	Montpellier, %	26.7	19.5	<0.001
	Bordeaux, %	18.8	21.9
	Dijon, %	54.4	58.6
	Occupational status status,
	High, %	50.0	35.0	<0.001
	Tobacco
	(current or former smoker), %	40.5	29.2	<0.001
	Alcohol
	No drinker%	18.7	23.6	<0.001
	moderate%	62.9	62.1
	Abuse (>21g H; >14g F)%	18.4	14.2
	Physical activity
	once a week or more%	37.2	24.2	<0.001
	Glycaemia
	Impaired Fasting Glucose%	3.8	3.1	0.595
	Type 2 Diabetes %	8.5	8.5
	Hypercholesterolemia, %	36.1	42.8	<0.001
	ApoE4, %	19.9	17.3	0.064
	High blood pressure, %	57.7	62.2	0.010
	Cardio-vascular history, %	7.7	10.8	0.002
	Insomnia, %	14.5	56.9	<0.001
	Depressive symptoms, %	13.7	40.7	<0.001
	High Anxiety*, %	20.7	39.4	<0.001
	Baseline neuropsychological scores
	MMSE, Mean(sd)	27.6 (1.8)	27.3 (1.9)	<0.001
	BVRT, Mean(sd)	11.8 (1.9)	11.2 (2.0)	<0.001
	IST, Mean(sd)	31.9 (5.2)	30.5 (5.6)	<0.001
	TMT A**, Mean(sd)	5.1 (1.7)	4.5 (1.5)	<0.001
	TMT B**, Mean(sd)	2.4 (1.2)	2.0 (1.1)	<0.001
	Anxiety**, %	39.4	20.7	<0.001

Table 1 :

 1 Characteristics of benzodiazepines chronic users and non users.

			Crude model (N=5195)	Adjusted model ‡ (N=5195)
		Variable	Estimate	SE	P-value	Estimate	SE	P-value
	Effect of the Covariate on latent cognitive process
		Intercept	62,676	0,543		57,497	0,587
		Time *	-0,125	0,029	<0,001	-0,290	0,074	<0,001
		First passing effect	-1,832	0,102	<0,001	-1,926	0,102	<0,001
	(a)	Age † at inclusion Age † at inclusion*Time *	-3,687 -0,395	0,194 0,029	<0,001 <0,001	-3,417 -0,356	0,182 0,029	<0,001 <0,001
		Cross-sectional effect Benzodiazepines	-2,508	0,247	<0,001	-1,789	0,248	<0,001
		Longitudinal effect Benzodiazepines * Time *	-0,065	0,041	0,106	0,010	0,044	0,812
	Cross-sectional test specific effect for benzodiazepines
		IST	0,188	0,261	0,471	0,245	0,257	0,340
		BVRT	-0,875	0,314	0,005	-0,838	0,320	0,009
	(b)	TMTA	-0,467	0,264	0,077	-0,618	0,284	0,030
		TMTB	0,079	0,227	0,727	0,061	0,306	0,843
		MMSE	1,075	0,288	<0,001	1,150	0,283	<0,001
	Longitudinal test specific effect for benzodiazepines*Time
		IST	0,057	0,053	0,282	0,057	0,053	0,288
		BVRT	0,119	0,084	0,158	0,120	0,093	0,197
	(b)	TMTA	-0,108	0,059	0,069	-0,114	0,066	0,082
		TMTB	0,003	0,070	0,970	0,005	0,129	0,972
		MMSE	-0,071	0,069	0,305	-0,067	0,068	0,324
	Results from nonlinear mixed models with latent process.

* Time variable represents the number of years after inclusion. † Age at inclusion was included in the model as decades after 65 years old: (age-65)/10; ‡ Estimates adjusted for center, gender, education, socio-professional status, depression, insomnia, high blood pressure, hypercholesterolemia, tobacco, alcohol and sports activity.

Table 2 :

 2 Effect of Chronic benzodiazepines use on (a) the latent cognitive process and its change with time in the study and on (b) each of the neuropsychological tests through contrasts

		Adjusted model (N=5070)
	Variable	Estimate ‡	SE ‡	P-value
	Effect of the Covariate on latent cognitive process
	Intercept	56,960	0,584
	Time *	-0,331	0,064	<0,001
	First passing effect	-1,978	0,105	<0,001
	Age † at inclusion	-3,514	0,186	<0,001
	Age † at inclusion*Time *	-0,360	0,029	<0,001
	(a)

Predicted mean trajectories, including a first passing effect, for a women aged of 65 years old, recruited in Bordeaux, with a level of education below 9 years, a low socio-professional Figure 4: Overall effect (c) of benzodiazepines (Bzd) chronic use on each specific neuropsychological test and predicted mean trajectories after inclusion status, no hypercholesterolemia, no insomnia, no depression, no alcohol abuse and no physical activity. Time variable represents the number of years after inclusion.