Andreas Hartmann
email: andreas.hartmann@math.u-bordeaux1.fr

Equipe D'analyse

G Éom Étrie

POINTWISE MULTIPLIERS IN HARDY-ORLICZ SPACES, AND INTERPOLATION

Keywords: 1991 Mathematics Subject Classification. 30H05, 46E30, 30E05 Multipliers, Hardy-Orlicz spaces, free interpolation, rearrangement invariant spaces. 1

We study multipliers of Hardy-Orlicz spaces H Φ which are strictly contained between p>0 H p and so-called "big" Hardy-Orlicz spaces. Big Hardy-Orlicz spaces, carrying an algebraic structure, are equal to their multiplier algebra, whereas in classical Hardy spaces H p , the multipliers reduce to H ∞ . For Hardy-Orlicz spaces H Φ between these two extremal situations and subject to some conditions, we exhibit multipliers that are in Hardy-Orlicz spaces the defining functions of which are related to Φ. Even if the results do not entirely characterize the multiplier algebra, some examples show that we are not very far from precise conditions. In certain situations we see how the multiplier algebra grows in a sense from H ∞ to big Hardy-Orlicz spaces when we go from classical H p spaces to big Hardy-Orlicz spaces. However, the multiplier algebras are not always ordered as their underlying Hardy-Orlicz spaces. Such an ordering holds in certain situations, but examples show that there are large Hardy-Orlicz spaces for which the multipliers reduce to H ∞ so that the multipliers do in general not conserve the ordering of the underlying Hardy-Orlicz spaces. We apply some of the multiplier results to construct Hardy-Orlicz spaces close to p>0 H p and for which the free interpolating sequences are no longer characterized by the Carleson condition which is well known to characterize free interpolating sequences in H p , p > 0.

INTRODUCTION

Let D = {z ∈ C : |z| < 1} be the unit disk of the complex plane. For a space of holomorphic functions on D, X ⊂ Hol(D), we define the multiplier algebra of X by Mult(X) := {g ∈ Hol(D) : ∀f ∈ X, gf ∈ X}.

We will consider spaces X containing the constants so that automatically Mult(X) ⊂ X. Multiplier algebras have been studied in different settings. They appear for instance in the context of cyclic functions (see e.g. [START_REF] Brown | Multipliers and cyclic vectors in the Bloch space[END_REF]). Here we will rather be interested in interpolation problems where multipliers come into play for example via the Nevanlinna-Pick property (see e.g. [Ko], [MS], [Se]). In this paper we will not consider the Nevanlinna-Pick property but focus on spaces for which the multiplier algebra is big in the sense that its trace on H ∞ -interpolating sequences contains more than only bounded sequences. (Recall that H ∞ is the space of bounded holomorphic functions on D.) In such a situation it is possible to interpolate bounded sequences on suitable non separated unions of H ∞ -interpolating sequences. This was done in [START_REF] Duren | Interpolation in H p -spaces[END_REF] for Hardy spaces, and a more general result can be derived from [START_REF] Hartmann | Traces of certain classes of holomorphic functions on finite unions of Carleson sequences[END_REF] in so-called (C)-stable spaces. Note (and this will be clear from the definitions below) that if we can interpolate bounded sequences by functions in the multiplier algebra then we can interpolate freely in the initial space.

The spaces we are interested in here are included in the Smirnov class N + . Recall that the Nevanlinna class on D is defined by

N = {f ∈ Hol(D) : sup 0<r<1 1 2π T log + |f (re it )|dt < ∞}.
Here a + = max(0, a) for a real number a. It is well known that functions in the Nevanlinna class admit non-tangential boundary values almost everywhere on T = ∂D. Then

N + = {f ∈ N : sup 0<r<1 1 2π T log + |f (re it )|dt = 1 2π T log + |f (e it )|dt}.
Hardy-Orlicz classes can then be defined by logarithmic convex functions Φ = ϕ • log where ϕ is a positif, increasing, convex function with ϕ(t)/t → ∞:

H Φ = {f ∈ N + : T Φ(|f |)dm < ∞}
(for more precise definitions, see Section 2). In the special situation when ϕ(t) = e pt we obtain the usual Hardy spaces, and when ϕ(t) = t p we obtain so-called big Hardy-Orlicz spaces. It is clear that in the first case Mult(H p ) = H ∞ and in the second case Mult(H Φ ) = H Φ since ϕ(t) = t p satisfies a quasi-triangular inequality so that H Φ is an algebra and hence equal to its multiplier algebra (see also [START_REF] Hasumi | Remarks on Hardy-Orlicz classes[END_REF]Theorem 3.2]). A natural question arising from this observation is to understand how the multiplier algebra changes from H ∞ for Hardy spaces H p (in a sense small Hardy-Orlicz spaces) to H Φ for big Hardy-Orlicz spaces.

Under certain conditions on the defining function ϕ of the Hardy-Orlicz space under consideration H Φ we will find so-called admissible functions allowing the construction of new Hardy-Orlicz spaces that are included (as well as the algebras they generate) in the multipliers of H Φ (Theorem 3.1), or that contain the multipliers of H Φ (Theorem 3.4). Corollary 3.7 shows that for certain scales of Hardy-Orlicz spaces the gap between both inclusions is small. Proposition 3.3 shows that Theorem 3.1 is optimal in a sense, and Proposition 5.1 exhibits a function g contained in the space found in Theorem 3.4 as an upper bound of the multiplier algebra of H Φ 1/2 (here Φ 1/2 (t) = e √ t ) and not multiplying on H Φ 1/2 , thereby showing that Theorem 3.4 is not optimal.

We will also discuss the ordering of the multiplier algebras. Under some technical condition we prove in Proposition 3.3 that the multiplier algebras conserve the ordering of their underlying Hardy-Orlicz spaces. However, Theorem 4.2 shows that this is not the general situation. Surprisingly it turns out that there are very big Hardy-Orlicz spaces for which the multipliers reduce to H ∞ . In particular there exist Hardy-Orlicz spaces for which the ordering of the multipliers is in the opposite direction with respect of the ordering of the initial Hardy-Orlicz spaces.

Let us mention that multipliers of Hardy-Orlicz spaces have previously been considered by Hasumi and Kataoka [START_REF] Hasumi | Remarks on Hardy-Orlicz classes[END_REF], where conditions for H ∞ to contain or to be contained in the multiplier algebra are given, and also by Deeb [START_REF] Deeb | Multipliers and isometries of Orlicz spaces[END_REF]. In [START_REF] Hasumi | Remarks on Hardy-Orlicz classes[END_REF] the authors also give some orderings of Hardy-Orlicz spaces that turn out to be useful in our situation.

The question of multipliers is strongly related in particular to free interpolating sequences. Indeed, if we can interpolate bounded sequences on a given sequence Λ = {λ n } n ⊂ D by functions in the multiplier algebra, then Λ is a free interpolating sequence (for this and the following comments, precise definitions and results can be found in Section 6). Let us recall some facts on interpolating sequences. It is well known that the Carleson condition inf λ |B Λ\{λ} (λ)| > 0 characterizes free interpolating sequences for H p , p ∈ (0, ∞], and Hardy-Orlicz spaces included in the scale of H p spaces (see [START_REF] Carleson | An interpolation problem for bounded analytic functions[END_REF], [ShHSh], [START_REF] Kabaȋla | Interpolation sequences for the H p classes in the case p < 1[END_REF], [START_REF]Free interpolation in Hardy-Orlicz spaces[END_REF]). We have already mentioned that in this situation Mult(H p ) = H ∞ . On the other hand, in N, N + , and in big Hardy-Orlicz spaces (e.g. ϕ(t) = t p ), which are actually algebras (and so equal to their multipliers), free interpolating sequences are characterized by the existence of a harmonic majorant of log(1/|B Λ\{λ} (λ)|) (see [START_REF] Hartmann | Interpolation in the Nevanlinna and Smirnov classes and harmonic majorants[END_REF], [START_REF]Interpolation and harmonic majorants in big Hardy-Orlicz spaces[END_REF]). This condition is much weaker than the Carleson condition (which can be restated as saying that log(1/|B Λ\{λ} (λ)|) admits in particular constants as harmonic majorants). For instance separated sequences (with some conditions if we are in big Hardy-Orlicz spaces) are interpolating in these classes.

Our starting point was to know whether there exist Hardy-Orlicz spaces beyond p>0 H p for which the Carleson condition still characterizes the interpolating sequences, which leads us to the following question.

Question 1. Let H Φ be a Hardy-Orlicz space. If the interpolating sequences of H Φ are characterized by the Carleson condition, is it true that H Φ is included in the scale p>0 H p ?

In the light of this question, a first step is to construct examples of Hardy-Orlicz spaces above p>0 H p which are very close to the latter union and which have interpolating sequences that are not Carleson. The key to such a construction is the multiplier algebra of the Hardy-Orlicz space under consideration when this multiplier algebra is strictly bigger than H ∞ . Corollary 5.3 exhibits multipliers of H Φ where e.g. Φ(t) = t 1/ log t is in a sense very close to the defining functions t -→ t p of H p , p > 0. In such a situation it is possible to use ideas of Douglas and Shapiro [START_REF] Duren | Interpolation in H p -spaces[END_REF] to interpolate bounded sequences on suitable non separated unions of Carleson sequences. This yields Corollary 6.9 which claims the existence of a non Carleson sequence which is free interpolating for H Φ when Mult(H Φ ) contains a Hardy-Orlicz space H Ψ that is strictly bigger than H ∞ .

Since there exist large Hardy-Orlicz spaces for which the multipliers reduce to H ∞ (see Theorem 4.2), we can give a more precise version of Question 1.

Question 2. If the multiplier algebra of a Hardy-Orlicz space containing strictly p>0 H p is equal to H ∞ , does it have interpolating sequences that are not Carleson?

The paper is organized as follows. In Section 2, we will introduce the necessary material on Orlicz and Hardy-Orlicz spaces as well as some facts on decreasing rearrangements. The main results on multipliers are presented in Section 3. More precisely we exhibit Hardy-Orlicz spaces that bound below and above the multiplier algebra of a given Hardy-Orlicz space. Orderings of multipliers will be discussed in Section 4. Under some technical condition we will prove that the multiplier algebra inherits the ordering of the underlying Hardy-Orlicz spaces. However we will prove that there are large Hardy-Orlicz spaces for which the multipliers reduce to H ∞ . An important example is discussed in Section 5 to show how far we are from a characterization of the multiplier algebra. Other examples of Hardy-Orlicz spaces coming very close to p>0 H p and having unbounded multipliers will be treated in Subsection 5.2. These examples are important in Section 6 where we apply the multiplier results to the interpolation problem. Using ideas in the spirit of [START_REF] Duren | Interpolation in H p -spaces[END_REF] we will construct Hardy-Orlicz spaces H Φ containing strictly p>0 H p but being very close to this union, and for which there exist non separated unions of Carleson sequences which are interpolating for H Φ .

Finally a word concerning notation. For two expressions u, v depending on the same discrete a continuous variable we will sometimes write

u << v if u = o(v). As usual, u ∼ v means that u = v(1 + ε) (or v = u(1 + ε)) where ε = o(1).
Acknowledgements. Part of this work was presented at a joint PICASSO-GDR AFHA meeting in Marseille. I would like to thank the participants of that meeting, in particular A. Borichev and P. Thomas, for some interesting questions that are maybe answered in this paper.

ORLICZ AND HARDY-ORLICZ SPACES

When discussing Hardy-Orlicz spaces which are strictly bigger than p>0 H p , one can consider logarithmic convex defining functions. This is very natural since convex functions conserve the subharmonicity of log |f | which makes it possible to define Hardy-Orlicz spaces via the existence of harmonic majorants (see [START_REF] Rosenblum | Hardy classes and operator theory[END_REF]). For this reason we will consider in all what follows defining functions of the form ϕ•log where ϕ : R -→ [0, ∞) is a convex, nondecreasing function with lim t→∞ ϕ(t)/t = ∞. To fix the ideas we should set ϕ(-∞) = 0. According to the terminology in [START_REF] Rudin | Function theory in polydiscs[END_REF] such a function is called strongly convex.

With such a function we will associate the Orlicz class on T defined by

L ϕ•log = L ϕ•log (T) = {f measurable on T : T ϕ(log |f |) < ∞}.
In order to simplify the notation, we will also write

Φ = ϕ • log,
and so

L Φ = L ϕ•log .
The functions ϕ or Φ are both called defining function for the Orlicz class (hopefully no confusion will arise in this paper).

It should be noted that the Orlicz class is in general not a vector space (see for instance the example 2 in [RosRov85, p.52] for the case of Hardy-Orlicz classes), and one can define two other spaces. According to the notation in [START_REF] Leśniewicz | On linear functionals in Hardy-Orlicz spaces, I[END_REF] we will call

L * Φ := {f measurable on T : ∃a > 0, T Φ |f | a < ∞}
the Orlicz space, and

L • Φ := {f measurable on T : ∀a > 0, T Φ |f | a < ∞}
the space of finite elements of L * Φ . In [START_REF] Lefèvre | Composition operators on Hardy-Orlicz spaces[END_REF], the latter space was called the Morse-Transue space. Note that L • Φ ⊂ L Φ ⊂ L * Φ , and in general these three classes are different. In order to ensure that L Φ is already a vector space, one sometimes adds another condition to that of a defining function of an Orlicz space: the function ϕ satisfies the ∆2 -condition if ϕ(t + 2) ≤ Mϕ(t) + K, t ≥ t 0 for some constants M, K ≥ 0 and t 0 ∈ R. This condition is formulated in such a way that Φ satisfies the usual ∆ 2 -condition: there exist constants M ′ , K ′ ≥ 0 and s 0 such that for all s ≥ s 0 we have

Φ(2s) ≤ M ′ Φ(s) + K ′ . (2.1) If ϕ satisfies the ∆2 -condition (or Φ satisfies the ∆ 2 -condition), then L Φ = L • Φ = L * Φ . On L *
Φ we can introduce the following functional. For f ∈ L * Φ , let

f Φ := inf{t > 0 : T Φ |f | t dm ≤ 1}.
If Φ is convex, then L * Φ equipped with • Φ is a Banach space (also if we replace T by other measure spaces), see [START_REF] Lindenstrauss | Classical Banach spaces II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]p.120]. The expression J Φ (f ) := T Φ(|f |)dt is sometimes called a modular. It does of course not define a norm in general.

Here are some facts on orderings of Orlicz spaces. Let ϕ 1 and ϕ 2 be two strongly convex functions and set [START_REF] Hasumi | Remarks on Hardy-Orlicz classes[END_REF]Theorem 1.3] where this result is proved for Hardy-Orlicz spaces, but the argument works for Orlicz spaces). The relation [START_REF] Krasnosel | Convex functions and Orlicz spaces[END_REF]Theorem 13.1] in case Φ l , l = 1, 2, convex). Also, if two functions Φ 1 and Φ 2 (or ϕ 1 and ϕ 2 ) are comparable, i.e. there are constants C 1 , C 2 with C 1 ϕ 1 (t) ≤ ϕ 2 (t) ≤ C 2 ϕ 1 (t) for big t, then the corresponding (Hardy-)Orlicz spaces are equal. This allows for instance to replace the defining functions by smooth ones. In all what follows we will thus suppose that the defining functions are sufficiently smooth.

Φ i = ϕ • log, i = 1, 2. Then lim sup t→∞ Φ 1 (t)/Φ 2 (t) < ∞ if and only if L Φ 2 ⊂ L Φ 1 (see
L * Φ 1 ⊂ L * Φ 2 follows from lim t→∞ Φ 1 (t)/Φ 2 (kt) = ∞ for every k > 0 (see
It should be noted that it is possible to construct strongly convex functions ϕ 1 and ϕ 2 for which lim sup ϕ 1 (t)/ϕ 2 (t) = +∞ and lim inf ϕ 1 (t)/ϕ 2 (t) = 0. In such a situation, by the above cited result, no one of the considered Orlicz spaces can be included in the other one.

As in the classical case of L p -spaces, one can associate with L Φ a subclass of boundary limits of a space of holomorphic functions on the disk. Recall that N + is the Smirnov class. The Hardy-Orlicz class is defined as

H Φ = H ϕ•log = {f ∈ N + : T ϕ(log |f (ζ)|) dσ(ζ) < ∞} = N + ∩ L Φ ,
where f (ζ) is the non-tangential boundary value of f at ζ ∈ T, which exists almost everywhere since f ∈ N + . By [START_REF] Rosenblum | Hardy classes and operator theory[END_REF]Theorem 4.18] this definition is equivalent to the definition via the existence of harmonic majorants that we mentioned in the introduction to this section. Also, since H Φ as well as its multiplier algebra are contained in the Smirnov class N + , we have a factorization. Recall that each f ∈ N + can be written as f = IF , where I is an inner function and F is outer in N + . More precisely

F (z) := [f ](z) := exp T ζ + z ζ -z log |f (ζ)|dm(ζ) , z ∈ D, (2.2) and log |f | ∈ L 1 (T). If f ∈ H Φ then F ∈ H Φ and moreover |f | ∈ L Φ (T).
The classical examples are the following. When ϕ(t) = e pt for some p > 0, then H Φ is simply the Hardy space H p , in which case Mult(H Φ ) is just the algebra H ∞ of bounded holomorphic functions on D.

The situation which has been considered in [START_REF]Interpolation and harmonic majorants in big Hardy-Orlicz spaces[END_REF] in connection with free interpolation is when ϕ satisfies a quasi-triangular inequality:

ϕ(a + b) ≤ c(ϕ(a) + ϕ(b)), (2.3)
for some constant c, and all reals a, b ≥ t 0 , t 0 also fixed. A simple example is ϕ(t) = t p . The condition (2.3) is of course related to the ∆ 2 condition for ϕ. In this situation, H Φ is an algebra and its multiplier algebra is of course the algebra itself: Mult(H Φ ) = H Φ . Analogously to the above definitions, we will write H * Φ for the Hardy-Orlicz space, H • Φ for the Hardy-Orlicz space of finite elements (or the Hardy-Morse-Transue space). Again, if Φ satisfies the ∆ 2 condition than all spaces are identical H Φ = H * Φ = H • Φ and we simply write H Φ . We will introduce some conditions for a strongly convex function ϕ. Since we will consider multipliers, we are interested in the integrability of ϕ(log |f | + log |g|). Hence we would like to know if we can add some growth to the argument t of ϕ without changing too much the growth of ϕ. Here is a precise definition.

Definition 2.1. A convex, strictly increasing function ϕ : R -→ R + with lim t→∞ ϕ(t)/t = +∞ is said to satisfy the ∆-condition if there is a c > 1, t 0 ∈ R and a strictly increasing concave function γ : R + -→ R + with lim t→∞ γ(t) = ∞ such that for all t ≥ t γ ϕ(t

+ γ(t)) ϕ(t) ≤ c. (2.4)
A function γ will be called ∆-admissible if (2.4) holds for suitable c and t 0 .

The requirement of γ being concave is not restrictive since if an increasing function γ satisfying (2.4) exists, then we can replace it by a concave one. This condition is stronger than the ∆2 -condition since instead of adding 2 in the argument of ϕ we add a function that can tend to infinity. If ϕ itself already satisfies the standard ∆ 2 -condition (2.1) (which leads us to big Hardy-Orlicz spaces), then we can choose γ(t) = t so that ϕ then satisfies the ∆-condition.

Our model case is

ϕ α (t) = e t α , t ≥ t 0 > 0,
where α ∈ (0, 1). In this case we can construct the optimal function γ: in order to have ϕ α (t + γ(t)) ≤ cϕ α (t) it is necessary and sufficient that t -→ (t + γ(t)) α -t α is bounded (observe that necessarily γ(t) ≤ t). By standard calculus, this is equivalent to

t α α γ(t) t + o γ(t) t ≤ c, t ≥ t γ ,
which happens if and only if

γ(t) ≤ Ct 1-α .
So, we can choose γ α,C (t) := Ct 1-α which meets the requirements of the function γ in the definition of the ∆-condition above, and no ∆ admissible function can grow faster than any γ α,C .

Note that the ∆-condition imposes a restriction on the growth of ϕ: clearly we cannot reach the function ϕ(t) = e t (defining H 1 ), which is natural in view of our results.

When ϕ satisfies the ∆-condition, we will see (Theorem 3.1) that the admissible functions γ allow us to construct subalgebras of multipliers, i.e. algebras which bound the multipliers of H Φ from below. So it is natural to ask whether something sensible can be said about the multipliers when condition (2.4) is not satisfied. Actually, it turns out that if γ is not admissible then the algebras constructed in Theorem 3.1 do no longer bound the multiplier algebra from below. However it seems too ambitious to hope for an upper bound in this situation. Still, under some mild growth condition on the quotient ϕ(t + γ(t))/ϕ(t) we can obtain such an upper bound. Definition 2.2. A convex, strictly increasing function ϕ : R -→ R + with lim t→∞ ϕ(t)/t = +∞ is said to satisfy the ∇-condition if there is a strictly increasing concave function γ : R + -→ R + with lim s→∞ γ(s) = ∞ and an ε > 0 such that for all s ≥ s γ ϕ(s + γ(s)) ϕ(s) ≥ log 1+ε ϕ(s). (2.5)

A function γ will be called ∇-admissible if (2.5) holds for suitable s γ and ε > 0.

Let us discuss the ∇-admissible functions for the model case ϕ α (s) = e s α . The condition (2.5) is equivalent to

e s α ((1+γ/s) α -1) ≥ s (1+ε)α , s ≥ s γ , so that for example γ(s) := γ (log)
α,η (s) := (1 + η)s 1-α log s, s ≥ s γ , with η > 0 works. Of course for "bigger" functions γ the estimate in (2.5) is more easily true. However, as we will see later on, we will use reciprocals of ∇-admissible functions to find upper bounds for the multipliers. Hence we will get more precise bounds with small ∇-admissible functions γ. The reader may check that the function γ (log) α,η is not ∇-admissible for η = 0.

2.1. Decreasing rearrangements. We will need some facts on decreasing rearrangements (for the material of this subsection see for instance [START_REF] Lindenstrauss | Classical Banach spaces II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]). Let us begin by recalling some basic facts.

Let (Ω, Σ, µ) be a measure space (we will only be concerned with T equipped with the usual normalized Lebesgue measure on Borel sets). With a measurable function f on Ω one associates the distribution function

µ f (t) = µ{ω ∈ Ω : |f (ω)| > t}, t > 0,
and the decreasing rearrangement

f * (s) = inf{t > 0 : µ f (t) ≤ s}, s ∈ (0, µ(Ω)).
Note that the decreasing rearrangement of f is a positive function. The main consequence on rearrangement invariant spaces that we will use in the context of (Hardy-)Orlicz spaces is that

T Φ(|f (t)|)dt = 1 0 Φ(f * (t))dt. (2.6) (We have used here that (Φ • |f |) * = Φ • f * since Φ is increasing.) We will also use the fact that when Φ is convexe, then L * Φ is rearrangement invariant [LT, p.120].
The reader should notice that the initial measure space we are interested in, i.e. T equipped with the Lebesgue measure, can be identified with the measure space [0, 1] (equipped with normalized Lebesgue measure) on which the decreasing rearrangement f * is defined. Thus f * is obtained from |f | by a measure preserving mapping α from Ω := T (i.e. Ω :

= [0, 1]) onto itself, so that f * (t) = |f (α(t))|.
Moreover, it is clear that if a function g multiplies on H Φ then so does its outer part (the modular J Φ in Hardy(-Orlicz) spaces does not "feel" the inner part). In all what follows we will thus assume that the multiplier is outer (it could even be assumed that |g| ≥ 1). Let g * be the decreasing rearrangement of a multiplier g, and let α g be a corresponding measure preserving mapping of T (or [0, 1]) onto itself. We have already mentioned that g is automatically in H Φ and so |g| ∈ L Φ . By (2.6) the function g * is also in L Φ , and so we can associate with it the outer function G in H Φ such that |G| = g * a.e. on T.

Lemma 2.3. If g ∈ Mult(H Φ ) then the outer function G defined by |G| = g * a.e. T is also a multiplier on H Φ .

More generally it can be said that for every outer multiplier g and every measure preserving mapping α : T → T, the outer function g α with |g α | = |g • α| a.e. on T is also a multiplier.

Proof. Let α be the measure preserving mapping such that g * = |g • α| a.e. on T. Let f ∈ H Φ with outer part F . Then the outer function with modulus |F • α -1 | is also in H Φ (with same modular J Φ as f ), and

T Φ(|f (ζ)G(ζ)|)dm(ζ) = T Φ(|f (ζ)g(α(ζ))|)dm(ζ) = T Φ(|f • α -1 (ζ)g(ζ)|)dm(ζ) = T Φ(|F (ζ)g(ζ)|)dm(ζ) < ∞
In the later discussions we can (and will) thus suppose that the multiplier is outer, its only singularity is in ζ = 1, and θ → |g(e iθ )| is decreasing in θ on (0, 2π) (2π corresponding to 1).

MULTIPLIERS -UPPER AND LOWER BOUNDS

In this section we will give a general construction to obtain multipliers of a Hardy-Orlicz space with a defining function ϕ satisfying the ∆-condition. More precisely, the ∆-admissible functions γ associated with ϕ allow the construction of defining functions Ψ γ of Hardy-Orlicz spaces contained in the multiplier algebra. Since Mult(H Φ ) is an algebra it is clear that when H Ψγ ⊂ Mult(H Φ ) then also Alg(H Ψγ ) ⊂ Mult(H Φ ). Here Alg(F ) denotes the algebra generated by a family of functions F Then, using the ∇-condition, we will give an inclusion of the multiplier algebra of H Φ in another Hardy-Orlicz space the defining function of which Ψ = ψ • log is associated with ∇admissible functions. Again, since Mult(H Φ ) is an algebra, if it contains f ∈ H Ψ then it contains also all powers f n , n ∈ N, and so does H Ψ . Hence, setting

Ψ [n] (t) = ψ(n log t), the inclusion Mult(H Φ ) ⊂ H Ψ implies that Mult(H Φ ) ⊂ n∈N * H Ψ [n] .
We will discuss both results in the model case ϕ(t) = ϕ α (t).

Let us begin with a lower bound on the multiplier algebra. Theorem 3.1. Let ϕ be a strongly convex function satisfying the ∆-condition and γ a ∆-admissible function. Then

Alg(H Ψγ ) ⊂ Mult(H Φ )
where

Ψ γ = ϕ • γ -1 • log.
Remarks 3.2. 1) Obviously, Mult(H Ψ ) contains the algebra generated by the union over all H Ψγ where γ is admissible for ϕ.

2) In general ψ γ := ϕ • γ -1 does not satisfy ∆2 and so we have to distinguish a priori in the theorem between H Ψγ , H • Ψγ and H * Ψγ . This is of no harm since all these spaces are of course included in Alg(H Ψγ ) (and we are of course interested in the biggest lower bound); see also some comments concerning the ∆2 -condition of ψ γ in the model case at the end of this section.

Proof. Let f ∈ H Φ and g ∈ H Ψγ . Let A := {ζ ∈ T : log |g(ζ)| ≤ γ(log |f (ζ)|)} and A 0 := {ζ ∈ A : log |f (ζ)| ≥ t γ }. Then A 0 ϕ(log |gf |)dm = A 0 ϕ(log |f | + log |g|) ≤ A 0 ϕ (log |f (ζ)| + γ(log |f (ζ)|)) dm ≤ c A 0 ϕ(log |f (ζ)|)dm,
and so the integral on A 0 converges. Since on A \ A 0 , |f | and |g| are bounded (so that ϕ(log |f g|) is bounded), the integral also converges on A.

We will now consider the part of the integral on

B := T \ A. Set B 0 := {ζ ∈ B : γ -1 (log |g(ζ)|) ≥ t γ }. Clearly log |f (ζ)| < γ -1 (log |g(ζ)|) on B. Hence B 0 ϕ(log |f (ζ)| + log |g(ζ)|)dm ≤ B 0 ϕ(γ -1 (log |g(ζ)|) x + log |g(ζ)|)dm ≤ c B 0 ϕ(γ -1 (log |g(ζ)|))dm.
Since by assumption g ∈ H Ψ where Ψ = ϕ • γ -1 • log, the last integral converges. Since on B \ B 0 the functions |f | and |g| are bounded, the integral converges also on B.

Note that if ϕ satisfies the ∆ 2 -condition (the case of big Hardy-Orlicz spaces), then, as we have already mentioned, we can choose γ(t

) = t. Hence Ψ γ (t) = ϕ•γ -1 •log(t) = ϕ(log(t)) = Φ(t),
which confirms that we are in the algebra situation.

In order to show that Theorem 3.1 is sharp we shall prove that if a function γ is not admissible for ϕ, then H Ψγ contains functions that do not multiply on H Φ . Recall that

Ψ γ = ϕ • γ -1 • log.
Proposition 3.3. Let ϕ be a strongly convex function, and let γ be a concave function on R strictly increasing to infinity such that

lim sup t→∞ ϕ(t + γ(t)) ϕ(t) = +∞.
Then there exists g ∈ H Ψγ such that g / ∈ Mult(H Φ ).

Proof. The proof follows some ideas of the proof of [HK88, Theorem 1.3]. By the hypotheses, there exists a sequence (t n ) n such that

     ϕ(t n + γ(t n )) ϕ(t n ) ≥ n, ϕ(t n ) ≥ 2 n n 2 . Set ε n = (n 2 ϕ(t n )) -1 . Clearly ε n ≤ 2 -n so
Then T Φ(|f |)dm = n ϕ(t n )|σ n | = n ϕ(t n )ε n = n 1 n 2 < ∞. Hence f ∈ H Φ .
In the same way, we let g be the outer function the modulus of which takes the values n e γ(tn) χ σn on n σ n and 1 elsewhere.

Then T ϕ(γ -1 (log |g|))dm = n ϕ(t n )|σ n | < ∞ (the reader might have observed that this is equal to T Φ(|f |)dm). Let us compute the modular of their product T Φ(|f g|)dm = T ϕ(log |f | + log |g|)dm = n ϕ(t n + γ(t n ))|σ n | ≥ n nϕ(t n )ε n = 1 n = +∞.
We shall discuss this proposition further on an example in Section 5.

The next result discusses an upper bound of the multiplier algebra via ∇-admissibility.

Theorem 3.4. Let ϕ be a strongly convex function satisfying the ∇-condition and γ a ∇-admissible function. Then

Mult(H Φ ) ⊂ n∈N * H Ψ [n] γ ,
where, as before,

Ψ γ = ϕ • γ -1 • log, and Ψ [n] γ (t) = ϕ(γ -1 (n log t)).
Before proving the theorem, we will cite the following well-known property.

Lemma 3.5. Every positive decreasing function on (0, 1] which is integrable on (0, 1) is necessarily bounded by the function t -→ 1/t on (0, t 0 ) for a suitable t 0 ∈ (0, 1).

Proof of Theorem 3.4. Let g ∈ Mult(H Φ ). As before we will suppose g outer and |g| equal to its decreasing rearrangement. This will allows us to test g against functions in H Φ that approach the maximal possible growth of the class. Since we have identified T with [0, 1], we will set

w(t) = e t log 1+η e t , t ∈ (0, 1], (3.1)
where η ∈ (0, ε) is fixed (ε being the value associated with the ∇-admissible function γ). It is clear that w ∈ L 1 . Let f be the outer function in H Φ such that log |f (e 2πit )| = ϕ -1 (w(t)) a.e. on (0, 1].

Since g is a multiplier on H Φ and |f |, |g| are decreasing on (0, 1], the function

φ := ϕ(log |f g|) = ϕ(log |f | + log |g|)
is decreasing on (0, 1] and integrable on this interval. By Lemma 3.5, we get

φ(t) ≤ e t , t ∈ (0, t 0 )
(where t 0 ∈ (0, 1) is fixed suitably).

Hence

ϕ(ϕ -1 (w) + log |g|) ≤ e t ,
and so

log |g| ≤ ϕ -1 (e/t) -ϕ -1 (w).
Hence

Ψ γ (|g|) = ϕ(γ -1 (log |g|)) ≤ ϕ(γ -1 (ϕ -1 (e/t) -ϕ -1 (w))). (3.2)
Since ϕ satisfies the ∇-condition and γ is admissible we have

ϕ(s + γ(s)) ≥ ϕ(s) log 1+ε ϕ(s), s ≥ s γ , so that s ≥ γ -1 (ϕ -1 (ϕ(s) log 1+ε ϕ(s)) -s), s ≥ s γ .
Applying ϕ to this inequality and choosing s such that w = w(t) = ϕ(s) we obtain w ≥ ϕ(γ -1 (ϕ -1 (w log 1+ε w) -ϕ -1 (w))).

(3.3)

We will check that 1/t ≤ w log 1+ε w. From (3.1), we get

w(t) log 1+ε w(t) = e t
w ≥ ϕ(γ -1 (ϕ -1 (e/t) -ϕ -1 (w))).
We recognize here the right hand side of (3.2) so that

Ψ γ (|g(e it )|) ≤ w(t) = 1 t log 1+ε (1/t)
.

Since w ∈ L 1 , we conclude g ∈ H ψ . By the remarks in the introduction to this section we also have g ∈ H Ψ [n] for every n ∈ N * .

Example.

Let us consider the model case ϕ α , 0 < α < 1. We have already constructed the optimal function γ α,C = Ct 1-α . Obviously, γ -1 α (t) = (t/C) 1/(1-α) , and -α) . This together with Theorem 3.1 yields the first inclusion of the proposition below.

ψ α,C (t) := ϕ α • γ -1 α,C (t) = e dt α/(1-α) = ϕ d α/(1-α) , where d = C -α/(1

Note that Alg(H

Φ α/(1-α) ) = d>0 H ϕ d α/(1-α) •log (one can use that f g = (1/2) (f + g) 2 -f 2 - g 2 and h ∈ H α/(1-α) implies h 2 ∈ H ϕ d α/(1-α) with d = (1/2) α/(1-α) ).
For the second one we introduce another defining function. Set

ϕ (log) α,δ (t) = e δ( t log t ) α/(1-α) , t ≥ t 0 , where δ > 0. Clearly, if β < α/(1 -α) then ϕ β (t) = e t β << ϕ (log)
α,δ (t), t → ∞. Hence by the remarks on orderings of (Hardy-)Orlicz spaces in Section 2

H ϕ (log) α,δ •log H Φ β . (3.4) Proposition 3.6. Let 0 < α < 1. Then Alg(H Φ α/(1-α) ) = d>0 H ϕ d α/(1-α) •log ⊂ Mult(H Φα ) ⊂ δ>0 H ϕ (log) α,δ •log .
Before proving this result, we give the following consequence which is maybe easier to state and follows immediately from this proposition and (3.4).

Corollary 3.7. Let 0 < α < 1 Then

Alg(H Φ α/(1-α) ) = d>0 H ϕ d α/(1-α) •log ⊂ Mult(H Φα ) ⊂ 0<β<α/(1-α) H Φ β .
Corollary 3.7 shows that Theorem 3.4 is optimal in the sense that it allows to separate those Hardy-Orlicz spaces contained in the scale (H Φα ) α>0 and multiplying on H Φ 1/2 from those contained in the scale that do not multiply on H Φ 1/2 . We could of course have replaced

H Φ β by n∈N * H Φ [n] β .
Proof of the proposition. As already indicated, the first inclusion is established by the above discussion. Let us consider the second inclusion. Recall that for ϕ α the function

γ(s) = (1 + η)s 1-α log s is ∇-admissible whenever η > 0. Set Ψ γ = ϕ • γ -1 • log. It can be checked that γ -1 (u) ∼ 1 -α 1 + η u log u 1/(1-α) , u → ∞. So Ψ γ (t) = exp   1 -α 1 + η log t log log t α/1-α (1 + o(t))   , t → ∞.
Since γ is ∇-admissible for arbitrary η > 0 and o(t) is arbitrarily small, we can take

Ψ γ (t) = exp   (1 -δ)C α log t log log t α/(1-α)   .
where δ > 0 is arbitrary and C α = (1-α) α/(1-α) , From Theorem 3.4 we deduce that Mult(H Φ ) ⊂ H Ψγ . And by the general remarks we also have

Mult(H Φ ) ⊂ H Ψ [n] γ
, where

Ψ [n] γ (t) = exp   (1 -δ)C α n log t log(n log t) α/(1-α)   = exp   c log t log n + log log t) α/(1-α)  
with a suitable constant c. Clearly there exist δ 1 , δ 2 such that ϕ

(log) α,δ 1 (log t) ≤ Ψ [n] γ (t) ≤ ϕ (log)
α,δ 2 (log t) from which the remaining inclusion of the proposition follows.

The example ϕ α is quite instructive concerning the behaviour of the multiplier algebra. Clearly the index α/(1-α) that we can associate with ϕ α increases with α (we will see in Proposition 4.1 that for reasonable strongly convex functions -and ϕ α are reasonable in our situation -that the multiplier algebra increases with the space). A crucial point is

α = 1/2. Then ψ 1/2,1 (t) = ϕ 1/2 • γ -1 1/2,1 (t) = ϕ 1 (t) = e t
which is the defining function for H 1 , so that the multiplier algebra of H Φ 1/2 contains Alg(H 1 ) = p>0 H p (and it is contained in H Φ β for any β < 1, and even in smaller Hardy-Orlicz spaces defined by ϕ

(log) α,δ ). When α > 1/2, then by the corollary we have Mult(H α ) ⊂ n∈N * H Φ [n] 1 = p>0 H p . Choos- ing β ∈ (1, α/(1 -α))
we can even deduce that Mult(H α ) ⊂ H Φ β which is extremely small and close to H ∞ . Conversely, if α < 1/2, then since α/(1 -α) < 1, we get ϕ α/(1-α) (t) = o(e pt ) which yields H p ⊂ H Ψ α/(1-α) for every p > 0 and hence p>0 H p ⊂ H Ψ α/(1-α) ⊂ Mult(H Φα ). So, in this case, the multiplier algebra is very big containing every H p , p > 0, and even bigger spaces. Corollary 3.7 tells us that in this example the multiplier algebras vary from very small spaces when H Φ is close to the classical Hardy spaces to very big ones when we approach the big Hardy-Orlicz spaces.

Another observation can be made concerning the critical value

α = 1/2. For α ≤ 1/2 the function ξ : t -→ (t + 2) α/(1-α) -t α/(1-α) is bounded so that ψ α,C = ϕ α • γ -1
α,C satisfies that ∆ 2 condition, whereas for α > 1/2 the function ξ is unbounded and so ψ α,C / ∈ ∆ 2 .

A similar observation can be made in the context of Theorem 3.4. By the above proof, the ∇-admissible function γ (log) α,η satisfies γ (log)

α,η -1 (t) ∼ c(t/ log t) 1/(1-α) for a suitable constant c. The function ξ (log) : t -→ ((t + 2)/ log(t + 2)) α/(1-α) -(t/ log t) α/(1-α) is bounded if and only if α ≤ 1/2 so that ψ γ = ϕ α • γ (log) α,η -1
satisfies the ∆ 2 condition if and only if α ≤ 1/2.

ORDERINGS ON MULTIPLIERS

4.1. A general result. We begin the section with a general fact. Pick Φ 1 = ϕ 1 • log and Φ 2 = ϕ 2 • log two defining functions of Hardy-Orlicz spaces, where ϕ 1 , ϕ 2 are strongly convex. In Section 2 we have mentioned that the condition

lim sup t→∞ Φ 2 (t) Φ 1 (t) < +∞ (4.1) is equivalent to H Φ 1 ⊂ H Φ 2 .
Replacing Φ i by ϕ i we get the same kind of estimate for ϕ 2 /ϕ 1 in (4.1). It is also possible to replace moreover ϕ 1 by ϕ 1 + ϕ 2 without changing H Φ 1 , so that we can suppose that h := ϕ 1 -ϕ 2 is strongly convex, and even that h ′ = ϕ ′ 1 -ϕ ′ 2 tends to infinity at infinity. This does unfortunately not always imply that ϕ -2 -ϕ -1 is increasing. However, if we assume the later to hold then the ordering of the Hardy-Orlicz spaces is inherited by their respective multipliers.

Proposition 4.1. Let ϕ 1 , ϕ 2 be strongly convex functions. If ϕ -1 2 -ϕ -1 1 is increasing then Mult(H Φ 1 ) ⊂ Mult(H Φ 2 ).
Proof. We can suppose that ϕ 1 and ϕ 2 are differentiable. By the hypothesis ϕ -1 2 -ϕ -1 1 is a strictly increasing function, so that

(ϕ -1 2 -ϕ -1 1 ) ′ ≥ 0. Hence ((ϕ -1 2 ) ′ (ϕ 1 (u))-(ϕ -1 1 ) ′ (ϕ 1 (u)))ϕ ′ 1 (u) ≥ 0 (note that obviously ϕ ′ 1 ≥ 0). Hence (ϕ -1 2 • ϕ 1 ) ′ (u) ≥ 1, for sufficiently big u. Setting ξ(u) := ϕ -1 2 • ϕ 1 • log(u) we deduce from this that ξ ′ (u) ≥ 1/u for big u. Define now Ξ = Φ -1 2 • Φ 1 . Then we get (log •Ξ) ′ (u) = ξ ′ (u) ≥ 1 u ,

and hence the function

Θ : t -→ Ξ(t) t = Φ -1 2 • Φ 1 (t) t is increasing.
After these preliminary remarks let us come to the proof of the proposition. Suppose g ∈ Mult(H Φ 1 ). Let f ∈ H Φ 2 . We have to check that gf ∈ H Φ 2 . Define a measurable function on

T by f 0 = Φ -1 1 (Φ 2 (|f |)).
Clearly there exists an outer function F the modulus of which is equal to f 0 almost everywhere on T, and by construction F ∈ H Φ 1 . Since g multiplies on H Φ 1 we have gF ∈ H Φ 1 . For the remaining argument we will suppose |g| ≥ 1 almost everywhere on T (we have already seen that g can be supposed outer; it is also clear that g is a mulitplier if and only if the outer function the modulus of which is equal to max(1, |g|) is a multiplier). With this assumption we have |F | ≤ |gF | and since Θ is increasing we get

Ξ(|F |) |F | ≤ Ξ(|gF |) |gF | , i.e. |g|Φ -1 2 (Φ 1 (|F |)) ≤ Φ -1 2 (Φ 1 (|gF |)), from where we get T Φ 2 (|gf |)dm = T Φ 2 (|g|Φ -1 2 (Φ 1 (|F |)))dm ≤ Φ 1 (|gF |)dm < ∞.
Any "reasonable" pair of strongly convex functions with H Φ 1 ⊂ H Φ 2 satisfies the hypothesis of Proposition 3.3. A simple example is ϕ 1 (t) = e t α and ϕ 2 (t) = e t β with α > β (this follows already from Corollary 3.7). Another example is given by ϕ 1 (t) = e t and ϕ 2 (t) = e t/ log t for which it is simple to check that (ϕ -1 2 -ϕ -1 1 ) ′ ≥ 0. A natural question raised by the preceding proposition is whether there exist Hardy-Orlicz spaces for which the ordering of the multipliers is in the opposite direction of that of the Hardy-Orlicz spaces themselves. The next subsection answers this question by giving examples where the ordering of the multipliers cannot be pulled back to the underlying Hardy-Orlicz spaces 4.2. Small multipliers on large Hardy-Orlicz spaces. Here we show that there are large Hardy-Orlicz spaces for which the multipliers reduce to H ∞ , so that in general the multipliers are not necessarily ordered as the Hardy-Orlicz spaces (when these can be ordered).

Let us make more precise what we mean by "large" here. In fact it turns out that the Hardy-Orlicz spaces we consider can be very far from p>0 H p . We have to introduce a new class of strongly convex functions. Set ϕ α (t) = e (ln t) α , t ≥ t 0 .

These functions define Hardy-Orlicz spaces H Φα , where Φ α = ϕ α • log, which are much bigger than those associated with ϕ(t) = e t α considered in Section 3. Let us observe that for every (concave) function γ strictly increasing to infinity and such that γ(t) = o(t) we have

ϕ α (t + γ(t)) ϕ α (t) = exp (ln t) α (1 + ln(1 + γ(t)/t) ln t ) α -1 = exp α γ(t) t(ln t) 1-α + o( γ(t) t(ln t) 1-α ) .
which is bounded when γ(t) ≤ Ct(ln t) α-1 . The latter expression suggests that we could attain a growth faster than the identity when α > 1. In this situation the above computations, which work under the assumption γ(t) = o(t), are of course false. Anyway, since we are only interested in concave γ it is not worth while seeking γ growing faster than the identity. So, ϕ α satisfies the ∆-condition and for instance γ p (t) = t p is admissible for every p ∈ (0, 1). Using Theorem 3.1 this implies that the multipliers of H Φα contain a very big space: H Ψα,p , where

Ψ α,p = ψ α,p • log, ψ α,p (t) = ϕ α • γ -1 p (t) = e (1/p) α (ln t) α = ϕ (1/p) α α .
We have the following result.

Theorem 4.2. For every β > 1 there exists a strongly convex function ϕ satisfying the ∆2condition such that H Φ contains H Φ β and

Mult(H Φ ) = H ∞ .
Proof. We begin by constructing the strongly convex function on R + . Suppose ϕ(1) = 1. Let (t n ) be a sequence of positive real numbers tending strictly to infinity and t 1 = 1. We will also assume that (t n+1 -t n ) goes to infinity. The construction of ϕ goes inductively. On each interval

I n = [t n , t n+1
) the function is affine with ϕ(t n ) = lim t→t - n ϕ(t) so that ϕ is continuous in t n and with slope ϕ(t n ) (the function doubles its values from t n to t n + 1). This yields of course a convex function the slope of which tends to infinity from where we deduce that it is strongly convex. (It is clear how to extend ϕ to R -.) Let us check that by a suitable choice of (t n ) we obtain a function ϕ tending more slowly to infinity than ϕ β . This will show that

H Φ β ⊂ H Φ . Fix γ > 1 β -1 . By construction ϕ(t n+1 ) = ϕ(t n )(1 + (t n+1 -t n )). Set t n+1 = t n + e n γ -1, so that ϕ(t n+1 ) = e n γ ϕ(t n ), and an immediate induction yields ϕ(t n ) = e n-1 k=1 k γ ϕ(t 1 ), where ϕ(t 1 ) = 1. It is well known that n-1 k=1 k γ ∼ (n-1) γ+1
γ+1 , from where we deduce that e (1-ε)(n-1) γ+1 /(γ+1) ≤ ϕ(t n ) ≤ e (1+ε)(n-1) γ+1 /(γ+1) for sufficiently big n (depending on ε). Now t n+1 -t n ∼ e n γ , so that

t n = t 1 + n-1 k=1 (t k+1 -t k ) ∼ n-1
k=1 e k γ ≥ e (n-1) γ . By assumption γ > 1 β -1 , so that γβ > γ + 1. Hence for sufficiently big n ϕ β (t n-1 ) ≥ e (ln e (n-2) γ ) β = e (n-2) γβ >> e (1+ε)(n-1) γ+1 /(γ+1) ≥ ϕ(t n ).

This implies that on the whole interval I n the function ϕ β dominates ϕ. Since this is true for every interval I n (n sufficiently big), we can deduce that

H Φ β ⊂ H Φ .
The remaining part of the proof is again built on the arguments of of [HK88, Theorem 1.3]. Suppose now that there exists an unbounded multiplier g for

H Φ . Let σ k = {ζ ∈ T : log |g(ζ)| ∈ [k, k + 1)
} which are of positive measure by assumption. Since ϕ tends to infinity, there exists a subsequence

(t n k ) k such that ϕ(t n k ) ≥ 1 k 2 |σ k | . Then we can find σ ′ k ⊂ σ k such that ϕ(t n k )|σ ′ k | = 1 k 2 .

Let f be the outer function the boundary values of which are in modulus equal to

k e tn k χ σ ′ k on k σ ′ k and 1 elsewhere. Then T Φ(|f |)dm = k ϕ(t n k )|σ ′ k | = k 1 k 2 < ∞. On the other hand, since for γ > 0 we have ϕ(t n k + γ) ≥ ϕ(t n k ) + ϕ(t n k )γ ≥ γϕ(t n k ), we obtain T Φ(|f g|)dm = k σ ′ k ϕ(log |f | + log |g|)dm ≥ k σ ′ k log |g|ϕ(t n k )dm ≥ k kϕ(t n k )|σ ′ k | = k 1 k = ∞.
So, g does not multiply f to a function in H Φ . We have reached a contradiction, and any multiplier in H Φ has to be bounded.

It is easily checked that, by construction, ϕ satisfies the ∆2 -condition, so that we also have

H ∞ ⊂ Mult(H Φ ).

SOME MORE EXAMPLES

5.1. Optimality of the conditions. We begin this section with an example discussing the optimality of the results of Section 3. We have already seen in Proposition 3.3 that the result of Theorem 3.1 is in a sense sharp: whenever a concave function γ is not admissible for ϕ then we can find a function in H Ψγ , where Ψ γ = ϕ • γ -1 • log, that does not multiply on H Φ .

We will discuss this more thouroughly here through the example ϕ 1/2 (t) = e √ t . Recall that in this situation our Theorem 3.1 gave the inclusion p>0 H p ⊂ Mult H Φ 1/2 . On the other side,

Theorem 3.4 shows that Mult H Φ 1/2 ⊂ H ϕ (log) 1/2,δ •log for every δ > 0. Recall that ϕ (log) 1/2,δ (t) = e δ t log t .
Here we will use Proposition 3.3 to show the existence of a function g not multiplying on H Φ 1/2 and which is in Hardy-Orlicz classes coming much closer to p>0 H p than do the spaces H ϕ (log) 1/2,δ •log , δ > 0. This shows that Theorem 3.4 is not optimal (even if Corollary 3.7 gave us some optimality; see the comments after that corollary).

We begin by introducing a new scale of Hardy-Orlicz spaces. In order to simplify the notation we will set for k ≥ 1

log k := log • • • • • log k times .
We will also set e 1 := e and e k+1 := e e k . Then for k ≥ 2 we introduce the functions ϕ (k) which are defined by

ϕ (k) (t) = exp t log k-1 (t)
, for t ≥ e 2k .

The functions are completed suitably for t < e 2k to convex functions.

The spaces H Φ (k) , where

Φ (k) = ϕ (k)
• log, come extremely close to p>0 H p when k → ∞ without ever atteining the latter union.

Proposition 5.1. There is a function g ∈ k≥1 H (k) that does not multiply on H Φ 1/2 . Proof. Using the numbers e k , we will define a function γ which is not admissible for ϕ 1/2 . Let ε : R + → R be continuous and piecewise affine such that

ε(e k ) = k, k ≥ 1.
The function ε is clearly concave on [1, +∞), and so will be γ defined by γ(t) = √ tε(t) on [1, +∞). The function γ is not admissible since

ϕ 1/2 (t + γ(t)) ϕ 1/2 (t) = e √ t+ √ tε(t)- √ t = e 1 2 ε(t)+o(ε(t))
tends to infinity (we had already mentioned in Section 2 that any ∆-admissible function for ϕ α can grow at most as t → Ct 1-α ). Hence by Proposition 3.3 there is a function in H Ψγ that does not multiply on H Φ .

We will show that

H Ψγ ⊂ H Φ (k) ,
for every k. For this it is sufficient to check that for every k ∈ N * there is a t k such that for every

t ≥ t k ϕ 1/2 • γ -1 (t) ≥ e t log k t .
Passing to logarithms and observing that γ is continuous and strictly increasing to +∞ so that we can change to the variable u = γ -1 (t), we are led to the verification of

log ϕ 1/2 (u) = √ u ≥ γ(u) log k (γ(u)) = √ uε(u) log k ( √ uε(u))
for u sufficiently big. This is of course equivalent to log k ( √ uε(u)) ≥ ε(u) for big u. The left hand side of this estimate behaves like log k u so that it remains to show that ε is neglectible with respect to log k at infinity. Fix such a k and let n > k. Then for t ∈ [e n , e n+1 ) we have log k (t) ≥ log k (e n ) = e n-k which goes "extremely" fast to infinity (one could observe that for k ≥ 1 we have e k+1 /e k = e e k /e k ≥ M := e e-1 since e t ≥ Mt for t ≥ e, so that e n-k grows at least exponentially in n), whereas ε(t) ≤ ε(e n+1 ) = n + 1.

Big multipliers in small

Hardy-Orlicz spaces. In this section we will show that there are Hardy-Orlicz spaces beyond p>0 H p coming very close to p>0 H p and containing unbounded multipliers. More precisely, such Hardy-Orlicz spaces contain Hardy-Orlicz spaces strictly bigger than H ∞ . This is of central interest in the interpolation problem since it will allow to conclude that such Hardy-Orlicz spaces admit interpolating sequences which are not Carleson, i.e. which are not interpolating for H ∞ .

The key result to our examples here is the following proposition.

Proposition 5.2. Let ϕ be a strongly convex function on R strictly increasing to +∞. Let (t n ) n be the sequence defined by

ϕ(t n ) = 2 n , n ∈ N.
If (t n+1 -t n ) n tends to infinity, then ϕ is ∆-admissible, i.e. there exists γ : [t 0 , +∞) → R concave, increasing with lim t→∞ γ(t) = +∞ such that

ϕ(t + γ(t)) ≤ 4ϕ(t), t ≥ t 0 . (5.1)
Proof. Since we are only interested in the estimate (5.1) for big t, we can normalize the function

ϕ such that ϕ(0) = 1. Split R into subintervals [t n , t n+1 ) (possibly adding (-∞, t 0 ]).
Let us construct a ∆-admissible function. To begin with let γ 0 be the continuous and piecewise affine function defined on each interval [t n , t n+1 ) by

γ 0 : [t n , t n+1 ) -→ [t n+1 , t n+2 ), t -→ t n+1 + t n+2 -t n+1 t n+1 -t n (t -t n ).
This is just the affine increasing bijection from [t n , t n+1 ) onto [t n+1 , t n+2 ). Define moreover γ 1 (t) = γ 0 (t) -t so that γ 1 (t n ) = t n+1 -t n for every n. This function is still continuous and piecewise affine. Moreover it tends to infinity since the sequence (t n+1 -t n ) n does and since it is bounded below on any interval [t n , t n+1 ) by the values γ(t n ) and γ(t n+1 ). It is clear that we can then bound below γ 1 by a function γ which is concave (one could construct such a function as a continuous piecewise affine function with decreasing growth coefficient on each interval).

Let us check that the so obtained function γ satisfies the ∆-admissibility type condition (5.1). Let t ∈ R and suppose t ∈ [t n , t n+1 ). Observe that then γ 0 (t) ∈ [t n+1 , t n+2 ). Hence

ϕ(t + γ(t)) ≤ ϕ(t + γ 1 (t)) = ϕ(γ 0 (t)) ≤ ϕ(t n+2 ) ≤ 2 2 ϕ(t n ) ≤ 2 2 ϕ(t).
As a consequence of the previous proposition and Theorem 3.1 we obtain Corollary 5.3. Let ϕ be as in the proposition. There exists a strongly convex function ψ such that

Alg(H Ψ ) ⊂ Mult(H Φ ), where Ψ = ψ • log.
Important examples of strongly convex functions for which the sequence (t n+1 -t n ) n tends to infinity are given by ϕ

(log) k (t) = e t/ log k t , k ∈ N * , ϕ(t) = e t/
√ log k (t) , and it is even possible to construct functions ϕ(t) that behave on intervals I n like ϕ log n n . Let us discuss more thouroughly the case of ϕ (log) k . This function defines a Hardy-Orlicz space that is very close to p>0 H p and having unbounded multipliers. We will check that γ k,c (t) = c log k (t) is admissible:

t + c log k t log k (t + c log k t) - t log k t = t log k t + c log 2 k t -t log k (t + c log k t) log k t log k (t + c log k t) = c log 2 k t -t(log k (t + c log k t) -log k t) log k t log k (t + c log k t) ≤ c log k (t) log k (t + log k t) ≤ c. Also γ -1 k,c (t) = exp k (t/c) where exp k = exp • • • • • exp k times . So Ψk,α (t) = ϕ • γ -1 k,c • log t = ϕ(exp k log t c ) = ϕ(exp k-1 t α ) = exp( exp k-1 t α log k exp k-1 t α ) = exp( exp k-1 t α α log t ).
Setting also

Ψ k,α (t) = exp k t α , we again get α>0 H Ψ k,α = α>0 H Ψk,α ⊂ Mult(H Φ k ).
The spaces H Ψα (and a fortiori the spaces H Ψ k,α ) are extremely small, by which we mean that they are very close to H ∞ . This can be expressed by the Boyd indices. For Orlicz spaces, [LT, Proposition 2.b.5] gives an explicit formula allowing the computation of these indices. It turns out that -not very surprisingly -p X = q X = +∞ for X = H Ψ k,α .

INTERPOLATION

In this section we will consider the interpolation problem in Hardy-Orlicz spaces beyond p>0 H p . We shall begin by recalling some definitions. The interpolation problem we would like to consider is that of free interpolation. Definition 6.1. A sequence Λ = {λ n } n ⊂ D is called a free interpolating sequence for a space of holomorphic functions on D, X = Hol(D), if for every f ∈ X, and for every sequence (b n ) n with

|b n | ≤ |f (λ n )|, n ∈ N, there exists a function g ∈ X such that g(λ n ) = b n , n ∈ N. Notation: Λ ∈ Int l ∞ X.
Another way of expressing that a sequence is of free interpolation is to say that l ∞ is contained in the multiplier algebra of X|Λ := {(f (λ n )) n : f ∈ X}: for every (a n ) n = (f (λ n )) n ∈ X|Λ and for every µ

= (µ n ) n ∈ l ∞ there is g ∈ X such that g(λ n ) = µ n a n , n ∈ N, i.e. (µ n a n ) n ∈ X|Λ.
It is clear that if we can interpolate the bounded sequences by functions in the multiplier algebra, i.e. l ∞ ⊂ Mult(X)|Λ, then Λ ∈ Int l ∞ X.

The definition of free interpolation originates in the work by Vinogradov and Havin in the 70s. It is very well adapted to the Hilbert space situation where it can be connected to the unconditionality of a sequence of reproducing kernels, see e.g. [Nik02, Theorem C3.1.4, Theorem C3.2.5] for a general source; see also [START_REF] Hartmann | Interpolation in the Nevanlinna and Smirnov classes and harmonic majorants[END_REF] or [START_REF]Interpolation and harmonic majorants in big Hardy-Orlicz spaces[END_REF] for more motivations for the non-Banach situation.

Let us recall that by a famous result of L. Carleson [START_REF] Carleson | An interpolation problem for bounded analytic functions[END_REF] the interpolating sequences for H ∞ , i.e. the sequences Λ for which H ∞ |Λ = l ∞ , are characterized by the Carleson condition:

inf λ∈Λ |B Λ\λ (λ)| = δ > 0. (6.1)
Here B E = λ∈E b λ is the Blaschke product associated with a discrete set E ⊂ D (supposed to satisfy the Blaschke condition λ∈E (1

-|λ| 2 ) < ∞). Recall that for λ ∈ D b λ (z) = |λ| λ λ -z 1 -λz , z ∈ D.
A sequence satisfying (6.1) will be called a Carleson sequence. It is clear that for X = H ∞ classical interpolation and free interpolation are the same.

The Carleson condition still characterizes interpolating sequences (free or classical) in a large class of Hardy-Orlicz spaces included in the scale of H p spaces (see [ShHSh] for H p , p ≥ 1; [START_REF] Kabaȋla | Interpolation sequences for the H p classes in the case p < 1[END_REF] Of course a big gap remains between big Hardy-Orlicz spaces considered in [START_REF]Interpolation and harmonic majorants in big Hardy-Orlicz spaces[END_REF] and p>0 H p . In particular an intriguing question is to know whether there are Hardy-Orlicz spaces beyond p>0 H p where the Carleson condition still characterizes the interpolating sequences. In the light of Theorem 4.2, this question is still more exciting since there are very large Hardy-Orlicz spaces for which the multipliers reduce to H ∞ . Here we will give examples of Hardy-Orlicz spaces which are close to the union p>0 H p and which have free interpolating sequences which are not Carleson.

We will consider the problem through the multiplier algebra of the Hardy-Orlicz space under consideration. As already explained, the idea is to solve the interpolation problem: find Λ =

{λ n } n ⊂ D such that l ∞ ⊂ Mult(H Φ )|Λ.
Then Λ is a free interpolating sequence for H Φ , and in our context we would like that Λ is not a Carleson sequence.

The situation we will consider here is that of a Hardy-Orlicz space H Φ the multiplier algebra of which contains H Ψ where Ψ = ψ • log and ψ : R → [0, ∞) is a strongly convex function. Examples of such a situation can be deduced from Corollary 5.3. In such a situation H Ψ contains not only H ∞ but also -and this will be important for us -unbounded functions such as for example the outer function g with |g| = Ψ -1 • v 1 a.e. on T, where v 1 (t) = 1 t log 1+ε (1/t) and ε > 0.

Let M := alg(H Ψ ) which is inluded in Mult(H Φ ).

We need two simple properties on M. Recall from (2.2) that for a function f in the Smirnov class we have written [f ] for its outer part. We will use more generally this notation for the outer function associated with a measurable function f on T with log |f | ∈ L 1 . Lemma 6.2. If f ∈ M then there exists n ∈ N * such that [f ] 1/n ∈ H Ψ Proof. We begin by checking the result for products and sums of functions in the generator

H Ψ of M. Observe first that if f 1 , f 2 ∈ H Ψ , then [w] ∈ H Ψ where w := max(|f 1 |, |f 2 |) (just split the integral T Ψ(|w|)dm into two parts where |f 1 | (respectively |f 2 |) has bigger modulus). So, if f = f 1 f 2 then |f | ≤ w 2 and [f ] 1/2 ∈ H Ψ .
By a simple induction this holds for finite products.

Of course, [f 1 + f 2 ] ∈ H Ψ whenever f 1 , f 2 ∈ H Ψ , and this extends obviously to finite sum of functions in H Ψ .

Let us now look whether the property holds for products and sums of functions in M. If

f 1 , f 2 ∈ M with [f 1 ] 1/n ∈ H Ψ , [f 2 ] 1/k ∈ H Ψ then [w] 1/N ∈ H Ψ ,
1 | ≤ 1 or |f 2 | ≤ 1 is of no relevance here). Hence, if f = f 1 f 2 then |f | ≤ |[w]| 2 , and T Ψ(|f | 1/(2N ) )dm ≤ T Ψ(|w| 1/N )dm < ∞, i.e. [f ] 1/(2N ) ∈ H Ψ .
By a simple induction this also holds for finite products.

For sums of functions in M, let f 1 , f 2 , w, N as above. In particular

[w] 1/N ∈ H Ψ . If now f = f 1 + f 2 , then |f | ≤ 2w so that |[f ] 1/N | ≤ |2w| 1/N from where we deduce that [f ] 1/N ∈ H Ψ .
By a simple induction this generalizes to finite sums.

Since the property of the lemma is true for functions in H Ψ and it is conserved by finite sums and products of functions in M = alg(H Ψ ) it holds for the algebra generated by H Ψ .

A simple consequence is the following.

Corollary 6.3. If f ∈ M, then [max(1, |f |)] ∈ M.
Proof. From the lemma we obtain that

[f ] 1/n ∈ H Ψ for a convenient n ∈ N * . Then clearly h := [max(1, |f |)] 1/n = [max(1, |f | 1/n )] ∈ H Ψ . Hence [max(1, |f |)] = h n ∈ M = Alg H Ψ .
We can add another consequence of Lemma 6.2. Corollary 6.4. We have Alg(H Ψ ) = n∈N H Ψn where Ψ n (t) = Ψ(t 1/n ).

It should be noted that Ψ n is not necessarily convex, but ψ n (t) := Ψ n •exp(t) = ψ(t/n) is still stongly convex in the terminology of [START_REF] Rudin | Function theory in polydiscs[END_REF] so that we still can define the corresponding Hardy-Orlicz classes (which are not necessarily vector spaces). In the case Ψ 1,α (t) = e t α , which defines a Hardy-Orlicz space contained in the multiplier algebra of H Φ (log) 1 (Φ (log) 1 (t) = e log t/ log log t for t sufficiently big), (Ψ 1,α ) n will be convex (we have taken the notation from the end of Subsection 5.2).

Like in [START_REF] Duren | Interpolation in H p -spaces[END_REF] our example of a free interpolating sequence will be constructed as a non separated union of two Carleson sequences (this is different to [START_REF] Taylor | Interpolation of l q sequences by H p functions[END_REF] where Carleson's method is used to interpolate l q -sequences by H p -functions). In order to do that we will use the results of [START_REF] Hartmann | Traces of certain classes of holomorphic functions on finite unions of Carleson sequences[END_REF] based on the so-called (C)-stability.

Let us recall the definition of (C)-stability (see [START_REF] Hartmann | Traces of certain classes of holomorphic functions on finite unions of Carleson sequences[END_REF]). Definition 6.5. Let X ⊂ Hol(D). If there exists δ 0 ∈ (0, 1) such that for every pair of Carleson sequences

Λ = {λ n } n ⊂ D and Λ = { λn } n ⊂ D with sup n |b λn ( λn )| = δ < δ 0 we have X|Λ = X| Λ then X is called (C)-stable.
Since H ∞ ⊂ H Ψ and M is an algebra containing H Ψ we also have H ∞ ⊂ M = Mult(M) which in particular implies that a Carleson sequence is a free interpolating sequence for M. Proposition 6.6. The space M is (C)-stable.

Proof. Pick f ∈ M, and let Λ, Λ as in the definition. Set a n = f (λ n ). We have to verify that {a n } n ∈ M| Λ. Put w = max(1, |f |) a.e. T. By Corollary 6.3,

F := [w] ∈ M. It is clear that |a n | ≤ |A n | where A n = F (λ n ). Note that log |F |
is by construction a positive harmonic function, and so by Harnack's inequality there is a constant c > 1 such that

|F ( λn )| 1/c ≤ |F (λ n )| ≤ |F ( λn )| c , n ∈ N. So |a n | ≤ |F c ( λn )|
Let n be a natural number bigger than c. Then |F c | ≤ |F n | and F n ∈ M by Lemma 6.2.

Since Λ is a Carleson sequence by assumption, and so a free interpolating sequence for M, there exists a function g ∈ M, such that g( λn ) = a n , n ∈ N.

Hence M|Λ ⊂ M| Λ. Since the problem is symmetric, we also have the reverse inclusion, and M is (C)-stable.

We will now examine the trace of H * Ψ . For our purpose it will be sufficient to know the restriction H * Ψ |Λ when Λ is a Carleson sequence. For this we will use the Jones-Vinogradov interpolation operator (see e.g. [Nik02, Vol.2, pp.179-180]), which with a sequence a = {a n } n associates a holomorphic function

T a(z) = n∈N a n f n (z), z ∈ D.
The exact form of the functions f n is not very interesting for our discussion here (we refer the reader to the above cited monograph, or to [START_REF] Hartmann | Traces of certain classes of holomorphic functions on finite unions of Carleson sequences[END_REF]). The family (f n ) is of course a Beurling-type family, by which we mean that f n (λ k ) = δ nk and

sup z∈D n∈N |f (z)| < ∞. The operator T is continuous from l 1 (1 -|λ| 2 ) = {a = (a n ) n : n∈N a l 1 (1-|λ| 2 ) := (1 - |λ n | 2 )|a n | < ∞} to H 1
and from l ∞ to H ∞ (see the above cited monograph). These results suggest the use of interpolation between Banach space (lattices). In order to do this we will adapt a Calderón interpolation theorem for rearrangement invariant subspaces (see e.g. [LT, Theorem 2.a.10]) to our situation.

The space

l * Ψ (1 -|λ n | 2 ) := {a = (a n ) n : ∃C > 0, n∈N (1 -|λ n | 2 )Ψ |a n | C < ∞},
equipped with the usual norm • Φ is a Banach space.

Proposition 6.7. Let Λ ∈ (C). The operator T is continuous from l

* Ψ (1 -|λ n | 2 ) to H * Ψ . Consequently, if Λ ∈ (C) then l * Ψ (1 -|λ n | 2 ) ⊂ H * Ψ |Λ.
Proof. We have already introduced the distribution function and the decreasing rearrangement of a function defined on a measure space. We now have to consider these notions in the sequence Φ is (C)-stable. This explains why we pass through M which we know to be (C)-stable.

Let us now turn to the construction of an interpolating sequence for H Φ not satisfying the Carleson condition. As already mentioned, for that it is sufficient to construct a sequence Λ which is not Carleson yet M|Λ contains l ∞ . We will use Theorem 1.4 of [START_REF] Hartmann | Traces of certain classes of holomorphic functions on finite unions of Carleson sequences[END_REF] (the idea of course goes back to [START_REF] Duren | Interpolation in H p -spaces[END_REF]. The latter condition guarantees that Λ = Λ 1 ∪ Λ 2 is not Carleson. The condition on the speed of convergence to zero of (|b λ n,1 (λ n,2 )|) n will be fixed later.

Let M(Λ 1 ) = M|Λ 1 (= M|Λ 2 ), and set M 2 (Λ) := {(a n,i ) n∈N,i=1,2 : (a n,1 ) n ∈ M(Λ 1 ), a n,1 -a n,2 b λ n,2 (λ n,1 ) n ∈ M(Λ 1 )}, which is a kind of inductive limit of first order discrete Sobolev-Orlicz spaces. Since M is (C)stable, we deduce from [Ha99, Theorem 1.4] that M 2 (Λ) ⊂ M|Λ := {(f (λ n,i )) n∈N,i=1,2 : f ∈ M} (the careful reader might have observed that we only use one half of that theorem, but this is sufficient for our purpose here since we are only interested in one inclusion). Set also Choosing Λ 2 such that (|b λ n,1 (λ n,2 )|) n goes to zero and

l * Ψ,
|b λ n,1 (λ n,2 )| ≥ 2 Ψ -1 (γ n ) -1 ,
we obtain for every a ∈ l ∞ with a ∞ ≤ 1,

n∈N (1 -|λ n,1 | 2 )Ψ |a n,1 | + a n,2 -a n,1 b λ n,1 (λ n,2 ) ≤ n∈N (1 -|λ n,1 | 2 )Ψ 1 + 2 |b λ n,1 (λ n,2 )| ≤ n∈N (1 -|λ n,1 | 2 )γ n < ∞.
Hence, the unit ball of l ∞ is in M|Λ, and so also the whole space l ∞ since M is a vector space. We are done.

As a consequence we obtain the following result. Corollary 6.9. Let ϕ be a strongly convex function, and Ψ a strictly increasing, convex, unbounded function such that

H Ψ ⊂ Mult(H ϕ•log ).
Then there exists Λ ∈ (C) such that Λ ∈ Int l ∞ H ϕ•log .

  for H p , p < 1 and [Har99] for more general Hardy-Orlicz spaces included in the scale of classical Hardy spaces H p ). The situation is intrinsically different in spaces close to the Nevanlinna and Smirnov classes. Here interpolating sequences are characterized by the existence of harmonic majorants of the function ϕ Λ defined by ϕ Λ (λ) = log 1 |B λ (λ)| when λ ∈ Λ and ϕ Λ = 0 otherwise. See [HMNT04] for precise results in the Nevanlinna and Smirnov classes and [Ha06] for big Hardy-Orlicz spaces where Mult(H Φ ) = H Φ .

  where w := max(|f 1 |, |f 2 |) and N = max(n, k) (just split the integral T Ψ(|w| 1/N )dm into two parts where |f 1 | (respectively |f 2 |) has bigger modulus; the case when |f

  space l * Ψ (1 -|λ n | 2 ) (the underlying measure space being N with the measure µ = n∈N (1 -|λ n | 2 )δ n ) and in the Lebesgue space L * Ψ . We start with a sequence a ∈ l * Ψ (1-|λ n | 2 ).Repeating the arguments of the proof of Calderón's theorem given in [LT, Theorem 2.a.10], we set for our sequence a and an s ∈ [0, L], L :=n∈N (1 -|λ n | 2 ) < ∞, b s n = (|a n | -a * (s)) an |an| if |a n | > a * (s) 0 if |a n | ≤ a * (s)and c s n = a n -b s n . Clearly, c s l ∞ ≤ a * (s).

Figure 2 : 0 a

 20 Figure 2: Decreasing rearrangement of (a n ) n

Proposition 6. 8 .

 8 There exists a sequenceΛ ∈ (C) such that M|Λ ⊃ l ∞ . Proof. Let Λ j = {λ n,j } ⊂ D be Carleson sequences, j = 1, 2, such that |b λ n,1 (λ n,2 )| ≤ δ 0 and such that lim n→∞ |b λ n,1 (λ n,2 )| = 0.

  that there exists a sequence (σ n ) n of disjoint measurable subsets of T with |σ n | = ε n , where | • | denotes the Lebesgue measure. Let f be the outer function the modulus of which is equal to n e tn χ σn on n σ n and 1 otherwise (χ E is the characteristic function of a measurable set E).

  2 (1 -|λ n,i | 2 ) := {(a n,i ) n∈N,i=1,2 : (a n,1 ) n ∈ l * Ψ (1 -|λ n,1 | 2 ), a n,1 -a n,2 b λ n,2 (λ n,1 ) n ∈ l * Ψ (1 -|λ n,1 | 2 )},and analogously l Ψ,2 (1 -|λ n,i | 2 ) by omitting the stars everywhere in the previous definition. By Proposition 6.7,l * Ψ (1 -|λ n,1 | 2 ) ⊂ H * Ψ |Λ ⊂ M(Λ 1 ), and so l * Ψ,2 (1 -|λ n,i | 2 ) ⊂ M 2 (Λ). In particular we can interpolate every sequence (a n,i ) n∈N,i=1,2 with -|λ n,1 | 2 )Ψ |a n,1 | + a n,2 -a n,1 b λ n,1 (λ n,2 ) < ∞by a function in M. Now, since Λ 1 is a Blaschke sequence, there exists an increasing sequence (γ n ) n of positive elements tending to infinity and such that n∈N (1 -|λ n,1 | 2 )γ n < ∞.

n∈N

(1


B Garcia

V Rouchon

J-P Deflandre

Renewable energies | Eco-friendly production | Innovative transport | Eco-efficient processes | Sustainable resources

Context and objectives

The exploitation of source rocks implies to monitor the surroundings aquifers and specially the potable aquifers present at the sub-surface, as required by USA and Canadian regulations. The acquisition of a representative aquifer baseline is an absolute necessity prior to the performing of a long term monitoring.

Indeed, a huge variety of geochemical species is present and we have to demonstrate that we can distinguish a natural geochemical signature from an anthropogenic geochemical signature due to the exploitation process. Hydraulic fracturing operations are needed to exploit these resources, that's why 3 main sources have to be considered: 1) the shale formation itself, 2) the fluids used for the hydraulic fracturing operations and 3) the aquifers.

The main geochemical species to monitor will be presented in this poster with a new geochemical monitoring methodology/technical approach associated with a notion of priority/interest and early warning systems.

A new geochemical monitoring methodology associated with specific techniques

The Shale Plays: (i) organic matter and hydrocarbons species (in liquid and gaseous forms) + (ii) non-hydrocarbon species have to be considered Parameters listed in recommendations by state environmental protection agencies [START_REF] Boyer | Shale Gas: a Global Resource[END_REF] The 3 main sources to monitor

• 3 /

 3 The surroundigs aquifers • 2/ The hydraulic fracturing fluids 4 Conclusions and perspectives A large baseline is necessary to evaluate the aquifers evolution and a possible leakage during the shale formation exploitation Specific methodologies and technologies are necessary to propose early warning systems 2 The 3 main sources and their associated geochemical signature Huc., 2013 • 1/ Shale plays = organic matter (O.M.) + a heterogeneous mixture of minerals (clay, silt and also carbonate and sandstone) + presence of oil and/or gas.Twenty most frequently used additives in shale gas production hydraulic fracturing fluids(US EPA, 2015)

From

  conditions Fluid transfer at preserved reservoir pressure conditions Interpretation of data with stand-alone software Priority and Early Warning Systems strategy in term of geochemical species 1. Better quantification of the bulk HC amount (minimize loss of light HC) 2. Information on the quality of occurring fluids 3. Better separation between HC fluids and kerogen 4. Better assessment of Tmax value

Major Geochemical species Minor Geochemical species

	General data	min	average	max
	Density	1.1	1.18	1.23
	pH	4	5.7	7.9
	Total carbon	1.58 mmol/L 2.86 mmol/L 5.31 mmol/L
	alcalinity	1.22 meq/L	3.12 meq/L	5.43 meq/L
	Cl -	20 mg/L	156.6 g/L	218 g/L
	Na +	21 mg/L	61.7 g/L	85.2 g/L
	Ca 2+	16 mg/L	29.2 g/L	43.1 g/L
	Mg 2+	23 mg/L	3.2 g/L	8.9 g/L
	K +	2.5 mg/L	1.8 g/L	6.6 g/L
	SO4 2-	26 mg/L	215 mg/L	670 mg/L
	Br	847 µg/L	4.4 g/L	6.6 g/L
	Sr	91 µg/L	1.6 g/L	2.7 g/L
	B	20 µg/L	125 mg/L	338 mg/L
	Li	55.5 µg/L	122.7 mg/L	396 mg/L
	Fe	23 µg/L	63.3 mg/L	400 mg/L
	SiO2	3.8 mg/L	28.7 mg/L	49 mg/L
	Zn	15.2 µg/L	26.5 mg/L	143 mg/L
	Mn	3.5 µg/L	16.2 mg/L	87 mg/L
	Ba	111 µg/L	16 mg/L	51 mg/L
	Pb	1 mg/L	14.7 mg/L	40 mg/L
	Rb	7 µg/L	7 mg/L	32.5 mg/L


Manfred Kufleitner

A Proof of the Factorization Forest Theorem

published or not. The documents may come

Introduction

Factorization forests where introduced by Simon [START_REF] Simon | Factorization forests of finite height[END_REF][START_REF] Simon | Factorization forests of finite height[END_REF]. An important property of finite semigroups is that they admit factorization forests of finite height. This fact is called the Factorization Forest Theorem. It can be considered as an Ramsey-type property of finite semigroups. There exists different proofs of this fact of different difficulty and with different bounds on the height. The first proof of the Factorization Forest Theorem is due to Simon [START_REF] Simon | Factorization forests of finite height[END_REF]. He showed that for every finite semigroup S there exists a factorization forest of height ≤ 9 |S|. The proof relies on several different techniques. It uses graph colorings, Green's relations, and a decomposition technique inspired by the Rees-Suschkewitsch Theorem on completely 0-simple semigroups. In [START_REF] Simon | A short proof of the factorization forest theorem[END_REF] Simon gave a simplified proof relying on the Krohn-Rhodes decomposition. The bound shown is 2 |S|+1 -2. A concise proof has been given by Chalopin and Leung [START_REF] Chalopin | On factorization forests of finite height[END_REF]. The proof relies on Green's relations and yields the bound 7 |S| on the height. Independently of this work, Colcombet has also shown a bound of 3 |S| for the height of factorization forests [START_REF] Colcombet | Factorisation forests for infinite words[END_REF]. He uses a generalization of the Factorization Forest Theorem in terms of Ramseyan splits. The proof also relies on Green's relations. A variant of our proof for the special case of aperiodic monoids has been shown in [START_REF] Diekert | On first-order fragments for words and Mazurkiewicz traces: A survey[END_REF] with a bound of 3 |S|. The main benefit of that proof is that is uses very little machinery. The proof in this paper can be seen as an extension of that proof. The main tool are again Green's relations. We only require basic results from the theory of finite semigroups which can be found in standard textbooks such as [START_REF] Pin | Varieties of Formal Languages[END_REF].

A lower bound of |S| was shown for rectangular bands in [START_REF] Simon | Properties of factorization forests[END_REF] and also in [START_REF] Chalopin | On factorization forests of finite height[END_REF]. The same bound has also been shown for groups [START_REF] Chalopin | On factorization forests of finite height[END_REF]. Therefore, the upper bound of 3 |S| reduces the gap between the lower and the upper bound.

The Factorization Forest Theorem

Let S be a finite semigroup. A factorization forest of a homomorphism ϕ : Γ + → S is a function d which maps every word w with length |w| ≥ 2 to a

factorization d(w) = (w 1 , . . . , w n ) of w = w 1 • • • w n with n ≥ 2 and w i ∈ Γ + and such that n ≥ 3 implies ϕ(w 1 ) = • • • = ϕ(w n ) is idempotent in S.
The height h of a word w is defined as

h(w) = 0 if |w| ≤ 1 1 + max {h(w 1 ), . . . , h(w n )} if d(w) = (w 1 , . . . , w n )
We call the tree defined by the "branching" d for the word w the factorization tree of w. The height h(w) is the height of this tree. The height of a factorization forest is the supremum over the heights of all words.

Factoriztion Forest Theorem (Simon [START_REF] Simon | Factorization forests of finite height[END_REF]). Let S be a finite monoid. Every homomorphism ϕ : Γ + → S has a factorization forest of height ≤ 3 |S|.

Proof: Let [w] = ϕ(w). We show that for every w ∈ Γ + there exists a factorization tree of height h(w) with a i ∈ Γ and w i ∈ Γ * satisfying the following two conditions:

≤ 3 |{x ∈ S | [w] ≤ J x}|.
∀ 1 ≤ i ≤ m : [a i w i ] J [w] and ∀ 0 ≤ i ≤ m : w i = ε ∨ [w] < J [w i ]
The idea is that we successively choose a i w i ∈ Γ + from right to left to be the shortest non-empty word such that [

a i w i ] J [w]. Let w ′ i = a i w i for 1 ≤ i ≤ m. For each 1 ≤ i < m define a pair (L i , R i ) where L i is the L-class of [w ′ i ] and R i is the R-class of [w ′ i+1 ].
Every such pair represents an H-class within the J -class of [w]. All H-classes within this J -class contain the same number n of elements. Let

h ′ (w) = h(w) -3 • |{x ∈ S | [w] < J x}|
We can think of h ′ as the height of a tree where we additionally allow words v ∈ Γ + with [w] < J [v] as leafs. Within the J -class of w we perform an induction on the cardinality of the set

{(L i , R i ) | 1 ≤ i < m} in order to show h ′ (w) ≤ 3n • |{(L i , R i ) | 1 ≤ i < m}| Since n • |{(L i , R i ) | 1 ≤ i < m}| ≤ |{x ∈ S | [w]
J x}| this yields the desired bound for the height h(w). If every pair (L, R) occurs at most twice then we have m -

1 ≤ 2 • |{(L i , R i ) | 1 ≤ i < m}|.
We define a factorization tree for w by

d(w) = (w 0 w ′ 1 , w ′ 2 • • • w ′ m ) d(w 0 w ′ 1 ) = (w 0 , w ′ 1 ) d(w ′ i • • • w ′ m ) = (w ′ i , w ′ i+1 • • • w ′ m ) for 2 ≤ i < m d(w ′ i ) = (a i , w i ) for 1 ≤ i ≤ m Since [w] < J [w i ],
by induction every w i has a factorization tree of height

h(w i ) < 3 |{x | [w i ] ≤ J x}| ≤ 3 |{x | [w] < J x}|. This yields: h ′ (w) ≤ m ≤ 3n • |{(L i , R i ) | 1 ≤ i < m}|
Note that the height does not increase if some of the w i are empty. Now suppose there exists a pair (L, R)

∈ {(L i , R i ) | 1 ≤ i < m} occurring (at least) three times. Let i 0 < • • • < i k be the sequence of all positions with (L, R) = (L ij , R ij ). By construction we have k ≥ 2. Let w j = w ′ ij-1+1 • • • w ′ ij for 1 ≤ j ≤ k. For all 1 ≤ j ≤ ℓ ≤ k we have • [ w j • • • w ℓ ] ≤ L [w ′ i ℓ ] L [w ′ i0 ]. • [ w j • • • w ℓ ] ≤ R [w ′ ij-1+1 ] R [w ′ i0+1 ]. • [w ′ i ℓ ] ≤ J [ w j • • • w ℓ ] ≤ J [w] J [w ′ i ℓ ] J [w ′ i0 ] J [w ′ i0+1
] by assumption on the factorization.

Thus for all 1 ≤ j ≤ ℓ ≤ k and 1 ≤ j ′ ≤ ℓ ′ ≤ k we get

• [ w j • • • w ℓ ] L [w ′ i1 ] L [ w j ′ • • • w ℓ ′ ] and • [ w j • • • w ℓ ] R [w ′ i1+1 ] R [ w j ′ • • • w ℓ ′ ] and therefore • [ w j • • • w ℓ ] H [ w j ′ • • • w ℓ ′ ]
Therefore, all [ w j ] denote elements in the same H-class H and since k ≥ 2 the class H is a group. We consider the following set of elements in H induced by proper prefixes

P ( w 1 • • • w k ) = {[ w 1 • • • w j ] | 1 ≤ j < k}
For the pair (L, R) we show by induction on |P (

w 1 • • • w k )| that h ′ (w) ≤ 3 |P ( w 1 • • • w k )| + 3n |{(L i , R i ) | 1 ≤ i < m}| Suppose every element x ∈ P ( w 1 • • • w k ) ⊆ H occurs at most twice. Then k -1 ≤ 2 |P ( w 1 • • • w k )|.
We construct the following factorization tree for w:

d(w) = (w 0 w ′ 1 • • • w ′ i1 , w ′ i1+1 • • • w ′ m ) d(w 0 w ′ 1 • • • w ′ i1 ) = (w 0 w ′ 1 • • • w ′ i0 , w 1 ) d(w ′ i0+1 • • • w ′ m ) = ( w 2 • • • w k , w ′ i k +1 • • • w ′ m ) d( w i • • • w k ) = ( w i , w i+1 • • • w k ) for 2 ≤ i < k
By induction on the number of pairs (L i , R i ) there exist factorization trees for the words

w 0 w ′ 1 • • • w ′ i0 , w ′ i k +1 • • • w ′ m , and all w i of height ≤ 3n |{(L i , R i ) | 1 ≤ i < m} \ {(L, R)}| + 3 |{x | [w] < J x}| This yields h ′ (w) -3n |{(L i , R i ) | 1 ≤ i < m}| ≤ k ≤ 3 |P ( w 1 • • • w k )|
Now suppose there exists an element x ∈ P ( w 1 • • • w k ) ⊆ H that occurs at least three times. Let j 0 < • • • < j t be the sequence of all positions with x =

[ w 1 • • • w ji ]. By construction we have t ≥ 2. It follows that [ w ji+1 • • • w ji+1 ] = e = e 2
where e is the neutral element of the group H. Let v i = w ji-1+1 • • • w ji for 1 ≤ i ≤ t. We construct the following factorization tree for w:

d(w) = (w 0 • • • w i0 , w i0+1 • • • w ′ m ) d( w i0+1 • • • w ′ m ) = (v 1 • • • v t , w it+1 • • • w ′ m ) d(v 1 • • • v t ) = (v 1 , . . . , v t ) We have x ∈ P ( w 1 • • • w k )\P ( w 1 • • • w i0 ) and xP ( w ji-1+1 • • • w ji ) ⊆ P ( w 1 • • • w k ) but x ∈ xP ( w ji-1+1 • • • w ji )
. Hence, by induction on the cardinality of the prefix sets, there exist factorization forests for w 0 • • • w i0 , w it+1 • • • w ′ m and the v i of height

≤ 3 |P ( w 1 • • • w k )| -3 + 3n |{(L i , R i ) | 1 ≤ i < m} \ {(L, R)}| + 3 |{x | [w] < J x}|
This yields a factorization tree for w with the desired height bound.

2

  First, we perform an induction on the cardinality of the set {x ∈ S | [w] ≤ J x}; then within one Jclass we refine this parameter. Let w ∈ Γ + with |w| ≥ 2. Then w has a unique factorization w = w 0 a 1 w 1 • • • a m w m

Acknowledgement. I would like to thank Volker Diekert for numerous discussions on this topic.


Nicolas Larrieu

Theobald De Riberolles
email: theobald.deriberolles@activus-services.fr

Guthemberg Da

Silva Silvestre

Design, Development and Implementation of a Network Intrusion Detection Tool for Air Traffic Management Systems

Keywords: Anomaly detection, intrusion, IDS, ATM

An Air Traffic Management (ATM) relies on a set of critical systems composed of control centers, sensors, communication means and radio navigation systems. These critical systems may be subject to different attacks and thus compromise their security. Indeed as there is a desire to open this system more and more outward and there is a gap between this world and the interconnected world, threats are increasing. The ATM system have particular characteristics as the fact that it is a very distributed system with a lot of real-time applications using proprietary and/or legacy protocols. Thus, the need to have an efficient Intrusion Detection System (IDS) is primordial in terms of reliability (a false negative rate as low as possible) and relevance (a lowest possible false-positive rate). The development of an IDS combining misuse detection (i.e., defining attack scenarios and finding traces of these scenarios in the traffic.) and anomaly detection ( i.e., the construction of a reference model of the behavior of the supervised entity to which we will be able to compare the observed behavior) based on wavelet theory is a promising approach as they are already shown for this type of systems. The detection capability for such complex system could be enhanced using the specific characteristics of its exchanges, use them to enrich its normal signature and reduce the probability of false positive and false negatives.

This paper describes the context and the state of the art of the current research direction of the authors with the aim to present the challenges and the future works that the student aims to perform in the next years.

I. INTRODUCTION

Areas such as the military, health or civil aviation are deploying more and more critical applications in their communication network. The security of these systems is therefore an essential issue. The objectives of this security are confidentiality, integrity and availability. As defined in [START_REF] Amand | Etude d'un système de détection d'intrusion comportemental pour l'analyse du trafic aéroportuaire[END_REF]: An intrusion is a violation of one of these three objectives, and therefore exhibits a different behavior than the normal behavior of a system. An attack is an attempt to violate the system's security policy. The purpose of intrusion detection and prevention is to report intrusions or attacks to the security administrator so that the security administrator can take appropriate action. DSNA (Direction des Services de la Navigation aérienne), the French ANSP (Air Navigation Service Provider), relies on a set of control centers (en-route, approaches, control tower), sensors (radars, multilateration systems, etc.), communication means (centers, air traffic control and management) and radionavigation systems, distributed throughout the territory (metropolitan and ultra-marine) to ensure its missions of control and management of air traffic. A private computer network connects the centers to each other and allows the exchange of large amount of data in a non-centralized way between the different elements constituting the monitoring and control system. These critical systems are increasingly open to the outside world to facilitate the exchange of data and information retrieval. Thus there is a need for secure these informations exchanges from intrusions and attacks in the most efficient way.

The work of this thesis focuses on the definition of an intrusion detection method specifically design for air traffic management (ATM) systems.

The paper is organized as follows. Section II presents the background and related work, Section III presents the stakeholders of the work, while in section IV we describe the problematic of the thesis.Finally, Section V summarizes some key aspects and defines the future works.

II. RELATED WORKS

In intrusion detection, according to Chandola et al. [START_REF] Chandola | Anomaly detection: A survey[END_REF] there exist two main approaches: the signature approach (misuse detection) and the behavioral approach (anomaly detection).

The signature approach consists of defining attack scenarios and looking for signatures of these attacks in data exchanges. This approach is ineffective when dealing with new attacks because their signatures are unknown.

The behavioral approach is the capability to build a reference model, called normal, of the monitored system and to compare it with the observed behavior. The differences between the two raise an alert to report an anomaly and a potential intrusion. It is not easy, however, to define what may be representative of the normal behavior of a system, and thus these methods often generate a large number of false positives or false negatives as the approach is under-fitting.

This limit can nevertheless be circumvented when the data flows are from specific networks for which we have a very good knowledge of the characteristics of the traffic exchanged on the latter is possible. This is particularly the case for industrial networks for which all the data flows exchanged are mapped accurately and explicitly, and where the data exchanges should be invariant over time (or with identifiable patterns, for example over a day or a week. Network traffic within DSNA systems falls into this category. As such, it is conceivable to propose intrusion detection techniques based on behavioral tools. These should help to detect attacks on the specific traffic that is exchanged in and between the various en-route control centers of air navigation (CRNA) and the control centers present in each French airport.

The specificities of the DSNA traffic (statistical determinism in particular) make it possible to use intrusion detection methods based on statistical analysis which gives very good results on deterministic and invariant traffic.

Several research studies have been conducted for traffic characterization using spectral analysis. We can highlight, signature-based detection techniques based on statistical methods such as wavelets [START_REF] Barford | A signal analysis of network traffic anomalies[END_REF], Kalman filters [START_REF] Soule | Combining filtering and statistical methods for anomaly detection[END_REF], hashing projection [START_REF] Krishnamurthy | Sketch-based change detection: methods, evaluation, and applications[END_REF], [START_REF] Brauckhoff | Anomaly extraction in backbone networks using association rules[END_REF], principal component analysis (PCA) [START_REF] Lakhina | Diagnosing network-wide traffic anomalies[END_REF] [8] or pattern recognition [START_REF] Fontugne | A Hough-transform-based anomaly detector with an adaptive time interval[END_REF]. Several of these earlier works deal with the classification of anomalies using events detected with a limited set of signatures (less than 10) [START_REF] Lakhina | Mining anomalies using traffic feature distributions[END_REF], [START_REF] Tellenbach | Accurate network anomaly classification with generalized entropy metrics[END_REF] . The method presented in [START_REF] Borgnat | Seven years and one day: Sketching the evolution of internet traffic[END_REF] seems promising because adapted to any type of network traffic and seems to bring very interesting results for the rest of this thesis work. The authors demonstrated how Long Range Dependence (LRD) can be an efficient parameter to quantify the level of variability of Internet traffic. They used wavelet analysis, via a toolbox, to process data traffic. Thanks to this method, they obtain the variability level of any data series for different time scales and different moments of analysis. In our work, we will use a version of the method introduced in [START_REF] Abry | Long-range Dependence: Revisiting Aggregation with Wavelets[END_REF] developed by H. Wendt more recently called Wavelet Leader Multi-fractal (WLM) analysis wich already show efficient results in terms of false positives/false negatives with a system with a similar characteristic, a drone fleet. [START_REF] Condomines | Network intrusion detection system for drone fleet using both spectral analysis and robust controller/observer[END_REF].

III. STAKEHOLDERS

This part presents the roles and responsibilities of each stakeholders in the work.

A. Activus Services

ACTIVUS-Service is a digital service company operating in the information systems consulting sector in several IT domains. It is developing an internal project called Connect'it with the aims to design and develop, among others, a network monitoring solution. Part of ACTIVUS-Services' research and development work revolves around the centralization of network data as well as the detection of anomalies in these networks.

With its experience in the application of behavioral methods, the work with ACTIVUS-Services is to test and implement these methods by focusing on network data in order to generate relevant alerts for the user. This is done in a Big Data context that is to say using technologies allowing the real time processing of a large volume of data. ACTIVUS-Services software solution expertise will integrate work into a Connect'it industrial systems and networks monitoring module.

B. DSNA

For air transport management, the International Civil Aviation Organization (ICAO) and Air Navigation Service Providers (ANSP) have put in place dynamic air traffic management which is the space based on highly interconnected systems. In France, the DSNA (Direction des Services de la Navigation Aérienne) has the role of ANSP. It is one of the components of the DGAC (Direction Générale de l'Aviation Civile).

The DSNA provides the network traffic records from the air traffic control and management systems as well as the contextual elements for analyzing these records.

C. ENAC

ENAC (Ecole Nationale de l'Aviation Civile), a french engineering school, has a teaching program for the security of information systems and a privileged link with the aeronautical field. The ReSCo (networks and systems communicating) research team has expertise in the field of network security applied to aeronautical networks but also civil aviation. In addition, the ReSCo team is working on various research problems with security problems during exchanges between aircraft in flight. It has solid expertise in the field of onboard aeronautical networks and in the various traffics that can be exchanged between an aircraft and the various ground communication systems of French and European air navigation.

IV. PROBLEM FORMULATION

The presence of exchanges between old and/or proprietary protocols with newer protocols, and the large amount of data to be processed in real time in an ATM network means that intrusion detection for this type of network raises new challenges. Indeed, according to the DSNA, classical methods do not show sufficiently satisfactory results for the criteria of civil aviation. This part, express the difficulties related to anomalies detection and the particularities of the aeronautical environment for this subject.

1) Anomaly Detection: The methods to detect anomalies are based on the observation of numerous of events and on the analysis of these. We need to collect data and to analyze them to find traces of intrusions and / or anomalies. This analysis can be done in several ways: after the facts, almost real-time or in real time. One goal of our work is to get an intrusion detection system that responds to criteria express by the DSNA:

• A criterion of reliability of the system: any intrusion must rise an alert which corresponds to a false negative rate as low as possible. • A criterion of relevance of the alerts: any alert must correspond to an effective intrusion which corresponds to a false-positive rate as low as possible. Then, the goal is to detect intrusions properly in DSNA networks for air traffic control. These networks are critics and as a result, they are eager to detect intrusions as early as possible to prevent further damage. However, the amount of network traffic generated by ATM can be a large amount of bytes per day, requiring extremely efficient methods of analysis to maintain a reasonable detection time and the use of centralized IDS to obtain efficient results. In this context, we conduct research in models to detect intrusions in real time. Several modeling methods and attributes are explored and combined to obtain relevant data processing. Based on the work of statistical models [START_REF] Chandola | Anomaly detection: A survey[END_REF], [START_REF] Gupta | Outlier detection for temporal data: A survey[END_REF], [START_REF]Twitter Engineering: Introducing practical and robust anomaly detection in a time series[END_REF], [START_REF] Laptev | Generic and scalable framework for automated time-series anomaly detection[END_REF], neural networks [START_REF] Gupta | Outlier detection for temporal data: A survey[END_REF], immunological approach [START_REF] Ahmad | Real-Time Anomaly Detection for Streaming Analytics[END_REF], basic graph [START_REF] Pham | Anomaly detection in bitcoin network using unsupervised learning methods[END_REF], unsupervised approach [START_REF] Gupta | Outlier detection for temporal data: A survey[END_REF], [START_REF] Pham | Anomaly detection in bitcoin network using unsupervised learning methods[END_REF], we are working on the development of a better anomaly detection model to respond to the criteria express by the DSNA.

2) Aeronautical: One of the particularities of data exchange in the DSNA ATM system is the use of many proprietary and/or old protocols, but also the statistical determinism of these exchanges. Indeed, they exhibit characteristics relatively stable and have cyclic dynamics. This determinism is induced by the repetitive nature of the various aeronautical movements, which is reflected in the information from the sensors (radars) as well as in the processing subsystems of the flight plans. Thus, there will be recognizable and regular variations in air traffic that will have an impact on the data traffic. This repetitive nature present in the network makes it possible to envisage the use of algorithms based on behavioral approach to detect events that would be abnormal. This unsupervised methodology is different to the more traditional method, since it does not require upstream definition of attack signatures. By using the reproducibility of events in time, we will define a nominal behavior. This nominal signature will then be a basis to which we will compara collected data to highlighting abnormal behavior and therefore intrusion attempts.

The aim of this thesis work is to establish a link between the spectral characterization works (based on wavelet analysis) and the behavioral intrusion detection algorithms being developed for Connect'it. In particular, we are considering the development of a new detection module that would rely on the analytics methods presented in the previous paragraph and use these as inputs for the wavelet analysis. This method should answer to the requests for efficiency, stability and adaptability from the DSNA

V. WORK IN PROGRESS

In our future work, we intend to identify several research perspectives. First of all, we plan to propose an evolution of the WLM model adapted to our system to have a better characterization of the traffic analyzed. Moreover, we plan to analyze different types of traffic: first only radar data caracteristics of ATM system then also other traffic present in ATM system to enrich our bank of knowledges. To address these last objectives, we would like to consider experiments with real traffics from DSNA.

A. Intrusion Detection Methodology

This first step is dedicated to traffic characterization. The objective is to get a specific signature of the traffic we want to analyse. It is possible with the Wavelet Leader Multi-fractal (WLM) analysis.

The WLM analysis is used to quantify the variability of any time series, as a series of networks, that we want to characterize. With this method, we are able to capture the complexity of the traffic for different time scales and analysis times. The result of this analysis is a graph (called spectral signature) that allows us to visually see the difference between legitimate traffic and traffic that contains an attack. Using this WLM methodology, we are able to quantify the variability of any time series from two complementary parameters: the time scale and the time of the analysis. The time scale allows us to see any repetition in the process over time, while the analysis time allows us to analyze the traffic data in different spectral representations. This second metric quantifies the traffic variation according to, for example, q = 1 (mean), q = 2 (variance) and so on.

With this method, we are able to highlights differences between different analyzed traffics and these differences are useful for traffic characterization.

B. Characterization of data radar

As a first step to our IDS for ATM, we decide to work with data radar provided by DSNA. These data are considered as characteristic of an ATM system. They are the most stable data, but also those they represent perfectly the cyclic determinism of network data caused by the repetitive nature of the different aeronautical movements. Moreover, they are part of the essential data for the proper functioning of the ATM system, which makes them very critical. They are also intended to be increasingly exchanged with partners and the outside world to share information as air routes, flight tracking. It is therefore interesting at first to focus on the characterization and profiling of this type of network data.

To realize this characterization, we conduct two types of analysis concerning these data:

• The first is an exploratory data analysis (EDA) of about two weeks of actual traffic data collected from different radars. The results of this first analysis will allow us to better understand the analyzed traffic. Then we can compare it with other types of traffic, and get an idea of the specificity / difficulty of using anomaly detection tools applied to this type of generic traffic. • The second analysis, with the same data, is a traffic signature identification using a spectral analysis (with the WLM analysis methodology explained in the previous paragraph). It represents a normal signature to which other potentially corrupted data signatures can be compared.

With this characterization, we will be able to generate false or corrupted radar data and inject them into real traffic. With the comparison of the spectral analysis of the normal traffic and the corrupted one, we will can see visually the consequences of intrusions and thus see cases and conditions where we can raise alerts.

The results of these first works will be a first step towards designing a future and effective IDS for ATM. After the radar study, we will be able to use our methodology on other data, as flight plan data, flight information data or other critical data from ATM systems. Once, the characterization of these data will be done, one challenge will to analyze traffic with all these mixed data and be able to raise alerts when there are effective intrusions. The final challenge will be to use our module directly plug to ATM system and have effective results in real time.