Sukumar Rajauria

P Gandit

T Fournier

F W J Hekking

B Pannetier

H Courtois

Andreev Current-Induced Dissipation in a Hybrid Superconducting Tunnel Junction

Keywords: numbers: 74, 50, +r, 74, 45, +c

We have studied hybrid superconducting micro-coolers made of a double Superconductor-Insulator-Normal metal tunnel junction. Under subgap conditions, the Andreev current is found to dominate the single-particle tunnel current. We show that the Andreev current introduces additional dissipation in the normal metal equivalent to Joule heating. By analyzing quantitatively the heat balance in the system, we provide a full description of the evolution of the electronic temperature with the voltage. The dissipation induced by the Andreev current is found to dominate the quasiparticle tunneling-based cooling over a large bias range.

In a tunnel junction between a normal metal (N) and a superconductor (S), the charge transfer occurs mainly through two different mechanisms. The tunneling of a single quasi-particle is possible for electrons or holes with an energy E (compared to the Fermi level E F ) larger than the superconductor gap ∆. At low energy, the charge transfer occurs through the Andreev reflection [1,2]. In the normal metal, an electron (a hole) impinging on the superconducting interface is reflected as a hole (an electron), enabling the transfer of a Cooper pair into (out of) the superconductor. As the energies of the involved electron and hole are located symmetrically around E F , the Andreev reflection does not carry heat through the interface at zero bias. The probability for an incident quasi-particle to follow an Andreev reflection, a specular reflection or a tunnel transfer is given in the ballistic regime (no disorder) by the BTK theory [3]. For a N-I-S tunnel junction with an insulator (I) of intermediate or low transparency, the Andreev reflection probability is predicted to be vanishingly small. Taking into account the quasi-particles confinement in the vicinity of the interface, this is no longer true. This confinement can be induced by the disorder or the presence of a second barrier in the normal metal. A single quasiparticle then experiences several collisions with the interface [4,5]. The actual Andreev reflection transmission coefficient corresponds to the coherent addition of many individual transmission probabilities. Therefore, the Andreev sub-gap current significantly exceeds the ballistic case prediction [6,7] and can be modulated by a magnetic flux [8].

A quasi-particle current in a N-I-S junction indeed carries both a charge current and a heat current. With a voltage bias smaller than the gap ∆/e, the tunnel current is selectively made out of high-energy electrons (or holes); this cools the electronic population of the normal metal [9]. In this way, (S-I-N-I-S) micro-coolers based on a double tunnel junction provide a significant temperature reduction which reaches an optimum at a voltage bias just below the gap. At a very low temperature, the thermal transport in such N-I-S tunnel junctions appears to be still little understood. For instance, an apparent reversal of the normal metal temperature evolution was observed in various experiments [10,11] and related to a non-BCS density of states of the superconductor [11]. A clear understanding of this behavior is still missing.

In this letter, we describe an experimental study of the heat transport in a S-I-N-I-S junction, focusing on the very low temperature regime. We show that the phasecoherent Andreev current introduces a significant dissipation in the normal metal. We provide a fully quantitative analysis of the heat transfer in the system which shows that although the Andreev current is a small effect in terms of charge current, the heat it creates has a dominating influence on the heat balance. Fig. 1 inset shows a typical sample which features a geometry similar to the one studied in Ref. [12]. It consists of a 50 nm thick, 4 µm long and 0.3 µm wide normal metal Cu electrode embedded between two 40 nm thick and 1.5 µm wide superconducting Al electrodes. The tunnel barriers at the two symmetric junctions of dimensions 1.5 × 0.3 µm 2 were prepared by oxidization in 0.2 mbar of O 2 pressure for 3 min and give a total normalstate resistance R n in the range 2-3 kΩ. In addition to the two cooling junctions, we added three Cu tunnel probes of area 0.33×0.43 µm 2 on one Al electrode (one is shown in Fig. 1 insets). These probes are strongly connected to both a Cu reservoir and an Al reservoir so that no cooling is expected there. These junctions thus provide a reference for an isotherm behavior. In the following, we will describe the behavior of one out of three investigated samples which all showed similar behavior.

We have measured the current-voltage I(V ) characteristic of every probe junction and of the two cooling junctions in series at temperatures down to 90 mK. The differential conductance dI/dV (V ) is obtained by numerical differentiation. We have taken special care to obtain accurately the sub-gap conductance of our current-biased samples down to a level of about 10 -4 of the normal state conductance. Fig. 1 displays on a logarithmic scale the differential conductance of the cooling junctions for a series of cryostat temperatures. The tunnel current in a N-I-S junction is given by:

I(V ) = 1 eR n ∞ 0 n s (E)[f N (E - eV 2 ) -f N (E + eV 2 )]dE, (1)
where V /2 is the voltage across the N-I-S junction, f N (E) is the electron energy distribution function in the normal metal and

n s (E) =| E | / √ E 2 -∆ 2
is the normalized BCS density of states in the superconductor. An isotherm N-I-S junction would then feature in the subgap regime a linear behavior of the differential conductance on a logarithmic scale. In contrast, we observe in the high temperature (T > 200 mK) data an upward curvature which constitutes a clear signature of the electron cooling [12].

In the low temperature regime (T < 200 mK), clearly a different characteristic is obtained with a differential conductance peak at zero bias. A similar behavior is obtained on every probe (not shown). This zero-bias peak cannot be accounted for by a single-particle tunnel current. The zero-bias differential conductance increases while the temperature is lowered below about 200 mK, which suggests that it is a phase-coherent effect.

We will ascribe the low bias differential conductance peak to an Andreev current, i.e. a double particle tunnel current created by Andreev reflections at the N-I-S junctions. In order to calculate the Andreev current I A , we used the theory of Ref. [5]. We took into account the finite gap ∆ and the disorder both in the normal metal and in the superconductor. We considered the 1D regime where the coherence length of an Andreev pair in the normal metal L E = hD/E [13] is much larger than the junction dimension. Let us first discuss the current-voltage characteristics of one probe junction, where the electronic temperature can be considered as constant and very close to the cryostat temperature. Fig. 2 displays the measured current-voltage characteristic of one probe junction (1.55 µm away from the cooler junctions) together with the calculated single quasi-particle and Andreev currents. We have fitted the current-voltage characteristics of the probe while taking the electron diffusion coefficient in Cu equal to the measured value D = 80 cm 2 .s -1 . The fit parameters are ∆ = 0.228 meV, the electronic temperature T e = 105 mK, which is found slightly higher than the bath temperature, and L ϕ = 1.5 µm, which agrees well with expectations for a pure metal at very low temperature [14]. In this fit, we had to scale the Andreev current by a multiplying factor M = 4.3 in a similar way to Ref. [8]. The fit describes very well the probe data. The cross-over between the low-bias Andreev current and the high-bias single quasi-particle current is clearly visible. We did not need to take into account the possible contribution of pinholes in the barrier, which means that our junctions can be considered as ideal. We conclude that the sub-gap current in our N-I-S junctions is the superposition of a single-particle tunnel current and a two-particle Andreev current.

We tried to fit the S-I-N-I-S cooler junction characteristic in a similar way. Fig. 2 also shows the currentvoltage characteristic of the cooler junction together with the calculated single quasi-particle and Andreev currents. A good fit to the theory is found only at small bias, where cooling or heating effect remain negligible. This fit is obtained with the same parameter set than for the probe, except for the gap 2∆ = 0.43 meV and the scaling factor M = 1.55. The difference in the M factor is presumably due to difference in geometry between the types of junctions.

In order to analyze quantitatively the behavior of our S-I-N-I-S cooler junctions, we need to consider the heat balance in the normal metal. Here we assume a quasiequilibrium situation: the electrons and the phonons follow a thermal energy distribution functions at a respective temperature T e and T ph which are in general different from the cryostat temperature T base . This is justified because the inelastic scattering time is of the order of the phase-coherence time of 300 ps (derived from Fig. 2 fit), which is much shorter than the mean escape time from the island estimated to about 100 ns. The tunnel current is responsible for a cooling power in the normal metal:

P cool = 1 e 2 R n ∞ -∞ (E- eV 2 )n s (E)[f N (E- eV 2 )-f S (E)]dE
(2) where f S is the energy distribution function in the superconductor. This cooling power is compensated by the electron-phonon coupling power so that: 2P cool +P e-ph = 0. Here the factor 2 accounts for the presence of two junctions. We used the usual expression for the electronphonon coupling P e-ph (T e , T ph ) = ΣU (T 5 e -T 5 ph ), where Σ is a material-dependent constant and U is the metal volume. If one considers now the normal metal phonons, the electron-phonon coupling power is compensated by the Kapitza power P K (T ph , T base ) = KA(T 4 base -T 4 ph ), where K is an interface-dependant parameter and A the contact area, so that P e-ph + P K = 0. The related Kapitza thermal resistance is significant only at intermediate temperature and above (T > 300 mK), in which case the normal metal phonons can be cooled below the substrate temperature [12].

Fig. 3 displays a series of current-voltage characteristics obtained at different temperatures together with calculated curves. We have first numerically solved the heat balance equations in the high temperature regime where the Andreev current contribution can be neglected. We took the well-accepted value Σ = 2 nW.µm -3 .K -5 [15] and we determined from the fitting procedure the Kapitza coupling parameter value K.A = 144 W.K -4 . Fig. 3 dash-dotted line displays the calculated currentvoltage characteristic calculated at a 90 mK cryostat temperature by including the charge and heat currents of only the single-particle channel effects. The agreement is poor, which confirms the need for including the Andreev current channel. As a second step, we included the Andreev charge current component. The dashed line shows the result of the calculation with again the same parameters set than in Fig. 2 and a cryostat temperature of 90 mK. It provides an acceptable fit at low bias but shows a clear discrepancy at intermediate voltage. Adding a leakage term in the heat balance with a linear resistance does not provide a good description of the data. Thus a significant thermal contribution is missing in the heat balance equations described above.

We investigated theoretically the heat transfer created by the Andreev current I A [16] flowing through an N-I-S tunnel junction [17], straightforwardly extending [5]. We have found that the work performed by the current source feeding the circuit with a current I A generates a Joule heating I A V that is deposited in the normal metal. The full heat balance equation for the normal metal electrons can then be written as:

2P cool + P e-ph + I A V = 0. (3)
With this complete heat balance equation taken into ac- count, we calculated the current-voltage characteristic at every cryostat temperature, see the dotted lines fits in Fig. 3. The agreement is very good for every accessible voltage and temperature. Assuming a thermalization of the phonons to the substrate temperature would change the current by less than 2% at 90 mK, which means that the phonon cooling has a negligible role in the data analysis presented here. Let us note that at the 90 mK cryostat temperature, the agreement covers 4 orders of magnitude for the current. This excellent agreement demonstrates that the heat balance in our S-I-N-I-S junctions can be fully understood by taking into account the contributions of both the single quasi-particle tunneling and the two-particle Andreev current. The numerical solution of the heat balance equations also provides the electron temperature at every bias. Fig. 4 shows the calculated dependence of the normal metal central island electron temperature with the voltage across the cooler. At very low temperature, the electron temperature first increases with the bias due to Andreev current heating, before decreasing due to the tunnel current-based cooling effect. At a cryostat temperature of 90 mK, the cooling effect overcomes the Andreev current thermal contribution only close to the optimum bias. This demonstrates the great importance of the Andreev current-induced dissipation in a N-I-S tunnel junction. Although the Andreev current is a small effect in a such junction if one considers the charge current, this is no longer true if one considers the heat current. The explanation is the following. Compared to a Joule power I.V , the Andreev current contributes fully to heating while the tunnel current cools with a moderate efficiency [9]. This efficiency is of the order of T e /∆ which is about 5 % at a 100 mK electron temperature.

In conclusion, we have devised a quantitative analysis of the current-voltage characteristics of S-I-N-I-S hybrid junctions at very low temperature. Our study demonstrates the importance in terms of heat transport of the Andreev current which arises from the confinement of phase-coherent quasi-particles in the vicinity of the interface. This Andreev current-induced dissipation is with no doubt also of great importance in the behavior of S-N-S junctions with transparent interfaces, where a significant electron heating can be observed [18].

FIG. 1 :

 1 FIG. 1: (Color online) Left inset: Geometry of the sample. Right inset: Scanning electron microscope micrograph of a typical cooler sample made of a normal metal Cu electrode (light grey) connected to two superconducting Al reservoirs (dark grey) through tunnel junctions. One of the additional probe junctions connected to one Al reservoir is visible at the bottom. Main figure: normalized differential conductance as a function of the voltage and at the cryostat temperatures of 90 (red curve), 230 (blue), 330 (green) and 440 mK (brown).

FIG. 2 :

 2 FIG. 2: (Color online) Current-voltage characteristic of the cooler junctions (blue symbols) and of the probe (red symbols) as a function of the voltage and at a cryostat temperature of 90 mK together with best-fit calculations of the Andreev current (dotted lines) and of the single-particle current (dashed lines) using the parameters D = 80 cm 2 .s -1 , Lϕ = 1.5 µm, Rn = 1.9 kΩ (cooler) and 2.76 kΩ (probe), 2∆ = 0.43 meV (cooler) and ∆ = 0.228 meV (probe). The fit temperature is 105 mK. Compared to the theoretical calculations based on Ref. 5, the Andreev current was multiplied by 1.55 (cooler) and 4.3 (probe). No cooling effect is included in the calculation.

FIG. 3 :

 3 FIG. 3: (Color online) Current-voltage characteristic of the S-I-N-I-S cooler junction as a function the voltage at cryostat temperature of 90 (red line), 230 (blue) and 330 mK (green) together with best-fit calculated curves. Dot-dashed line: model including the cooling by the tunnel current, with parameters 2∆ = 0.43 meV, K.A = 144 W.K -4 . Dashed line: model including in addition the Andreev current, but not the related heating, with the parameters D = 80 cm 2 .s -1 , Lϕ =1.5 µm, M = 1.55. Dotted lines: full model taking into account the Andreev current and the related heating,

FIG. 4 :

 4 FIG. 4: (Color online) Dependence of the calculated electronic temperature with the voltage as derived from the fit of the experimental result and for a series of cryostat temperature: 90 (red symbols), 140 (green) and 230 mK (blue).

The authors are grateful to Nanofab-CNRS for sample fabrication. We acknowledge useful discussions with J. P. Pekola, M. Houzet and A. Vasenko. This work was funded by EU STREP project 'SFINx' and by NanoSciERA project 'Nanofridge'.


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Protoplast-to-tree systems for Prunus cerasifera and Prunus spinosa as a tool for phylogenetic studies within the Prunoideae, Rosaceae

Sergio Ochatt

To cite this version:

Sergio Ochatt. Protoplast-to-tree systems for Prunus cerasifera and Prunus spinosa as a tool for phylogenetic studies within the Prunoideae, Rosaceae. 8. International Protoplasts Symposium, 1991, Uppsala, Sweden. 1 p., 1991. hal-02843827


Frédéric Bernicot
email: frederic.bernicot@math.u-psud.fr

Uniform estimates for paraproducts and related multilinear multipliers

Keywords: Paraproducts, uniform estimate, multilinear operators, Littlewood-Paley theory, Calderón-Zygmund decomposition. MSC classification : 42B15, 42B20, 42B25

In this paper, we prove some uniform estimates between Lebesgue and Hardy spaces for operators closely related to the multilinear paraproducts on R d . We are looking for uniformity with respect to parameters, which allow us to disturb the geometry and the metric on R d .

Introduction

The purpose of this article is to prove uniform estimates on paraproducts and similar multilinear operators. Let us first recall what is a paraproduct. A n-linear paraproduct Π on R d is a n-linear operator of the following form :

Π(f 1 , .., f n )(x) := ∞ 0 n i=1 π i t * f i (x) dt t ,
or of the discrete form

Π(f 1 , .., f n )(x) := j∈Z n i=1 π i 2 j * f i (x) dt t .
1

Here the π i t are smooth functions which Fourier transform π i t are bump functions adapted to the ball {ξ ∈ R d , |ξ| ≤ 1} and we assume that there exists one index i ∈ {1, .., n} such that ∀t > 0, π i t (0). In all the sequel, a smooth function φ is said to be "adapted to a set" I ⊂ R d if it is supported on this set and satisfies : for all order n ∈ N d φ (n) ∞ ≤ |I| -|α| . Then for such a paraproduct, the classical Calderón-Zygmund theory gives us that for all exponents 1 < p 1 , .., p n < ∞ such that

0 < 1 p := n i=1 1 p i < 1,
there exists a constant C = C(p i ) such that for all functions f i ∈ S(R d ),

Π(f 1 , .., f n ) p ≤ C n i=1 f i p i .
These estimates in Lebesgue spaces depend on the functions π i t . We would like to understand how can we modify these functions, keeping uniform estimates.

The paraproducts are the first studied class of singular bilinear operators. Their study began by the works of J.M. Bony in [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] and of R. Coifman and Y. Meyer in [START_REF] Coifman | On commutators of singular integrals and bilinear singular integrals[END_REF][START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF][START_REF] Coifman | Commutateurs d'intégrales singulières et opérateurs multilinéaires[END_REF], where in particular continuities in Lebesgue spaces are shown. The first uniform result is the following one (from [START_REF] Coifman | Ondelettes et opérateurs III, Oérateurs multilinéaires[END_REF]) : Theorem 1.1. Let (M i ) 1≤i≤n be integers and (π i j ) j∈Z 1≤i≤n be smooth functions such that π i j is adapted to the rectangle [-2 j+M i , 2 j+M i ] d . Assume that there exist an integer N and an index i ∈ {1, .., n} such that for all j,

∀η ∈ [-2 j+M i -N , 2 j+M i -N ] d , π i j (η) = 0. (1.1)
Then for all exponents 1 < p 1 , ..., p n ≤ ∞ satisfying

0 < 1 p := 1 p 1 + .. + 1 p n < 1,
there exists a constant C = C(N, p i ), which does not depend on (M i ) i such that

∀f i ∈ S(R d ), j n i=1 π i j * f i p ≤ C n i=1 f i p i .
About this result, there are two different questions : what is the maximal range of exponents with uniform estimates ? May we weaken the assumption (1.1) ?

The second question was solved by C. Muscalu, T. Tao and C. Thiele in [START_REF] Muscalu | Uniform estimates on paraproducts[END_REF], where they prove the stronger result :

Theorem 1.2. Let (M i ) 1≤i≤n be integers and π i j be smooth functions such that π i j be adapted to the rectangle [-2 j+M i , 2 j+M i ] d . Assume that for all j, there exists an index i ∈ {1, .., n} with

π i j (0) = 0. (1.2)
Then for all exponents 1 < p 1 , ..., p n < ∞ satisfying

0 < 1 p := 1 p 1 + .. + 1 p n < 1,
there exists a constant C = C(N, p i ), which does not depend on (M i ) i such that

∀f i ∈ S(R d ), j n i=1 π i j * f i p ≤ C n i=1 f i p i .
It is even shown a little stronger version (a maximal version) than this one.

The assumption (1.2) is much weaker than (1.1). The proof of Theorem 1.2 is a mixture of the proof of Theorem 1.1 and arguments from graph theory. Such a result was motivated by the paper [START_REF] Muscalu | Uniforms estimates on multi-linear operators with modulation symmetry[END_REF] from the same authors. In this article, they study some uniform estimates for multilinear operators far more singular than the paraproducts, closely related to the bilinear Hilbert transforms. The "classical" time-frequency analysis, to decompose these kind of operators, uses some information and estimates on operators, which look like paraproducts. That is why they have first shown in [START_REF] Muscalu | Uniform estimates on paraproducts[END_REF] uniform estimates for paraproducts.

In this paper, we are interested in answering to the first question. Mainly we want to obtain uniform estimates with infinite exponents and some exponents lower than 1. In [START_REF] Li | Uniform estimates for some paraproducts[END_REF], X. Li has shown uniform estimates when 1 < p 1 , ..., p n < ∞ and p may be lower than one. We would like to extend his result for some exponent p i < 1 or p i = ∞. The continuities for this range of exponents have already been proved (for example in [START_REF] Grafakos | The Marcinkiewicz multiplier condition for bilinear operators[END_REF] by L. Grafakos and N. Kalton). Here we would like to improve these continuities with uniform estimates.

A second motivation for the study of paraproducts is this one : we know how to decompose a multilinear multiplier, satisfying Hörmander's condition, with multilinear paraproducts. A n-linear multiplier T is given by its symbol σ ∈ S(R dn ), with the formula :

T (f 1 , .., f n )(x) := R dn e ix.(ξ 1 +..+ξn) σ(ξ) n i=1 f i (ξ i )dξ. (1.
3)

The Hörmander condition corresponds to the following assumption :

∀m i ∈ N d , n i=1 ∂ m i ξ i σ(ξ 1 , .., ξ n ) ≤ 1 (|ξ 1 | + .. + |ξ n |) |m 1 |+..|mn| . (1.4)
Note the appearance of the quantity |ξ 1 | + .. + |ξ n |, which corresponds to the distance d(ξ, 0) in the frequency plane. We are now interested in disturbing the metric. We would like study the following distance

d λ (ξ, 0) := n i=1 |λ i ξ i |,
given by non vanishing reals λ i . In fact it is easy to see that our parameters λ i have the same function than the parameters M i of Theorems 1.1 and 1.2 (we have the relation λ i ≃ 2 -M i ). So we would like to have uniform estimates with respect to the new distance d λ . The problem of disturbing the metric appeared for example in the study of bilinear Hilbert transforms along polynomial curves ( [START_REF] Fan | A bilinear oscillatory integrals along parabolas[END_REF]) and was one of the X. Li motivations to study uniform estimates for paraproducts.

We will also prove the following result :

Theorem 1.3. Let σ be an x-independent symbol such that

∀m i ∈ N d ∂ m 1 ξ 1 ..∂ mn ξ n-1 σ(ξ 1 , .., ξ n ) n i=1 |λ i | |m i | d λ (ξ, 0) |m 1 |+..+|m n-1 | . (1.5) Let 0 < p i , p ≤ ∞ exponents satisfying 1 p = n i=1 1 p i .
Let us denote the three disjoint sets (which may be empty) S 1 , S 2 and S 3 such that

{1, .., n} = S 1 ⊔ S 2 ⊔ S 3 , with ∀i ∈ S 1 , p i = 1, ∀i ∈ S 2 , p i = ∞ and ∀i ∈ S 3 , p i ∈ {1, ∞} c .
Then we know that the multilinear multiplier T defined by (1.3) can be continuously extended from ⊗ n i=1 F i to G in the three following cases :

• if 0 < p < ∞ with G = L p , F i = H p i for i ∈ S 1 ∪ S 3 and F i = L ∞ c for i ∈ S 2 , • if 0 < p < ∞ with G = L p,∞ , F i = L 1 for i ∈ S 1 , F i = H p i for i ∈ S 3
and

F i = L ∞ c for i ∈ S 2 , • if p = ∞ (
and also for all i ∈ {1, .., n} p i = ∞) with G = BMO and F i = L ∞ c for all i ∈ {1, ..n}. In addition we have the two following improvements : Part 1) : All these continuities are uniformly bounded with respect to the parameters λ i under one of the two following assumptions :

a-) ∀1 ≤ i ≤ n, p i < ∞ b-) γ := j∈{1,..,n} |λ j |≃max{|λ l |, 1≤l≤n} 1 p j ≥ 1 2 .
Else the continuity bound depends on the ratio

max{|λ k |, 1 ≤ k ≤ n} min{|λ k |, 1 ≤ k ≤ n} .
Part 2) : We don't know if the conditions a-) or b-) are sufficient to get uniform bounds, however we will show that if p < ∞ and γ = 0 then we cannot have a uniform bound.

In this result we write L p = L p (R d ) for the "classical" Lebesgue spaces and H p = H p (R d ) for the Hardy spaces (which is equivalent to the Lebesgue spaces L p if 1 < p < ∞) and BMO = BMO(R d ) the space of functions of "bounded mean oscilaation". We write L ∞ c for the set of bounded compactly supported functions, equipped with the L ∞ -norm.

Remark 1.4. Our proof, show that in particular case, we can obtain the continuity with the whole space L ∞ instead of L ∞ 0 . For convenience and technical difficulties (see the proof of Corollary 4.11), we prefer to only work with the space L ∞ 0 . Remark 1.5. By Taking λ i ≃ 2 -M i , the paraproducts of Theorems 1.1 and 1.2 verify (1.5) uniformly with respect to λ because the symbol is given by

σ(ξ 1 , .., ξ n ) = j n i=1 ζ i,j (ξ i ).
So Theorem 1.3 improves the uniform estimates of Theorem 1.2 and answer to the asked question.

Remark 1.6. By using time-frequency tools such as "tiles" and "trees" as in [START_REF] Muscalu | A short proof of the Coifman-Meyer multilinear theorem[END_REF][START_REF] Li | Uniform estimates for some paraproducts[END_REF], it should be possible to prove some uniform "weak type restricted estimates" in L p , which are stronger than our continuity in L p for p < 1. Remark 1.7. The continuities are already known from the papers [START_REF] Grafakos | The Marcinkiewicz multiplier condition for bilinear operators[END_REF] and [START_REF] Grafakos | Multilinear Calderón-Zygmund operators on Hardy spaces[END_REF] of L. Grafakos and N. Kalton. In fact our operators are multilinear Calderón-Zygmund operators and so their continuities are a consequence of the paper [START_REF] Grafakos | Multilinear Calderón-Zygmund theory[END_REF] of L. Grafakos and R. Torres. The improvement is the fact that we can have uniform bounds and we must be careful because the constants, as multilinear Calderón-Zygmund operators, are not uniformly bounded. So we will use the ideas of the Calderón-Zygmund theory with a few improvements.

There is an other interest to study such uniform estimates. The symbols verifying (1.5) uniformly with respect to λ satisfy the Marcinkiewicz condition :

∀m i ∈ N d ∂ m 1 ξ 1 ..∂ mn ξ n-1 σ(ξ 1 , .., ξ n ) n i=1 |ξ i | -|m i | . (1.6)
However, from [START_REF] Grafakos | The Marcinkiewicz multiplier condition for bilinear operators[END_REF] we know that the condition (1.6) is in general not sufficient to guarantee continuity, as in the previous Theorem. So our result allows us to almost describe the "limit case" between (1.4) and (1.5) to get these continuities.

To prove our Theorem, we will use model operators, which generalize and are more symmetric than the paraproducts. In the definition of paraproducts, there has to be one (or more) index i ∈ {1, .., n} such that (1.1) or (1.2) is satisfied, so there is a lack of symmetry in their definition (see Remark 2.5).

The plan of this paper is the following one. In Section 2, we define notations and our model operators. We first prove Theorem 1.3 for our model operators : in the case where all exponents belong to (1, ∞) in Section 3 (this part only uses Littlewood-Paley theory) and after for others exponents in Section 4 (this part uses Carleson measures and an improved Calderon-Zygmund theory). Then we complete the proof of Theorem 1.3 for general multipliers in Section 5.

2 Definition of our model operators.

For the rest of this paper, we use the well-known notations : let ζ be a function on R d , t = 0 be a real and q ∈ R d be a vector. We set ζ t and ζ t,q for the L 1 -normalized functions defined by

ζ t (x) := 1 |t| d ζ(t -1 x) and ζ t,q (x) := 1 |t| d ζ(t -1 (x -q)).
We will work with the Hardy spaces on R d , so let us first recall one of its definitions.

Definition 2.1. Let Ψ be a smooth function. We define S Ψ to be the continuous or the discrete Littlewood-Paley square function, given by

S Ψ (f ) := |Ψ t * f | 2 dt t 1/2 or S ∆ Ψ (f ) := n∈Z |Ψ 2 n * f | 2 1/2
.

We use these functionals to get the following definition of Hardy spaces (See [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF]) :

Definition 2.2. Let Ψ be a non null smooth function whose spectrum is contained in a corona around 0. For 0 < p < ∞, we define the Hardy space H p = H p (R d ) as the set of distributions f ∈ S ′ (R d ) satisfying :

f H p := S Ψ (f ) p < ∞.
From the book [START_REF] Stein | Harmonic analysis : Real variable Methods, Orthogonality, and Oscillatory Integrals[END_REF] we know that for 1 < p < ∞ the Hardy space H p corresponds to the Lebesgue space L p . In addition, we have the choice to keep a discrete or a continuous square function : the definition of the space does not depend on it or on the choice of the function Ψ.

We have to control norms in the Schwartz space, so we set for an integer K c K (ζ) := sup

x∈R d (1 + |x|) K sup α∈N d |α|≤K |∂ α x ζ(x)| . (2.1)
Now we define our model operators.

Definition 2.3. Let Ψ be a smooth function on R d whose spectrum is contained in a corona around 0 and let Φ i be smooth functions whose spectrum is bounded. Let L be a bounded function on Z, λ = (λ 1 , .., λ n ) ∈ (R * ) n and ρ = (ρ 1 , .., ρ n ) ∈]0, 1] n be parameters. Then we define the following operator :

T ρ,λ,L (f 1 , .., f n )(x) = k∈Z L(k) R d Ψ 2 k (y) n i=1 [Φ i λ i 2 k * f i ](x -ρ i λ i y)dy.
We also have the continuous version with a bounded function L on R + , defined by

U ρ,λ,L (f 1 , .., f n )(x) = ∞ 0 L(t) R d Ψ t (y) n i=1 [Φ i λ i t * f i ](x -ρ i λ i y)dy dt t .
It is easy to see that these operators continuously act from S(R d ) ⊗n to S(R d ).

In addition, the operator T ρ,λ,L is associated to the following symbol σ :

σ(ξ 1 , .., ξ n ) := k∈Z L(k) Ψ 2 k (ρ 1 λ 1 ξ 1 + .. + ρ n λ n ξ n ) n i=1 Φ i (λ i 2 k ξ i ),
which satisfies (1.5) uniformly on λ.

We want now to make the link with the "classical" paraproducts.

Proposition 2.4. The parameters ρ i allow us to get the "classical" paraproducts as limit of our previous operators : for all f 1 , ..,

f n ∈ S(R d ) U ρ,λ,L (f 1 , .., f n )(x) ρ 1 =1 ρ i →0 ---→ ∞ 0 L(t)[Ψ * Φ 1 ] λ 1 t * f 1 (x) n i=2 [Φ i λ i t * f i ](x) dt t . (2.2)
Here the convergence is in the S(R d ) sense.

We do not write the details of this result. With the good assumptions about the functions f i , it is easy to prove this convergence.

Remark 2.5. Our model operators have a symmetry : the definition is invariant by permutations on the n functions, which is not the case for the "classical" paraproducts. For example in the bilinear case, we want to estimate in L 2 the two different paraproducts (for f ∈ L ∞ and g ∈ L 2 ) :

∞ 0 [Ψ λ 1 t * f ] [Φ λ 2 t * g] L(t) t dt and ∞ 0 [Φ λ 1 t * f ] [Ψ λ 2 t * g] L(t) t dt, uniformly on (λ 1 , λ 2 ) with |λ 2 | > |λ 1 |.
These two paraproducts are a little different and so their study ask some different arguments.

That is why we prefer working with our model operators, which own symmetry invariance and allow us to get by a limit argument these two kinds of paraproducts.

Remark 2.6. It is quite easy to show that our model operators satisfy the assumptions of Theorem 1.3 with uniform bounds with respect to λ and ρ.

We let to the reader the details of this claim.

Before to prove the positive part (part 1) of Theorem 1.3 for our model operators, we would like to explain the negative claim of this Theorem (part 2) in the bilinear case :

Proposition 2.7. Let ρ 1 = ρ 2 = 1 be fixed and |λ 1 | ≪ |λ 2 | be reals.
There exists operators U ρ,λ,m (also satisfying the assumptions of Theorem 1.3) which cannot be continuous from L p × L ∞ into L p for 1 < p < ∞ with an uniform bound with respect to λ.

Proof : Let us choose Φ i = ζ a smooth and nonnegative function whose integral is equal to 1 and set

U ǫ,λ (f, g)(x) := ǫ -1 ǫ R d Ψ t (y)ζ λ 1 t * f (x -λ 1 y)ζ λ 2 t * g(x -λ 2 y) dydt t .
When λ 1 tends to 0, we have

∀f ∈ S(R d ), x ∈ R d lim λ 1 →0 ζ λ 1 t * f (x -λ 1 y) = f (x).
Due to the presence of the ǫ > 0, we have for

f, g ∈ S(R d ) ∀x ∈ R d lim λ 1 →0 U ǫ,λ (f, g)(x) = f (x) ǫ -1 ǫ R d Ψ t (y)ζ λ 2 t * g(x -λ 2 y) dydt t .
We can now take ǫ → 0 and we get

lim ǫ→0 ǫ -1 ǫ R d Ψ t (y)ζ λ 2 t * g(x-λ 2 y) dydt t = ∞ 0 R d Ψ t (y)ζ λ 2 t * g(x-λ 2 y) dydt t .
Then we can choose good functions Ψ and ζ in order to find the linear Hilbert transform H. With these ones, we conclude

∀x ∈ R d , lim ǫ→0 lim λ 1 →0 U ǫ,λ (f, g)(x) = f (x)H(g)(x).
So if we have uniform estimates on U ǫ,λ from L p × L ∞ into L p , by using Fatou's lemma, we get :

∀f, g ∈ S(R d ), f H(g) p f p g ∞ .
Such an estimate implies the boundedness of H on L ∞ which is not possible. So we cannot have uniform estimates for the operators U ǫ,λ .

⊓ ⊔

After these remarks, we are going to prove Theorem 1.3 for our model operators.

N. Kalton used in [START_REF] Grafakos | Multilinear Calderón-Zygmund operators on Hardy spaces[END_REF], we can have boundedness of our operators on the sets of atoms associated to the considered Hardy spaces. For several years, many papers (see for example [START_REF] Bownik | Boundedness of operators on Hardy spaces via atomic decompositions[END_REF][START_REF] Meda | On the H 1 -L 1 boundedness of operators[END_REF][START_REF] Meyer | Some functional analytic properties of the spaces B q generated by blocks[END_REF]) emphasize the following problem : how can we extend a linear operator bounded on the set of atoms to the whole Hardy space ? This abstract question is a really problem and does not admit a general positive answer. For example there is a counter-example in [START_REF] Meyer | Some functional analytic properties of the spaces B q generated by blocks[END_REF] for the classical Hardy space.

For this reason, we prefer to describe an other proof, which does not use the atomic decomposition of Hardy spaces. That is why, we are going to directly work with the Littlewood-Paley square functions.

For convenience, we deal only with the bilinear case : n = 3. First remember the definition of our operator : we choose two smooth functions Φ 1 and Φ 2 with bounded spectrum and we choose a smooth function Ψ whose the spectrum is included in a corona around 0. Then we construct the operator

T ρ,λ,L (f, g)(x) := k∈Z L(k) R d Ψ 2 k (y)[Φ 1 λ 1 2 k * f ](x -ρ 1 λ 1 y)[Φ 2 λ 2 2 k * g](x -ρ 2 λ 2 y)dy.
To study this last one, we decompose the two functions f and g with the classical wavelets decomposition :

Lemma 3.1. Let ψ a smooth function such that c(ψ) := ∞ 0 ψ(tξ) 2 dt t
be a nonnegative constant independent with respect to ξ (for example, we can just assume that the function ψ is odd and radial). Then we have the decomposition :

f = c(ψ) -1 R d ∞ 0 f, ψ t,q ψ t,q dtdq t . (3.1)
In addition, the integral is absolutely convergent for a function f ∈ S(R d ).

Proof : The result is well-known for f ∈ L 2 (R d ), it is shown in the book [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF] at the chapter 5.6. When f ∈ S(R d ), integrations by parts give us fast decay for f, ψ t,q and so permit us to prove the absolute convergence.

⊓ ⊔

From now, we will choose a smooth function ψ which verifies the assumption of the previous Lemma and whose the spectrum is included in a corona around 0. We decompose also the two functions f and g with the previous lemma and we have also to study the following quantity :

F (k, u, v, q, s, x) := L(k) R d Ψ 2 k (y)[Φ 1 λ 1 2 k * ψ u,q ](x -ρ 1 λ 1 y)[Φ 2 λ 2 2 k * ψ v,s ](x -ρ 2 λ 2 y)dy.
With the inverse Fourier transform, we get :

F (k, u, v, q, s, x) = L(k) R 2d Ψ 2 p (ξ) Φ 1 λ 1 2 k ((ξ -α)ρ -1 1 λ -1 1 ) ψ u,q ((ξ -α)ρ -1 1 λ -1 1 )e i(ξ-α)xρ -1 1 λ -1 1 Φ 2 λ 2 2 k (λ -1 2 ρ -1 2 α) ψ v,s (ρ -1 2 λ -1 2 α)e iαx(ρ 2 λ 2 ) -1 (ρ 1 ρ 2 |λ 1 λ 2 |) -d dαdξ.
Due to the spectral conditions, we have a dependence for the three frequency parameters :

F (k, u, v, q, s, x) = 0 =⇒    max{|ρ 1 λ 1 |u -1 , |ρ 2 λ 2 |v -1 } ≃ 2 -k or |ρ 1 λ 1 | -1 u ≃ |ρ 2 λ 2 | -1 v ≤ 2 k .
In addition the product Φ 1 λ 1 2 k Ψ u,q is non vanishing only if |λ 1 |u -1 2 -k and similarly for v. As the coefficients ρ i are bounded by 1, we are always in the first case i.e.

max{|ρ 1 λ 1 |u -1 , |ρ 2 λ 2 |v -1 } ≃ 2 -k .
In addition, we have shown the stronger condition

max{|λ 1 |u -1 , |λ 2 |v -1 } ≃ 2 -k
uniformly with respect to the parameters ρ i ∈]0, 1].

After having study the frequency properties of F (k, u, v, q, s, x), we will remember spatial estimates : Proposition 3.2. We have the following estimate :

F k,v,s (x) := |Φ λ 2 2 k * ψ v,s (x)| inf{2 k |λ 2 |, v} d v 2d 1 + |x -s| max{|λ 2 |2 k , v} -M
, for any exponent M as large as we want. This estimate is uniform with respect to k and λ 2 .

Proof : The proof is essentially written in Appendix K-2 of [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF] and we only give the sketch of the proof. Let ψ be an other smooth function, whose the spectrum is included in a corona around 0 and such that ψ = 1 on the spectrum of ψ. We set ζ = Φ λ 2 2 k * ψv,s . It is also easy to check that

|ζ(x)| 1 v d 1 + |x| max{2 k |λ 2 |, v} -M
, for all exponent M as large as we want. Due to the spectral properties of ψ and ψ, we get :

F k,v,s (x) = |ζ * ψ v,s (x)| .
Then we can directly estimate the convolution product and prove what we want.

⊓ ⊔

After this study, we decompose our operator :

T ρ,λ,L (f, g)(x) = k R 2d ∞ 0 ∞ 0 f, ψ u,q g, ψ v,s F (k, u, v, q, s, x) dv v du u dqds.
We have seen by a spectral analysis that we can restrict this double integral over u and v on the set

(u, v), max{|λ 1 |u -1 , |λ 2 |v -1 } ≃ 2 -k .
In the study of paraproducts (see paragraph 8.4 of [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF]), we decompose the product as f g = Π f (g) + Π g (f ) + D(f, g) where the two paraproducts and the diagonal terms have different estimates. For the same reasons here we have to singly study the two following terms :

T ρ,λ,L = T 1 ρ,λ,L + T 2 ρ,λ,L with T 1 ρ,λ,L (f, g)(x) := k R 2d ∞ 0 ∞ 0 f, ψ u,q g, ψ v,s F (k, u, v, q, s, x)1 A k (u, v) dvdu vu dqds and T 2 ρ,λ,L (f, g)(x) := k R 2d ∞ 0 ∞ 0 f, ψ u,q g, ψ v,s F (k, u, v, q, s, x)1 B k (u, v) dvdu vu dqds.
We write A k and B k for the two sets :

A k := (u, v), max |λ 1 |u -1 , |λ 2 |v -1 ≃ 2 -k and min{u, v} ≤ C -1 max{u, v} and
B k := (u, v), max |λ 1 |u -1 , |λ 2 |v -1 ≃ 2 -k and u ≃ v ,
where C is a numerical constant, we later choose. Due to this constant, we can use spectral separation to study T 1 ρ,λ,L with the Littlewood-Paley square functions for f and g. Theorem 3.3. For 0 < ρ i ≤ 1 and λ i ∈ R * , the operator T 1 ρ,λ,L can been continuously extended from H r 1 ×H r 2 to H r 3 , if the exponents 0 < r 1 , r 2 , r 3 < ∞ satisfy the homogeneity condition

1 r 1 + 1 r 2 = 1 r 3 .
In addition we control the continuity bounds, uniformly with respect to λ and ρ by the quantity

c N (Ψ)c N (Φ 1 )c N (Φ 2 ) L ∞ ,
for N a large enough integer.

Proof : To estimate T 1 ρ,λ,L in the Hardy space H r 3 , we have to study its square function : S Ψ (T 1 ρ,λ,L (f, g)). We can compute the Fourier transform and get :

F (k, u, v, q, s, .)(ξ) =L(k) R d Ψ 2 k (ξρ 1 λ 1 -(ρ 2 λ 2 -ρ 1 λ 1 )α) ψ u,q (ξ -α) Φ 1 λ 1 2 k (ξ -α) Φ 2 λ 2 2 k (α) ψ v,s (α)dα. ( 3.2)
Consequently by writing ξ = (ξα) + α, the spectrum of F (k, u, v, q, s, .) is contained in

1 u spectrum(ψ) + 1 v spectrum(ψ) ⊂ ξ, |ξ| ≃ max{u -1 , v -1 } . (3.3)
For the last inclusion, we have used a large enough constant C in the definition of the set A k and so min{u, v} ≪ max{u, v}.

By symmetry we may assume u ≤ v and then by choosing a continuous square function S Ψ , we have

S Ψ (T 1 ρ,λ,L (f, g))(x) ∞ 0 k∈Z R 2d ∞ 0 | f, ψ u,q g, ψ v,s | |Ψ u * F (k, u, v, q, s, .)(x)| 1 A ′ k (u, v) dv v dqds 2 du u 1/2
, where

A ′ k := {(u, v) ∈ A k , u ≪ v}.
We have also to estimate the following quantity

Q := ∞ 0 k R 2d ∞ 0 | f, ψ u,q g, ψ v,s | |Ψ u * F (k, u, v, q, s, .)(x)| 1 A ′ k (u, v) dv v dqds 2 du u 1/2 r 3 ,dx
.

With the notations of Proposition 3.2, we have that for all

z ∈ R d |F (k, u, v, q, s, z)| ≤ |L(k)| R d |Ψ 2 k (y)| F k,u,q (z -λ 1 ρ 1 y)F k,v,s (z -ρ 2 λ 2 y)dy.
By using the estimates of Proposition 3.2 and the fast decay of Ψ, we get :

|F (k, u, v, q, s, z)| L ∞ R d 2 -dk 1 + |y| 2 k -M 1 + |z -ρ 1 λ 1 y -q| u -M |λ 1 | d 2 k u -2d |λ 2 | d 2 k v 2d 1 + |z -ρ 2 λ 2 y -s| v -M
dy.

As the parameters ρ i ≤ 1 and (u, v) ∈ A ′ k , we obtain with an other exponent :

|F (k, u, v, q, s, z)| (3.4) L ∞ 1 + |z -s| v -M 1 + |z -q| u -M |λ 1 ||λ 2 |2 2k u 2 v 2 d . (3.5)
The exponent M is not always the same, but it always corresponds to an integer as large as we want. Now by estimating the convolution product, we get

|Ψ u * F (k, u, v, q, s, .)(x)| L ∞ 1 + |x -s| v -M 1 + |x -q| u -M |λ 1 ||λ 2 |2 2k u 2 v 2 d .
Computing this estimate in the expression of Q, we have

Q L ∞ k∈Z R 2d ∞ 0 | f, ψ u,q g, ψ v,s | 1 + |x -s| v -M 1 + |x -q| u -M 1 A ′ k (u, v) |λ 1 ||λ 2 |2 2k u 2 v 2 d dv v dqds 2,du/u r 3 ,dx
.

We change the two variables

x -q u → a and x -s v → b, to get Q k R 2d ∞ 0 | f, ψ u,x-ua g, ψ v,x-vb | (1 + |a|) -M (1 + |b|) -M 1 A ′ k (u, v) |λ 1 ||λ 2 |2 2k uv d dv v dadb 2,du/u r 3 ,dx
.

We write ψ z for ψ z (y) := ψ(zy) and ψ z t := (ψ z ) t . With these notations, we have :

Q k R 2d ∞ 0 |ψ a u * f (x)||ψ b v * g(x)| (1 + |a|) -M (1 + |b|) -M 1 A ′ k (u, v) |λ 1 ||λ 2 |2 2k uv d dv v dadb 2,du/u r 3 ,dx
.

The definition of the set A ′ k allows us to have a finite number of choice for k. Therefore we have

Q R 2d ∞ 0 |ψ a u * f (x)||ψ b v * g(x)| (1 + |a|) -M (1 + |b|) -M 1 u≤C -1 v |λ 1 ||λ 2 | inf{|λ 1 | -1 u, |λ 2 | -1 v} 2 uv d dv v dadb 2,du/u r 3 ,dx
.

Then we use the Cauchy-Schwarz inequality with

(1) := ∞ Cu |λ 1 ||λ 2 | inf{|λ 1 | -1 u, |λ 2 | -1 v} 2 uv 2d dv v ∞ Cu inf{|λ 1 | -1 u, |λ 2 | -1 v} max{|λ 1 | -1 u, |λ 2 | -1 v} 2d dv v ∞ |λ 2 λ -1 1 |u |λ 1 | -1 u |λ 2 | -1 v 2d dv v + max{C,|λ 2 λ -1 1 |}u Cu |λ 2 | -1 v |λ 1 | -1 u 2d dv v 1 + |λ 2 | -1 |λ 1 | -1 2d max{C, |λ 2 λ -1 1 |} 2d -C 2d 1 + |λ 1 | |λ 2 | 2d |λ 2 λ -1 1 | 2d -min{C, |λ 2 λ -1 1 |} 2d 1 + 1 -min{C|λ -1 2 λ 1 |, 1} 2d 1, to finally get Q R 2d |ψ a u * f (x)| S ψ b (g)(x) (1 + |a|) -M (1 + |b|) -M dadb 2,du/u r 3 ,dx
.

With Minkowski inequality, we may write the last inequality as

Q R 2d S ψ a (f )(x)S ψ b (g)(x) (1 + |a|) -M (1 + |b|) -M dadb r 3 ,dx . (3.6)
We must have a pointwise estimate on the square functions when r 3 < 1, because of the lack for the triangle inequality in L r 3 . We also use this lemma :

Lemma 3.4. Let ζ be a smooth function satisfying ∀ξ = 0, ∞ 0 ζ(tξ) 2 dt t = 1. (3.7)
For ψ an other function satisfying (3.7) too, we have the pointwise estimate : for all r > 0, there exists a constant C r and an integer N such that

∀f ∈ S(R d ) S ψ (f ) ≤ C r c N (ζ)c N (ψ) M 2 r (ζ t * f ) 2 dt t 1/2
.

Here M HL is the Hardy-Littlewood maximal operator, M 2 HL corresponds to

M HL • M HL and M 2 r (g) = M HL 2 (|g| r ) 1/r .
Let assume first this Lemma. By applying it with ψ = ψ a and ψ = ψ b , we get :

Q R 2d (1 + |a|) N (1 + |b|) N ∞ 0 M 2 r (ζ t * f ) 2 dt t 1/2 ∞ 0 M 2 r (ζ t * g) 2 dt t 1/2 (1 + |b|) -M (1 + |a|) -M dadb r 3 .
Also by choosing a large enough integer M, we have :

Q ∞ 0 M 2 r (ζ t * f ) 2 dt t 1/2 ∞ 0 M 2 r (ζ t * g) 2 dt t 1/2 r 3
.

With Hölder inequality, we obtain :

Q ∞ 0 M 2 r (ζ t * f ) 2 dt t 1/2 r 1 ∞ 0 M 2 r (ζ t * g) 2 dt t 1/2 r 2 . (3.8)
We study only the first term with r 1 , the other one is identical. By definition,

∞ 0 M 2 r (ζ t * f ) 2 dt t 1/2 r 1 = ∞ 0 M 2 HL (|ζ t * f | r ) 2/r dt t r/2 1/r r 1 /r .
For r small enough such that min{r 1 /r, 2/r} > 1, the Fefferman-Stein inequality (Theorem 4.6.6 of [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF]) in L 2/r applied to the operator M 2 HL gives us :

∞ 0 M 2 r (ζ t * f ) 2 dt t 1/2 r 1 ∞ 0 [|ζ t * f | r ] 2/r dt t r/2 1/r r 1 /r .
In other words :

∞ 0 M 2 r (ζ t * f ) 2 dt t 1/2 r 1 S ζ (f ) r 1 .
By replacing this estimate in (3.8), we obtain the desired result :

Q f H r 1 g H r 2
uniformly with respect to λ i = 0 and 0 < ρ i ≤ 1.

⊓ ⊔

We have now to show the Lemma 3.4. This Lemma is "quite classical", it permits to understand for example that Definition of Hardy spaces (Definition 2.2) does not depend on the used function Ψ. It is almost proved in a discrete version in the book [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF], from which we take the notations. We only give the sketch of the proof. Proof : We define the maximal operator :

M b,t (f, ψ) = sup y∈R d |ψ t * f (x -y)| (1 + t -1 |y|) b .
Then it is obvious that

|ψ t * f | ≤ M b,t (f, ψ). (3.9)
In the proof of Theorem 6.5.6. of [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF], one may choose a function Θ satisfying (3.7) and such that Θ ≥ 0. Then it is shown that :

M b,t (f, ψ) c N (ψ)c N (Θ) inf{|t -s|, |t -s| -1 }M b,s (f, Θ) ds s .
Consequently with (3.9), we get :

S ψ (f ) c N (ψ)c N (Θ) [M b,s (f, Θ)] 2 ds s 1/2 .
(3.10) Now Lemma 6.5.3. of [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF] with b = n/r gives us,

M b,t (f, Θ) M r (Θ t * f ).
To substitute the function ζ to the function Θ, we use the spectral condition and the fact that

Θ t * f = 2 1/2 Θ t * ζ tu * f du u .
Then with the estimate (6.5.8) of [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF] :

|Θ t * ζ tu * f (x)| c N (Θ)M r (ζ t * f )(x),
we obtain that

M b,t (f, Θ) M r (Θ t * f ) c N (Θ)c N (ζ)M r M r (ζ t * f ) c N (ζ)M 2 r (ζ t * f ).
By computing this estimate in (3.10), we get the Lemma.

⊓ ⊔

To finish the study of T ρ,λ,L , we have to estimate the second operator T 2 ρ,λ,L : Theorem 3.5. For 0 < ρ i ≤ 1 and λ i = 0, the operator T 2 ρ,λ,L is continuous from H r 1 × H r 2 to L r 3 , for all exponents 0 < r 1 , r 2 , r 3 < ∞ satisfying

1 r 1 + 1 r 2 = 1 r 3 .
In addition we may control the continuity bounds, uniformly with respect to λ and ρ by the quantity

c N (Ψ)c N (ψ)c N (Φ 1 )c N (Φ 2 ) L ∞ ,
for N a large enough integer.

Proof : The operator T 2 ρ,λ,L is defined as

T 2 ρ,λ,L (f, g) := k∈Z R 2d ∞ 0 ∞ 0 f, ψ u,q g, ψ v,s F (k, u, v, q, s, x)1 B k (u, v) dvdu vu dqds, with B k := (u, v), max{|λ 1 |u -1 , |λ 2 |v -1 } ≃ 2 -k and u ≃ v .
By using the previous estimate, we have to control

Q := T 2 ρ,λ,L (f, g) r 3 k∈Z R 2d ∞ 0 ∞ 0 | f, ψ u,q g, ψ v,s | 1 + |x -s| v -M 1 + |x -q| u -M 1 B k (u, v) |λ 1 ||λ 2 |2 2k u 2 v 2 d dvdu uv dqds r 3 ,dx
.

In this case (3.3) is not satisfied. We compute the same changes of variables as in the end of the proof for Theorem 3.3. By using Cauchy-Schwarz inequality and the definition of the set B k , we obtain the same estimate as (3.6) and so we can conclude by the same arguments as before.

⊓ ⊔

Finally we get the following result :

Theorem 3.6. Let 0 < ρ i ≤ 1 and λ i = 0 be reals, then the operator

T ρ,λ,L is continuous from H r 1 × H r 2 to L r 3 for all exponents 0 < r 1 , r 2 , r 3 < ∞ satisfying 1 r 1 + 1 r 2 = 1 r 3 .
In addition we can estimate the continuity bound uniformly with respect to λ and ρ by the quantity

c N (Ψ)c N (ψ)c N (Φ 1 )c N (Φ 2 ) L ∞ ,
for N a large enough integer.

Proof : We have decomposed the operator T ρ,λ,L as

T ρ,λ,L = T 1 ρ,λ,L + T 2 ρ,λ,L .
The embedding H r 3 ֒→ L r 3 (see Theorem 2.5 of [START_REF] Uchiyama | Hardy Spaces on the Euclidean Space[END_REF]), Theorems 3.3 and 3.5 allow us to prove the desired result.

⊓ ⊔

This result proves the first part of Theorem 1.3 : under the assumption 1-) we have uniform estimates.

In the next section, we are going to prove a similar result for some infinite exponents with the concept of Carleson measure and ideas based on Calderón-Zygmund theory.

4

The study of T ρ,λ,L with Carleson measures and Calderón-Zygmund decompositions .

We use ideas of the book [START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF], where R. Coifman and Y. Meyer have already studied paraproducts with a Carleson measure. We adapt here their arguments to our model operators. As we have seen in Remark 2.5, our operators permit us to understand all the "different kinds" of paraproducts. In [START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF], the authors studied only one "kind" of paraproducts (which with other and extra arguments is sufficient to study the other ones).

That is why the use of our model operators seems interesting as we obtain a (only one) direct proof simultaneously for all the paraproducts.

We will (for convenience) work on the continuous version of them :

U ρ,λ,L (f 1 , .., f n )(x) := ∞ 0 L(t) R d Ψ t (y) n i=1 Φ i λ i t * f i (x -ρ i λ i y)dy dt t ,
where L is a bounded measurable function. By symmetry, we can assume that λ n satisfies :

|λ n | = max{|λ i |, 1 ≤ i ≤ n}. (4.1)
In this case we have the following result :

Theorem 4.1. Under the assumption (4.1), the operator U ρ,λ,L can be continuously extended from (L ∞ ) ⊗n-1 × L 2 to L 2 . In addition the continuity bound is controlled by L ∞ with c M (Ψ) and c M (Φ i ) (for a large enough integer M) independently with respect to the parameters ρ i ∈]0, 1] and λ.

Proof : By symmetry on the (n -1) first coordinates, we can assume that

|λ 1 | := min{|λ j |, 1 ≤ j ≤ n -1}. (4.2)
In [START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF] (Chap. VI prop 3), the following result (that we call the ( * )-result) is shown :

the operator V is continuous from (L ∞ ) ⊗n-1 × L 2 to L 2
, where V is defined by

V (f 1 , .., f n )(x) := U (ρ 1 ,0,..,0),λ,L (f 1 , .., f n )(x) = ∞ 0 Ψ ρ 1 λ 1 t * Φ 1 λ 1 t * f 1 (x) n j=2 Φ j λ j t * f j (x) L(t) t dt.
The estimate on V is independent on λ and ρ 1 due to the assumptions (4.1) and (4.2). Our idea is also to disturb our coefficients (ρ j ) 2≤j≤n and to bring them to 0. We temporarily forget in the notation the dependence on ρ, λ and L, by writing :

U ρ,λ,L (f 1 , .., f n ) = V (f 1 , .., f n ) + J ⊂{2,..,n} J =∅ 0≤s j ≤ρ j j∈J W s,J (f 1 , ..f n )ds,
where

W s,J (f 1 , .., f n )(x) := ∞ 0 R d Ψ t (y)Φ 1 λ 1 t * f 1 (x + ρ 1 λ 1 y) n j=2 j∈J λ j y λ j t .(∇Φ j ) λ j t * f j (x + s j λ j y)Π J (x, y, t) L(t) t dydt with Π J (x, y, t) := n j=2 j∈J c (Φ j ) λ j t * f j (x + λ j y).
Since V is estimated by the ( * )-result, ρ i ≤ 1 and the set of J being finite, we have only to bound the operators W s,J . We now decompose the gradient in the d coordinates :

W s,J (f 1 , .., f n )(x) = l∈{1,..,d} |J | ∞ 0 R d Ψ t (y)Φ 1 λ 1 t * f 1 (x + ρ 1 λ 1 y) n j=2 j∈J y l j t (∂ x l j Φ j ) λ j t * f j (x + s j λ j y)Π J (x, y, t) L(t) t dydt.
By setting

Θ l (x) = j∈J x l j Ψ(x),
the function Θ l is again a smooth function whose the spectrum is far away from 0. We have

W s,J (f 1 , .., f n-1 )(x) = l ∞ 0 R d Θ l t (y) ζ 1 λ 1 t * f 1 (x + ρ 1 λ 1 y) j∈J (∂ x l j Φ j ) λ j t * f j (x + λ j s j y)Π J (x, y, t) L(t) t dydt.
Now the interest of this operation is that J being not empty there exists an index j for which ∂ x l j Φ j (0) = 0 (what is false for the initial function Φ j ) 1 . The ( * )-result for V is based on the following quadratic estimate (due to the notion of Carleson measure, see [START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF]) :

[Ψ ρ 1 λ 1 t * f ] (x) ζ 2 λ 2 t * g (x) 2, dtdx t f ∞ g 2 ,
uniformly on λ and ρ for 0

< ρ i ≤ 1 and 0 < |λ 1 | ≤ |λ 2 |.
We are going also to produce the same proof for our operator W s,J . We have to show a quadratic estimate : for an index l ∈ {1, .., d} |J| :

(♠) l := R d Θ l t (y)Φ 1 λ 1 t * f 1 (x -ρ 1 λ 1 y) j∈J (∂ x l j Φ j ) λ j t * f j (x + λ j s j y) j∈J c Φ λ j t * f j (x + λ j y)dy 2, dxdt t f n 2 n-1 i=1 f i ∞ . (4.3) * First case : n ∈ J.
The convolution operators are bounded on L ∞ , so we get :

(♠) l ≤ i =n f i ∞ R d |Θ l t (y)| (∂ x ln Φ n ) λnt * f n (x -λ n s n y) dy 2, dxdt t .
We use Minkowski inequality for the norm in L 2 (dt/t) and after we can compute the integral over y. Then with the reminder (1), we get (4.3). * Second case : n ∈ J c . We use a Carleson estimate by keeping an other function f k with k ∈ J (due to J = ∅).

(♠) l ≤ i ={n,k} f i ∞ R d |Θ l t (y)| Φ n λnt * f n (x -λ n y) (∂ x l k Φ k ) λ k t * f k (x -λ k s k ) dy 2, dxdt t .
After changing the variable on y, we have :

(♠) l ≤ i ={n,k} f i ∞ R d |Θ l (y)| Φ n λnt * f n (x -λ n ty) (∂ x l k Φ k ) λ k t * f k (x -λ k ts k y) dy 2, dxdt t .
We write Φ k,a for Φ k (•a) and also get

(♠) l ≤ i ={n,k} f i ∞ R d |Θ l (y)| Φ n,y λnt * f n (x) (∂ x l k Φ k,s k y ) λ k t * f k (x) dy 2, dxdt t .
We now use Minkowski inequality on the measure dxdt/t and after the Carleson estimate to finally obtain :

(♠) l ≤ i =n f i ∞ f n 2 |Θ l (y)|(1 + |s k y|) 2d+2 dy.
The function Θ l is smooth and 0 ≤ s k ≤ 1, consequently we have shown (♠) l in this last case. All the estimates are uniform on λ due to

|λ k | ≤ |λ n |.
Hence (4.3) is shown in the two cases. We have now just to copy the proof of the ( * )-result of [START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF] by putting the previous quadratic estimate instead of the Carleson estimate. The details of this part of the proof are left to the reader.

⊓ ⊔

As for "classical" Calderón-Zygmund operators, we use a Calderón-Zygmund decomposition to obtain continuity results with other Lebesgue exponents.

Our multilinear operators are multilinear Calderón-Zygmund operators (as defined in [START_REF] Grafakos | Multilinear Calderón-Zygmund theory[END_REF]), however the bounds are not uniformly controlled with respect to λ i . By using the main result of [START_REF] Grafakos | Multilinear Calderón-Zygmund theory[END_REF], we obtain our desired continuities for U ρ,λ,L with a certain dependence on λ. The rest of this section is based on an improvement of the "classical" arguments, adapted to our problem.

Definition 4.2. A function K defined on R d × R d \ {(x, x), x ∈ R d } is called a "standard kernel of order N" if for all x = y ∀α, β ∈ {1, .., d}, |α|, |β| ≤ N ∂ α x ∂ β y K(x, y) ≤ A α,β 1 |x -y| d+|β|+|α| .
A linear operator T , continuously acting from S(R d ) to S ′ (R d ) and satisfying the integral representation

∀f ∈ C ∞ 0 , ∀x / ∈ supp(f ) T (f )(x) = R d K(x, y)f (y)dy,
is said to be associated to the kernel K. Such an operator is called a "Calderón-Zygmund operator of order N" if it is bounded on L (R d ) and associated to a "standard kernel of order N".

We have also the well-known following proposition (see for example the book [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF]) :

Proposition 4.3. Let T be a Calderón-Zygmund operator of order N. Then T admits a continuous extension from L p to L p for 1 < p ≤ 2, from L 1 to L 1,∞ and from H p to L p for d/(N + d) < p ≤ 1. In addition the continuity bounds only depend on the constants T L 2 →L 2 and (A 0,β ) β .

Remark 4.4. The other constants A α,β with α = 0 are useful to study the dual operator T * and also to get boundedness for T on L q with 2 < q < ∞.

We will use this proposition for our problem.

Proposition 4.5. Let f 1 , .., f n-1 be smooth fixed functions (considered in L ∞ ). Then the operator :

V := f n → U ρ,λ,L (f 1 , .., f n )
is a Calderón-Zygmund operator at any order. In addition the constants A(0, β) are uniformly bounded with respect to λ and ρ for 0 < ρ i ≤ 1.

Proof : The boundedness on L 2 of V is given by Theorem 4.1. We have only to check the desired estimates on the kernel. Let K be the kernel of V , which is given by :

K(x, z) = ∞ 0 R d Ψ t (y) n-1 j=1 Φ j λ j t * f j (x -ρ j λ j y)Φ n λnt (x -ρ n λ n y -z) L(t) t dydt.
We can differentiate the kernel and directly obtain :

∂ α x ∂ β z K(x, z) 1 (min i |λ i |) |α| ∞ 0 n-1 j=1 f j ∞ L ∞ R d 1 + |y| t -M 1 + |x -ρ n λ n y -z| λ n t -M |λ n t| -d-|β| dydt t |α|+d+1 .
By using ρ n-1 ≤ 1, we get :

∂ α x ∂ β z K(x, z) |λ n | min i |λ i | |α| ∞ 0 1 |λ n |t |β|+|α| 1 + |x -z| λ n t -M |λ n t| -d L ∞ dt t .
For M a large enough exponent, we can conclude that

∂ α x ∂ β z K(x, z) |λ n | min i |λ i | |α| |x -z| -d-|α|-|β| L ∞ .
We have also the desired estimates on the kernel and for α = 0 the estimates do not depend on λ.

⊓ ⊔

With the two previous propositions, we get the following corollary.

Corollary 4.6. The operator U ρ,λ,L is continuous from (L ∞ ) ⊗(n-1) × H p into L p for all exponent 0 < p ≤ 2 and from (L ∞ ) ⊗(n-1) × L 1 into L 1,∞ . The continuity bounds are uniformly controlled with respect to 0 < ρ i ≤ 1 and λ satisfying (4.1).

Here we do not know if a similar result for p > 2 is possible. Now we would like to get continuities with finite exponents instead of infinite exponents. To do this, we first prove the abstract following result :

Theorem 4.7. Let T be a linear operator, continuously acting from L p 1 to L p with 1 < p 1 ≤ ∞ and 0 < p ≤ p 1 . We set r > 0 the exponent defined by

1 p = 1 p 1 + 1 r .
We assume that T is associated to a kernel K (see definition 4.2) satisfying

∀α, |α| ≤ 1 |∂ α z K(x, z)| ≤ 1 |x -z| d+|α| h(x)
with a function h ∈ L r,∞ . Then T can be continuously extended from L q 1 to L q,∞ for all exponents (q 1 , q) such that

1 q = 1 q 1 + 1 r and 1 ≤ q 1 ≤ p 1 .
In addition the continuity bounds are controlled by h r,∞ . By real interpolation, we obtain the strong type (q 1 , q) when 1 < q 1 ≤ p 1 .

Proof : We follow the "classical proof" for r = ∞ and h = 1 R d . So let f be a normalized function of L q : f q = 1. We want to show

|{x, |T (f )| > α}| α -q . (4.4)
We use a Calderón-Zygmund decomposition of the function f at the scale α q/q 1 . We have also the following decomposition

f = g + b,
with a "good" function g and a "bad" function b satisfying :

g q 1 f q 1 = 1, g ∞ α q/q 1 , b = k b k , supp(b k ) ⊂ Q k , b k q 1 α q/q 1 |Q k | 1/q 1 , b k = 0, k |Q k | α -q f q 1 q 1 α -q and k 1 10Q k 1.
The (Q k ) k is a collection of balls (of R d ), associated to the "bad" function b. By linearity, we have

|{x, |T (f )| > α}| ≤ |{x, |T (g)| > α/2}| + |{x, |T (b)| > α/2}| .
1-) The case of the function g. This is the easiest case. We use the continuity of T with the exponents p 1 and p to get

|{x, |T (g)| > α/2}| α -p T (g) p p α -p g p p 1 .
By the assumption on g and the fact that q 1 ≤ p 1 :

g p 1 g q 1 /p 1 q 1 g 1-q 1 /p 1 ∞ α q(1-q 1 /p 1 )/q 1 .
We also obtain |{x, |T (g)| > α/2}| α -p α pq(1-q 1 /p 1 )/q 1 α -p α pq(1/q 1 -1/p 1 ) α -p α pq(1/q-1/p) α -q , which corresponds to the desired result (4.4). 2-) The case of the function b.

First we have

k 5Q k α -q .
In order to show (4.4), we can also assume that

x ∈ ∩ k (5Q k ) c and just estimate |{x ∈ ∩ k (5Q k ) c , |T (b)| > α/2}| .
Let also x be fixed and use

|T (b)(x)| ≤ k≥0 |T (b k )(x)|.
With the vanishing moment of the function b k , we have :

T (b k )(x) = K(x, z)b k (z)dz = [K(x, z) -K(x, c k )] b k (z)dz.
Here we write c k for the center of the cube Q k . As x is far away the support of b k , the integral representation has really a sense. Then by using the estimates of the kernel, we have

|K(x, z) -K(x, c k )| |z -c k | 1 0 |∇K(x, z + t(c k -z))| dt |z -c k | 1 0 1 |x -c k | d+1 h(x)dt |z -c k | |x -c k | d+1 h(x). Therefore |T (b k )(x)| h(x) |Q k | 1/d 1 |x -c k | d+1 b k (z)dz. With b k 1 ≤ |Q k | 1-1/q 1 b k q 1 |Q k |α q/q 1 , we obtain |T (b k )(x)| h(x)|Q k | 1+1/d α q/q 1 1 |x -c k | d+1 .
By computing the sum over the index k, we finally have

|T (b)(x)| h(x)α q/q 1 k 1 1 + |x-c k | |Q k | 1/d d+1 .
We find the Marcinkiewicz function associated to the collection (Q k ), we write it

M (Q k ) k . So we have |T (b)(x)| h(x)α q/q 1 M (Q k ) k (x).
However the collection (10Q k ) k is a bounded covering on the whole space, so we know (see [START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF]) that for 1 ≤ q 1 < ∞, M (Q k ) k is of weak type (q 1 , q 1 ). By using Hölder inequality on the weak Lebesgue spaces, we get

T (b) q,∞ h r,∞ α q/q 1 M (Q k ) k (x) q 1 ,∞ | ∪ Q k | 1/q 1 h r,∞ α q/q 1 .
We obtain also the desired estimate :

T (b) q,∞ h r,∞ .

⊓ ⊔

We now prove a similar result for the Hardy spaces :

Theorem 4.8. Let T be a linear operator, continuously acting from L p 1 to L p with 1 < p 1 ≤ ∞ and p ≤ p 1 . We set r > 0 the exponent satisfying

1 p = 1 p 1 + 1 r .
We assume that T is associated (see Definition 4.2) to a kernel K verifying :

∀α, |α| ≤ N |∂ α z K(x, z)| ≤ 1 |x -z| d+|α| h(x)
with a function h ∈ L r,∞ . Then for all exponents (q 1 , q) such that

1 q = 1 q 1 + 1 r and d N + d ≤ q 1 ≤ 1,
there is a constant C such that for all atoms a ∈ H q 1 T (a) L q,∞ ≤ C. (4.5)

In addition the continuity bounds are controlled by h r,∞ . By real interpolation, T can be continuously extended from H q 1 to L q when d N +d < q 1 ≤ 1.

Proof : We use the atomic decomposition of the Hardy spaces H q 1 (See Theorem 6.6.10 of [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF]). Let a be an atom of H q 1 , that is meaning there exists

a cube Q such that supp(a) ⊂ Q, a 2 ≤ |Q| 1/2-1/q 1 ∀α, |α| ≤ [ d q 1 -d],
x α a(x)dx = 0.

We write [] for the integer part. We want to estimate T (a). Assume first that x ∈ (5Q) c . By assumption

q 1 > d(N + d) -1 so N q 1 := [ d q 1 -d] ≤ N -1. We have also T (a)(x) = R d K(x, y)a(y)dy = R d   K(x, y) - |α|≤Nq 1 (y -c(Q)) α α! ∂ α y K(x, c(Q))   a(y)dy,
where c(Q) is the center of the cube Q. We can estimate the difference between the square brackets by

K(x, y) - |α|≤Nq 1 (y -c(Q)) α α! ∂ α y K(x, c(Q)) |α|=Nq 1 +1 (y -c(Q)) α α! ∂ α y K(x, y) ∞,y∈Q |Q| (Nq 1 +1)/d h(x) |x -c(Q)| d+Nq 1 +1 . We also get |T (a)(x)| |Q| (Nq 1 +1)/d h(x) |x -c(Q)| d+Nq 1 +1 |a(y)|dy |Q| (Nq 1 +1)/d h(x) |x -c(Q)| d+Nq 1 +1 |Q| 1-1/q 1 .
Therefore with the Hölder inequality on the weak Lebesgue spaces L p,∞ and by integrating x ∈ (5Q) c , we obtain

T (a) q,∞,(5Q) c |Q| (Nq 1 +1)/d h r,∞ 1 |Q| 1-1/q 1 +(Nq 1 +1)/d |Q| 1-1/q 1 1.
We compute the proof by studying the case x ∈ 5Q with the Hölder inequality and the L p -boundedness of T : T (a) q,∞,(5Q) |Q| 1/q-1/p T (a) p |Q| 1/q-1/p a p 1 |Q| 1/q-1/p+1/p 1 -1/2+1/2-1/q 1 1.

Here we have assumed that p 1 ≤ 2. If it is not the case, we have to consider the L s -atoms of H q 1 with s ≥ p 1 or use first our previous theorem to have continuities for T with p 1 = 1. By consequence, we have shown that T is bounded on all the atoms of H q 1 into L q,∞ .

⊓ ⊔ Remark 4.9. Nowadays, it is well known that an operator, which is bounded on whole the set of atoms, does not always admit a continuous extension to the whole Hardy space. There is a counterexample for the Hardy space H 1 in [START_REF] Meyer | Some functional analytic properties of the spaces B q generated by blocks[END_REF].

We now apply this abstract result to our operator.

Proposition 4.10. Let f 1 , .., f n-2 be fixed and smooth functions belonging to S(R d ) ⊂ L ∞ and f n be a smooth function belonging to H q with q ≤ 2 (or L 1 ). Then the operator

V := f n-1 → U ρ,λ,L (f 1 , .., f n )
satisfies the assumptions of Theorems 4.

Proof :

The assumption of the boundedness is given by Corollary 4.6. So we have just to check the assumption about the kernel K(x, z), which is given by

K(x, z) = ∞ 0 R d Ψ t (y)Φ n-1 λ n-1 t (x -ρ n-1 λ n-1 y -z) n-2 j=1
Φ j λ j t * f j (xρ j λ j y)Φ n λnt * f n (xρ n λ n y) L(t) t dydt.

We can differentiate the kernel and we obtain

∂ β z K(x, z) ∞ 0 n-2 j=1 f j ∞ R d 1 + |x -ρ n-1 λ n-1 y -z| λ n-1 t -M |λ n-1 t| -d-|β| 1 + |y| t -2M 1 + |y| t -d/q+1 M * * d/q+1 (f n , Φ n )(x)
L ∞ dydt t d+1 .

Decomposition of multipliers with our model operators

In this section we will prove how to decompose a multilinear multiplier of Theorem 1.3 with our model operators. This reduction will also conclude the proof of this Theorem. The way to decompose a multilinear multiplier with paraproducts is well known (see for example [START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF]). We quickly remember this operation and check that we keep the uniformity with respect to the important parameter λ.

So let T be an operator of Theorem 1.3. It is also associated to a symbol σ which satisfies

∀m i ∈ N d ∂ m 1 ξ 1 ..∂ mn ξ n-1 σ(ξ 1 , .., ξ n ) n i=1 |λ |m i | i | d λ (ξ, 0) |m 1 |+..+|m n-1 | .
(5.1)

As we have seen in Proposition 2.4, our model operators allow us to get the paraproducts. So we use the "classical" decomposition of an Hörmander multiplier with paraproducts. Let us recall it (we use the ideas of [START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF]). We have also a partition of the unity on the whole frequency plane and we get τ u l Ψ(2 k λ l ξ l ) f l (ξ l ) j =l τ u j Φ(2 k λ j ξ j ) f j (ξ j )dξdu.

T (f 1 , .., f n )(x) =

7 and 4 .8 for p 1 =

 41 ∞, p = q = r and h M * * d/q+1 (f n , Φ n ) at any order N. In addition the bounds can be uniformly controlled with respect to ρ and λ under the condition (4.1). Here we set M * * b for the following maximal operator :M * * b (f, Φ n )(x) := sup t>0 sup y∈R d 1 + t -1 |y| -b |Φ n t * f (xy)| .

ForΨ(2 k η) 2 = 1 . ( 5 . 2 )

 2152 any index l ∈ {1, .., n}, we choose a smooth homogeneous function ζ l on (R d ) n supported in the cone :ξ ∈ (R d ) n , |λ l ξ l | ≃ max j |λ j ξ j | .We can choose them in order that∀ξ ∈ (R d ) n , n l=1 ζ l (ξ) = 1.Let Ψ be a real and smooth function on R d whose the spectrum is contained in a corona around 0 and such that∀η ∈ R d \ {0} k∈Z dLet φ be a smooth function on R d whose the spectrum is bounded and such that ∀l ∈ {1, .., n}, ∀ξ ∈ supp(ζ l ) Ψ(2 k λ l ξ l ) = 0 =⇒ ∀j = l, Φ(2 k λ j ξ j ) = 1.

e 1 1 ξ 1 ,( 1 +

 111 e ix.(ξ 1 +..+ξn) Ψ 2 (2 k λ l ξ l ) f l (ξ l ) j =l Φ 2 (2 k λ j ξ j ) f j (ξ j )σ(ξ)dξ.Let us define new symbolsσ l,k (ξ) := σ( ξ 1 λ 1 2 k , .., ξ n λ n 2 k ) Ψ(ξ l ) j =l Φ(ξ j ).Hence the operator T can be written byT (f 1 , .., f n )(x) = k∈Z n l=1 R dn e ix.(ξ 1 +..+ξn) Ψ(2 k λ l ξ l ) f l (ξ l ) j =l Φ(2 p λ j ξ j ) f j (ξ j )σ l,k (2 k λ 1 ξ 1 , .., 2 k λ n ξ n )dξ.With the assumption (5.1), we get thatσ l,k ∈ L 1 and ∆ N σ l,k ∈ L 1uniformly with respect to k and l for an integer N as large as we want. So the symbols σ l,k satisfyσ l,k (ξ) = R dn e iξ.u L(l, k, u) (1 + |u| 2 ) N du with a function L ∈ L ∞ ({1, .., n} × Z × R dn ). Then we have T (f 1 , .., f n )(x) = ix.(ξ 1 +..+ξn) L(l, k, u) (1 + |u| 2 ) N Ψ(2 k λ l ξ l ) f l (ξ l ) j =l Φ(2 k λ j ξ j ) f j (ξ j )e i2 -k (λ -|u| 2 ) NR dn e ix.(ξ 1 +..+ξn)

The study of T ρ,λ,L with Littlewood-Paley square functions.In this section, we obtain the uniform bounds of Theorem 1.3 with the Hardy spaces when all the exponents p i are finite. As we will see in Section

4, our model operators can be considered as multilinear Calderón-Zygmund operators. Consequently, with similar arguments to those of L.Grafakos and

In this case, we know from[START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF] that (∂ x l j Φ j ) t * f

2, dxdt t f 2

With the arguments, used in the proof of Proposition 4.3, we can estimate the integrals. We also get the desired result :

uniformly with respect to λ. ⊓ ⊔ Corollary 4.11. The operator U ρ,λ,L can be continuously extended from (L ∞ c ) ⊗(n-2) × H p × H q into L s for all exponents 0 < q ≤ 2 and 0 < p ≤ ∞ such that

In addition if p = 1 or q = 1, we are allowed to substitute the Hardy space H 1 by the Lebesgue space L 1 with changing the final space L s,∞ instead of L s . All these continuity bounds are uniform on 0 < ρ i ≤ 1 and λ satisfying (4.1).

Proof : It is a direct consequence of the previous Proposition and the two previous Theorems. We use the fact the maximal operator M * * d/q+1 is continuous from H q to L q for all exponent 0 < q < ∞ and from L 1 to L 1,∞ . This claim is proved in Theorem 6.4.4 of [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF]. So for f 1 , .., f n-2 fixed bounded and compactly supported functions and f n ∈ H q , we obtain that the operator

is bounded on all the H p -atoms into L s . Now we use that U ρ,λ,L is bounded from (L 2 ) ⊗(n-2) × H p × H q into L t (for the corresponding exponent t, see Theorem 3.6) and that the functions f i are in L 2 (beeing compactly supported) in order to be able to extend V on the whole Hardy space H p . This is a classical argument (see for example the proof of Theorem 6.7.1 in [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF]).

We also obtain the continuity of U ρ,λ,L from (L ∞ c ) ⊗(n-2) × H p × H q into L s . We use the same ideas for the space L 1 instead of H p with p = 1.

⊓ ⊔

By producing the same reasoning over each component and by using interpolation results, we can prove Theorem 1.3 for our model operators.

⊓ ⊔

Here we are writing τ y for the translation of the vector y ∈ R d . We also obtain

For l and u being fixed, we also find "classical" paraproducts.

We now can finish the proof of Theorem 1.3 :

End of the proof of Theorem 1.3 :

We have seen in Proposition 2.4 that the "classical" paraproducts are obtained as limit objects of our models operators when some ρ i tends to 0. So the uniform results of Theorem 1.3, proved for our model operators (at the end of the previous section), are also satisfied for the paraproducts. So for each l and u fixed, the operator

satisfies all the continuities of Theorem 1.3. These continuities are bounded by a weight (1 + |u|) K for a large enough integer K (uniformly with respect to λ). So if the exponent of the final space p is bigger than 1, by using the triangular inequality with an integer N ≫ K, we get the same continuities for the operator T .

If p < 1 we cannot use the triangular inequality. * If all the exponents are finite (first part with the Littlewood-Paley square functions) : we exactly use the same proof. The spectral study is identical due to the fact that the parameters u and l have no importance. With Lemma 3.4 we can have a pointwise estimates of the different square functions which permit us to obtain the result. * If some exponent are infinite (second part with Carleson measure and Calderón-Zygmund decomposition), the proof is based on the first continuity from (L ∞ ) n-1 × L 2 into L 2 (which is satisfied for T by the triangular inequality) and on estimates about the multilinear kernel (which are again satisfied for T by the linear correspondance between the kernel and the operator).

In the two cases, the continuities of Theorem 1.3 are proved for T .

⊓ ⊔

We have also finish the proof of our Theorem 1.3. A question stays open : is one of our condition a-) or b-) (in Theorem 1.3) necessary to have uniform estimates ?


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