Geralf H Ütter

Karam Sab

Samuel Forest

Geralf Hütter
email: geralf.huetter@imfd.tu-freiberg.de

Kinematics and constitutive relations in the stress-gradient theory: interpretation by homogenization

Keywords: stress-gradient theory, generalized continua, homogenization, negative size effect

The stress-gradient theory has a third order tensor as kinematic degree of freedom, which is work-conjugate to the stress gradient. This tensor was called micro-displacements just for dimensional reasons. Consequently, this theory requires a constitutive relation between stress gradient and micro-displacements, in addition to the conventional stress-strain relation. The formulation of such a constitutive relation and identification of the parameters therein is difficult without an interpretation of the micro-displacement tensor.

The present contribution presents an homogenization concept from a Cauchy continuum at the micro-scale towards a stress-gradient continuum at the macro-scale. Conventional static boundary conditions at the volume element are interpreted as a Taylor series whose next term involves the stress gradient. A generalized Hill-Mandel lemma shows that the micro-displacements can be identified with the deviatoric part of the first moment of the microscopic strain field. Kinematic and periodic boundary conditions are provided as alternative to the static ones. The homogenization approach is used to compute the stress-gradient properties of an elastic porous material. The predicted negative size effect under uni-axial loading is compared with respective experimental results for foams and direct numerical simulations from literature.

Introduction

The classical Cauchy theory of continuum mechanics requires a constitutive relation between stress and strain. The constitutive parameters appearing therein can, for dimensional reasons, consequently always be grouped into those with dimension of a stress and dimensionless ones. Lacking an intrinsic length scale, this theory predicts a power-law scaling behavior when considering self-similar specimens of different size, the integer scaling exponent just depending on whether stress, strains, forces or displacements are considered.

Deviations from this scaling behavior are termed size effects and have been observed for numerous physical phenomena, cf. [START_REF] Aifantis | Update on a class of gradient theories[END_REF]. That is why certain generalized theories of continuum mechanics have been proposed in the literature. A classification of the generalizations was given by Maugin [START_REF] Maugin | A historical perspective of generalized continuum mechanics[END_REF]. Most of the generalized theories fall into the class of micro-morphic continua, which were established by Mindlin [START_REF] Mindlin | Micro-structure in linear elasticity[END_REF] and Eringen [START_REF] Eringen | Nonlinear theory of simple micro-elastic solids-i[END_REF]. Therein, the (dimensionless) micro-deformation is introduced as additional kinematic degree of freedom. Certain sub-classes of theories, like the micro-polar theory (Cosserat theory) or the strain-gradient theory can be obtained by imposing kinematic constraints to the micro-deformation. As alternative approach, Forest and Sab [START_REF] Forest | Stress gradient continuum theory[END_REF] imposed a kinetic constraint to obtain a stressgradient theory. Therein, a kinematic degree of freedom Φ ijk appears as work-conjugate quantity to the gradient

R ijk := ∂Σ ij ∂X k . (1)
of the stress tensor Σ ij . Though, the stress gradient cannot take arbitrary values but it is restricted by the equilibrium conditions as will be detailed below. Due the presence of R ijk in the respective potentials, Neumann boundary conditions do not involve only the tractions as normal component of Σ ij , but all components of Σ ij need to be prescribed at a surface. Alternative Dirichlet boundary conditions involve additional terms as well, cf. [START_REF] Sab | Stress gradient elasticity theory: Existence and uniqueness of solution[END_REF]. The third-rank tensor Φ ijk has the dimension of length, which is why it was termed "micro-displacements". Like all generalized theories of continuum mechanics, the stressgradient requires additional constitutive relations. Their formulation and the interpretation of the boundary conditions is difficult without an interpretation of the tensor of micro-

displacements Φ ijk .
The scope of the present contribution is to provide a homogenization methodology from a classical, but heterogeneous, continuum at the micro-scale towards a homogeneous stressgradient theory at the macro-scale.

The present contribution is structured as follows: Section 2 presents the homogenization theory, before this theory is employed in Section 3 to compute the macroscopic non-classical constitutive parameters of a plane elastic micro-structure with pores. These constitutive parameters are used in Section 4 to predict the size effect under uni-axial tension. Finally, Section 5 closes with a summary and conclusions.

Homogenization theory

In a homogenization procedure, a material with heterogeneous micro-structure is replaced by an homogeneous continuum with (more or less) equivalent macroscopic properties. For this purpose, a volume element ∆V is considered, which contains the relevant heterogeneities of the micro-structure as sketched in Figure 1. In the following, capital symbols refer to macroscopic quantities, and lower-case symbols to microscopic ones. For instance, σ ij and ε ij are the microscopic stress and strain, respectively, whereas Σ ij and E ij refer to their macroscopic counterparts.

In the classical theory of homogenization by Hill [START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF], either kinematic boundary conditions u i = E ij y j can be prescribed for the displacements on ∂∆V (X), or static ones

n i σ ij = n i Σ ij
for the tractions. Therein, y j = x j -X j refers to the position vector of a point x j relative to the center X j = x j of the volume element, cmp. Figure 1. The operator (•) computes the volume average over the volume element ∆V .

Gologanu, Kouznetsova et al. [START_REF] Gologanu | Recent extensions of Gurson's model for porous ductile metals -part II: A Gurson-like model including the effect of strong gradients of the macroscopic field[END_REF][START_REF] Kouznetsova | Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme[END_REF] interpreted the kinematic boundary conditions as a Taylor series. In this sense, they incorporated an additional term to Hill's expression to obtain a homogenization scheme for the strain-gradient theory. Mühlich et al. [START_REF] Mühlich | Estimation of material properties for linear elastic strain gradient effective media[END_REF] argued that an analogous expansion of Hill's static boundary condition would yield the homogenization for a stress-gradient theory. This proposal shall be exploited here in detail. Using the notation of the stress-gradient theory, an expanded static boundary condition thus reads

σ ij n i = n i [Σ ij + R ijk y k ] ∀y k ∈ ∂∆V (X) . (2)
Purely static boundary conditions are prone to the condition that, in absence of volume forces σ ij,i = 0, prescribed tractions need to be self-equilibrating (statically admissible):

∂∆V σ ij n i dS = 0, ∂∆V σ ij n i y k jkl dS = 0 . (3)
For the particular tractions (2) these conditions require

R iji = Σ ij,i = 0, Σ ij = Σ ji , (4)
corresponding to the macroscopic equilibrium conditions. 1 Consequently, the stress gradient is symmetric R ijk = R jik and deviatoric in the sense R ijj = R jij = 0. The loading to a volume element by stress gradients according to Eq. ( 2) is shown schematically in Figure 2.

x 1

x 2 (a) Furthermore, a homogenization theory requires a condition of macro-homogeneity (Hill-Mandel condition), which defines the macroscopic mechanical power P int as average over its microscopic pendant:

x 1 x 2 (b)
σ ij εij = P int (X k ) . (5)
By partial integration, the left-hand side of Eq. ( 5) can be transformed to a surface integral over the boundary ∂∆V (X) of the volume element ∆V into which boundary condition [START_REF] Andrews | Size effects in ductile cellular solids. part ii: experimental results[END_REF] can be inserted. After rearrangement and application of the divergence theorem, the lefthand side of Eq. ( 5) becomes

σ ij εij = 1 ∆V ∂∆V σ ij n i uj dS = Σ ij εij +R ijk εij y k - 1 n + 1 εim y m δ jk + εjm y m δ ik (6)
Therein, n = δ kk refers to the dimension of space (n = 2 or n = 3). From Eq. ( 6), a strain tensor E ij and a third-order tensor Φ ijk , called tensor of micro-displacements [START_REF] Forest | Stress gradient continuum theory[END_REF],

can be introduced as work-conjugate macroscopic deformation measures to Σ ij and R ijk , respectively, as

E ij = ε ij = 1 2∆V ∂∆V u i n j + u j n i dS (7)
Φ ijk = ε ij y k - 1 n + 1 ε im y m δ jk + ε jm y m δ ik (8) = 1 2∆V ∂∆V (u i n j + u j n i ) y k - 1 n + 1 (u i n m + u m n i ) y m δ jk - 1 n + 1 (u j n m + u m n j ) y m δ ik dS .
Thereby, it was taken into account that the stress and stress gradient exhibit symmetries, and so do their work-conjugate quantities (E ij = E ji , Φ ijk = Φ jik and Φ ijj = 0). Equation [START_REF] Hütter | Homogenization of a Cauchy continuum towards a micromorphic continuum[END_REF] indicates that the micro-displacement tensor Φ ijk corresponds to the deviatoric part of the first moment of the local strain field. Furthermore, it shall be mentioned that micro-macro relations [START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF] and ( 8) are objective, i. e., that they are invariant to superimposed rigid-body motions.

For a hyperelastic material σ ij εij = Ẇ, Eq. ( 6) can be integrated in time to a macroscopic strain energy potential

W(E ij , Φ ijk ) = W(ε ij ) (9)
with

Σ ij = ∂W ∂E ij , R ijk = ∂W ∂Φ ijk . (10)
Furthermore, it is required that the energy at the macroscopic scale is conserved. This means, that it must be possible to convert the internal power

P int = Σ ij Ėij + R ijk Φijk to the divergence of a flux Q mech i
of mechanical power:

P int = Q mech i,i . (11)
By partial integration of P int using the equilibrium conditions (4) and the definition (1) of the stress gradient, it turns out that the flux of mechanical work has to be identified as

Q mech i = Σ ij Uj + Σ jk Φjki . (12)
Therein, U j (X k ) is the macroscopic displacement field. Furthermore, the kinematic relation for the strain becomes

E ij = U (i,j) + Φ ijk,k . (13)
The round brackets (ij) are used here and in the following to indicate the symmetric part of a tensor with respect to indices i and j. The kinematic relation [START_REF] Liebenstein | Size and disorder effects in elasticity of cellular structures: From discrete models to continuum representations[END_REF] involves the divergence of the micro-displacements at the right-hand side, in addition to the symmetric part of the displacement gradient.

In this context, it may be recalled, that it is an (implicit) ad-hoc postulate of the classical homogenization theory of Hill [START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF], that the strain field E ij is macroscopically compatible, i. e., that it it is related to a macroscopic displacement field via a kinematic relation, and that the field of macroscopic stresses satisfies equilibrium conditions. In the present approach, both, the macroscopic equilibrium conditions (4) as well as the kinematic relation ( 13) are an outcome of the homogenization procedure.

Alternatively, relations ( 12) and ( 13) can be written in terms of a "generalized displacement tensor" [18]

Ψ ijk := 1 2 (U i δ jk + U j δ ik ) + Φ ijk (14)
in short as

E ij = Ψ ijk,k and Q mech i = Σ jk Ψjki , respectively. The trace Ψ ijj = (n + 1)U i /2 is
directly related to the macroscopic displacement vector, whereas the deviatoric part of Ψ ijk corresponds to the micro-displacement tensor Φ ijk . In view of Eq. ( 8) 2 , the micro-macro relation for the generalized displacement tensor is formulated as

Ψ ijk = 1 2∆V ∂∆V (u i n j + u j n i ) y k dS = ε ij y k + 1 2 u i δ jk + u j δ ik . (15)
The deviatoric part of Eq. ( 15) is identical to Eq. ( 8) for the micro-displacement. Furthermore, for a superimposed rigid translation the right-hand side of (15) transforms according to Eq. ( 14).

In classical homogenization, kinematic or periodic boundary are usually favored over static ones for several reasons. In order to construct kinematic boundary conditions for the present stress-gradient homogenization, it has firstly be noted that the kinematic micromacro relations ( 7) and ( 8) can be transformed to pure surface integrals. Thus, it is possible at all to prescribe E ij and Φ ijk exclusively by suitable boundary conditions (in contrast to micromorphic theory, cf. e. g. [START_REF] Forest | Cosserat overall modeling of heterogeneous materials[END_REF][START_REF] Hütter | Homogenization of a Cauchy continuum towards a micromorphic continuum[END_REF][START_REF] Jänicke | Minimal loading conditions for higher order numerical homogenisation schemes[END_REF]). In particular, an additional quadratic term is added to conventional kinematic boundary conditions

u i = U i + E ij y j + C ijk y j y k (16)
as proposed in [START_REF] Gologanu | Recent extensions of Gurson's model for porous ductile metals -part II: A Gurson-like model including the effect of strong gradients of the macroscopic field[END_REF][START_REF] Kouznetsova | Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme[END_REF]. It can be verified easily that ansatz [START_REF] Mühlich | Estimation of material properties for linear elastic strain gradient effective media[END_REF] satisfies the classical micromacro relation [START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF] ad hoc. Furthermore, Eq. ( 15) yields a set of 18 equations for the microdisplacements Ψ ijk in terms of the 18 independent componentes of C ijk . These equations involve the second geometric moment G ij = y i y j . For simply shaped volume elements, the second geometric moment is a spherical tensor G ij = Gδ ij . In this case, the system of equations for C ijk can be solved, cf. [START_REF] Hütter | Homogenization of a Cauchy continuum towards a micromorphic continuum[END_REF]. After reinserting Eq. ( 15), the kinematic boundary condition for the stress-gradient theory reads

u i = U i + E ij y j + 1 2G Φ ijk + Φ ikj -Φ kji + 1 n + 2 Φ mmi δ jk y j y k . ( 17
)
This boundary condition can be inserted to the left-hand side of the generalized Hill-Mandel condition [START_REF] Forest | Stress gradient continuum theory[END_REF]. A comparison with the right-hand side of Eq. ( 5) shows that the kinetic micro-macro relations read

Σ ij = 1 ∆V ∂∆V n k σ k(i y j) dS = σ ij (18)
R ijk = 1 2∆V G ∂∆V 2n p σ p(i y j) y k -n p σ pk y i y j + 1 n + 2 n p σ pk δ ij -2 n + 3 n + 1 σ p(i δ j)k y m y m dS = 1 G σ ij y k + 1 n + 2 δ ij σ km -2 n + 3 n + 1 σ (im δ j)k y m . (19)
It can be verified, that the extended static boundary condition (2) satisfies these kinetic micro-macro relations. The quadratic deformation modes are illustrated in Figure 3 for certain components of the micro-displacement tensor Φ ijk . It seems to be plausible, that the loading and deformation modes in figures 2 and 3, respectively, belong to each other.

Periodic boundary conditions can be constructed by amending a fluctuation ∆u i (y k ) to the kinematic boundary condition (17)

u i = U i + E ij y j + 1 2G Φ ijk + Φ ikj -Φ kji + 1 n + 2 Φ mmi δ jk y j y k + ∆u i (y k ) . ( 20
)
x 1

x 2 (a)

x 1

x 2 (b) This fluctuation field is assumed to be periodic

∆u i (y + k ) = ∆u i (y - k ) . (21)
Therein, y + k and y - k refer to homologeous points of the boundary ∂∆V (X), i. e., to points with opposing normal n i (y - k ) = -n i (y + k ) as sketched in Figure 1b. In order to formulate a boundary-value problem for the microscopic displacement field u i (y k ), the fluctuations are eliminated in Eq. ( 21) by Eq. ( 20), yielding

u i (y + k )-u i (y - k ) = E ij y + j -y - j + 1 2G Φ ijk + Φ ikj -Φ kji + 1 n + 2 Φ mmi δ jk y + j y + k -y - j y - k (22)
The periodicity of the fluctuation field, Eq. ( 21) or ( 22), satisfies ad hoc the kinematic micromacro relation [START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF] for the strain, but not Eq. ( 8) for the micro-displacements. Thus, Eqs. [START_REF] Hütter | Homogenization of a Cauchy continuum towards a micromorphic continuum[END_REF] and ( 22) have to be imposed as global constraints at the micro-scale [START_REF] Hütter | On the micro-macro relation for the microdeformation for the homogenization of heterogeneous materials towards micromorphic and micropolar continua[END_REF]. For a hyper-elastic material with strain-energy density W(ε ij ), the corresponding Lagrangian thus reads

L = W - 1 ∆V ∂∆V + λ i (y + p ) u i (y + k )-u i (y - k )-E ij y + j -y - j - 1 2G 2Φ i(jk) -Φ kji + 1 n+2 Φ mmi δ jk y + j y + k -y - j y - k dS + λ ijk   Φ ijk - 1 ∆V ∂∆V u (i n j) y k - 1 n+1 u (i n m) y m δ jk - 1 n+1 u (j n m) y m δ ik dS   .
Therein, the first surface integral is taken over one half of the boundary y + k ∈ ∂∆V + and the respective homologous points y - k have to be given as a function in terms of y + k ∈ ∂∆V + . Correspondingly, the field of scalar Lagrange multipliers λ i is defined in terms of y + p . The functional L is to be optimized with respect to the microscopic displacement field u i (y k ) and to the Lagrange multipliers λ i (y + p ) and λ ijk . The corresponding stationarity conditions are the local equilibrium conditions σ ij,i = 0 and σ ij = σ ji , as well as the enforced relations [START_REF] Hütter | Homogenization of a Cauchy continuum towards a micromorphic continuum[END_REF] and ( 22) and the boundary conditions

n i σ ij = ±λ j (y k ) + n i λ ijk y k . ( 23
)
The plus sign +λ j (y k ) in the first term applies to points y k ∈ ∂∆V + , whereas the minus sign applies to respective homologeous points y - k . Thus, the tractions at the boundary, Eq. ( 23), involve the anti-periodic part λ j (y k ) with an superimposed linear term with λ ijk .

Correspondingly, the classic case is recovered in absence of stress-gradients. For irreversible material behavior, the stationarity conditions are generalized to hold without existence of a Lagrangian function L (principle of virtual power).

Insering Eq. ( 23) to the kinetic micro-macro relations [START_REF] Sab | Stress gradient elasticity theory: Existence and uniqueness of solution[END_REF] and [START_REF] Tekoğlu | Size effects in foams: Experiments and modeling[END_REF] yields

Σ ij = 1 ∆V ∂∆V + y + (i -y - (i λ j) (y + k ) dS (24)
R ijk =λ ijk + 1 2∆V G ∂∆V + 2λ (i y + j) y + k -y - j) y - k -λ k y + i y + j -y - i y - j + 1 n+2 λ k δ ij -2 n+3 n+1 λ (i δ j)k y + m y + m -y - m y - m dS (25)
These terms coincide with the coefficients of Ėij and Φijk when evaluating the left-hand side of the generalized Hill-Mandel condition [START_REF] Forest | Stress gradient continuum theory[END_REF], so that the latter is satisfied.

Homogenization of an elastic porous medium

A circular (or spherical) volume element as shown in Figure 4 can be used as approximation to a material with a regular hexagonal arrangement of pores. The circular volume element has firstly the advantage, that this geometry does not posses preferred directions. Consequently, isotropic behavior of the microscopic constituents will result in an isotropic homogenized behavior. Secondly, certain analytical solutions can be found for this simple geometry. That is why, this geometry has been used within numerous studies on fundamental aspects of homogenization, e. g. [START_REF] Gologanu | Recent extensions of Gurson's model for porous ductile metals -part II: A Gurson-like model including the effect of strong gradients of the macroscopic field[END_REF][START_REF] Hütter | On the micro-macro relation for the microdeformation for the homogenization of heterogeneous materials towards micromorphic and micropolar continua[END_REF][START_REF] Mühlich | Estimation of material properties for linear elastic strain gradient effective media[END_REF].

In the present study, the effective properties of the stress-gradient continuum shall be computed for linear elastic behavior σ ij = λδ ij ε kk +2µε ij of the matrix material r i ≤ |y i | ≤ r a using periodic boundary conditions. For the circular volume element, the homologeous points are located opposite to each other y - i = -y + i with respect to the center of the volume element. Thus, Eq. (25) reduces to R ijk = λ ijk . Effectively, this means that the problem (23) can be interpreted as superposition of static boundary conditions for the stress-gradient terms with the conventional periodic conditions for classical behavior, i. e., for the effective Lamé's constants λ (eff) and µ (eff) in a relation

Σ ij = λ (eff) δ ij E kk + 2µ (eff) E ij . ( 26
)
The solution for λ (eff) and µ (eff) is well-known. It remains to address the non-classical terms.

In the plane case, the stress gradient tensor has four independent components

R 111 = -R 122 = -R 212 , R 221 , R 222 = -R 211 = -R 121 , R 112
, and so does have the tensor of microdisplacements Φ ijk [START_REF] Forest | Stress gradient continuum theory[END_REF].

Favorably, the circular volume element is treated in polar coordinates r, ϕ. In particular, the part of the boundary condition (23), which is related to the stress gradient, reads

σ rr (r a ) = r a 4 [-(R 221 -3R 111 ) cos(3ϕ)+(R 111 +R 221 ) cos(ϕ)+(R 112 +R 222 ) sin(ϕ)+(R 112 -3R 222 ) sin(3ϕ)] (27)
σ rϕ (r a ) = r a 4 [(R 221 -3R 111 ) sin(3ϕ)+(R 111 +R 221 ) sin(ϕ)-(R 112 +R 222 ) cos(ϕ)+(R 112 -3R 222 ) cos(3ϕ)] (28)
The problem can be solved with an ansatz for the Airy stress function F (r, ϕ), which involves respective terms of the Mitchell series:

F = [(R 111 + R 221 ) cos(ϕ) + (R 112 + R 222 ) sin(ϕ)] A 1 r 3 + A 2 r + [(R 221 -3R 111 ) cos(3ϕ) -sin(3ϕ)(R 112 -3R 222 )] A 3 r 5 + A 4 r + A 5 r 3 + A 6 r 3 . ( 29
)
The coefficients A 1 to A 6 can be determined from boundary conditions ( 27) and ( 28), and the trivial natural boundary condition σ rr (r i ) = σ rϕ (r i ) = 0 at the surface of the pore. Instead of evaluating the kinematic micro-macro relation [START_REF] Hütter | Homogenization of a Cauchy continuum towards a micromorphic continuum[END_REF], the corresponding micro-displacements can be computed equivalently by Castigliano's method. For this purpose, the complementary strain energy is computed as

W * = 1 4µ σ ij σ ij -νσ 2 kk = b1 2 (R 111 +R 221 ) 2 +(R 112 +R 222 ) 2 + b2 2 (R 221 -3R 111 ) 2 +(R 112 -3R 222 ) 2 (30) with b1 = r 2 a 32µ 3 -4ν + c 2 1 -c 2 , b2 = r 2 a 32µ 1 + c + 9c 3 -7c 2 + (3 -4ν)(1 + c)(1 + c 2 )c 2 (1 + 4c + c 2 )(1 -c) 3 (31)
for the plane strain case. Therein, c = r 2 i /r 2 a refers to the porosity of the material. The procedure can be performed analogously for kinematic boundary conditions as described in the appendix. The resulting compliance moduli are plotted in Figure 5. Plausibly, their value tends to infinity as c tends to 1. The same b1 is obtained for both types of boundary conditions. The values of b2 coincide for homogeneous material c = 0, but for porous material c > 0 periodic boundary conditions yield more compliant behavior than kinematic boundary conditions. This behavior is known from classical homogenization. For the plane stress case, ν in Eq. 31 has to be replaced by ν/(1 + ν).

X 1 X 2 H Σ ij = 0 Σ ij = 0 U 1 = εX 1
Forest and Sab [START_REF] Forest | Stress gradient continuum theory[END_REF] wrote the non-classical linear-elastic constitutive relation of an isotropic and centro-symmetric material in compliance form in a Voigt-type notation as

  3Φ 111 Φ 221   = [B] •   R 111 R 221   ,   3Φ 222 Φ 112   = [B] •   R 222 R 112   (32)
The factor 3 in front of Φ 111 and Φ 222 was introduced such that the Voigt-type column vectors are work-conjugate to each other. Correspondingly, [B] is a symmetric and positive definite compliance matrix. A comparison of Eq. 31 with Eq. (32) shows, that the compliance matrix for the stress gradients has to be identified as

[B] =   b1 + 9 b2 b1 -3 b2 b1 -3 b2 b1 + b2   . (33)

Uni-axial tension

Stress gradient theory

As an example, the predictions of the stress-gradient theory for uni-axial tension shall be investigated as sketched in Figure 6. The stress-gradient theory requires extended boundary conditions in form of a second order tensor, cf. [START_REF] Forest | Stress gradient continuum theory[END_REF]. Here, the trivial natural boundary condition

Σ ij (X 2 = ±H/2) = 0 (34)
is prescribed at the lateral free surfaces. Consequently, a state of constant stress Σ 11 = const. is not a solution to the uni-axial tension problem since it would violate the boundary condition (34), in contrast to classical Cauchy continuum theory or even (first order) micromorphic or strain-gradient theories.

For a sufficiently long specimen, the stress state depends only on X 2 and the only nonvanishing components of stress and its gradient are Σ 11 (X 2 ) and R 112 (X 2 ), respectively.

Inserting the latter to the constitutive relation (32

) yields Φ 222 = 1/3 B12 R 112 and Φ 112 = B22 R 112 .
Correspondingly, the components of the strain tensor, Eq. ( 13), are

E 11 =U 1,1 + Φ 112,2 , E 22 =U 2,2 + Φ 222,2 . (35)
Therein , U 1,1 equals the applied strain ε. Furthermore, the constitutive relation (26) between Σ 11 and strains E 11 and E 22 is required. Favorably, it is used in compliance form eff) , wherein Y (eff) refers to (macroscopic) Young's modulus. Together with the constitutive law for Φ 112 , Eq. ( 35) 1 yields the ODE

E 11 = Σ 11 /Y (
Σ 11 -Y (eff) B22 Σ 11,22 = Y (eff) ε , (36)
whose coefficient introduces the intrinsic length = Y (eff) B22 . Under boundary conditions (34), the solution is

Σ 11 = Y (eff) ε 1 - cosh X 2 cosh H 2 (37)
as plotted in Figure 7a for some parameter sets. Subsequently, Eq. (35) 2 could be solved for the lateral displacements U 2 (X 2 ). Finally, the strain energy within a single cross section

X 1 = const is computed as 1 2 H/2 -H/2 Σ 11 E 11 + R 112 Φ 112 dX 2 = 1 2 ε2 H Y (eff) 1 - 2 H tanh H 2 =:Yapp , (38)
from which the apparent Young's modulus Y app of the specimen can be extracted. The square bracket in Eq. (38) reflects the size effect. Figure 7b shows that the apparent Young's modulus of smaller samples is smaller than that of sufficiently large samples. Such negative loading conditions [START_REF] Tran | Mori-Tanaka estimates of the effective elastic properties of stress-gradient composites[END_REF].

The size effect depends on the single intrinsic length only. The predicted values of this intrinsic length from the homogenization in Section 3 are depicted in Figure 8. Thereby, the required effective value of Young's modulus Y (eff) from [START_REF] Hütter | On the micro-macro relation for the microdeformation for the homogenization of heterogeneous materials towards micromorphic and micropolar continua[END_REF] has been used to compute . The figure shows firstly that Poisson's ratio ν of the matrix material has a very weak influence on . Secondly, the intrinsic length has an approximately constant and small value ≈ 0.5r a for small porosities c 0.6. For larger values of c, the value of increases strongly and even tends to infinity as c goes to one. This behavior is attributed to the fact that the classical properties like Y (eff) tend to zero as 1c, whereas the stress gradient compliance, Eq. (31), has a (1c) 3 singularity. Furthermore, Figure 8 shows that the predicted size effect does not vanish completely for homogeneous material c = 0. Though, this was neither the case for the strain-gradient theory [START_REF] Gologanu | Recent extensions of Gurson's model for porous ductile metals -part II: A Gurson-like model including the effect of strong gradients of the macroscopic field[END_REF][START_REF] Kouznetsova | Size of a representative volume element in a second-order computational homogenization framework[END_REF][START_REF] Mühlich | Estimation of material properties for linear elastic strain gradient effective media[END_REF].

Comparison with experiments and direct numerical simulations

It is known that foam materials exhibit size effects when the specimen size becomes comparable to the cell size of the foam. In particular, negative size effects under uni-axial loading have been observed in experiments with foams [START_REF] Andrews | Size effects in ductile cellular solids. part ii: experimental results[END_REF] and direct numerical simulations with discretely resolved strut structure [START_REF] Liebenstein | Size and disorder effects in elasticity of cellular structures: From discrete models to continuum representations[END_REF][START_REF] Tekoğlu | Size effects in foams: Experiments and modeling[END_REF]. The observed negative size effect was attributed to a surface layer of incomplete cells which do not carry any load [START_REF] Andrews | Size effects in ductile cellular solids. part ii: experimental results[END_REF][START_REF] Rueger | Experimental study of elastic constants of a dense foam with weak Cosserat coupling[END_REF][START_REF] Wheel | Is smaller always stiffer? On size effects in supposedly generalised continua[END_REF]. This surface layer can be seen as physical explanation of the boundary condition (34) for the stress-gradient theory in the previous section. It was shown that the stress-gradient theory can describe the negative size effect qualitatively. The subsequent question is whether the present homogenization approach allows quantitative predictions of this size effect. Figure 9 compares the experimental results of Andrews et al. [START_REF] Andrews | Size effects in ductile cellular solids. part ii: experimental results[END_REF] and the direct numerical simulations (DNS) of Tekoğlu et al. [START_REF] Tekoğlu | Size effects in foams: Experiments and modeling[END_REF] and Liebenstein et al. [START_REF] Liebenstein | Size and disorder effects in elasticity of cellular structures: From discrete models to continuum representations[END_REF] with the predictions of the present homogenization theory. Andrews et al. [START_REF] Andrews | Size effects in ductile cellular solids. part ii: experimental results[END_REF] investigated two materials ("Alporas", "Duocell").

Tekoğlu et al. [START_REF] Tekoğlu | Size effects in foams: Experiments and modeling[END_REF] modeled these foams by plane, Voronoi-tesselated beam networks. They specified a "cell size d", which is taken here as d ≈ 2r a . Liebenstein et al. [START_REF] Liebenstein | Size and disorder effects in elasticity of cellular structures: From discrete models to continuum representations[END_REF] investigated honeycomb structures for which r a is identified with the radius of a circle of equal area.

The relative density of the foams was specified to be 7-10%, corresponding to a porosity of c = 0.90 . . . 0.93. Figure 9 shows that the trend of the experimental data and direct numerical simulations is captured quite well by the present homogenized stress-gradient theory ("SG").

However, the absolute size effect is slightly overestimated by the homogenized theory if the actual porosity is used. Rather, the average of experimental results and direct numerical simulations comply best with the predictions of the present stress-gradient approach for c ≈ 0.80. This deviations might be attributed to the simple representation of the pores by circles. Further studies are required on the effect of the topology of foam micro-structures on size effects during elastic and inelastic deformations.

Summary and conclusions

The stress-gradient theory requires a constitutive relation between the tensor of microdisplacements Φ ijk and the stress gradient R ijk . In the present contribution, a homogeniza- The homogenization procedure was employed to compute the stress-gradient parameters of an elastic material with pores. These parameters were used to predict the negative size effect of foam materials under uni-axial loading. A comparison with respective experiments and direct numerical simulations from literature exhibited a reasonable agreement.

It shall be mentioned that similar non-classical terms in static or kinematic boundary conditions appear in homogenization approaches towards strain-gradient or (first order) micromorphic theories [START_REF] Gologanu | Recent extensions of Gurson's model for porous ductile metals -part II: A Gurson-like model including the effect of strong gradients of the macroscopic field[END_REF][START_REF] Hütter | Homogenization of a Cauchy continuum towards a micromorphic continuum[END_REF]. The latter theories predict positive size effects, in contrast to the stress-gradient theory. This means that the choice of the generalized continuum theory to be used at the macro-scale, is an important constitutive assumption itself.

Figure 1 :

 1 Figure 1: Homogenization procedure: (a) volume element, (b) heterogeneous microstructure[START_REF] Hütter | Homogenization of a Cauchy continuum towards a micromorphic continuum[END_REF][START_REF] Hütter | On the micro-macro relation for the microdeformation for the homogenization of heterogeneous materials towards micromorphic and micropolar continua[END_REF]

Figure 2 :

 2 Figure 2: Loading to the volume element by macroscopic stress gradients: (a) R 111 = -R 122 = -R 212 , (b) R 221

Figure 3 :

 3 Figure 3: Non-classical kinematic boundary conditions: (a) Φ 111 = -Φ 122 = -Φ 212 , (b) Φ 221

Figure 4 :Figure 5 :

 45 Figure 4: Circular volume element with pore

Figure 6 :

 6 Figure 6: Tensile test with stress-gradient material

Figure 7 :

 7 Figure 7: Stress-gradient medium under uni-axial tension: (a) stresses over cross section, (b) size effect in apparent Young's modulus

1 Figure 8 :Figure 9 :

 189 Figure 8: Intrinsic length of stress-gradient theory from homogenization (plane stress)

  tion framework was developed to identify this constitutive relation from the microstructure of a material. For this purpose, the static boundary conditions of classical homogenization have been interpreted as a Taylor series, whose subsequent term involves the stress gradient. A condition of macro-homogeneity (generalized Hill-Mandel condition) yields a kinematic micro-macro relation for Φ ijk . It turned out that Φ ijk can be identified with the deviatoric part of the first moment of the microscopic strain field. Based on the kinematic micro-macro relations, kinematic boundary conditions for the micro-scale have been identified, where the micro-displacements Φ ijk appear as coefficients of the non-classical quadratic term. Furthermore, generalized periodic boundary conditions have been formulated. The proposed homogenization framework with generalized boundary conditions at the microscale and micro-macro relations for all involved kinematic and kinematic quantities allows to use linear or nonlinear constitutive relations at the micro-scale. It is thus well-suited for numerical implementations like FE 2 .

This approach is used in many textbooks and lectures to derive the equilibrium conditions (4) for the Cauchy theory.

Acknowledgments

The financial support by the Deutsche Forschungsgemeinschaft (DFG) under contract HU 2279/2-1 (GH) is gratefully acknowledged.

Appendix A. Solution for kinematic higher-order boundary conditions

For the circular volume element, kinematic BCs [START_REF] Rueger | Experimental study of elastic constants of a dense foam with weak Cosserat coupling[END_REF]

The problem can be solved with an Airy ansatz analogous to Eq. ( 29)

Finally, a macroscopic strain-energy density of

is obtained for the plane strain case with shows that stiffness and compliance coefficients are related by

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Geralf Hütter: Funding acquisition, Investigation, Writing -original draft Samuel Forest:

Resources, Supervision, Writing -review & editing Karam Sab: Supervision, Writing -review & editing


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

Local estimates and global continuities in Lebesgue spaces for bilinear operators

Keywords: Local estimates, multilinear operators, time-frequency analysis. MSC classification : 42B15-42A20-42A99

In this paper, we first prove some local estimates for bilinear operators (closely related to the bilinear Hilbert transform and similar singular operators) with truncated symbol. Such estimates, in accordance with the Heisenberg uncertainty principle correspond to a description of "off-diagonal" decay. In addition they allow us to prove global continuities in Lebesgue spaces for bilinear operators with spatial dependent symbol.

Introduction

The simplest bilinear operator is the product Π , defined by ∀ f, g ∈ S(R), Π(f, g)(x) := f (x)g(x).

Hölder's inequalities give us the continuities on Lebesgue spaces for this operator. So for all exponents 0 < p, q, r ≤ ∞ such that

1 p + 1 q = 1 r , (1.1)
the operator Π is continuous from L p (R) × L q (R) into L r (R). Also a natural question appears : how can we modify this bilinear operation and simultaneous keep these continuities ? First let T be a bilinear operator, acting from S(R) × S(R) into S ′ (R).

It is well-known that we have a spatial representation of T with a kernel K ∈ S ′ (R 3 ) and a frequential representation with a symbol σ ∈ S ′ (R 3 ) such that (in distributional sense)

∀ f, g ∈ S(R), T (f, g)(x) = In the rest of this paper, we denote by T σ the operator associated to the symbol σ. The kernel and the symbol are related by the relation

K(x, y, z) = R 2
e i[α(x-y)+β(x-z)] σ(x, α, β)dαdβ.

(1.3)

For example, the product operator Π is given by the symbol σ(x, α, β) = 1.

A first class of bilinear symbols is the one satisfying the "bilinear Hörmander's condition" :

∀ a, b, c ≥ 0, ∂ a x ∂ b α ∂ c β σ(x, α, β) (1 + |α| + |β|) -b-c . (1.4)
The operators T σ were already studied par R. Coifman and Y. Meyer in [START_REF] Coifman | Au delà des opérateurs pseudodifférentiels[END_REF][START_REF] Coifman | On commutators of singular integrals and bilinear singular integrals[END_REF], C. Kenig and E.M. Stein in [START_REF] Kenig | Multilinear estimates and fractional integration[END_REF] and recently by L. Grafakos and R. Torres in [START_REF] Grafakos | Multilinear Calderón-Zygmund theory[END_REF]. We know that under (1.4), the operator T σ is bounded from L p (R) × L q (R) in L r (R) for all exponents p, q, r verifying (1.1) and 1 < p, q < ∞. In fact if the symbol is x-independent, one can just assume an homogeneous decay in (1.4) (i.e. with (|α| + |β|) -b-c ) and then these operators can be decomposed with paraproducts, which were first exploited by J.M. Bony in [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] and R. Coifman and Y. Meyer in [START_REF] Coifman | Au delà des opérateurs pseudodifférentiels[END_REF]. The paraproducts are studied with the linear tools (the Calderón-Zygmund decomposition, the Littlewood-Paley theory and the concept of Carleson measure). In order to get the continuities for x-dependent symbols, pointwise estimates of the bilinear kernel are used. Mainly for a symbol σ satisfying (1.4), integrations by parts allow us to obtain

|K(x, y, z)| (1 + |x -y| + |x -z|) -M (1.5)
for any large enough integer M. This estimate is very useful and precisely describes the "off-diagonal" decrease of the operator. Such an information helps us to reduce the study of x-dependent symbols to the study of xindependent symbols (and so to the study of paraproducts). Through these ideas, this first class of symbols are well understood nowadays. We note that this reduction (using pointwise estimates on the kernel) has already been used in the linear case to study the pseudo-differential operators of the well-known class op(S 0 1,0 ). Thus "off-diagonal" estimates play an important role.

Since the work of A.Calderón about the L 2 boundedness of commutators and Cauchy integrals ( [START_REF] Calder Ón | Commutators of singular integral operators[END_REF][START_REF] Calder Ón | Cauchy integrals on Lipschitz curves and related operators[END_REF] in the 70's), more singular bilinear operators have appeared. Mainly, he shew that the commutators and Cauchy integrals can been decomposed by using the bilinear Hilbert transforms. The bilinear Hilbert transform H λ 1 ,λ 2 is defined by

∀ f, g ∈ S(R), H λ 1 ,λ 2 (f, g)(x) := p.v. R f (x -λ 1 y)g(x -λ 2 y) dy y .
The x-independent symbol is

σ(α, β) = iπsign(λ 1 α + λ 2 β)
and so is singular on a whole line in the frequency plane. A.Calderón conjectured that these operators are continuous on Lebesgue spaces. This famous conjecture was first partially solved by M. Lacey and C. Thiele in [START_REF] Lacey | L p estimates on the Bilinear Hilbert Transform[END_REF][START_REF] Lacey | L p estimates on the bilinear Hilbert transform for 2 < p < ∞[END_REF][START_REF] Lacey | On the Calderón conjectures for the bilinear Hilbert transform[END_REF][START_REF] Lacey | On Calderón's conjecture[END_REF]. Then L. Grafakos and X. Li show in [START_REF] Grafakos | Uniform bounds for the bilinear Hilbert transforms I[END_REF][START_REF] Li | Uniform Bounds for the Bilinear Hilbert Transforms[END_REF] some uniform (with respect to the parameters λ 1 and λ 2 ) continuities. These proofs use a technical time frequency analysis, which was proven by C. Muscalu, T. Tao and C. Thiele in [START_REF] Muscalu | Multi-linear operators given by singular multipliers[END_REF][START_REF] Muscalu | Uniforms estimates on multilinear operators with modulation symmetry[END_REF][START_REF] Muscalu | L p estimates for the biest II : The Fourier case[END_REF] and independently by J. Gilbert and A. Nahmod in [START_REF] Gilbert | Bilinear operators with non smooth symbols : part 1[END_REF][START_REF] Gilbert | l p -boundedness for Time-Frequency Paraproducts : part 2[END_REF]. They also get a very important result in the study of bilinear operators : continuities in Lebesgue spaces for more singular operators than those of the first class. We are interested by these bilinear operators and we will deal with them and some "smooth spatial perturbations". So we replace in (1.4) the quantity |α| + |β| = d((α, β), 0) by the lower quantity d((α, β), ∆), where ∆ is a line in the frequency plane :

∆ := (α, β) ∈ R 2 , λ 1 α + λ 2 β = 0 .
We assume that ∆ is nondegenerate, i.e. λ 1 and λ 2 are non vanishing reals and not equal, in order that ∆ be a graph over the three variables α, β and α + β. We assume that the symbol σ satisfies

∀ a, b, c ≥ 0, ∂ a x ∂ b α ∂ c β σ(x, α, β) (1 + |λ 1 α + λ 2 β|) -b-c . (1.6)
In the previous mentioned papers, the main result is the following one : if σ is x-independent and satisfies (1.6) (or the homogeneous version) then

T σ is continuous from L p (R) × L q (R) in L r (R) for every exponents 0 < p, q, r ≤ ∞ verifying 0 < 1 r = 1 p + 1 q < 3 2 and 1 < p, q ≤ ∞.
So there is a natural question (asked in [START_REF] Benyi | Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators[END_REF]) : if an x-dependent symbol satisfies (1.6), is the operator T σ continuous from L p (R) × L q (R) into L r (R) with the same exponents p, q, r ? In [START_REF] Benyi | Modulation invariant bilinear T(1)-Theorem[END_REF], A. Benyi, C. Demeter, A. Nahmod, R. Torres, C. Thiele and P. Villarroya proved a general result for singular integrals kernels. As an example, they can apply their result to pseudo-differential operators associated to symbols σ(x, α, β) = τ (x, λ 1 α + λ 2 β) with τ in the class S 0 1,0 because of a modulation invariant condition imposed. Here we are able to treat general symbols satisfying (1.6) and complete the answer to the question in [START_REF] Benyi | Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators[END_REF]. These operators do not fall under the scope of [START_REF] Benyi | Modulation invariant bilinear T(1)-Theorem[END_REF] because they do not have modulation invariance. On the other hand, the general operators in [START_REF] Benyi | Modulation invariant bilinear T(1)-Theorem[END_REF] cannot be realized as pseudo-differential bilinear operators with symbols verifying (1.6) because of the minimal regularity assumptions required in the kernels.

With this aim, we would like to use the same arguments as for the symbols satisfying (1.4), where we have seen the important role of the "off-diagonal" decay of the bilinear kernel, obtained with integrations by parts. For our more singular operators, integration by parts does not work : to obtain a description of "off-diagonal" estimates is the most important difficulty.

We now come to our main result. For notation, we denote the norm in L p (E) for any measurable set E ⊂ R by p,E,dx (or p,E if there is no confusion for the variable). For an interval I, we set

C k (I) := x ∈ R, 2 k ≤ 1 + d(x, I) |I| < 2 k+1 (1.7)
the scaled corona around I. So we have C 0 (I) = 2I and C k (I) ⊂ 2 k+1 I. We will first prove the following result :

Theorem 1.1. Let ∆ be a nondegenerate line of the frequency plane. Let p, q be exponents such that

1 < p, q ≤ ∞ and 0 < 1 r = 1 q + 1 p < 3 2 .
Then for all δ ≥ 1, there is a constant C = C(p, q, r, ∆, δ) such that for all interval

I ⊂ R, for all symbol σ ∈ C ∞ (R 3 ) satisfying ∀ a, b, c ≥ 0 ∂ a x ∂ b α ∂ c β σ(x, α, β) |I| -1 + d((α, β), ∆) -b-c , (1.8)
we have the following local estimate : for all functions f, g ∈ S(R)

1 |I| I |T σ (f, g)(x)| r dx 1/r ≤ C k≥0 2 -kδ 1 |2 k+1 I| C k (I) |f (x)| p dx 1/p k≥0 2 -kδ 1 |2 k+1 I| C k (I) |g(x)| q dx 1/q .
In particular, with the Hardy-Littlewood's operator M HL , we have

|I| -1 I |T σ (f, g)| r 1/r inf I M HL (|f | p ) 1/p inf I M HL (|g| q ) 1/q f ∞ g ∞ .
The weight (|I| -1 +d((α, β), ∆)) -N is not integrable over the whole frequency plane (even if N is large enough due to the modulation invariance) and therefore we cannot have a pointwise estimate of the bilinear kernel (such (1.5) when we assume (1.4)). So such a result is interesting because it precisely describes "off-diagonal" estimates for the bilinear operator :

Corollary 1.2. With the same notations as Theorem 1.1, for all large enough δ, there exists a constant C = C(p, q, r, ∆, δ) such that for any measurable sets E, F ⊂ R we have for all functions f ∈ L p (E) and g ∈ L q (F ) :

T σ (f, g) r,I ≤ C 1 + d(I, E) |I| -δ 1 + d(I, F ) |I| -δ f p,E g q,F . (1.9)
This corollary is a direct application of the previous Theorem. So in spite of the fact that the symbol could be much more singular than those satisfying only (1.4), we almost obtain the pointwise estimate (1.5). Here we have a description of the same fast decrease for the bilinear kernel, not with a pointwise estimate, but with local estimates at the scale |I|. These local estimates are less precise than the pointwise estimate but we will see that they are sufficient and they can have the same role.

We note that Theorem 1.1 is in accordance with the Heisenberg's uncertainty principle, which tell us that if we want to localize at the scale |I| in the spatial domain, we cannot localize in the frequency domain at a lower scale than |I| -1 . For example, ou Theorem 1.1 applies if the symbol is supported in the domain {(α, β), d((α, β), ∆) ≥ |I| -1 } and it is this case that we consider first in the proof. In fact in (1.8), we allow instead a nice behavior around the line ∆. With this point of view, we could call Theorem 1.1 : an "high frequency estimate". In this expression, the term "frequency" corresponds to the distance between the point (α, β) to the line of singularity ∆. We prefer the expression "local estimates", because we will use the spatial fast decay in order to get the following result.

Theorem 1.3. Let ∆ be an nondegenerate line of the frequency plane. Let p, q be exponents such that

1 < p, q ≤ ∞ and 0 < 1 r = 1 q + 1 p < 3 2 . For all symbol σ ∈ C ∞ (R 3 ) satisfying ∀ a, b, c ≥ 0 ∂ a x ∂ b α ∂ c β σ(x, α, β) (1 + d((α, β), ∆)) -b-c , the associated operator T σ is bounded from L p (R) × L q (R) into L r (R).
This result answers a question of [START_REF] Benyi | Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators[END_REF]. In addition it will allow us to define a bilinear pseudodifferential calculus, based on these operators : in a next paper [START_REF] Bernicot | Bilinear Pseudodifferential Calculus[END_REF], we will define classes for bilinear pseudo-differential operators of order (m 1 , m 2 ) and study their action on Sobolev spaces. In order to carry on the work of [START_REF] Benyi | Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators[END_REF], we will give rules of symbolic calculus for the duality and the composition and also complete the construction of a bilinear pseudodifferential calculus.

Remark 1.4. The proof of Theorem 1.1 is a shake between a localization argument and the "classical" time-frequency analysis used for these bilinear operators. So it is quite easy to obtain a version of our Theorems 1.1 and 1.3 for (n -1)-linear operators T σ with a nondegenerate space ∆ of dimension k < n/2, by following the ideas of [START_REF] Muscalu | Multi-linear operators given by singular multipliers[END_REF]. By using the results of [START_REF] Terwilleger | Boundedness for a bilinear model sum operator on R n[END_REF], we are able to obtain the same results for a multidimensional problem ; and by using the uniform estimates of [START_REF] Muscalu | Uniforms estimates on multilinear operators with modulation symmetry[END_REF], it seems possible to obtain uniform (with respect to the nondegenerate line ∆) local estimates.

The plan of this article is as follows. We first prove Theorem 1.1 in Section 2 for x-independent symbols. Then in Section 3 we get the same result for maximal bilinear operators and we conclude the proof of Theorem 1.1 in the general case. Then in Section 4, we use these local estimates to obtain global continuities for bilinear operators in weighted Lebesgue and Sobolev spaces and in particular we prove Theorem 1.3.

2 Proof of Theorem 1.1 for x-independent symbol.

In this section, we assume that the symbol σ is x-independent and is supported on the domain :

(α, β), d((α, β), ∆) ≥ |I| -1 .
We divide the proof in two subsections. First, we will quickly recall the decomposition of our bilinear operator T σ by combinatorial model sums. So we will reduce the problem to a study of the "restricted weak type" for some localized trilinear forms. We will study them in the second subsection.

Reduction to the study of discrete models.

First of all, we define and recall the time-frequency tools (see for example [START_REF] Muscalu | L p estimates for the biest II : The Fourier case[END_REF]) :

Definition 2.1. A tile is a rectangle (i.e. a product of two intervals) I × ω of area one. A tri-tile s is a rectangle s = I s × ω s of bounded area, which contains three tiles

s i = I s i × ω s i for i = 1, 2, 3 such that ∀i, j ∈ {1, 2, 3}, I s i = I s and i = j =⇒ ω s i ∩ ω s j = ∅. A set {I} I∈I of real intervals is called a grid if for all k ∈ Z I∈I 2 k-1 ≤|I|≤2 k+1 1 I 1 R , (2.1)
where the implicit constant is independent of k and of the grid. So a grid has the same structure than the dyadic grid. Let Q be a set of tri-tiles. It is called a collection if

• {I s , s ∈ Q} is a grid, • J := {ω s , s ∈ Q} 3 i=1 {ω s i , s ∈ Q} is a grid, • ω s i ̟ ∈ J =⇒ ∀ j ∈ {1, 2, 3}, ω s j ⊂ ̟.
Now we can define the wave packet for a tile. Definition 2.2. Let Φ be a smooth function such that

Φ 2 = 1 and supp( Φ) ⊂ [-1/2, 1/2].
For P = I × ω a tile, we set

Φ P (x) := |I| -1/2 Φ x -c(I) |I| e 2iπxc(ω) ,
where for U an interval we denote c(U) its center. So Φ P is normalized in the L 2 (R) space, concentrated in space around I and its spectrum is exactly contained in ω.

Nowadays it is well known (see for example [START_REF] Bilyk | A new way of looking at distributional estimates; applications for the bilinear Hilbert transform[END_REF] and [START_REF] Bilyk | Distributional estimates for the bilinear hilbert[END_REF]) that the operator T σ of Theorem 1.1 admits a decomposition :

T σ (f, g)(x) := u=(u 1 ,u 2 ,u 3 )∈Z 3 (1 + |u| 2 ) -N s∈Su |I s | -1/2 ǫ s (u) (τ u 1 φ) s 1 , f (τ u 2 φ) s 2 , g (τ u 3 φ) s 3 (x), (2.2)
where S u is a collection of tri-tiles depending on u, (ǫ s (u)) s∈Su are bounded reals and N is an integer as large as we want. We write τ v for the translation operator :

τ v (f )(x) = f (x -v)
. The coefficients ǫ s (u) are uniformly bounded with respect to the parameter u and the implicit constant in (2.1) (for the definition of a grid) is bounded by the estimates of the symbol σ. By using the assumption that σ is supported in {(α, β), |α -β| ≥ |I| -1 }, we have the very important property

∀u ∈ Z 3 , ∀s ∈ S u , |ω s | |I| -1 (which is equivalent to |I s | |I|). (2.3) So Theorem 1.
1 is a consequence of the following one :

Theorem 2.3. Let S be a collection of tri-tiles satisfying the property (2.3), (ǫ s ) s∈S bounded reals and (φ i ) i=1,2,3 smooth functions whose the spectrum is contained in [-1/2, 1/2]. We denote T S the bilinear operator defined by

T S (f, g)(x) := s∈S |I s | -1/2 ǫ s φ 1 s 1 , f φ 2 s 2 , g φ 3 s 3 (x).
For the exponents (p, q, r) of Theorem 1.1 and for all δ ≥ 1, we have the local estimate

I |T S (f, g)| r 1/r k≥0 2 -k(1/p+δ) f p,C k (I) k≥0 2 -k(1/q+δ) g q,C k (I) .
In addition the implicit constant depends on the functions φ i by the parame-

ters c M (φ i ) := sup x∈R 0≤k≤M (1 + |x|) M (φ i ) (k) (x)
for M = M(p, q, r, δ) a large enough integer.

We have also to prove this result. In order to show it, we need to decompose the functions f and g around the interval I. The interval I being fixed, we omit it in the notation for convenience and for i ∈ N, we set the corona

C i := C i (I). With the property (2.
3), we have the following decomposition

T S (f, g) = k 1 ,k 2 ≥0 T k 1 ,k 2 S,0 (f, g) + k 1 ,k 2 ≥0 l≤0 T k 1 ,k 2 ,l S,1 (f, g) (2.4) with T k 1 ,k 2 S,0 (f, g)(x) := s∈S Is⊂2I |I s | -1/2 ǫ s φ 1 s 1 , f 1 C k 1 φ 2 s 2 , g1 C k 2 φ 3 s 3 (x)
and

T k 1 ,k 2 ,l S,1 (f, g)(x) := s∈S Is 2I 2 l |I|≤|Is|<2 l+1 |I| |I s | -1/2 ǫ s φ 1 s 1 , f 1 C k 1 φ 2 s 2 , g1 C k 2 φ 3 s 3 (x).
Due to the important property (2.3), we only have to consider tiles s with |I s | ≤ |I|. The other corresponding terms (l > 0) cannot be studied as we are going to do, according to the Heisenberg uncertainty principle.

In the next subsection, we shall prove the following theorem :

Theorem 2.4. Let (p, q, r) be exponents of Theorem 1.1. The operators T j S,i are continuous from L p (R) × L q (R) into L r (I). For convenience, we denote C(T j S,i ) := T j S,i L p ×L q →L r and we omit the exponents. Then these continuity bounds satisfy

C(T k 1 ,k 2 S,0 ) c M (φ 1 )c M (φ 2 )c M (φ 3 )2 -δ ′ (k 1 +k 2 ) C(T k 1 ,k 2 ,l S,1 ) c M (φ 1 )c M (φ 2 )c M (φ 3 )2 -δ ′ (|l|+k 1 +k 2 )
for any large enough real δ ′ , with an integer M = M(p, q, r, δ ′ ).

We claim that Theorem 2.3 is a consequence of Theorem 2.4. Proof of Theorem 2.3 : By using Theorem 2.4 and the decomposition (2.4), we have that for all functions f, g ∈ S(R) if r ≥ 1 then

T S (f, g) r,I k 1 ,k 2 ≥0 C(T k 1 ,k 2 S,0 ) f 1 C k 1 p g1 C k 2 q + k 1 ,k 2 ≥0 l≤0 C(T k 1 ,k 2 ,l S,1
)

f 1 C k 1 q g1 C k 2 r ,
and if r < 1 then

T S (f, g) r r,I k 1 ,k 2 ≥0 C(T k 1 ,k 2 S,0 ) r f 1 C k 1 r p g1 C k 2 r q + k 1 ,k 2 ≥0 l≤0 C(T k 1 ,k 2 ,l S,1 ) r f 1 C k 1 r p g1 C k 2 r q .
The case r ≥ 1.

With the estimate of

C(T k 1 ,k 2 S,0 ) and C(T k 1 ,k 2 ,l S,1
) given by Theorem 2.4, we obtain

T S (f, g) r,I k 1 ,k 2 ≥0 2 -δ ′ (k 1 +k 2 ) f 1 C k 1 p g1 C k 2 q + k 1 ,k 2 ≥0 l≤0 2 -δ ′ (k 1 +k 2 +|l|) f 1 C k 1 p g1 C k 2 q k 1 ,k 2 ≥0 2 -δ ′ (k 1 +k 2 ) f 1 C k 1 p g1 C k 2 q .
Hence by using that δ ′ is as large as we want, we can conclude.

The case r ≤ 1. We have

T S (f, g) r r,I k 1 ,k 2 ≥0 2 -rδ ′ (k 1 +k 2 ) f 1 C k 1 r p g1 C k 2 r q + k 1 ,k 2 ≥0 l≤0 2 -rδ ′ (k 1 +k 2 +|l|) f 1 C k 1 r p g1 C k 2 r q k 1 ,k 2 ≥0 2 -rδ ′ (k 1 +k 2 ) f 1 C k 1 r p g1 C k 2 r q .
By using Hölder's inequality and ρ > 0 such that p -1 + ρ, q -1 + ρ < 1, we obtain :

T S (f, g) r,I k 1 ≥0 2 -k 1 p(δ ′ -1)(ρ+1/p) f 1 C k 1 p p 1/p k 2 ≥0 2 -k 2 q(δ ′ -1)(ρ+1/q) g1 C k 2 q q 1/q k 1 ≥0 2 -k 1 (δ ′ -1)(ρ+1/p) f 1 C k 1 p k 2 ≥0 2 -k 2 (δ ′ -1)(ρ+1/q) g1 C k 2 q .
This corresponds to the desired result (the real δ ′ being as large as we want). So Theorem 2.3 is proved in the two cases.

⊓ ⊔

We have also reduced the proof of Theorem 1.1 (for our particular symbol σ) to the proof of Theorem 2.4.We will prove it in the next subsection.

2.2 Proof of Theorem 2.4.

By using "duality", to prove Theorem 2.4, we have to estimate the trilinear form defined on S(R) × S(R) × S(R) by

Λ j i (f 1 , f 2 , f 3 ) := T j S,i (f 1 , f 2 ), f 3 1 I (2.5) = s∈Q j i |I s | -1/2 ǫ s φ 1 s 1 , f 1 1 C k 1 φ 2 s 2 , f 2 1 C k 2 φ 3 s 3 , f 3 1 I ,
where Q j i is a collection of tri-tiles, depending on T j S,i . We need to define usual tools of time-frequency analysis. Definition 2.5. We have already defined the tri-tiles. For j ∈ {1, 2, 3} an index and t ∈ S a tri-tile, a collection T of tri-tiles is called a j-tree with top t if ∀ s ∈ T, I s ⊂ I t and ω t j ⊂ ω s j .

Then we set I T := I t the time-interval of the tree T. A collection T of tri-tiles is called a tree if there exists an index j ∈ {1, 2, 3} such that T is a j-tree. For T a j-tree, we define the size of the function f j over this tree by

size j (T) := 1 |I T | s∈T f j , φ j s j 2 1/2 .
For Q a collection of tri-tiles, we define the global size by :

size * j (Q) = sup T⊂Q T=k-tree k =j size k (T).
The quantity |I T | 1/2 size j (T) corresponds to the norm of the function f j in the space L 2 , after having restricted it on the tree T in the time-frequency space.

We recall the (abstract) Proposition 6.5 of [START_REF] Muscalu | L p estimates for the biest II : The Fourier case[END_REF] (where we use Lemma 6.7 of [START_REF] Muscalu | L p estimates for the biest II : The Fourier case[END_REF] to estimate the quantities energy j ) :

Proposition 2.6. Let (θ j ) 1≤j≤3 be three exponents of [0, 1[ satisfying

θ 1 + θ 2 + θ 3 = 1.
Then there exists a constant C = C(θ i ) such that for all collection Q of tri-tiles, we have :

s∈Q 1 |I s | 1/2 3 i=1 φ i s i , f i ≤ C 3 i=1 size * i (Q) θ i f i 1-θ i 2 .
This result is the main idea of this time-frequency analysis. To prove it, we use a stopping-time argument in order to build an "orthogonal" covering of the time-frequency space with trees of Q.

Now we recall the notion of restricted weak type for trilinear forms :

Definition 2.7. For E a borelian set of R, we write :

F (E) := {f ∈ S(R), ∀x ∈ R, |f (x)| ≤ 1 E (x)} .
Let Λ be a trilinear form, defined on S(R) × S(R) × S(R). Let p 1 , p 2 , p 3 be exponents of R * , possiblynegative. We say that Λ is of restricted weak type (p 1 , p 2 , p 3 ) if there exists a constant C such that for all measurable sets E 1 , E 2 , E 3 of finite measure, we can find a substantial subset

E ′ β ⊂ E β (i.e. |E ′ β | ≥ |E β |/2) for β ∈ {1, 2, 3} with ∀ f β ∈ F (E ′ β ), |Λ(f 1 , f 2 , f 3 )| ≤ C 3 β=1 |E β | 1/p β (2.6)
and

E ′ β = E β if p β > 0. The best constant in (2.6
) is called the bound of restricted type and will be denoted by C(Λ).

By the real interpolation theory for trilinear forms of restricted weak type (Lemmas 3.6, 3.7, 3.8, 3.9, 3.10 and 3.11 of [START_REF] Muscalu | Uniforms estimates on multilinear operators with modulation symmetry[END_REF]), Theorem 2.4 is a consequence of the following result (which is a stronger continuity result) :

Theorem 2.8. Let p 1 , p 2 , p 3 be non vanishing reals such that

1 p 1 + 1 p 2 + 1 p 3 = 1
and there exists a unique index α ∈ {1, 2, 3} with -1/2 < p -1 α < 0 and 1/2 < p -1 β < 1 for β = α. Then the trilinear forms Λ j i (defined by (2.5)) are of restricted weak type (p 1 , p 2 , p 3 ). In addition the bounds of restricted type

C(Λ j i ) satisfy C(Λ k 1 ,k 2 0 ) c M (φ 1 )c M (φ 2 )c M (φ 3 )2 -δ ′ (k 1 +k 2 ) C(Λ k 1 ,k 2 ,l 1 ) c M (φ 1 )c M (φ 2 )c M (φ 3 )2 -δ ′ (|l|+k 1 +k 2 )
for any real δ ′ ≥ 1 with M = M(δ ′ , p i ) a large enough integer.

Proof : The exponents (p β ) β and the index α ∈ {1, 2, 3} are fixed for the proof. Let E 1 , E 2 and E 3 measurable sets of finite measure. First we construct the substantial subset

E ′ α ⊂ E α . Denote U := 3 i=1 x ∈ R, M HL (1 E i )(x) > η |E i | |E α | .
By using Hardy-Littlewood's Theorem, there exists a numerical constant η such that

|U| ≤ |E α |/2.
We set also

E ′ α = E α \ U.
It is interesting to note that the set E ′ α does not depend on the form Λ j i . Now we fix the functions f β ∈ F (E ′ β ) for β ∈ {1, 2, 3} and we shall prove the inequality (2.6). The proof is divided in three parts : in the first step we use general estimates for collections of tri-tiles, in the second step we will use specific estimates adapted to the above collections of tri-tiles and then we will conclude in the third step.

First step : A general estimate. Let P an "abstract" collection of tri-tiles, then for k ≥ 0 we set P k the following sub-collection

P k := s ∈ P, 2 k ≤ 1 + d(I s , U c ) |I s | < 2 k+1 .
These collections form a partition of P : P = k≥0 P k .For each k ≥ 0, we can apply Proposition 2.6 to the collection Q = P k . So for any choice of exponents 0 < θ 1 , θ 2 , θ 3 < 1 with

3 β=1 θ β = 1,
we obtain

Λ(P k ) := s∈P k |I s | -1/2 ǫ s 3 β=1 f β , φ β s β 3 β=1 (size * β (P k )) θ β f β 1-θ β 2
.

In order to estimate the quantities size * β (P k ), we recall Lemma 7.8 of [START_REF] Muscalu | Uniforms estimates on multilinear operators with modulation symmetry[END_REF] :

Lemma 2.9. For all integer N as large as we want, there exists a constant C = C(N) such that for all collection Q of tri-tiles, for all β ∈ {1, 2, 3}, we have :

size * β (Q) ≤ C sup s∈Q 1 |I s | R 1 + d(x, I s ) |I s | -N |f β (x)|dx.
Then for Q = P k , by using the definition of the sets U and E ′ α , we have :

∀β = α size * β (P k ) 2 k |E β | |E α | and size * α (P k ) 2 -N k . (2.7) As f β belongs to F (E β ), we have f β 2 ≤ |E β | 1/2
. So for 0 < ǫ < 1 and N an integer as large as we want, we get :

Λ(P k ) β =α 2 k |E β | |E α | θ β (1-ǫ) |E β | (1-θ β )/2 2 N kθα(1-ǫ) |E α | (1-θα)/2 3 β=1 (size * β (P k )) θ β ǫ 2 -k β =α |E β | (1+θ β )/2-ǫθ β |E α | (θα-1)/2+ǫ(1-θα) 3 β=1 (size * β (P k )) θ β ǫ .
By definition of size * β , P k is a sub-collection of P so

∀β ∈ {1, 2, 3}, size * β (P k ) ≤ size * β (P).
We can also compute the sum over k ≥ 0 and we obtain

Λ(P ) := s∈P |I s | -1/2 ǫ s 3 β=1 f β , φ β s β ≤ k≥0 P k β =α |E β | (1+θ β )/2-ǫθ β |E α | (θα-1)/2+ǫ(1-θα) 3 β=1 (size * β (P)) θ β ǫ . (2.8)
The first term is "good", according to the wished global continuity. In the next step, we will use an other estimate of the quantities size * β , which will be adapted to our specific trilinear forms Λ j i and which allow us to obtain the desired decays.

Second step : Use of the specificity of our trilinear forms Λ j i . First case : the forms Λ j 1 . In this case, we use an other decomposition :

Λ k 1 ,k 2 ,l 1 (f 1 , f 2 , f 3 ) ≤ I 0 2I 2 l-1 |I|≤|I 0 |≤2 l+1 |I| Λ k 1 ,k 2 ,l 1 (I 0 )(f 1 , f 2 , f 3 )
where I 0 is an interval of R and

Λ k 1 ,k 2 ,l 1 (I 0 )(f 1 , f 2 , f 3 ) := s∈S Is=I 0 |I s | -1/2 ǫ s f 1 1 C k 1 , φ 1 s 1 f 2 1 C k 2 , φ 2 s 2 1 I f 3 , φ 3 s 3 .
Let I 0 be fixed and denote 2 l = |I 0 |/|I|. The collection of tri-tiles associated to Λ k 1 ,k 2 ,l 1 (I 0 ) is also P := {s ∈ S, I s = I 0 } For all s ∈ P, (by using

f 3 ∈ F (E ′ 3 )) we have 1 |I s | I |f 3 (x)| 1 + d(x, I s ) |I s | -N dx ≤ 1 |I s | I 1 + d(x, I s ) |I s | -N dx ≤ |I| |I 0 | 1 + d(I, I 0 ) |I 0 | -N .
Then Lemma 2.9 gives us

size * 3 (P) 2 -l 1 + d(I, I 0 ) |I 0 | -N
.

By the same reasoning, we obtain for f 1 ∈ F (E ′ 1 ) and s ∈ P

1 |I s | C k 1 |f 1 (x)| 1 + d(x, I 0 ) |I s | -N dx ≤ 1 |I 0 | C k 1 1 + d(x, I 0 ) |I 0 | -N dx ≤ 2 k 1 -l 1 + d(C k 1 , I 0 ) |I 0 | -N
.

And so we get

size * 1 (P) 2 k 1 -l 1 + d(C k 1 , I 0 ) |I 0 | -N
.

Likely, we have

size * 2 (P) 2 k 2 -l 1 + d(C k 2 , I 0 ) |I 0 | -N
.

With θ 1 + θ 2 + θ 3 = 1 and Lemma 2.9, we can estimate :

size * 1 (P) θ 1 size * 2 (P) θ 2 size * 3 (P) θ 3 2 θ 1 k 1 +θ 2 k 2 -l A(I 0 ), (2.9)
where A(I 0 ) is the product of three terms

A(I 0 ) := 1 + d(I, I 0 ) |I 0 | -N θ 3 1 + d(C k 1 , I 0 ) |I 0 | -N θ 1 1 + d(C k 2 , I 0 ) |I 0 | -N θ 2 .
We are going to get four different estimates for A(I 0 ). To keep the information about the position of I 0 , we first have

A(I 0 ) ≤ 1 + d(I, I 0 ) |I 0 | -N θ 3 .
(2.10)

By using d(I, I 0 ) + d(C k 1 , I 0 ) d(I, C k 1 ) 2 k 1 |I| ≃ 2 k 1 -l |I 0 |
and the fact that 2 l ≤ 1, we obtain :

A(I 0 ) 1 + 2 k 1 -l -N min{θ 1 ,θ 3 } 2 -k 1 N min{θ 1 ,θ 3 } (2.11)
and likely

A(I 0 ) 2 -k 2 N min{θ 2 ,θ 3 } .
(2.12)

As I 0 2I and 2 l ≤ 1 then d(I 0 , I)

≥ |I| hence 1 + d(I, I 0 ) |I 0 | -N |I 0 | |I| N .
So we get

A(I 0 ) |I 0 | |I| N θ 3 2 lN θ 3 . (2.13)
Taking the geometric mean of (2.10),(2.11), (2.12) and (2.13) (with an other exponent N which is as large as we want), we obtain :

A(I 0 ) 2 -(k 1 +k 2 +|l|)N 1 + d(I, I 0 ) |I 0 | -N . (2.14)
With the help of (2.8) and (2.9), we finally estimate

Λ k 1 ,k 2 ,l 1 (f 1 , f 2 , f 3 ) I 0 Λ k 1 ,k 2 ,l 1 (I 0 )(f 1 , f 2 , f 3 ) I 0 β =α |E β | (1+θ β )/2-ǫθ β |E α | (θα-1)/2+ǫ(1-θα) 2 ǫ(k 1 +k 2 +|l|) A(I 0 ).
The sum over the interval I 0 is bounded with |I 0 | = 2 l |I| by (2.14). For N a large enough exponent (not exactly the same), we have

Λ k 1 ,k 2 ,l 1 (f 1 , f 2 , f 3 ) β =α |E β | (1+θ β )/2-ǫθ β |E α | (θα-1)/2+ǫ(1-θα) C(Λ k 1 ,k 2 ,l 1 ), with C(Λ k 1 ,k 2 ,l 1
) := 2 -N ǫ(k 1 +k 2 +|l|) . (2.15)
Second case : the forms Λ j 0 . We use the same principle. We are interested by

Λ k 1 ,k 2 0 (f 1 , f 2 , f 3 ) := s∈S Is⊂2I |I s | -1/2 ǫ s f 1 1 C k 1 , φ 1 s 1 f 2 1 C k 2 , φ 2 s 2 f 3 , φ 3 s 3 .
So now we choose the collection

P := {s ∈ S, I s ⊂ 2I}.
For all s ∈ P,

1 |I s | I 1 + d(x, I s ) |I s | -N dx ≤ 1
and so with Lemma 2.9 we have size * 3 (P) 1. For f 1 , we use that

1 |I s | C k 1 1 + d(x, I s ) |I s | -N dx 1 + d(C k 1 , I) |I| -(N -2)
to conclude that size * 1 (P) 2 -k 1 (N -2) . By the same argument for f 2 , we have 2) . In this case, we can also estimate (with N an other large enough integer)

size * 2 (P) 2 -k 2 (N -
size * 1 (P) θ 1 size * 2 (P) θ 2 size * 3 (P) θ 3 ≤ 2 -(k 1 +k 2 )N ǫ . With (2.8), we finally obtain Λ k 1 ,k 2 0 (f 1 , f 2 , f 3 ) β =α |E β | (1+θ β )/2-ǫθ β |E α | (θα-1)/2+ǫ(1-θα) C(Λ k 1 ,k 2 0 ), with C(Λ k 1 ,k 2 0 ) := 2 -N (k 1 +k 2 )ǫ . (2.16)
Third step : End of the proof.

For the trilinear form Λ j i , we have obtain a bound C = C(Λ j i ) such that for all functions f β ∈ F (E ′ β ) we have

Λ j i (f 1 , f 2 , f 3 ) C(Λ j i ) β =α |E β | (1+θ β )/2-ǫθ β |E α | (θα-1)/2+ǫ(1-θα) .
Let (p β ) β be the exponents of Theorem 2.8. Then we shall show that we can find (θ 1 , θ 2 , θ 3 ) and ǫ > 0 such that

∀β = α (1 + θ β )/2 -ǫθ β = 1 p β and (θ α -1)/2 + ǫ(1 -θ α ) = 1 p α .
Let γ > 0 be a real satisfying

∀β = α 1 2 - 1 p β < 1 2 + γ .
This is possible because 1 < p β < 2 for β = α. We begin to choose

θ α ∈]0, 1[ such that 1 > θ α > max θ 0 α := p α + (2 + γ) p α , 0 and min -1 2 + γ = 1 p α (1 -θ 0 α ) , 1 p α > 1 p α (1 -θ α ) > -1 2 .
This is possible because p α is negative and satisfies

1 p α > - 1 2 .
Then we get ǫ by

ǫ := 1 2 + 1 p α (1 -θ α ) ∈]0, 1 2 [⊂]0, 1[.
We now define θ β for β = α by

θ β := 1 p β -1 2 1 2 -ǫ .
We have 1 < p β < 2 and 0 < ǫ < 1/2, so 0 < θ β and

0 < θ β = 1 p β -1 2 -1 pα(1-θα) < 1 2+γ 1 2+γ = 1.
Consequently, we have solved the system of equations for the exponents. With this choice, we obtain

∀f 1 ∈ F (E ′ 1 ), f 2 ∈ F (E ′ 2 ), f 3 ∈ F (E ′ 3 ), Λ j i (f 1 , f 2 , f 3 ) C(Λ j i ) 3 β=1 |E i | 1/p β ,
where C(Λ j i ) are defined in (2.15) and (2.16). So Λ j i is of restricted weak type and we have the following estimate about C(Λ j i ) :

C(Λ j i ) C(Λ j i ).
In addition the parameter N in (2.15) and (2.16) is as large as we want, and we have also obtained the desired estimates on C(Λ j i ).

⊓ ⊔

The proof of Theorem 2.8 is now completed. By using the concept of "restricted weak type", we can have a "stronger" result than Theorem 1.1 :

Theorem 2.10. Let T, p, q, r be an operator and exponents of Theorem 1.1.

Then for all δ ≥ 1, there exist a constant C = C(p, q, r, δ) (independent on the interval I) such that for all sets E 3 of finite measure, there exists a substantial subset E ′ 3 ⊂ E 3 satisfying that for all functions f ∈ S(R), g ∈ S(R) and

h ∈ F (E ′ 3 ) we have | T (f, g), h1 I | ≤ C k≥0 2 -k(1/p+δ) f 1 2 k I p k≥0 2 -k(1/q+δ) g1 2 k I q |E 3 | 1/r ′ .
When r > 1, this result is stronger than Theorem 1.1 but less practicable. We now prove it because it will be useful in the sequel.

Proof : The proof is exactly the same as the previous one, so we shall only explain the modifications. We always study the trilinear form Λ(f, g, h) := T (f, g), h1 I .

In the section 2.1 we have seen that the study of Λ can be reduced to the study of the following model sum

Λ(f, g, h) = s∈S |I s | -1/2 ǫ s φ s 1 , f φ s 2 , g φ s 3 , h1 I ,
where S is a general collection of tri-tiles. Then we have decomposed this sum with (2.4) by :

Λ(f, g, h) = k 1 ,k 2 ≥0 Λ k 1 ,k 2 0 (f, g, h) + k 1 ,k 2 ≥0 l≤0 Λ k 1 ,k 2 ,l 1 (f, g, h).
By Theorem 2.4, we have shown that the trilinear forms Λ j i are of restricted weak type (p, q, r ′ ) and we have obtained estimates on their bounds. The construction of the substantial subset E ′ α = E ′ 3 does not depend on the trilinear form Λ j i , so we can deduce that our trilinear form Λ is always of restricted weak type. Also for measurable sets E 1 , E 2 , E 3 of finite measure, there exists a substantial subset E ′

3 ⊂ E 3 such that for all functions f ∈ F (E 1 ), g ∈ F (E 2 ) and h ∈ F (E ′

3 ) we have :

|Λ(f, g, h)| |E 3 | 1/r ′ k 1 ,k 2 ≥0 2 -δ ′ (k 1 +k 2 ) |E 1 ∩ C k 1 | 1/p |E 2 ∩ C k 2 | 1/q .
Here δ ′ is an exponent as large as we want. Over each corona, by using the real interpolation on the exponents p and q (so r is fixed), we obtain also the desired result.

⊓ ⊔

Having obtained our main result for the x-independent symbols, we will extend our result for maximal operators and for x-dependent symbols in the next section.

3 More general bilinear operators.

Let us name our "off-diagonal" estimates for convenience.

Definition 3.1. Let T be an operator (maybe non bilinear) acting from S(R) × S(R) into S ′ (R). For 0 < p, q, r ≤ ∞ exponents such that

1 r = 1 p + 1 q ,
we say that T satisfies "off-diagonal" estimates at the scale L and at the order δ, in short T ∈ O L,δ (L p ×L q , L r ), if there exists a constant C = C(p, q, r, L, δ) such that for all functions f, g ∈ S(R) and all interval I of length |I| = L, we have

T (f, g) r,I ≤ C k≥0 2 -k(δ+1/p) f p,2 k+1 I k≥0 2 -k(δ+1/q) g q,2 k+1 I . (3.1)
Remark 3.2. Equivalently, an operator T satisfies "off-diagonal" at the scale L and at the order δ if there exists a constant C = C(p, q, r, L, δ) with for all functions f, g ∈ S(R) and all interval I of length |I| = L, we have

T (f, g) r,I ≤ C k≥0 2 -k(δ+1/p) f p,C k (I) k≥0 2 -k(δ+1/q) g q,C k (I) .
This is a better way to describe the "off-diagonal" decay of an operator T and these properties can been describe as in Corollary 1.2.

First we generalize the previous result for maximal operators.

3.1 "Off-diagonal" estimates for maximal bilinear operators.

We have the three following theorems :

Theorem 3.3.
Let ∆ be a nondegenerate line in the frequency plane. Let 1 < p, q ≤ ∞ be exponents such that

0 < 1 r = 1 q + 1 p < 3 2 .
For all δ ≥ 1, L > 0, for all symbol σ supported in

{(α, β), d((α, β), ∆) ≥ L -1 } satisfying ∀b, c ≥ 0 ∂ b α ∂ c β σ(α, β) |d((α, β), ∆)| -b-c
and for all smooth function φ, which is equal to 1 around 0, the maximal bilinear operator

T max (f, g)(x) := sup r>0 e ix(α+β) f (α) g(β)σ(α, β) 1 -φ(r(α -β)) dαdβ (3.
2) satisfies "off-diagonal" estimates at the scale L and at the order δ : T max ∈ O L,δ (L p ×L q , L r ). In addition the implicit constant can be uniformly bounded in L > 0.

Theorem 3.4. For the same exponents, we have the same continuities for the maximal bilinear operator (at the scale L) :

M L (f, g)(x) := sup 0<r≤L 1 r |t|≤r |f (x -t)g(x + t)| dt.
Theorem 3.5. Let K be a kernel on R satisfying the Hörmander's conditions, then the maximal bilinear operator

T L max (f, g)(x) := sup 0<ǫ<r<L ǫ≤|y|≤r f (x -y)g(x + y)K(y)dy (3.3)
verifies the same local estimates : T L max ∈ O L,δ (L p ×L q , L r ) (for the exponents p, q, r of Theorem 3.3).

Proof : The proof of these three Theorem is a shake between the proof of our Theorem 1.1 and an additional maximality argument. The maximal truncation in the physical space (Theorems 3.4 and 3.5) is a little more complex than the maximal truncation in the frequency space (Theorem 3.3). So we deal with the two last theorems and just explain the modifications to prove them. The maximal version of the different arguments have first shown in [START_REF] Lacey | The bilinear maximal functions map into L p for 2/3 < p ≤ 1[END_REF] by M. Lacey and then improved in [START_REF] Demeter | Maximal multilinear operators[END_REF] by C. Demeter, T. Tao and C. Thiele. In these articles, the authors study the behaviour of the maximal averages (like Theorem 3.4). The remark 1.6 of [START_REF] Demeter | Maximal multilinear operators[END_REF] specifies the similarity between the operators of Theorem 3.4 and 3.5. So in fact our three previous theorems are an illustration of the same ideas, we will not detail. The subsection 2.1 is based on the decomposition of the bilinear operator by discrete models. For our maximal operators, the same reduction is shown in [START_REF] Demeter | Maximal multilinear operators[END_REF] (Theorem 4.4) and the important condition (2.3) for the tiles are always satisfied. Then the maximal version of Proposition 2.6 is written in [START_REF] Demeter | Maximal multilinear operators[END_REF] too (there is a new factor in the different inequalities but it is not important). We have exactly the same version of Lemma 2.9 for maximal bilinear operators (see Proposition 6.2 of [START_REF] Demeter | Maximal multilinear operators[END_REF]). Using these few technical modifications, we can compute our proof of Theorem 1.1 and obtain its maximal versions : Theorems 3.3,3.4 and 3.5. ⊓ ⊔

3.2 Proof of Theorem 1.1 for x-dependent symbols.

In this subsection, we prove the "off-diagonal" estimates of Theorem 1.1 in the case where the symbol σ depends on the spatial variable x and also we complete the proof of our main result.

Theorem 3.6. Let ∆ be an nondegenerate line of the frequencial space. Let σ ∈ C ∞ (R 3 ) be a symbol satisfying : ∀a, b, c ≥ 0

∂ a x ∂ b α ∂ c β σ(x, α, β) (1 + d((α, β), ∆)) -b-c .
Then the bilinear operator T σ (defined on S(R) × S(R) by (1.2)) verifies T ∈ O 1,δ (L p × L q , L r ) for any δ ≥ 0 and any exponents p, q, r such that

0 < 1 r = 1 p + 1 q < 3 2
and 1 < p, q ≤ ∞.

Our assumptions for the symbol corresponds to the class BS 1,0 θ of [START_REF] Benyi | Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators[END_REF] where the angle θ ∈]π/2, π/2[\{0, -π/4} is given by the line ∆ :

∆ := {(α, β), β = tan(θ)α} .
For convenience, we will deal in the proof only with the case θ = π/4. The important fact is that the singular quantity βtan(θ)α does not correspond to the quantity α + β, which appears in the exponential term of (1.2). The limit and particular case θ = -π/4 is studied in [START_REF] Benyi | Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators[END_REF].

Proof :

The proof is quite technical. We will also assume that r ≥ 1 (which allows us to simplify a few arguments). Then we will explain in Remark 3. 1-) The case of the symbol σ ∞ .

We have an operator associated to this symbol :

T ∞ (f, g)(x) := R 2 e ix(α+β) f (α) g(β)σ(x, α, β)[1 -Φ(α -β)]dαdβ,
which can been written

T ∞ (f, g)(x) = U x (f, g)(x),
with U defined by

U y (f, g)(x) := R 2 e ix(α+β) f(α) g(β)σ(y, α, β)[1 -Φ(α -β)]dαdβ.
By using the Sobolev's imbedding W 1,r (I) ֒→ L ∞ (I) (because r ≥ 1), we get

∀x ∈ I, |T ∞ (f, g)(x)| ≤ U y (f, g)(x) ∞,y∈I 1 k=0 ∂ k y U y (f, g)(x)1 I (y) r,dy .
Then by integrating for x ∈ I and using Fubini's Theorem, we obtain

T ∞ (f, g) r,I 1 k=0
∂ k y U y (f, g) r,I r,I,dy .

We can fix k ∈ {0, 1} and y ∈ I. Then we have

∂ k y U y (f, g) r,dx V (f, g) r,I ,
where V is the bilinear operator defined by

V (f, g)(x) := R 2 e ix(α+β) f (α) g(β)∂ k y σ(y, α, β)[1 -Φ(α -β)]dαdβ.
So V = T τ is the bilinear operator associated to the x-independent symbol

τ (α, β) := ∂ k y σ(y, α, β)[1 -Φ(α -β)].
From the assumptions about σ, the symbol τ satisfies : for all b, c ≥ 0

∂ b α ∂ c β τ (α, β) |α -β| -n-p .
In addition τ is supported in the domain : {|α -β| ≥ 1}. We can also apply Theorem 1.1 proved in section 2 for x-independent symbol. For all δ ≥ 1, we have an "off-diagonal" estimate at the scale 1 :

V (f, g) r,I k 1 ≥0 2 -k 1 (1/p+δ) f p,2 k 1 I k 2 ≥0
2 -k 2 (1/q+δ) g q,2 k 2 I .

All theses estimates are uniform with respect to k ∈ {0, 1} and y ∈ I, so we get

T ∞ (f, g) r,I k 1 ≥0 2 -k 1 (1/p+δ) f p,2 k 1 I k 2 ≥0
2 -k 2 (1/q+δ) g q,2 k 2 I .

(3.4)

So we have shown the desired estimates for this first term. 2-) The case of the symbol σ 0 . The associated operator is given by

T 0 (f, g)(x) := R 2
e ix(α+β) f (α) g(β)σ(x, α, β)Φ(α, β)dαdβ.

We use the same arguments as for the first point. So we have to study the operator V defined by

V (f, g)(x) := R 2 e ix(α+β) f(α) g(β)∂ k y σ(y, α, β)Φ(α -β)dαdβ.
The parameters k ∈ {0, 1} and y ∈ I are fixed. The symbol associated to this operator is supported on {(α, β), |α -β| ≤ 2}. That is why, we use modulations to move this support.

V (f, g)(x) = R 2 e ix(α+β) f (α + 3) g(β -3)∂ k y σ(y, α + 3, β -3)Φ(α -β + 6)dαdβ = R 2 e ix(α+β) e 3i. f (α) e -3i. g(β)∂ k y σ(y, α + 3, β -3)Φ(α -β + 6)dαdβ.
Also V is now the bilinear operator, applied to the functions e 3i. f and e -3i. g, whose (x-independent) symbol

τ (α, β) := ∂ k y σ(y, α + 3, β -3)Φ(α -β + 6). is supported on {(α, β), |α -β + 6| ≤ 2} ⊂ {(α, β), 1 ≤ |α -β| ≤ 8} and satisfies ∀b, c ≥ 0, ∂ b α ∂ c β τ (α, β) max 0≤j≤b max 0≤i≤c (1 + |α -β + 6|) -i-j 1 1≤|α-β|≤8 1 1≤|α-β|≤8 1 1≤|α-β|≤8 |α -β| -b-c .
Also we can again use Theorem 1.1 (proved in Section 2 for x-independent symbol) and we obtain :

V (f, g) r,I k 1 ≥0 2 -k 1 (1/p+δ) f p,I k 2 ≥0
2 -k 2 (1/q+δ) g q,I .

We have also finished the proof. ⊓ ⊔ Remark 3.7. In this remark, we want to explain how modify the previous proof when r < 1. When we want to study bilinear operators with r < 1, we have to use the associated trilinear form and the concept of "restricted weak type" (defined in Definition 2.7). These two arguments allow us to get round the lack of the triangular inequality in the space L r . Let Λ(f, g, h) := T (f, g), h .

We have

Λ(f, g, h) = R R 2 e ix(α+β) σ(x, α, β) f(α) g(β)h(x)dαdβdx.
We use the same decomposition of σ, getting the trilinear forms Λ ∞ and Λ 0 . Let us study first Λ ∞ and fix an interval I of length |I| = 1. We take a function h ∈ S(R), which is supported on I. We use again the Sobolev imbedding W 1,1 (I) ֒→ L ∞ (I). By writing

|Λ ∞ (f, g, h)| ≤ R U y (f, g)(x)1 I (y) ∞,I,dy |h(x)| 1 I (x)dx.
We can also obtain

|Λ ∞ (f, g, h)| I I |U y (f, g)(x)| |h(x)| dxdy+ I I |∂ y U y (f, g)(x)| |h(x)| dxdy.
Then when y ∈ I and k ∈ {0, 1} are fixed, we find again the quantities

I ∂ k y U y (f, g)(x) |h(x)| dx.
Now the bilinear operator ∂ k y U y is associated to an x-independent symbol, which verifies the good assumptions. We can also use Theorem 2.10 in order to obtain the wished estimates (3.4) in a "restricted weak type sense" for the exponent r. We produce the same modifications to study Λ 0 . By noticing that the way to construct the substantial subset (in the definition of restricted weak type) does not depend on the trilinear form, we can deduce that the trilinear form Λ satisfies (3.4) in a "restricted weak type sense" too. Then we use interpolation on the exponent r, to obtain exactly (3.4), which allows us to conclude.

4 Continuities for bilinear operators, satisfying "off-diagonal" estimates.

Recall that in the linear case, by using the maximal sharp function, we can prove weighted continuities for linear operator with the Muckenhoupt's weights. In the bilinear case, we do not have a good substitute to the maximal sharp function. That is why we shall use the previous "off-diagonal" estimates to obtain weighted global continuities on Lebesgue spaces and in particular prove Theorem 1.3.

We first want to give an application of these off-diagonal estimates. Recall that in the previous sections, we have proved that our bilinear operators (and maximal bilinear operators) satisfy these "off-diagonal" estimates at any order. The time-frequency analysis does not work for functions in the L ∞ space. So we don't know if our operators T are bounded from L ∞ ×L ∞ in BMO. However these local estimates give a weak result about the behavior of T (f, g) when the two functions f and g belong to L ∞ .

Proposition 4.1. Let f, g be two functions of L 1 (R) ∩ L ∞ (R) and fix 1 < r < ∞. If there exist L > 0, δ ≥ 1 and p, q > 1 such that an operator T ∈ O δ,L (L p × L q , L r ), then we have

lim sup |I|→∞ 1 |I| I |T (f, g)| r 1/r f ∞ g ∞ .
Here we take the limit when I is an interval with |I| → ∞ and the implicit constant does not depend on the two functions f and g and on the parameter L.

Proof : We set I i := [iL, (i + 1)L[ for all i ∈ Z. Then for I with |I| >> L, we get

I |T (f, g)| r ≤ i∈Z I i ∩I =∅ I i |T (f, g)| r .
However the number of index i which appears in the sum is bounded by 29 |I|/L, so by using the local estimate we get

I |T (f, g)| r i∈Z I i ∩I =∅ L |I i | I i |T (f, g)| r i∈Z I i ∩I =∅ L f r ∞ g r ∞ |I| f r ∞ g r ∞ .
The second inequality is due to the fact that :

|I i | 1/r T (f, g) r,I i inf x∈I i M HL (f )(x) inf x∈I i M HL (g)(x) f ∞ g ∞ .
So we obtain 1

|I| I |T max (f, g)| r 1/r f ∞ g ∞ ,
uniformly with L for |I| large enough.

⊓ ⊔

Let us now define our weights. So we claim that a function ω ∈ P θ (l) seems to be likely a polynomial function whose the degree is less than θ and it is almost constant at the scale l. We show in the following example that these classes are not empty. We cannot compare these weights with the Muckenhoupt's weights, because for ω ∈ P θ (l) we have informations only at the scale l.

The second term is studied by the same way. By summing over i ∈ Z, we get T (f, g) r,wdx k≥0 2 -k(1/p+δ-θ/p) f p,wdx,2 k I i k≥0 2 -k(1/q+δ-θ/q) g q,wdx,2 k I i r,i∈Z

.

With the help of Hölder's and Minkowski's inequalities, we get T (f, g) r,wdx k≥0 2 -k(1/p+δ-θ/p) f p,wdx,2 k I i p,i∈Z k≥0 2 -k(1/q+δ-θ/q) g q,wdx,2 k I i q,i∈Z .

However the collection of sets (2 k I i ) i is a 2 k -covering, so T (f, g) r,wdx k≥0 2 -k(δ-θ/p) f p,wdx k≥0 2 -k(δ-θ/q) g q,wdx .

Then we conclude with the fact that p, q > 1 hence max θ p , θ q ≤ θ r < δ if r ≥ 1 θ < δ if r ≤ 1 .

⊓ ⊔ Remark 4.5. From the fact that the weight ω(x) = 1 belongs to the class P θ (L), we have also proved that the operators of Theorem 1.1 and the maximal operators of Theorems 3.3,3.4, 3.5 are bounded in classical Lebesgue spaces.

We complete this result by the following proposition in Sobolev spaces. Let p, q, r be exponents satisfying

0 < 1 r = 1 p + 1 q < 3 2
and 1 < p, q ≤ ∞.

Then the bilinear operator T σ (defined on S(R) × S(R) by (1.2)) satisfies : for all integer n ≥ 0 ∀f, g ∈ S(R), D (n) T σ (f, g) L r (ω)

0≤i,j≤n i+j≤n D (i) f L p (ω) D (j) g L q (ω) .

(4.3) Here we write D (i) for the derivative operator of order i. Also T σ is continuous from W m,p (ω) × W m,q (ω) in W m,r (ω) for all real m ≥ 0.

Proof : Let us begin to prove (4.3). The two functions f and g are smooth so we can differentiate the integral defining T σ (f, g). It is also easy to check that D (1) T σ (f, g) = T σ (D (1) f, g) + T σ (f, D (1) g) + T ∂xσ (f, g).

Then for higher orders, we get D (n) T σ (f, g) = 0≤i,j,k≤n i+j+k=n T ∂ k x σ (D (i) f, D (j) g).

By using the previous Theorems 1.1 and 4.4, we obtain (4.3). We can also deduce a weaker estimate :

∀f, g ∈ S(R), D (n) T σ (f, g) r,ω f W n,p (ω) g W n,q (ω) .

By density (see Lemma 4.8), the operator T σ can be continuously extended from W n,p (ω) × W n,q (ω) into W n,r (ω). Then we will use interpolation to extend this result when n is not an integer. The exponents p, q, r are fixed and we study the bilinear operator T σ . We have shown that T σ is continuous from W n,p (ω) × W n,q (ω) into W n,r (ω), for all integer n. By using the bilinear interpolation (with Lemme 4.8)1 on n, we finish the proof. ⊓ ⊔

R 2 K 2 e

 22 (x, y, z)f (y)g(z)dydz = R ix(α+β) σ(x, α, β) f(α) g(β)dαdβ. (1.2)

  7 how modify the proof to obtain the same result when r < 1. So we fix an interval I of length |I| = 1. We use a decomposition of the symbol σ. Let Φ be a smooth function on R such that |x| ≤ 1 =⇒ φ(x) = 1 and supp(φ) ⊂ [-2, 2]. We have also σ(x, α, β) = σ(x, α, β)[1 -Φ(αβ)] + σ(x, α, β)Φ(α, β) := σ ∞ (x, α, β) + σ 0 (x, α, β).

Definition 4 . 2 .

 42 Let θ > 0 and l > 0 be fixed. We set that a nonnegative function ω belongs to the class P θ (l) if there exists a constant C such that for all interval I of length |I| = l and for all integer k ≥ 0, we have 2 -kθ sup x∈I ω(x) ≤ C inf

Example 1 .

 1 For all θ > 0 and α ∈ [0, θ[, the functionsx → 1, x → (1 + |x|) α and x → (1 + |x|) -αbelong to the class P θ (1). The proof is easy and left to the reader. Remark 4.3. In fact, it is easy to prove that a weight ω belongs to the class P θ (l) if and only if there exists a constant C such that ∀x, y ∈ R, ω(x) ≤ C 1 + |x -y| l θ ω(y).

Definition 4 . 6 . 2 . 4 . 7 .

 46247 Let ω be a weight on R. For all m ≥ 0 and p ∈]1, ∞[, we set W m,p (ω) for the Sobolev space on R with the weight ω, defined as the set of distributions f ∈ S ′ (R) such that J m (f ) ∈ L p (ω), where J m := (Id -∆) m/Proposition Let ∆ be a nondegenerate line, ω be a weight in ∪ θ≥0 P θ (1) and σ ∈ C ∞ (R 3 ) be a symbol satisfying∀a, b, c ≥ 0, ∂ a x ∂ b α ∂ c β σ(x, α, β) (1 + d((α, β), ∆)) -b-c .

The theory of the multilinear interpolation is studied in the chapter 4 of[START_REF] Lions | Sur une classe d'espaces d'interpolation[END_REF] for the real interpolation and in Theorem 4.4.1 of[START_REF] Bergh | Interpolation Spaces : An Introduction[END_REF] for the complex interpolation.

We have the following result : Theorem 4.4. Let T be a bilinear operator and 0 < p, q, r < ∞ be exponents satisfying 1 r = 1 p + 1 q and 1 ≤ p, q.

For δ > 0 and l > 0, if T satisfies "off-diagonal" estimates at the order δ and at the scale l, then for all ω ∈ P θ (l) with 0 ≤ θ < δ max{r, 1}, the operator

Proof : To check this, recall that for all interval I of length |I| = l,

2) So we decompose the whole space R with the disjoint intervals I i defined by

.

Let i ∈ Z be fixed, we use (4.1) and (4.2) to obtain

We estimate the first sum with

Lemma 4.8. For all weight ω ∈ ∪ θ≥0 P θ (1), all exponent 1 < p < ∞ and all real s ≥ 0, the space S(R) is a dense subspace in W s,p (ω). In addition, the collection of Sobolev spaces (W s,p (ω)) s≥0 form an interpolation scale.

Proof : Let ω be a fixed weight in ∪ θ≥0 P θ (1). We have seen in Remark 4.3 that ω has a polynomial growth. Due to the fact that J s (S(R)) = S(R), we have the inclusion S(R) ⊂ W s,p (ω). Recall that J s := (Id -∆) s/2 . In addition, we have that L p (ω) ⊂ S ′ (R), so we can compute the operator J -s on the space L p (ω). We finally obtain that J s is an automorphism from W s,p (ω) to L p (ω) and an isomorphism on S(R). As S(R) is dense in L p (ω), we get the density of S(R) into the Sobolev space W s,p (ω).

For the interpolation claim, we omit the details. The classical proof for complex interpolation with ω = 1 can easily been extended to the general case. ⊓ ⊔ Remark 4.9. From the fact that the weight ω(x) = 1 belongs to the class P θ (1), we have also proved that the operators of Theorem 1.3 satisfy an Hölder's inequality in Sobolev spaces.

Remark 4.10. Also with the notation of [START_REF] Benyi | Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators[END_REF], we have proved continuities for all the operators associated to symbols σ ∈ BS 0 1,0;θ . In addition, we have described the action of these operators on Sobolev spaces. This is an interesting improvement of this article and it incite us to obtain new results in order to continue the construction of a bilinear pseudodifferential calculus. We will do it in a next paper [START_REF] Bernicot | Bilinear Pseudodifferential Calculus[END_REF] by introducing new larger symbolic classes of bilinear symbols of order (m 1 , m 2 ). About continuities in Lebesgue spaces, a question is still open : what about the class BS 0 ρ,δ;θ (defined in [START_REF] Benyi | Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators[END_REF]) ?