F R Éd

Éric Bernicot

Diego Maldonado
email: dmaldona@math.ksu.edu

AND Kabe Moen
email: kmoen@as.ua.edu

Virginia Naibo
email: vnaibo@math.ksu.edu

Frédéric Bernicot
email: frederic.bernicot@univ-nantes.fr

BILINEAR SOBOLEV-POINCAR É INEQUALITIES AND LEIBNIZ-TYPE RULES

Keywords: October 7, 2012. 2010 Mathematics Subject Classification. Primary 26D10, 31B10, 35S05 47G30; Secondary 42B15, 42B20, 46E35 Bilinear operators, Poincaré inequalities, pseudodifferential operators, fractional Leibniz rules

The dual purpose of this article is to establish bilinear Poincaré-type estimates associated to an approximation of the identity and to explore the connections between bilinear pseudo-differential operators and bilinear potential-type operators. The common underlying theme in both topics is their applications to Leibniz-type rules in Sobolev and Campanato-Morrey spaces under Sobolev scaling.

Introduction

Leibniz-type rules quantify the regularity of a product of functions in terms of the regularity of its factors. In this sense, Leibniz-type rules are represented by inequalities of the form

(1.1) f g Z f X 1 g Y 1 + f X 2 g Y 2 ,
where X 1 , X 2 , Y 1 , Y 2 , and Z are appropriate functional spaces. Along these lines, perhaps the better-known Leibniz-type rules correspond to the fractional Leibniz rules, pioneered by Kato-Ponce [START_REF] Kato | Commutator estimates and the Euler and Navier-Stokes equations[END_REF], Christ-Weinstein [START_REF] Christ | Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation[END_REF] and Kenig-Ponce-Vega [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF] in their work on PDEs, where the spaces X 1 , X 2 , Y 1 , Y 2 , and Z belong to the scale of Sobolev spaces W m,p ; namely,

(1.2) f g W m,q f W m,p 1 g L p 2 + f L p 1 g W m,p 2 ,
where m ≥ 0 and

(1.3) 1 q = 1 p 1 + 1 p 2 with 1 < p 1 , p 2 < ∞, 1 ≤ q.
The estimates (1.2) follow as a consequence of interpolation and the boundedness properties on products of Lebesgue spaces of bilinear Coifman-Meyer multipliers ( [START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF][START_REF] Grafakos | Multilinear Calderón-Zygmund theory[END_REF]): If σ satisfies

(1.4) |∂ α ξ ∂ β η σ(ξ, η)| ≤ C α,β (|ξ| + |η|) -(|α|+|β|) , ξ, η ∈ R n , α, β ∈ N n 0 , |α| + |β| ≤ C n ,
where C n is a certain constant depending only on n, and

T σ (f, g)(x) := R 2n
σ(ξ, η) f (ξ)ĝ(η)e ix•(ξ+η) dξ dη, x ∈ R n , f, g ∈ S(R n ), then T σ is bounded from L p 1 × L p 2 into L q , where p 1 , p 2 , and q conform to the Hölder scaling (1.3). Then, inequalities (1.2) are obtained from this result after observing that, by frequency decoupling, the identity

(1.5) J m (f g)(x) = T σ 1 (J m f, g)(x) + T σ 2 (f, J m g)(x),
holds true for some bilinear symbols σ 1 and σ 2 and

J m (h)(ξ) := (1 + |ξ| 2 ) m/2 ĥ(ξ), ξ ∈ R n , m > 0, h ∈ S(R n ).
For m sufficiently large, depending only on dimension, the symbols σ 1 and σ 2 satisfy (1.4) and interpolation with the case m = 0 (notice that here q ≥ 1) gives (1.2) for any m > 0. Two immediate conclusions can be derived from this approach. First, since the symbol σ 0 ≡ 1 satisfies (1.4) and yields, through T σ 0 , the product of two functions, the Hölder scaling (1.3) occurs naturally. Second, the identity (1.5) can be exploited to produce Leibniz-type rules (1.1) involving function spaces that interact well with J m (for example, Besov and Triebel-Lizorkin spaces) provided that mapping properties for bilinear multipliers T σ are established for such spaces. Indeed, implementations of this program (see, for instance, [START_REF] Bernicot | Sobolev space estimates for a class of bilinear pseudodifferential operators lacking symbolic calculus[END_REF][START_REF] Gulisashvili | Exact smoothing properties of Schrödinger semigroups[END_REF][START_REF] Maldonado | On the boundedness of bilinear operators on products of Besov and Lebesgue spaces[END_REF]), produce Besov, Triebel-Lizorkin, and mixed Besov-Lebesgue Leibniz-type rules.

A Littlewood-Paley-free path towards Leibniz-type rules was introduced in [START_REF] Maldonado | Weighted multilinear Poincaré inequalities for vector fields of Hörmander type[END_REF] in the scales of Campanato-Morrey spaces. In this context, the role of the identity (1.5) is played by the inequality (1.6) |f (x)g(x) -

f B g B | I 1 (|∇f |χ B , |g|χ B ) + I 1 (|f |χ B , |∇g|χ B ), x ∈ B,
where B ⊂ R n is a ball, f, g ∈ C 1 (B), I 1 is a bilinear potential operator, and f B := 1 |B| B f (x) dx. Inequality (1.6) arises as a bilinear interpretation of the linear inequality

(1.7) |f (x) -f B | I 1 (|∇f |χ B ), x ∈ B, f ∈ C 1 (B),
where I 1 denotes the Riesz potential of order 1. Inequality (1.7) is usually referred to as a representation formula (for the oscillation |f (x) -f B |). In the linear setting, representation formulas and Poincaré inequalities imply embeddings of Campanato-Morrey spaces (see, for instance, [START_REF] Lu | Embedding theorems on Campanato-Morrey spaces for vector fields of Hörmander type[END_REF] for such embeddings in the Carnot-Carathéodory framework). As proved in [START_REF] Maldonado | Weighted multilinear Poincaré inequalities for vector fields of Hörmander type[END_REF], via (1.6), the bilinear analogs to these embeddings come in the form of Campanato-Morrey Leibniz-type rules. More precisely, in the scale of Campanato-Morrey spaces (L p,λ (w) and L q,λ (w) below), a typical weighted Leibniz-type rule takes the form (see [START_REF] Maldonado | Weighted multilinear Poincaré inequalities for vector fields of Hörmander type[END_REF])

(1.8) f g L q,λ (w) ∇f L p 1 ,λ 1 (u) g L p 2 ,λ 2 (v) + f L p 1 ,λ 1 (u) ∇g L p 2 ,λ 2 (v) ,
for (a large class of) weights u, v, w and indices q, λ, p 1 , λ 1 , p 2 , and λ 2 . In the unweighted case, the natural scaling for (1.8) turns out to be the bilinear Sobolev scaling (1.9)

1 q = 1 p 1 + 1 p 2 - 1 n with 1 < p 1 , p 2 < ∞.
From (1.6), it now becomes apparent that the prevailing tools for obtaining inequalities (1.8) rely on boundedness properties of suitable bilinear potential-type operators. Thus, in the scale of Campanato-Morrey spaces, bilinear potential-type operators play the role that paraproducts and the bilinear Coifman-Meyer multipliers play in the proofs of the Sobolev-based Leibniz-type rules (1.2) and their Besov and Triebel-Lizorkin counterparts. Accordingly, the time-frequency Fourier-based tools in the latter are replaced by real-analysis methods in the former. The purpose of this article is to further develop time-frequency and real-analysis approaches that allow to prove new Leibniz-type rules in Sobolev and Campanato-Morrey spaces. In the rest of this introduction we feature some of the main results as we explain the organization of the manuscript.

In Section 2 we recall some definitions and known results on boundedness properties of bilinear fractional integrals in weighted and unweighted Lebesgue spaces that will be useful for our proofs.

In Section 3 we explore the behavior of the bilinear oscillation |f (x)g(x) -f B g B | when the mean-value operator is replaced by an approximation of the identity {S t }. Our exposition includes the case of the infinitesimal generator L of an analytic semigroup {S t } t>0 on L 2 (R n ) (i.e. S t = e -tL ) whose kernel p t (x, y) has fast-enough offdiagonal decay. The quantity S t f = e -tL f can be thought of as an average version of f at the scale t and plays the role of f B for some t = t B , when defining function spaces, such as BMO L and H 1 L , which better capture properties of the solutions to Lu = 0; see for instance [START_REF] Duong | Duality of Hardy and BMO spaces associated with operators with heat kernel bounds[END_REF]. In the linear case, the new study of Sobolev-Poincaré inequalities associated to the oscillation |f -S t B f | has been successfully carried out in [START_REF] Badr | L p self-improvement of generalized Poincaré inequalities in spaces of homogeneous type[END_REF][START_REF] Jiménez-Del-Toro | Self-improvement of Poincaré-type inequalities associated with approximations of the identity and semigroups[END_REF] (see also [START_REF] Badr | New Calderón-Zygmund decompositions[END_REF]), yielding Sobolev-Poincaré type inequalities such as

(1.10) 1 |B| B |f -S t B f | q 1/q k∈N 0 α k r(2 k B) 1 |2 k B| 2 k B |∇f | p 1/p
, for suitable choices of indices 1 < p < q and sequences {α k } ⊂ [0, ∞). As described in [START_REF] Badr | L p self-improvement of generalized Poincaré inequalities in spaces of homogeneous type[END_REF][START_REF] Jiménez-Del-Toro | Self-improvement of Poincaré-type inequalities associated with approximations of the identity and semigroups[END_REF], the presence of the series expansion on the right-hand side of (1.10) accounts for the lack of localization of the approximation of the identity {S t }. In this vein, we study bilinear oscillations of the type |f g -S t B f S t B g| and establish bilinear Poincarétype inequalities in the Euclidean setting associated to a general approximation of the identity {S t }. We prove:

Theorem 1. Let S := {S t } t>0 and S ′ := {t∂ t S t } t>0 be approximations of the identity in R n of order m > 0 and constant ε in (3.28), 1 < p 1 , p 2 < ∞, q > 0, and 0 < α < min{1, ε} such that

1 q = 1 p 1 + 1 p 2 -1-α n . B |f g -S r(B) m (f )S r(B) m (g)| q 1/q r(B) α l≥0 2 -l(ε-α) 2 l+1 B |∇f | p 1 1/p 1 2 l+1 B |g| 1/p 2 + 2 l+1 B |f | p 1 1/p 1 2 l+1 B |∇g| 1/p 2 .
If fact, our full result is a more general weighted version of Theorem 1 (see Theorem 4). The proof consists in establishing a bilinear representation formula tailored to the semigroup {S t }, which, as expected, turns out to be an expanded version of (1.6) (see Theorem 5). This bilinear representation formula involves logarithmic perturbations of the bilinear fractional integral used in [START_REF] Maldonado | Weighted multilinear Poincaré inequalities for vector fields of Hörmander type[END_REF], whose kernels are proved to still satisfy appropriate growth conditions that guarantee boundedness of the operator on products of weighted Lebesgue spaces.

In Section 4 we define bilinear Campanato-Morrey spaces associated to {S t } and use the results of section 3 to produce associated (weighted) Leibniz-type rules.

In Section 5 we point out relevant extensions to the contexts of doubling Riemannian manifolds and Carnot groups.

In Section 6 we close the circle of ideas developed in Sections 1-3 by relating bilinear pseudo-differential operators and bilinear potential operators. More specifically, we study bilinear pseudo-differential operators of the form

T σ (f, g)(x) = R 2n e ix(ξ+η) σ(x, ξ, η) f (ξ) g(η)dξdη, f, g ∈ S(R n ), x ∈ R n . (1.11)
We relate such operators to potential operators via the inequalities

(1.12) |T σ (f, g)| B s (|f |, |g|) and |T σ (f, g)| I s (|f |, |g|), f, g ∈ S(R n ),
where B s is the bilinear fractional integral of order s, introduced and studied in [START_REF] Grafakos | On multilinear fractional integrals[END_REF] and [START_REF] Kenig | Multilinear estimates and fractional integration[END_REF], I s is the bilinear Riesz potential of order s introduced in [START_REF] Kenig | Multilinear estimates and fractional integration[END_REF], and σ belongs to standard classes of bilinear symbols of order -s. As a consequence of these bonds between bilinear pseudo-differential and potential operators, we obtain the following (see Sections 2 and 6 for pertinent definitions):

Theorem 2. Suppose n ∈ N and consider exponents p 1 , p 2 ∈ (1, ∞) and q, s > 0 that satisfy

(1.13) 1 q = 1 p 1 + 1 p 2 - s n . (a) If s ∈ (0, 2n), 0 ≤ δ < 1, and σ ∈ BS -s 1,δ (R n ) ∪ ḂS -s 1,δ (R n ) then T σ is bounded from L p 1 × L p 2 into L q . (b) If s ∈ (0, n), θ ∈ (0, π)\{π/2, 3π/4}, 0 ≤ δ < 1 and σ ∈ BS -s 1,δ;θ (R n )∪ ḂS -s 1,δ;θ (R n ) then the bilinear operator T σ is bounded from L p 1 × L p 2 into L q .
We actually prove a more general weighted version of Theorem 2, for which we refer the reader to Theorem 12 for a more precise statement. In section 7 we present some consequences of Theorem 2, such as Sobolev-based fractional Leibniz rules of the form

(1.14) f g W m,q f W s+m,p 1 g L p 2 + f L p 1 g W s+m,p 2 ,
for a range of indices m and s, with the novelty 0 < q < 1, under the bilinear Sobolev scaling of equation (1.13). The relation (1.13) sheds additional light onto the balance between integrability and smoothness built into inequalities of the type (1.2). Notice that inequality (1.14) cannot be obtained by applying Sobolev embedding and then the fractional Leibniz rule (1.2), because equation (1.13) allows for 0 < q < 1 in which case the Sobolev embedding W m+s,r ⊆ W m,q fails for any choice of r with 1/r = 1/p 1 + 1/p 2 and q as in (1.13).

The bilinear Poincaré estimates introduced in [START_REF] Maldonado | Weighted multilinear Poincaré inequalities for vector fields of Hörmander type[END_REF] rely on the oscillation of the pointwise product of two functions (i.e. T σ with σ ≡ 1); in turn, they give rise to bilinear Sobolev inequalities of the form

(1.15) f g L q ∇f L p 1 g L p 2 + f L p 1 ∇g L p 2 ,
for exponents p 1 , p 2 > 1 and q > 0 satisfying the Sobolev relation (1.9). These results correspond to the limit of bilinear Poincaré inequalities on balls, by making the radius of the ball tend to infinity. We direct the reader to [START_REF] Maldonado | Weighted multilinear Poincaré inequalities for vector fields of Hörmander type[END_REF][START_REF] Moen | Weighted inequalities for multilinear fractional integral operators[END_REF][START_REF] Moen | Higher-order multilinear Poincaré inequalities in Carnot groups[END_REF] for other versions of (1.15), including weights and higher order derivatives in the context of Hörmander vector fields. The results presented in Section 7 further substantiates inequalities of the type (1.15) under Sobolev scaling and unifies their study in the language of bilinear pseudo-differential operators.

Throughout the paper, we use upper-case letters to label theorems corresponding to known results while we use single numbers (with no reference to the section) for theorems, propositions and corollaries that are new and proved in this article.

Bilinear fractional integrals and their boundedness properties on weighted Lebesgue spaces

Given a weight w defined on R n and p > 0, the notation L p w will be used to refer to the weighted Lebesgue space of all functions f : R n → C such that f p L p w := R n |f (x)| p w(x) dx < ∞, when w ≡ 1 we will simply write L p .

If w 1 , w 2 are weights defined on R n , 1 < p 1 , p 2 < ∞, q > 0, and w := w

q/p 1 1 w q/p 2 2
, we say that (w 1 , w 2 ) satisfies the A (p 1 ,p 2 ),q condition (or that (w 1 , w 2 ) belongs to the class

A (p 1 ,p 2 ),q ) if [(w 1 , w 2 )] A (p 1 ,p 2 ),q := sup B 1 |B| B w(x) dx 2 j=1 1 |B| B w j (x) 1-p ′ j dx q p ′ j < ∞,
where the supremum is taken over all Euclidean balls B ⊂ R n and |B| denotes the Lebesgue measure of B.

The classes A (p 1 ,p 2 ),q are inspired in the classes of weights A p,q , 1 ≤ p, q < ∞, defined by Muckenhoupt and Wheeden in [START_REF] Muckenhoupt | Weighted norm inequalities for fractional integrals[END_REF] to study weighted norm inequalities for the fractional integral: a weight u defined on R n is in the class

A p,q if sup B 1 |B| B u q p dx 1 |B| B u (1-p ′ ) dx q p ′ < ∞.
The classes A (p 1 ,p 2 ),q for 1/q = 1/p 1 + 1/p 2 were introduced in [START_REF] Lerner | New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory[END_REF] to study characterizations of weights for boundedness properties of certain bilinear maximal functions and bilinear Calderón-Zygmund operators in weighted Lebesgue spaces. As shown in [START_REF] Moen | Weighted inequalities for multilinear fractional integral operators[END_REF], the classes A (p 1 ,p 2 ),q characterize the weights rendering analogous bounds for bilinear fractional integral operators .

Remark 2.1. If (w 1 , w 2 ) satisfies the A (p 1 ,p 2 ),q condition then w = w

q/p 1 1 w q/p 2 2
and w [START_REF] Lerner | New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory[END_REF]Theorem 3.6 ] and [START_REF] Moen | Weighted inequalities for multilinear fractional integral operators[END_REF]Theorem 3.4].

1-p ′ i i , i = 1, 2, are A ∞ weights as shown in
For α > 0, we consider bilinear fractional integral operators on R n of order α > 0 defined by

B α (f, g)(x) := R n f (x -s 1 y)g(x -s 2 y) |y| n-α dy, x ∈ R n . (2.16) I α (f, g)(x) := R 2n f (y)g(z) (|x -y| + |x -z|) 2n-α dydz, x ∈ R n , (2.17)
where s 1 = s 2 are nonzero real numbers. In the following theorem we summarize results concerning boundedness properties on weighted and unweighted Lebesgue spaces for the operators B α and I α , which will be useful in some of our proofs.

Theorem A. In R n : (a) [START_REF] Kenig | Multilinear estimates and fractional integration[END_REF][START_REF] Moen | Weighted inequalities for multilinear fractional integral operators[END_REF] Let α ∈ (0, 2n), 1 < p 1 , p 2 < ∞ and q > 0 such that

1 q = 1 p 1 + 1 p 2 -α n . Then I α is bounded from L p 1 w 1 × L p 2 w 2 into L q w for w := w q/p 1 1 w q/p 2 2
and pairs of weights (w 1 , w 2 ) satisfying the A (p 1 ,p 2 ),q condition. (b) [START_REF] Grafakos | On multilinear fractional integrals[END_REF][START_REF] Kenig | Multilinear estimates and fractional integration[END_REF]

Let α ∈ (0, n), 1 < p 1 , p 2 < ∞ and q > 0 such that 1 q = 1 p 1 + 1 p 2 -α n . Then B α is bounded from L p 1 × L p 2 into L q . (c) [Remark 2.2] Let α ∈ (0, n), 1 < p 1 , p 2 < ∞ such that 1/p := 1/p 1 + 1/p 2 < 1 and q > 1 such that 1/q = 1/p -α/n. Then B α is bounded from L p 1 w 1 × L p 2 w 2 into L q w for w := w q/p 1 1 w q/p 2 2
and weights w 1 , w 2 in A p,q .

Remark 2.2. Part (c) of Theorem A follows from the following observations. Muckenhoupt and Wheeden [START_REF] Muckenhoupt | Weighted norm inequalities for fractional integrals[END_REF] showed that the linear fractional integral operator

I α f (x) := R n f (x -y) |y| n-α dy satisfies R n |I α f (x)| q u q p dx 1/q ≤ C R n |f (x)| p u dx 1/p
for 1/q = 1/p -α/n, u ∈ A p,q and p, q > 1. Using p and q as in the statement of part (c) of Theorem A, let r = p 1 /p and s = p 2 /p, so that r, s > 1 and 1/r + 1/s = 1. By Hölder's inequality

|B α (f, g)| I α (|f | r ) 1/r I α (|g| s ) 1/s ,
and

R n |B α (f, g)| q w 1/q ≤ R n I α (|f | r ) q/r I α (|g| s ) q/s w q p 1 1 w q p 2 2 1/q ≤ R n I α (|f | r ) q w q p 1 dx 1/qr R n I α (|g| s ) q w q p 2 1/sq .
Using the result of Muckenhoupt and Wheeden, the last inequality is bounded by

C R n |f | rp w 1 1/rp R n |g| sp w 2 1/sp = C R n |f | p 1 w 1 1/p 1 R n |g| p 2 w 2 1/p 2 ,
which is the desired result.

Multilinear potential operators, of which I α is a particular case, were studied in [START_REF] Maldonado | Weighted multilinear Poincaré inequalities for vector fields of Hörmander type[END_REF] in the context of spaces of homogeneous type. We now briefly recall some those results, as they will be used in the proofs in the next sections.

Let (X, ρ, µ) be a space of homogenous type. That is, X is a non-empty set, ρ is a quasi-metric defined on X that satisfies the quasi-triangle inequality (2.18) ρ(x, y) ≤ κ(ρ(x, z) + ρ(z, y)), x, y, z ∈ X, for some κ ≥ 1, and µ is a Borel measure on X (with respect to the topology defined by ρ) such that there exists a constant L 0 ≥ 0 verifying

(2.19) 0 < µ(B ρ (x, 2r) ≤ L 0 µ(B ρ (x, r)) < ∞
for all x ∈ X and 0 < r < ∞, and where B ρ (x, r) = {y ∈ X : ρ(x, y) < r} is the ρ-ball of center x and radius r. Given a ball B = B ρ (x, r) and θ > 0 we will usually write r(B) to denote the radius r and θB to denote B ρ (x, θr). In the Euclidean setting, this is, when X = R n , ρ is Euclidean distance and µ is Lebesgue measure, we use the notation B(x, r) instead of B ρ (x, r).

The measure µ is said to satisfy the reverse doubling property if for every η > 1 there are constants c(η) > 0 and γ > 0 such that

(2.20) µ(B ρ (x 1 , r 1 )) µ(B ρ (x 2 , r 2 )) ≥ c(η) r 1 r 2 γ , whenever B ρ (x 2 , r 2 ) ⊂ B ρ (x 1 , r 1 ), x 1 , x 2 ∈ X and 0 < r 1 , r 2 ≤ η diam ρ (X).
We consider bilinear potential operators of the form

(2.21) T (f, g)(x) = X 2 f (y)g(z)K(x, y, z) dµ(y)dµ(z),
where the kernel K is the restriction of a nonnegative continuous kernel K(x 1 , x 2 , y, z) (i.e. K(x, y, z) = K(x, x, y, z) for (x, y, z) ∈ X × X × X) that satisfies the following growth conditions: for every c > 1 there exists

C > 1 such that K(x 1 , x 2 , y, z) ≤ C K(v, w, y, z) if ρ(v, y) + ρ(w, z) ≤ c (ρ(x 1 , y) + ρ(x 2 , z)), and (2.22) K(x 1 , x 2 , y, z) ≤ C K(y, z, v, w) if ρ(y, v) + ρ(z, w) ≤ c (ρ(x 1 , y) + ρ(x 2 , z)).
The functional ϕ associated to K is defined by

ϕ(B) := sup{K(x, y, z) : (x, y, z) ∈ B × B × B, ρ(x, y) + ρ(x, z) ≥ c r(B)}
for a sufficiently small positive constant c and for B a ρ ball such that r(B) ≤ η diam ρ (X), for some fixed η > 1. The functional ϕ associated to K will be assumed to satisfy the following property: there exists δ > 0 such that for all C 1 > 1 there exists

C 2 > 0 such that (2.23) ϕ(B ′ )µ(B ′ ) 2 ≤ C 2 r(B ′ ) r(B) δ ϕ(B)µ(B) 2 for all balls B ′ ⊂ B, with r(B ′ ), r(B) < C 1 diam ρ (X).
We note that, in the Euclidean setting, the operator I α defined in (2.17) has kernel and associated functional given, respectively, by

K(x, y, z) = 1 (|x -y| + |x -z|) 2n-α and ϕ(B) ∼ r(B) α-2n ,
and both satisfy (2.22) and (2.23).

Theorem B ([24]

). Suppose that 1 < p 1 , p 2 ≤ ∞, 1 p = 1 p 1 + 1 p 2 and 1 2 < p ≤ q < ∞. Let (X, ρ, µ) be a space of homogeneous type that satisfies the reverse doubling property (2.20) and let K be a kernel such that (2.22) holds with ϕ satisfying (2.23). Furthermore, let u, v k , k = 1, 2 be weights defined on X that satisfy condition (2.24)

if q > 1 or condition (2.25) if q ≤ 1, where (2.24) sup B ρ-ball ϕ(B)µ(B) 1 q + 1 p 1 ′ + 1 p ′ 2 1 µ(B) B u qt dµ 1/qt 2 j=1 1 µ(B) B v -tp ′ i i dµ 1/tp ′ i < ∞, for some t > 1, (2.25) sup B ρ-ball ϕ(B)µ(B) 1 q + 1 p 1 ′ + 1 p ′ 2 1 µ(B) B u q dµ 1/q 2 j=1 1 µ(B) B v -tp ′ i i dµ 1/tp ′ i < ∞,
for some t > 1, with the supremum taken over ρ-balls with r(B) diam ρ (X). Then there exists a constant C such that

X (|T (f 1 , f 2 )|u) q dµ 1/q ≤ C 2 k=1 X (|f k |v k ) p k dµ 1/p k for all (f 1 , f 2 ) ∈ L p 1 v p 1 1 (X) × L p 2 v p 2
T op C 1 -D -δ W,
where W is the constant from (2.24) or (2.25), C = max(C 1 , C 2 ) and δ > 0 are the constants from (2.23), and D > 1 is a structural constant.

Remark 2.4. In the Euclidean setting, consider weights w 1 , w 2 ∈ A (p 1 ,p 2 ),q and w = w

q/p 1 1 w q/p 2 2 , for some 1 < p 1 , p 2 < ∞, 0 < 1 q < 1 p 1 + 1 p 2 and suppose that T is an operator of the form (2.21) such that sup B ϕ(B)|B| 1 q + 1 p 1 ′ + 1 p ′ 2 ∼ sup B ϕ(B)r(B) n q + n p 1 ′ + n p ′ 2 < ∞.
It then follows that u := w 1 q and v k := w [START_REF] Maldonado | Weighted multilinear Poincaré inequalities for vector fields of Hörmander type[END_REF]) and (2.25). Indeed, the second factor in (2.24) is given by sup

1 p k k , k = 1, 2, satisfy (2.
x∈R n ,r>0 1 |B(x, r)| B(x,r) w t dx 1/qt 2 j=1 1 |B(x, r)| B(x,r) w -t p i -1 i dx. 1/tp ′ i (2.26)
Since w, w

-1 p 1 -1 1 , w -1 p 2 -1 2 are A ∞ weights (see Remark 2.1), there exists t > 1 such that (2.26) is bounded by sup B 1 |B| B w dx 1/q 2 j=1 1 |B| B w -1 p i -1 i dx 1/p ′ i = [(w 1 , w 2 )] A (p 1 ,p 2 ),q < ∞,
where finiteness is due to ( w 1 , w 2 ) satisfying the A (p 1 ,p 2 ),q condition. A similar reasoning applies to (2.25).

The last two remarks imply the following Corollary 3. In the n-dimensional Euclidean setting, consider weights w 1 , w 2 ∈ A (p 1 ,p 2 ),q and w = w

q/p 1 1 w q/p 2 2
, for some

1 < p 1 , p 2 < ∞, 0 < 1 q < 1 p 1 + 1 p 2 .
Suppose that T is an operator of the form (2.21) such that its kernel satisfies (2.22), the associated functional ϕ satisfies (2.23), and

sup B ϕ(B)r(B) n q + n p 1 ′ + n p ′ 2 < ∞.
Then there exists a constant A such that

R n |T (f 1 , f 2 )| q w dx 1/q ≤ A 2 k=1 R n |f k | p k w k dx 1/p k for all (f 1 , f 2 ) ∈ L p 1 w 1 × L p 2 w 1 . The constant A satisfies A ≤ c sup B ϕ(B)r(B) n q + n p 1 ′ + n p ′ 2 ,
where c depends only on [(w 1 , w 2 )] A (p 1 ,p 2 ),q and other absolute constants.

Bilinear Poincaré-type inequalities relative to an approximation of the identity

An approximation of the identity of order m > 0 in R n is a collection of operators S := {S t } t>0 acting on functions defined on R n ,

S t f (x) = R n p t (x, y)f (y) dy, x ∈ R n ,
such that for each t > 0 the kernels p t satisfy R n p t (x, y) dy = 1 for all x and the scaled Poisson bound

(3.27) |p t (x, y)| ≤ t -n/m γ |x -y| t 1/m , x, y ∈ R n , where γ : [0, ∞) → [0, ∞) is a bounded, decreasing function for which (3.28) lim r→∞ r 2n+ε γ(r) = 0, for some ε > 0.
As examples, it is well-known that if a sectorial operator L generates a holomorphic semigroup {e -zL } z whose kernels satisfy suitable pointwise bounds, then S t = e -tL gives rise to an approximation of the identity. The resolvents

S t = (1 + tL) -M or S t = 1 -(1 -e -tL
) N can be considered as well. We refer the reader to [START_REF] Duong | Duality of Hardy and BMO spaces associated with operators with heat kernel bounds[END_REF] and [START_REF] Mcintosh | Operators which have an H ∞ -calculus, Miniconference on operator theory and partial differential equations[END_REF] for more details concerning holomorphic functional calculus. Other examples can be built on a second-order divergence form operator L = -div(A∇) with an elliptic matrix-valued function A. Since L is maximal accretive, it admits a bounded H ∞ -calculus on L 2 (R n ). Moreover, when A has real entries or when the dimension n ∈ {1, 2}, then the operator L generates an analytic semigroup on L 2 with a heat kernel satisfying Gaussian upper-bounds.

The main result of this section is the following: Theorem 4. Let S := {S t } t>0 and S ′ := {t∂ t S t } t>0 be approximations of the identity in R n of order m > 0 and constant ε in (3.28), 1 < p 1 , p 2 < ∞, q > 0, and 0 < α < min{1, ε} such that

1 q = 1 p 1 + 1 p 2 -1-α n . If (w 1 , w 2 ) satisfy the A (p 1 ,p 2 ),q condition and w := w q/p 1 1 w q/p 2 2
, then there exists a constant C such that for all Euclidean balls

B f g -S r(B) m (f )S r(B) m (g) L q w (B) ≤ C r(B) α l≥0 2 -l(ε-α) ∇f L p 1 w 1 (2 l+1 B) g L p 2 w 2 (2 l+1 B) + f L p 1 w 1 (2 l+1 B) ∇g L p 2 w 2 (2 l+1 B) .
Remark 3.1. It is possible to consider two collections of operators S 1 := {S 1 t } t>0 and S 2 := {S 2 t } t>0 , then the proof of Theorem 4 holds true when estimating the oscillation f g -S 1 r(B) m (f )S 2 r(B) m (g) L q w (B) . Remark 3.2. Note that condition (3.28) assumes exponent 2n + ε rather than n + ε. This is quite natural in our context since the proof of Theorem 4 involves the semigroup P t := S t ⊗S t which is expected to have decay for 2n-dimensional variables.

Remark 3.3. The scaling of the result in Theorem 4 is in accordance with the classical situation corresponding to α = 0 and obtained in [START_REF] Maldonado | Weighted multilinear Poincaré inequalities for vector fields of Hörmander type[END_REF]. More precisely, a particular case of [24, Theorem 1] reads

(3.29) f g -f B g B L q w (B) ≤ C ( ∇f L p 1 w 1 (B) g L p 2 w 2 (B) + f L p 1 w 1 (B) ∇g L p 2 w 2 (B) ) for 1 q = 1 p 1 + 1 p 2 -1 n , with 1 p 1 + 1 p 2 < 2, (w 1 , w 2 ) ∈ A (p 1 ,p 2 )
,q and w = w q/p 1 1 w q/p 2 2 . Hölder's inequality and the conditions on the weights imply

f g -f B g B L q w (B) ≤ C r(B) α ( ∇f L p 1 w 1 (B) g L p 2 w 2 (B) + f L p 1 w 1 (B) ∇g L p 2 w 2 (B) ) for 1 q = 1 p 1 + 1 p 2 -1-α n , (w 1 , w 2 ) ∈ A (p 1 ,p 2 ),q and w = w q/p 1 1 w q/p 2 2
. For instance, let p 1 , p 2 , q, α, w 1 , w 2 , w be as in the statement of Theorem 4. Define q by 1 q = 1 p 1 + 1 p 2 -1 n = 1 q -α n and assume that q > 0 and that the pair (w 1 , w 2 ) is in

A (p 1 ,p 2 ),q . Setting w = w q/p 1 1 w q/p 2 2
and using Hölder's inequality and (3.29) we obtain

f g -f B g B L q w (B) ≤ B w w q-q p 1 1 w q-q p 2 2 ( q q ) ′ 1 q( q/q) ′ f g -f B g B L q w(B) r(B) α ( ∇f L p 1 w 1 (B) g L p 2 w 2 (B) + f L p 1 w 1 (B) ∇g L p 2 w 2 (B)
). Note, however, that Theorem 4 does not include the case α = 0. Remark 3.4. Since we do not require spatial regularity on the kernels p t in (3.27), our results can be extended to every subset of R n (not necessarily Lipschitz) by considering truncations as used in [START_REF] Duong | New function spaces of BMO type, the John-Niremberg inequality, Interplation and Applications[END_REF].

Our proof of Theorem 4 is based on an appropriate representation formula for the bilinear oscillations associated to the approximation of the identity and the boundedness properties of operators studied in [START_REF] Maldonado | Weighted multilinear Poincaré inequalities for vector fields of Hörmander type[END_REF]. We present the details in the next two subsections.

3.1. Representation formula. We start by introducing the collection of bilinear operators that shape our representation formula. For a ball B ⊂ R n , the operator J B is defined as

(3.30) J B (f 1 , f 2 )(x) := B×B K(x, (a, b))f 1 (a)f 2 (b) da db x ∈ B, with kernel K(x, (a, b)) := 1 (|x -a| + |x -b|) 2n-1 log 8 r(B) |x -a| + |x -b| , x, a, b ∈ B.
Theorem 5 (Bilinear representation formula). Let S = {S t } t>0 and S ′ = {t∂ t S t } t>0 be approximations of the identity in R n of order m > 0 and constant ε in (3.28).

There exists a constant C > 0 such that for every ball B ⊂ R n and x ∈ B,

f (x)g(x) -S r(B) m (f )(x)S r(B) m (g)(x) ≤ C l≥0 2 -lε [J 2 l+1 B (|∇f |χ 2 l+1 B , |g|χ 2 l+1 B )(x) + J 2 l+1 B (|f |χ 2 l+1 B , |∇g|χ 2 l+1 B )(x)] .
Remark 3.5. As mentioned in the Introduction, since the approximation operator S r(B) m is not a local operator, we cannot expect perfectly localized estimates as for the "classical" Poincaré inequality.

Proof. We consider the operator on R 2n given by P t := S t ⊗ S t , that is,

P t (F )(x, x) = R n R n p t (x, y)p t (x, z)F (y, z) dydz.
For given functions f and g defined on R n , let F (y, z) := f (y)g(z). Fix B of radius r(B), x ∈ B and for each t ∈ (0, r(B) m ) let B t be the ball of radius t 1/m centered at x ∈ R n . Then

F (x, x) -P r(B) m (F )(x, x) = - r(B) m 0 t∂ t P t (F )(x, x) dt t = - r(B) m 0 t∂ t P t (F -F Bt×Bt )(x, x) dt t ,
where we used that F Bt×Bt = f Bt g Bt is a constant and ∂ t S t (1) = 0 for all t > 0. The pointwise bounds (3.27) for the kernels p t (x, y) give

|t∂ t P t (F -F Bt×Bt )(x, x)| R n R n t -2n m 1 + |x -y| t 1 m -2n-ε 1 + |x -z| t 1 m -2n-ε |f (y)g(z) -f Bt g Bt | dydz Bt×Bt t -2n m 1 + |x -y| t 1 m -2n-ε 1 + |x -z| t 1 m -2n-ε |f (y)g(z) -f Bt g Bt | dydz + l∈N C l (Bt×Bt) t -2n m 1 + |x -y| t 1 m -2n-ε 1 + |x -z| t 1 m -2n-ε |f (y)g(z) -f Bt g Bt | dydz =: I 0 (f, g, t)(x) + l∈N I l (f, g, t)(x),
where for l ≥ 1, C l (B t × B t ) denotes the annulus

C l (B t × B t ) := 2 l (B t × B t ) \ 2 l-1 (B t × B t ).
We now proceed to estimating each of the terms I l (f, g, t), l ≥ 0.

The bound for I 0 (f, g, t). Notice that for all y, z ∈ B t , For a, b ∈ B t , we have

(3.31) |f (y)g(z) -f Bt g Bt | Bt×Bt |∇f (
I(a, b, t) ≤ |y-a|≤2t 1/m |z-b|≤2t 1/m 1 (|y -a| + |z -b|) 2n-1 dy t n/m dz t n/m 2t 1/m 0 2t 1/m 0 1 (u + v) 2n-1 u n-1 v n-1 du t n/m dv t n/m t (-2n+1)/m 1 0 1 0 u n-1 v n-1 (u + v) 2n-1 dudv t (-2n+1)/m , (3.32)
where the last integral is controlled by separately estimating for v ≥ u and for u ≥ v. We conclude that, for a, b

∈ B t , Bt×Bt 1 (|y -a| + |z -b|) 2n-1 dy t n/m dz t n/m t (-2n+1)/m 1 (|x -a| + |x -b|) 2n-1 , (3.33)
and therefore

I 0 (f, g, t)(x) Bt×Bt |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 da db.
Integration with respect to the variable t ∈ (0, r(B) m ) yields,

r(B) m 0 I 0 (f, g, t)(x) dt t r(B) m 0 Bt×Bt |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 da db dt t r(B) m 0 |x-a|≤t 1/m |x-b|≤t 1/m |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 da db dt t 2B×2B 0≤t≤r(B) m |x-a|≤t 1/m |x-b|≤t 1/m |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 dt t da db 2B×2B |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 log 1 + r(B) m max{|x -a| m , |x -b| m } da db 2B×2B |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 log 16r(B) |x -a| + |x -b| da db J 2B (|∇f |, |g|)(x) + J 2B (|f |, |∇g|)(x),
where the operator J 2B was defined in (3.30). It remains to treat the terms I l (f, g, t)(x) with l ≥ 1.

The bound for I l (f, g, t) with l ≥ 1. Recall that I l is given by

I l (f, g, t)(x) := C l (Bt×Bt) t -2n/m 1 + |x -y| t 1 m -2n-ε 1 + |x -z| t 1 m -2n-ε × |f (y)g(z) -f Bt g Bt | dydz,
where B t = B(x, t 1/m ) (and therefore x ∈ B t ) and C l (B t ×B t ) := 2 l (B t ×B t )\2 l-1 (B t × B t ). We have to estimate the oscillation |f (y)g(z)-f Bt g Bt |, with (y, z) ∈ C l (B t ×B t ), for which we consider the intermediate averages as follows:

|f (y)g(z) -f Bt g Bt | ≤ |f (y)g(z) -f 2 l Bt g 2 l Bt | + l-1 k=0 |f 2 k+1 Bt g 2 k+1 Bt -f 2 k Bt g 2 k Bt | .
For all k ∈ 0, ..., l -1, we use

|f 2 k+1 Bt g 2 k+1 Bt -f 2 k Bt g 2 k Bt | (2 k t 1/m ) -2n 2 k Bt×2 k Bt |f (u)g(v) -f 2 k+1 Bt g 2 k+1 Bt |dudv (2 k t 1/m ) -2n 2 k+1 Bt×2 k+1 Bt |f (u)g(v) -f 2 k+1 Bt g 2 k+1 Bt |dudv.
As done in (3.31) applied to the ball 2 k+1 B t , we obtain that for (u, v)

∈ 2 k+1 B t × 2 k+1 B t |f (u)g(v) -f 2 k+1 Bt g 2 k+1 Bt | 2 k+1 Bt×2 k+1 Bt |∇f (a)||g(b)| + |f (a)||∇g(b)| (|u -a| + |v -b|) 2n-1 dadb.
Proceeding as in (3.33), by replacing the ball B t with 2 k+1 B t , we have that for (a, b) ∈ 2 k+1 B t × 2 k+1 B t and (u, v) ∈ 2 k+1 B t × 2 k+1 B t , (3.34)

2 k+1 Bt×2 k+1 Bt 2 -2kn t -2n/m dydz (|u -a| + |v -b|) 2n-1 (2 k t 1/m ) -(2n-1) (|x-a| + |x-b|) 1-2n .
Combining everything we have

|f 2 k+1 Bt g 2 k+1 Bt -f 2 k Bt g 2 k Bt | 2 2kn t 2n/m (2 k t 1/m ) 2n 2 k+1 Bt×2 k+1 Bt |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 dadb 2 k+1 Bt×2 k+1 Bt |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 dadb.
We conclude that for (y, z) ∈ 2 l+1 (B t × B t ) (actually for any y and z) Consequently,

|f (y)g(z) -f Bt g Bt | |f (y)g(z) -f 2 l Bt g 2 l Bt | + l-1 k=0 2 k+1
I l (f, g, t)(x) I 1 l (f, g, t)(x) + I 2 l (f, g, t)(x) with I 1 l (f, g, t)(x) := C l (Bt×Bt) 1 + |x -y| t 1 m 1 + |x -z| t 1 m -2n-ε × |f (y)g(z) -f 2 l Bt g 2 l Bt | dydz t 2n/m and I 2 l (f, g, t)(x) := l k=0 C l (Bt×Bt) 2 k+1 Bt×2 k+1 Bt 1 + |x -y| t 1 m -2n-ε 1 + |x -z| t 1 m -2n-ε |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 dadb dydz t 2n/m .
The first term I 1 l (f, g, t)(x) can be estimated in the same way as the quantity I 0 (f, g, t) by replacing B t with 2 l B t . Since (y, z) ∈ C l (B t × B t ) and x ∈ B t , the term

1 + |x -y| t 1 m -2n-ε 1 + |x -z| t 1 m -2n-ε
provides an extra factor 2 -l(ε+2n) which partially compensates the normalization coefficient 2 2ln . So we have

r(B) m 0 I 1 l (f, g, t)(x) dt t 2 -lε J 2 l+1 B (|∇f |χ 2 l+1 B , |g|χ 2 l+1 B )(x) + J 2 l+1 B (|f |χ 2 l+1 B , |∇g|χ 2 l+1 B )(x) .
We now study the term related to

I 2 l (f, g, t)(x). Since x ∈ B t , C l (Bt×Bt) 1 + |x -y| t 1 m -2n-ε 1 + |x -z| t 1 m -2n-ε dydz t 2n/m 2 -l(ε+n) .
Integrating in the variable t ∈ (0, r(B) m ), we obtain

r(B) m 0 I 2 l (f, g, t)(x) dt t r(B) m 0 2 -l(ε+n) l-1 k=0 2 k+1 Bt×2 k+1 Bt |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 dadb dt t l2 -l(ε+n) 2 l B×2 l B    0≤t≤r(B) m |x-a|≤2 l t 1/m |x-b|≤2 l t 1/m dt t    |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 dadb l2 -l(ε+n) 2 l B×2 l B |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 × log 1 + r(B) 2 -l (|x -a| + |x -b|) dadb l2 -l(ε+n) 2 l B×2 l B |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 log 8 • 2 l+1 r(B) (|x -a| + |x -b|) dadb l2 -l(ε+n) J 2 l+1 B (|∇f |χ 2 l+1 B , |g|χ 2 l+1 B )(x) + J 2 l+1 B (|f |χ 2 l+1 B , |∇g|χ 2 l+1 B )(x) .
Having obtained pointwise estimates both for I 0 (f, g, t) and I l (f, g, t), we can now conclude the proof of the theorem.

End of the proof of Theorem 5. Using the estimates for I 0 (f, g, t), I 1 l (f, g, t) and I 2 l (f, g, t), we finally obtain that

f (x)g(x) -S r(B) m (f )(x)S r(B) m (g)(x) l≥0 2 -lε (1 + l2 -ln ) × J 2 l+1 B (|∇f |χ 2 l+1 B , |g|χ 2 l+1 B )(x) + J 2 l+1 B (|f |χ 2 l+1 B , |∇g|χ 2 l+1 B )(x) l≥0 2 -lε J 2 l+1 B (|∇f |χ 2 l+1 B , |g|χ 2 l+1 B )(x) + J 2 l+1 B (|f |χ 2 l+1 B , |∇g|χ 2 l+1 B )(x) .
3.2. Boundedness properties of the operator J B . Boundedness properties of the operators J B follow from results for multilinear potential operators in the context of spaces of homogeneous type studied in [START_REF] Maldonado | Weighted multilinear Poincaré inequalities for vector fields of Hörmander type[END_REF]. We use those results, which were recalled in Section 2, to prove the following proposition.

Proposition 6. Let p 1 , p 2 > 1, q > 0, 0 < α ≤ 1 and 1 q = 1 p 1 + 1 p 2 -1-α n .
If (w 1 , w 2 ) belongs to the class A (p 1 ,p 2 ),q then the operator J B defined in (3.30) satisfies

J B L p 1 w 1 (B)×L p 2 w 2 (B)→L q w (B) [r(B)] α ,
with a constant uniform in B.

Proof. Following the results in [START_REF] Maldonado | Weighted multilinear Poincaré inequalities for vector fields of Hörmander type[END_REF], we work in the space of homogeneous type ( x,r) is the Euclidean ball in R n of radius r centered at x. Since for x ∈ B and r r(B), |B(x, r) ∩ B| ∼ |B(x, r)| = c n r n , we then have to prove that there exists δ > 0 such that given C 1 > 1 there is C 2 > 0 independent of B for which

B, | • -• |, dx) is B(x, r) ∩ B where B(
ϕ(B 1 ∩ B) ϕ(B 2 ∩ B) ≤ C 2 r 2 r 1 2n-δ , for all balls B i = B(x i , r i ), x i ∈ B, r i ≤ C 1 r(B), B 1 ∩ B ⊂ B 2 ∩ B, where
ϕ(B i ∩ B) = sup{K(x, a, b) : x, a, b ∈ B i ∩ B, |x -a| + |x -b| ≥ c r i }
for some fixed positive small constant c and i = 1, 2. We have ϕ(

B i ∩ B) = ( 1 cr i ) 2n-1 log( 8r(B) cr i ) which gives ϕ(B 1 ∩ B) ϕ(B 2 ∩ B) ∼ r 2 r 1 2n-1 log 8r(B) cr 1 log 8r(B) cr 2 r 2 r 1 2n-δ , 0 < δ < 1, since log(t ′ ) log(t)
( t ′ t ) γ for 2 ≤ t ≤ t ′ and 0 < γ < 1. We now check that the assumptions on the weights w 1 , w 2 , and w imply (2.24) if q > 1 and (2.25) if q ≤ 1 with u = w 1/q = w

1/p 1 1 w 1/p 2 2 , v k = w 1/p k k , k = 1, 2.
This means that we have to prove that there exists t > 1 such that (3.36)

sup Q ϕ(Q)|Q| 1 q + 1 p 1 ′ + 1 p ′ 2 1 |Q| Q w t dx 1/qt 2 j=1 1 |Q| Q w -t p i -1 i dx 1/tp ′ i < ∞, q > 1, and (3.37) sup Q ϕ(Q)|Q| 1 q + 1 p 1 ′ + 1 p ′ 2 1 |Q| Q wdx 1/q 2 j=1 1 |Q| Q w -t p i -1 i dx 1/tp ′ i < ∞, q ≤ 1,
where the sup is taken over all balls Q in the space (B,

| • -• |, dx) with r(Q) r(B).
The proofs follow using the same ideas as in Remark 2.4 . Let Q be a ball in the space (B, | • -• |, dx) with r(Q) r(B); then Q = B ∩ B(x, r) for some x ∈ B and r > 0, r(Q) = r r(B) and |Q| ∼ |B(x, r)|. Moreover, using the relation between p 1 , p 2 , q and α as in the statement of the proposition,

ϕ(Q)|Q| 1 q + 1 p 1 ′ + 1 p ′ 2 ∼ 1 r(Q) 2n-1 log 8r(B) c r(Q) r(Q) 2n-1+α = r(Q) α log 8r(B) c r(Q) r(B) α .
In addition, the second factor in (3.36) is bounded by

sup x∈R n ,r>0 1 |B(x, r)| B(x,r) w t dx 1/qt 2 j=1 1 |B(x, r)| B(x,r) w -t p i -1 i dx 1/tp ′ i . (3.38) Since w, w -1 p 1 -1 1 , w -1 p 2 -1 2 are A ∞ weights (see Remark 2.1), there exists t > 1 such that (3.38) is bounded by sup x∈R n ,r>0 1 |B(x, r)| B(x,r) w dx 1/q 2 j=1 1 |B(x, r)| B(x,r) w -1 p i -1 i dx 1/p ′ i < ∞,
where finiteness is due to ( w 1 , w 2 ) satisfying the A (p 1 ,p 2 ),q condition. A similar reasoning applies to (3.37). We conclude that (3.36) and (3.37) are bounded by a multiple (independent of B) of r(B) α .

By Theorem B and Remark 2.3 we have that J B is bounded from L p 1 w 1 (B) ×L p 2 w 2 (B) into L q w (B) and the operator norm is bounded by a multiple (uniform on B) of r(B) α .

3.3.

Proof of Theorem 4. Let p 1 , p 2 , q, w 1 , w 2 and w be as in the statement of Theorem 4. By Proposition 6 we have

J 2 l B L p 1 w 1 (B)×L p 2 w 2 (B)→L q w (B) 2 l r(B) α ,
uniformly in B and l ≥ 0, this and Theorem 5 imply

f g -S r(B) m (f )S r(B) m (g) L q w (B) l≥0 2 -lε 2 α(l+1) r(B) α ∇f L p 1 w 1 (2 l+1 B) g L p 2 w 2 (2 l+1 B) + f L p 1 w 1 (2 l+1 B) ∇g L p 2 w 2 (2 l+1 B) ,
which concludes the proof of Theorem 4.

Applying an analogous proof to that of Theorem 4, we obtain the following result:

Theorem 7. Under the same assumptions of Theorem 4,

f g -S r(B) m S r(B) m (f )S r(B) m (g) L q w (B) ≤ C r(B) α l≥0 2 -l(ε-α) ∇f L p 1 w 1 (2 l+1 B) g L p 2 w 2 (2 l+1 B) + f L p 1 w 1 (2 l+1 B) ∇g L p 2 w 2 (2 l+1 B) .
We will leave it to the reader to check the details for the fact that the proof of Theorem 4 still holds after noting that a similar representation formula can be used as we can write

f g -S r(B) m S r(B) m (f )S r(B) m (g) = - r(B) m 0 t∂ t S t [P t (F )] dt t ,
since t∂ t S t [P t ] satisfies the same estimates as t∂ t P t and the cancellation property t∂ t S t [P t (1)] = 0.

Leibniz-type rules in Campanato-Morrey spaces associated to generalized approximations of identity

In this section we apply Theorem 4 to prove a Leibniz-type rule of the form (1.1) where the spaces X 1 , X 2 , Y 1 , Y 2 belong to the scale of the classical Campanato-Morrey spaces and the space Z quantifies the oscillation

|f g -S r(B) m (f )S r(B) m (f )| of the product f g in L q (B) where B ⊂ R n is a Euclidean ball in R n (compare to (1.8)).
In this context, it will become clear how, as announced in the Introduction, the bilinear potential operators introduced in Section 2 play the role that paraproducts and the bilinear Coifman-Meyer multipliers play in the proofs of the Sobolev-based Leibniz-type rules (1.2).

Next, we recall the definition of the classical Campanato-Morrey spaces and introduce notions of bilinear Campanato-Morrey spaces associated to approximations of the identity and semigroups.

For p > 0 and λ ≥ 0 we say that f

∈ L 1 loc (R n ) belongs to the Campanato-Morrey space L p,λ (R n ) if (4.39) f L p,λ (R n ) := sup B⊂R n 1 |B| λ 1 |B| B |f (x)| p dx 1 p
is finite. For f, g ∈ L 1 (R n ) we say that the pair (f, g) belongs to the bilinear Campanato-Morrey space L p,λ S⊗S (R n ) associated to an approximation of the identity

S = {S t } t>0 of order m > 0 if (4.40) (f, g) L p,λ S⊗S (R n ) := sup B⊂R n 1 |B| λ 1 |B| B |f (x)g(x) -S r(B) m (f )(x)S r(B) m (g)(x)| p dx 1 p
is finite. We use the notation S ⊗S to signify that the oscillation in question coincides with the tensorial oscillation

|(f ⊗ g)(x, y) -(S ⊗ S) r(B) m (f ⊗ g)(x, y)|, for x, y ∈ B,
restricted to the diagonal x = y. These new spaces L p,λ S⊗S (R n ) arise as natural bilinear counterparts to the Campanato-Morrey spaces L p,λ S (R n ) associated to S introduced by Duong and Yan in [START_REF] Duong | Duality of Hardy and BMO spaces associated with operators with heat kernel bounds[END_REF][START_REF] Duong | New function spaces of BMO type, the John-Niremberg inequality, Interplation and Applications[END_REF]. In this case,

f ∈ L p,λ S (R n ) if (4.41) f L p,λ S (R n ) := sup B⊂R n 1 |B| λ 1 |B| B |f (x) -S r(B) m (f )(x)| p dx 1 p < ∞.
Theorem 8. Let S := {S t } t>0 and S ′ := {t∂ t S t } t>0 be approximations of the identity of order m > 0 in R n and constant ε in (3.28), 1 < p 1 , p 2 < ∞, 0 < α < min(ε, 1) and q > 0 such that

1 q = 1 p 1 + 1 p 2 -1-α n . Given λ 1 , λ 2 ≥ 0 set λ = 1 n + λ 1 + λ 2 and assume that ε > n λ + 1 q .
Then there exists a structural constant C > 0 such that the following Leibniz-type rule holds true

(4.42) (f, g) L q,λ S⊗S (R n ) ≤ C ∇f L p 1 ,λ 1 (R n ) g L p 2 ,λ 2 (R n ) + f L p 1 ,λ 1 (R n ) ∇g L p 2 ,λ 2 (R n ) . Proof. From Theorem 4 we have f g -S r(B) m (f )S r(B) m (g) L q (B) r(B) α l≥0 2 -l(ε-α) ∇f L p 1 (2 l B) g L p 2 (2 l B) + f L p 1 (2 l B) ∇g L p 2 (2 l B) .

By writing

∇f L p 1 (2 l B) = |2 l B| λ 1 + 1 p 1 1 |2 l B| λ 1 1 |2 l B| 2 l B |∇f | p 1 1 p 1 ≤ |2 l B| λ 1 + 1 p 1 ∇f L p 1 ,λ 1 (R n ) and g L p 2 (2 l B) = |2 l B| λ 2 + 1 p 2 1 |2 l B| λ 2 1 |2 l B| 2 l B |g| p 2 1 p 2 ≤ |2 l B| λ 2 + 1 p 2 g L p 2 ,λ 2 (R n ) ,
and similarly with f L p 1 (2 l B) and ∇g L p 2 (2 l B) , and by setting s

:= λ 1 + λ 2 + 1 p 1 + 1 p 2 , we obtain r(B) α l≥0 2 -l(ε-α) ∇f L p 1 (2 l B) g L p 2 (2 l B) + f L p 1 (2 l B) ∇g L p 2 (2 l B) ≤ |B| α n +s l≥0 2 -l(ε-α-ns) ∇f L p 1 ,λ 1 (R n ) g L p 2 ,λ 2 (R n ) + f L p 1 ,λ 1 (R n ) ∇g L p 2 ,λ 2 (R n ) ≤ C|B| α n +s ∇f L p 1 ,λ 1 (R n ) g L p 2 ,λ 2 (R n ) + f L p 1 ,λ 1 (R n ) ∇g L p 2 ,λ 2 (R n ) , since α +ns = n α n + 1 p 1 + 1 p 2 + λ 1 + λ 2 = n λ + 1 q < ε. Consequently, using that λ + 1 q = α n + s, 1 |B| λ 1 |B| B |f (x)g(x) -S r(B) m (f )(x)S r(B) m (g)(x)| q dx 1 q = 1 |B| α n +s f g -S r(B) m (f )S r(B) m (g) L q (B) ≤ C ∇f L p 1 ,λ 1 (R n ) g L p 2 ,λ 2 (R n ) + f L p 1 ,λ 1 (R n ) ∇g L p 2 ,λ 2 (R n ) ,
and (4.42) follows.

In relation with (4.41), we define another suitable notion of Campanato-Morrey spaces associated to an approximation of the identity S = {S t }: a function f belongs to Lp,λ

S (R n ) if f Lp,λ S (R n ) := sup B⊂R n inf h∈L 1 loc 1 |B| λ 1 |B| B |f (x) -S r(B) m (h)(x)| p dx 1 p < ∞,
where the supremum is taken over all Euclidean balls B ⊂ R n . Then, we have the following Leibniz-type rule:

Theorem 9. Let S := {S t } t>0 and S ′ := {t∂ t S t } t>0 be approximations of the identity in R n of order m > 0 and constant ε in

(3.28), 1 < p 1 , p 2 < ∞, 0 < α < min(ε, 1) and q > 0 such that 1 q = 1 p 1 + 1 p 2 -1-α n . Given λ 1 , λ 2 ≥ 0 set λ = 1 n + λ 1 + λ 2 and assume that ε > n λ + 1 q .
Then there exists a structural constant C > 0 such that the following Leibniz-type rule holds true

(4.43) f g Lq,λ S (R n ) ≤ C ∇f L p 1 ,λ 1 (R n ) g L p 2 ,λ 2 (R n ) + f L p 1 ,λ 1 (R n ) ∇g L p 2 ,λ 2 (R n ) .
The proof follows by estimating the norm sup

B⊂R n inf h∈L 1 loc 1 |B| λ 1 |B| B |f (x)g(x) -S r(B) m (h)(x)| p dx 1 p
with h = S r(B) m (f )S r(B) m (g) and following the arguments in Theorem 8 by invoking Theorem 7 instead of Theorem 4.

5.

Extensions to doubling Riemannian manifolds and Carnot groups 5.1. Doubling Riemannian manifolds. Let (M, ρ, dµ) be a doubling Riemannian manifold, this is a space of homogeneous type with a gradient vector field ∇ (e.g. a complete Riemannian manifold with nonnegative Ricci curvature). An approximation of the identity of order m > 0 in M is a collection of operators S := {S t } t>0 acting on functions defined on M, S t f (x) = M p t (x, y)f (y) dµ(y), such that for each t > 0 the kernels p t satisfy M p t (x, y) dµ(y) = 1 for all x and the scaled Poisson bound

(5.44) |p t (x, y)| ≤ µ(B ρ (x, t 1/m )) -1 γ ρ(x, y) t 1/m ,
where γ : [0, ∞) → [0, ∞) is a bounded, decreasing function such that (5.45) lim r→∞ r 2n+ε γ(r) = 0, for some ε > 0.

Theorem 10. Assume (M, ρ, µ) is a doubling Riemannian manifold. Let S := {S t } t>0 and S ′ := {t∂ t S t } t>0 be approximations of the identity in M of order m > 0 and constant ε in (5.45), 1 < p 1 , p 2 < ∞, q > 0, and 0 < α < min{1, ε} such that

1 q = 1 p 1 + 1 p 2 -1-α n .
Then there exists a constant C such that for all balls B ⊂ M f g -S r(B) m (f )S r(B) m (g) L q (B)

≤ C r(B) α l≥0 2 -l(ε-α) ∇f L p 1 (2 l+1 B) g L p 2 (2 l+1 B) + f L p 1 (2 l+1 B) ∇g L p 2 (2 l+1 B) .
The proof of this theorem follows from that of Theorem 4 after minor modifications. The Leibniz rules in Campanato/Morrey spaces, obtained in Section 4, can be extended to this framework as well. 5.2. Carnot groups. In this section we provide a description of how to extend our results of section 3 in the context of Carnot groups. Let Ω be an open connected subset of R n and X = {X k } M k=1 be a family of infinitely differentiable vector fields with values in R n . We identify X k with the first order differential operator acting on continuously differentiable functions defined on Ω by the formula

X k f (x) = X k (x) • ∇f (x), k = 1, • • • , M,
and we set Xf = (X 1 f, X 2 f, • • • , X M f ) and |Xf (x)| = M k=1 |X k f (x)| 2 1/2 , x ∈ Ω.
Given two vector fields X i and X j define the commutator or Lie bracket by [X i , X j ] = X i X j -X j X i . We will assume that X satisfies Hörmander's condition in Ω; that is, there is some finite positive integer M 0 such that the commutators of the vector fields in X up to length M 0 span R n at each point of Ω. The function ρ is in fact a metric in Ω called the Carnot-Carathéodory metric on Ω associated to X.

Let G be a Lie group on R n , that is a group law on R n such that the map (x, y) → xy -1 is C ∞ . The Lie algebra associated to G, denoted g, is the collection of all left invariant vector fields on G. A Carnot group is a Lie group whose Lie algebra admits a stratification

g = V 1 ⊕ • • • ⊕ V l , where [V 1 , V i ] = span{[Y, Z] : Y ∈ V 1 , Z ∈ V i } = V i+1 , i = 1, • • • , l -1, and [V 1 , V i ] = {0} for i ≥ l.
A basis for V 1 generates the whole Lie algebra. We will often denote this family as {X 1 , . . . , X n 1 } and refer to it as a family of generators for the Carnot group. In particular, a system of generators {X 1 , . . . , X n 1 } satisfies Hörmander's condition, and hence we have the notion of a Carnot-Caratheodory metric.

Set

n i = dim(V i ), then n = n 1 + • • • + n l ,
and the number Q = l i=1 in i is called the homogeneous dimension of G . The dilation operators δ λ x = (λx (1) , λ 2 x (2) , . . . , λ l x (l) )

x (i) ∈ R n i form automorphisms of G for each λ > 0. Furthermore, if B is a metric ball of radius r(B) with respect to the Carnot-Carathéodory metric then |B| = c r(B) Q , which shows that (R n , ρ, Lebesgue measure) is a space of homogenous type. We refer the reader to [START_REF] Bonfiglioli | Stratified Lie groups and potential theory for their sub-Laplacians[END_REF] for more information about analysis on Carnot groups. An approximation of the identity of order m > 0 in G is a collection of operators S := {S t } t>0 acting on functions defined on R n , S t f (x) = R n p t (x, y)f (y) dy, such that for each t > 0 the kernels p t satisfy R n p t (x, y) dy = 1 for all x and the scaled Poisson bound

|p t (x, y)| ≤ t -Q/m γ ρ(x, y) t 1/m
, where γ : [0, ∞) → [0, ∞) is a bounded, decreasing function such that lim r→∞ r 2Q+ε γ(r) = 0, for some ε > 0.

Theorem 11. Suppose G is a homogeneous Carnot group of dimension Q with generators X = {X 1 , . . . , X n 1 } and ρ is the Carnot-Carathéodory metric on R n associated to X. Suppose further that S = {S t } t>0 and S ′ = {t∂ t S t } t>0 are approximations of the identity in G of order m and ε as given above. If p 1 , p 2 > 1, 0 < α < min(ε, 1) and q > 0 are such that 1

q = 1 p 1 + 1 p 2 -1-α Q , then, for every ρ-ball B, f g -S r(B) m (f )S r(B) m (g) L q (B) r(B) α l≥0 2 -l(ε-α) (l + 1) Xf L p 1 (2 l+1 B) g L p 2 (2 l+1 B) + f L p 1 (2 l+1 B) Xg L p 2 (2 l+1 B) .
Sketch of Proof. We will take the same approach as the proof of Theorem 4. The multilinear representation formula is given by

|f (x)g(x) -S r(B) m f (x)S r(B) m g(x)| l≥0 2 -l(ε-Q) [J 2 l+1 B (|Xf |, |g|)(x) + J 2 l+1 B (|f |, |Xg|)(x). (5.46) where J B (f, g)(x) = B×B f (y)g(z) (ρ(x, y) + ρ(x, z)) 2Q-1 log cr(B) ρ(x, y) + ρ(x, z) dydz x ∈ B
and B is a ball in R n with respect to the metric ρ. The operator J B satisfies the necessary growth bounds on its kernel and hence (5.47)

J B L p 1 (B)×L p 2 (B)→L q (B) [r(B)] α .
The inequalities (5.46) and (5.47) prove the desired result. The proof of inequality (5.46) follows that of Theorem 5 with the Euclidean distance replaced by ρ(x, y) and the dimension n replaced by Q. We just highlight the analog to inequality (3.33),

(5.48) Let B = B ρ be a ball in R n with respect to the metric ρ, x ∈ B, r(B) be the radius of B. Suppose 0 < t < r(B) m and a, b

∈ B t = B ρ (x, t 1/m ) then Bt×Bt 1 (ρ(y, a) + ρ(z, b)) 2Q-1 dydz Bρ(a,2t 1m )×Bρ(b,2t 1/m ) 1 (ρ(y, a) + ρ(z, b)) 2Q-1 dydx k≥0 D k 1 (ρ(y, a) + ρ(z, b)) 2Q-1 dydx where D k := {(y, z) : 2 -k t 1/m ≤ ρ(a, y) < 2 -k+1 t 1/m , 2 -k t 1/m ≤ ρ(b, z) < 2 -k+1 t 1/m }.
We continue estimating each term in the series

D k 1 (ρ(y, a) + ρ(z, b)) 2Q-1 dydz (2 k t -1/m ) 2Q-1 |B ρ (a, 2 -k+1 t 1/m )| • |B ρ (b, 2 -k+1 t 1/m )| 2 -k t 1/m which leads to Bt×Bt 1 (ρ(y, a) + ρ(z, b)) 2Q-1 dydz Bρ(a,2t 1m )×Bρ(b,2t 1/m ) 1 (ρ(y, a) + ρ(z, b)) 2Q-1 dydx t 1/m 1 (ρ(x, a) + ρ(x, b)) 2Q-1
This estimate contributes to the first term on the right side of inequality (5.46), the other terms are obtained in a similar manner.

Boundedness of bilinear pseudodifferential operators under

Sobolev scaling

Let BS m ρ,δ (R n ) and BS m ρ,δ;θ (R n ), where m ∈ R, 0 ≤ δ ≤ ρ ≤ 1, θ ∈ (0, π), be the classes of symbols σ ∈ C ∞ (R 3n ) satisfying, (6.49) ∂ α x ∂ β ξ ∂ γ η σ(x, ξ, η) ≤ C α,β,γ (1 + |ξ| + |η|) m-ρ(|β|+|γ|)+δ|α| , respectively, (6.50) ∂ α x ∂ β ξ ∂ γ η σ(x, ξ, η) ≤ C α,β,γ (1 + |ξ -tan(θ) η|) m-ρ(|β|+|γ|)
+δ|α| , for all x, ξ, η ∈ R n , all multi-indices α, β, γ ∈ N n 0 and some constants C α,β,γ , with the convention that θ = π 2 corresponds to decay in terms of 1 + |ξ|. We will use the notation ḂS m 1,δ (R n ) and ḂS m 1,0;θ (R n ) for the homogeneous versions of the above classes, defined by replacing 1+|ξ|+|η| by |ξ|+|η| and 1+|ξ-tan(θ) η| by |ξ-tan(θ) η| in (6.49) and (6.50), respectively. Also, we will use σ α,β,γ to denote the smallest constant C α,β,γ in (6.49) or (6.50).

These classes can be regarded as bilinear counterparts to the linear Hörmander classes S m ρ,δ (R n ) (and their homogeneous analogs

Ṡ m ρ,δ (R n )) which consists of symbols σ ∈ C ∞ (R 2n ) such that ∂ α x ∂ β ξ σ(x, ξ) ≤ C α,β (1 + |ξ|) m-ρ|β|+δ|α| , for all x, ξ ∈ R n ,
all multiindices α, β, and some constants C α,β .

Our results in this section assume symbols in the classes

BS m 1,δ (R n ) or ḂS m 1,δ (R n ), as well as those symbols in BS m 1,δ;θ (R n ) or ḂS m 1,δ;θ (R n ) of the form (6.51) σ(x, ξ, η) = σ 0 (x, ξ -tan(θ) η),
where σ 0 ∈ S m 1,δ (R n ) or Ṡ m 1,δ (R n ), respectively. For a number of properties of the Hörmander classes BS m ρ,δ (R n ), including symbolic calculus and boundedness properties of the associated bilinear operators with indices related by Hölder scaling, see [START_REF] Bényi | On the Hörmander classes of bilinear pseudodifferential operators[END_REF][START_REF] Bényi | On the Hörmander classes of bilinear pseudodifferential operators II[END_REF][START_REF] Bényi | Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators[END_REF] and references therein. The classes BS m ρ,δ;θ (R n ) were first introduced in [START_REF] Bényi | Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators[END_REF] inspired by their x-independent versions which originated in work on the bilinear Hilbert transform in [START_REF] Lacey | L p -bounds for the bilinear Hilbert transform[END_REF] and were extensively studied in [START_REF] Gilbert | L p -boundedness for time-frequency paraproducts[END_REF][START_REF] Bernicot | Local estimates and global continuities in Lebesgue spaces for bilinear operators[END_REF][START_REF] Bernicot | Sobolev space estimates for a class of bilinear pseudodifferential operators lacking symbolic calculus[END_REF] and references therein.

In this section we prove boundedness properties on Lebesgue spaces for bilinear pseudodifferential operator with symbols of negative order where the indices relation is now dictated by the Sobolev scaling. More precisely, the main result in this section is the following: Theorem 12. Suppose n ∈ N and consider exponents p 1 , p 2 ∈ (1, ∞) and q, s > 0 such that (6.52)

1 q = 1 p 1 + 1 p 2 - s n . (a) If s ∈ (0, 2n), 0 ≤ δ ≤ 1, and σ ∈ BS -s 1,δ (R n ) ∪ ḂS -s 1,δ (R n ) then T σ is bounded from L p 1 w 1 × L p 2 w 2 into L q
w for every pair of weights (w 1 , w 2 ) satisfying the A (p 1 ,p 2 ),q condition and w := w q/p 1 1 w q/p 2 2 . (b) If s ∈ (0, n), θ ∈ (0, π)\{π/2, 3π/4}, 0 ≤ δ ≤ 1 and σ ∈ BS -s 1,δ;θ (R n )∪ ḂS -s 1,δ;θ (R n ) is of the form (6.51) then the bilinear operator T σ is bounded from L p 1 × L p 2 into L q . If in addition 1 p := 1 p 1 + 1 p 2 < 1, then T σ is bounded from L p 1 w 1 × L p 2 w 2 into L q w for weights w 1 , w 2 in the class A p,q and w := w q/p 1 1 w q/p 2 2

.

Proof. We start with the proof of part (a). Let 0 ≤ δ < 1, s ∈ (0, 2n), and σ ∈ BS -s 1,δ (R n ) ∪ ḂS -s 1,δ (R n ). The results will follow from part (a) of Theorem A once we have proved that the operator T σ is controlled by the bilinear fractional integral I s as defined in (2.17). T σ is given by the spatial representation T σ (f, g)(x) = R n ×R n k(x, x -y, x -z)f (y)g(z)dydz where the kernel k is defined by k(x, u, v) := σ(x, •, •)(u, v).

We will prove that, (1 + |ξ| + |η|) -s dξdη t -2n+s , and (6.54) follows from this last inequality and (6.55). Then, (6.54) and integration over t ∈ (0, ∞) yield

|k(x, u, v)| ∞ 0 Ψ(t•)σ(x, •)(u, v) dt t ∞ 0 t -2n+s (1 + t -1 |(u, v)|) N dt t |(u, v)| -2n+s ∞ 0 t 2n-s (1 + t) N dt t |(u, v)| -2n+s ,
which ends the proof of (6.53). We now turn to the proof of part (b) of the theorem. If s ∈ (0, n), θ ∈ (0, π) \ {π/2, 3π/4} and σ ∈ BS -s 1,δ;θ (R n ) ∪ ḂS -s 1,δ;θ (R n ) is of the form σ(x, ξ, η) = σ 0 (x, ξtan(θ) η) with σ 0 ∈ S -s 1,δ (R n ) or σ 0 ∈ Ṡ -s 1,δ (R n ) as appropriate, we consider the following spatial representation for T σ : Remark 6.1. We note that pointwise decay properties of the kernels (and their derivatives) of pseudodifferential operators with symbols in the Hörmander classes have been studied in [START_REF] Bényi | On the Hörmander classes of bilinear pseudodifferential operators[END_REF]Theorem 5.1]. In particular, it is proved there that if σ ∈ BS -s 1,δ (R n ), then (6.53) holds. Remark 6.2. We observe that the proof of Theorem 12 uses the fact that the symbol σ satisfies conditions (6.49), (6.50), or their homogenous counterparts, only for a certain number of derivatives c n depending only on the dimension n.

T σ (f, g)(x) =

Leibniz-type rules in Sobolev spaces

In the following, we consider the inhomogeneous and homogeneous Sobolev spaces for indices s > 0 and 0 < p < ∞,

W s,p (R n ) = {f ∈ S ′ (R n ) : J s f ∈ L p (R n )} and Ẇ s,p (R n ) = {f ∈ S ′ (R n ) : D s f ∈ L p (R n )},
where F -1 denotes the inverse Fourier transform, J s is the operator with Fourier multiplier (1 + |ξ| 2 ) s 2 , and D s is the operator with Fourier multiplier |ξ| s . We use the notation f W s,p := J s f L p and f Ẇ s,p := D s f L p .

Corollary 13 (Leibniz-type rules). Let n ∈ N and consider exponents p 1 , p 2 ∈ (1, ∞) and q, s > 0 such that 1 q = 1 p 1 + 1 p 2 -s n . (a) If s ∈ (0, 2n), 0 ≤ δ < 1, and σ ∈ BS m 1,δ (R n ) for some m ≥ -s then T σ (f, g) L q f W m+s,p 1 g L p 2 + f L p 1 g W m+s,p 2 .

  a)||g(b)| + |f (a)||∇g(b)| (|y -a| + |z -b|) 2n-1 dadb.Indeed, the usual representation formula for a linear oscillation in (R n )2 gives|F (y, z) -F Bt×Bt | ≤ C Bt×Bt |∇F (a, b)| |(y, z) -(a, b)| 2n-1 dadbwhich yields (3.31). Hence, we getI 0 (f, g, t)(x) Bt×Bt t -2n/m |f (y)g(z) -f Bt g Bt | dydzBt×Bt (|∇f (a)||g(b)| + |f (a)||∇g(b)|)I(a, b, t) da db, where I(a, b, t) := Bt×Bt t -2n/m (|y -a| + |z -b|) 2n-1 dy dz.

  Bt×2 k+1 Bt |∇f (a)||g(b)| + |f (a)||∇g(b)| (|x -a| + |x -b|) 2n-1 dadb. (3.35)

  Suppose that X = {X k } M k=1 satisfies Hörmander's condition in Ω. Let C X be the family of absolutely continuous curvesζ : [a, b] → Ω, a ≤ b, such that there exist measurable functions c j (t), a ≤ t ≤ b, j = 1, • • • , M, satisfying M j=1 c j (t) 2 ≤ 1 and ζ ′ (t) = M j=1 c j (t)Y j (ζ(t))for almost every t ∈ [a, b]. If x, y ∈ Ω define ρ(x, y) = inf{T > 0 : there exists ζ ∈ C X with ζ(0) = x and ζ(T ) = y}.

  y, a) + ρ(z, b)) 2Q-1 dydz 1 (ρ(x, a) + ρ(x, b)) 2Q-1 .

( 6 .C

 6 53) |k(x, u, v)| 1 (|u| + |v|) 2n-s , uniformly in x, which gives |T σ (f, g)(x)| R n ×R n |f (y)||g(z)| (|x -y| + |x -z|) 2n-s dydz = I s (|f |, |g|)(x), and therefore the boundedness properties of T σ follow from part (a) of Theorem A. Let Ψ(ξ, η) be a smooth function in R 2n supported on the annulus 1 ≤ |(ξ, η)| ≤ 2, and such that ∞ 0 Ψ(tξ, tη) dt t = 1, (ξ, η) = (0, 0).So for each scale t > 0, we have to estimate Ψ(t•)σ(x, •). Now, integration by parts and the hypothesis σ∈ BS -s 1,δ (R n ) ∪ ḂS -s 1,δ (R n ) yield (6.54) Ψ(t•)σ(x, •)(u, v) t -2n+s (1 + t -1 |(u, v)|) Nfor every large enough integer N. Indeed, suppose that |v| ≤ |u| ∼ u j , so that|(u, v)| ∼ |u| ∼ u j , then Ψ(t•)σ(x, •)(u, v) = R n ×R n Ψ(tξ, tη)σ(x, ξ, η)e -i(u•ξ+v•η) dξdη = R n ×R n Ψ(tξ, tη)σ(x, ξ, η) 1 (-i) N u N j ∂ N ξ j e -i(u•ξ+v•η) dξdη = 1 (-i) N u N j |ξ|+|η|∼t -1 ∂ N ξ j (Ψ(tξ, tη)σ(x, ξ, η))e -i(u•ξ+v•η) dξdη.But, by the usual Leibniz rule and using the condition on the support of Ψ (which implies t -1 ∼ |ξ| + |η| ≤ 1 + |ξ| + |η|), we have|∂ N ξ j (Ψ(tξ, tη)σ(x, ξ, η))| = | N k=0 C N,k ∂ N -k ξ j Ψ(tξ, tη)∂ k ξ j σ(x, ξ, η)| ≤ N k=0 C N,k t N -k |(∂ N -k ξ j Ψ)(tξ, tη)| σ 0,k,0 (1 + |ξ| + |η|) -s-k N,k t N -k t s+k =: C σ,N t N +s .Consequently, (6.55)| Ψ(t•)σ(x, •)(u, v)| t N +s u N j |ξ|+|η|∼t -1 dξdη ∼ t -2n+s (t -1 |(u, v)|) N .On the other hand, again by the hypothesis σ ∈ BS -s 1,δ (R n ) ∪ ḂS -s 1,δ (R n ), we have| Ψ(t•)σ(x, •)(u, v)| ≤ R n ×R n |Ψ(tξ, tη)||σ(x, ξ, η)|dξdη ≤ Ψ L ∞ σ 0,0,0 |ξ|+|η|∼t -1

  , y)f (x + y)g(x -tan(θ) y)dy where the kernel k is defined byk(x, y) := σ 0 (x, •)(y).Following the same reasoning as above, we obtain |k(x, y)| |y| s-n , uniformly in x, and therefore|T σ (f, g)| B s (f, g),with B s defined in(2.16). The result then follows from parts (b) and (c) of Theorem A.

  Remark 2.3. A careful examination of the proof of Theorem B yields

2 (X). The constant C depends only on the constants appearing in (2.18), (2.19), (2.20), (2.22), (2.23), (2.24) and (2.25).

  Ch(t ′ ) if t ′ ≤ c t and t, t ′ ≤ 4r(B), for some C > 0 independent of B. Regarding condition (2.23), recall that the ball with center x ∈ B and radius r > 0 in the space (

	We will consider the kernel
	K((x, y), (a, b)) :=	1 (|x -a| + |y -b|) 2n-1 log	8 r(B) |x -a| + |y -b|	,	x, y, a, b ∈ B
	and check that K satisfies (2.22) and (2.23). For condition (2.22), note that for
	any c > 1 the function h(t) = 1 t 2n-1 log( 8r(B)

B, | • -• |, dx) noting that the constants in (2.18), (2.19), (2.20) are independent of B. t ) satisfies h(t) ≤

First author is supported by the ANR under the project AFoMEN no. 2011-JS01-001-01. Second, third, and fourth authors supported by the NSF under grants DMS 0901587, DMS 1201504, and DMS 1101327, respectively.

(b) If n ∈ N, s ∈ (0, 2n), 0 ≤ δ < 1, and σ ∈ ḂS m 1,δ (R n ) for some m ≥ -s then T σ (f, g) L q f Ẇ m+s,p 1 g L p 2 + f L p 1 g Ẇ m+s,p 2 .

Proof. Part (a) of Corollary 13 follows from Theorem 12 and composition with J m+s , along the lines of [5, Theorem 2] (see also [START_REF] Gulisashvili | Exact smoothing properties of Schrödinger semigroups[END_REF]Theorem 1.4] and [START_REF] Bényi | On the Hörmander classes of bilinear pseudodifferential operators[END_REF]Corollary 8]). Indeed, let σ ∈ BS m 1,δ (R n ) and consider φ ∈ C ∞ (R) such that 0 ≤ φ ≤ 1, supp(φ) ⊂ [-2, 2] and φ(r) + φ(1/r) = 1 on [0, ∞), then, the symbols σ 1 and σ 2 defined by

and σ 2 (x, ξ, η) = σ(x, ξ, η)φ 1 + |η| 2 1 + |ξ| 2 (1 + |ξ| 2 ) -(m+s)/2 are symbols in the class BS -s 1,δ (R n ), and the operators T σ , T σ 1 , and T σ 2 are related through T σ (f, g) = T σ 1 (J m+s f, g) + T σ 2 (f, J m+s g). Part (b) of Corollary 13 follows in the same way using the operators D m+s instead of J m+s .

We end this section by presenting particular cases related to Theorem 12 and Corollary 13.

• Fractional Leibniz rule under Sobolev scaling.

Proof. The case q ≥ 1 of the above inequality follows from the Sobolev imbedding W m,q ⊂ W m+s,r , 1 r = 1 p 1 + 1 p 2 , and the well-known fractional Leibniz rule (1.2). For the case q < 1 we proceed as follows:

Consider

where

The symbols σ 1 and σ 2 belong to the class BS -s 1,0 (1+|ξ +η| ∼ 1+|η| and 1+|ξ +η| ∼ 1 + |ξ|, in the respective supports) and therefore Corollary 13 imply that

Since 1 + |ξ + η| is not comparable to 1 + |η| or 1 + |ξ| in the support of σ 3 , we cannot expect to prove that this symbol belongs to a suitable class. We will then split σ 3 into elementary symbols. Choose smooth cut-off functions (ζ j ) 1≤j≤3 , such that ζ j is supported on B(0, 4) \ B(0, 1) and

=:

Now choose Ψ 1 , Ψ 2 smooth functions verifying the same support properties as the ζ j 's with Ψ j ≡ 1 on the support of ζ j , so that

where Ψ l stands for the usual dilation of Ψ and we identify Ψ l with the multiplier it produces. Now we focus on K k,l , the bilinear kernel of T m k,l , that is

Then,

First we notice that m k,l is supported on the set {(ξ, η), |ξ| ≃ |η| ≃ 2 l , |ξ +η| ≃ 2 k } whose measure is bounded by 2 n(k+l) . After the change of variables u := (ξ + η) and v := (ξ -η) we get

Next, integration by parts and the bounds

By Lemma 15 below, with m > n -s, we deduce that

Then the proof follows from the boundedness of the bilinear operator I s and Littlewood-Paley characterizations of Lebesgue spaces, since p 1 , p 2 ∈ (1, ∞).

Lemma 15. For l ∈ N 0 , a, b, s > 0 and m, n ∈ N 0 with m > n -s, we have

where the implicit constants depend only on n, m, and s.

In the case b > 2 k 0 +l we do

Remark 7.1. A shorter proof of Corollary 14 for q < 1 can be obtained as follows if we assume m > c n where c n is as in Remark 6.2 (note that

. By Remark 6.2 we can use Theorem 12 and conclude that T σ 1 and T σ 2 are bounded from L p 1 × L p 2 into L q if m > c n and therefore

from which the desired result follows.

• Paraproduct estimates under Sobolev scaling. Let n ∈ N, s ∈ (0, 2n), p 1 , p 2 ∈ (1, ∞) and q > 0 such that 1 q = 1 p 1 + 1 p 2 -s n . Consider a radial, real-valued function ϕ ∈ S(R n ) such that φ(ξ) = 1 for |ξ| ≤ 1 and ϕ(ξ) = 0 for |ξ| ≥ 3/2. Let ψ be given by ψ(ξ) = φ(ξ/2) -φ(ξ). For f ∈ L 1 (R n ) we set

where ϕ j (x) = 2 jn ϕ(2 j x), j ∈ Z. We also define ψ j (x) := 2 jn ψ(2 j x) and note that supp( ψ j ) ⊂ {ξ : 2 j ≤ |ξ| ≤ 3 2 j }. For f, g ∈ S(R n ) we define the Bony paraproduct of f and g by Π(f, g) := j∈Z ∆ j (f )S j-1 (g).

Straightforward computations show that

The symbol σ of the paraproduct Π is x-independent,

and belongs to the class ḂS 0 1,0 . As a consequence of Corollary 13, we have Π(f, g) L q f Ẇ s,p 1 g L p 2 + f L p 1 g Ẇ s,p 2 .

• Lowering the exponents for linear embeddings. It is well-known that in R n , for s ∈ (0, 1), W s,p is continuously embedded into L q as soon as p < d/s and q ≥ 1 with

By the previous approach, we get bilinear analogs: indeed we have proved that (f, g) → f g is continuous from W s,p 1 × W s,p 2 into L q as soon as p < d/s (where p is the harmonic mean value of p 1 , p 2 ) and q > 1/2. It is then possible to use this bilinear approach to give extensions of the linear inequalities for q < 1.

Proposition 16. Let consider s ∈ (0, 1) and p = t/2 < d/s and q ≤ 1 with

Then for every nonnegative smooth function h, we have h L q h 1/2 2 W s,t = h 1/2 2 W s,2p . Proof. We just write h = h 1/2 h 1/2 and apply the bilinear inequalities to the functions f = g = h 1/2 with the exponents p 1 = p 2 = t.

Such inequalities are of interest since they allow for an exponent q ≤ 1. To do that we have to pay the cost of estimating the regularity of √ h.

Acknowledgement

The authors would like to thank the anonymous referee for his/her careful reading of the manuscript and useful corrections.


T Laloë
email: laloe@unice.fr

R Servien
email: remi.servien@toulouse.inra.fr

Nonparametric estimation of regression level sets using kernel plug-in estimator

Keywords: Regression function, Level set, Plug-in estimator, Kernel estimator

 for the density function level sets.

Introduction

In this paper, we consider the problem of estimating the level sets of a regression function. More precisely, consider a random pair (X, Y ) taking values in R d ×J, where J ⊂ R is supposed to be bounded. The goal of this paper is then to build an estimator of the level sets of the regression function r of Y on X, dened for all x ∈ R d by

r(x) = E [Y |X = x].
For t > 0, a level set for r is dened by

L(t) = {x ∈ R d : r(x) > t}.
Assume that we have an independent and identically distributed sample (i.i.d.) ((X 1 , Y 1 ), . . . , (X n , Y n )) with the same distribution as (X, Y ). We then consider a plug-in estimator of L(t). More precisely, we use a consistent estimator rn of r, in order to estimate L(t) by L n (t) = {x ∈ R d : rn (x) > t}.

Most of the research works on the estimation of level sets concern the density function. One can cite the works of Cadre [START_REF] Cadre | Kernel estimation of density level sets[END_REF], Cuevas and Fraiman [START_REF] Cuevas | A plug-in approach to support estimation[END_REF], Hartigan [START_REF] Hartigan | Estimation of a convex density contour in two dimensions[END_REF], Polonik [START_REF] Polonik | Measuring mass concentrations and estimating density contour clustersan excess mass approach[END_REF], Tsybakov [START_REF] Tsybakov | On nonparametric estimation of density level sets[END_REF], Walther [START_REF] Walther | Granulometric smoothing[END_REF]. This large number of works on this subject is motivated by the high number of possible applications. Estimating these level sets can be useful in mode estimation (Müller and Stawitzki [7],

Polonik [START_REF] Polonik | Measuring mass concentrations and estimating density contour clustersan excess mass approach[END_REF]), or in clustering (Biau, Cadre and Pelletier [START_REF] Biau | A graph-based estimator of the number of clusters[END_REF], Cuevas, Febrero and Fraiman [START_REF] Cuevas | Cluster analysis: a further approach based on density estimation[END_REF][START_REF] Cuevas | Estimating the number of clusters[END_REF]). In particular, Biau, Cadre and Pelletier [START_REF] Biau | A graph-based estimator of the number of clusters[END_REF] use an estimator of the level sets of the density function to determine the number of clusters.

The same applications are possible with the regression function. Moreover, it is for instance possible to use an estimator of the level sets of the regression function to determine the path of water ow from a digital representation of an area. In the same vein, in medical imaging, people want to estimate the areas where some function of the image exceeds a xed threshold. In medical decision making, we can also nd a lot of applications. For instance, the severity of the cancer is characterized by a variable Y which directly impacts the choice of standard or aggressive chemotherapy. For osteosarcoma [START_REF] Man | Expression proles of osteosarcoma that can predict response to chemotherapy[END_REF], Y is the percent necrosis in the tumor after a rst round of treatment. If Y > 0.9 (this threshold has been xed by experts and is now the convention), the aggressive chemotherapy will be chosen. The problem is that Y is measured using an invasive biopsy. If we can collect from the patient a feature vector X (which acquisition is easier), such as gene expression levels, knowledge of the regression level sets would allow the choice of an ecient treatment planning without a biopsy. Note that, in these examples, the use of a compact set J is fully justied. This is generally the case in most practical situations, particularly in image analysis.

Despite the many potential applications, the estimation of the level sets of the regression function has not been widely studied. Müller [START_REF] Müller | The excess mass approach in statistics[END_REF] mentioned it briey in his survey. Willett and Nowak [START_REF] Nowak | Minimax optimal level-set estimation[END_REF] obtained minimax rates (for dierent smoothness classes) for estimators based on recursive dyadic partitions. Scott and Davenport [START_REF] Scott | Regression level set estimation via costsensitive classication[END_REF] use a cost sensitive approach and a dierent measure of risk. Cavalier [START_REF] Cavalier | Nonparametric estimation of regression level sets[END_REF] and Polonik and Wang [START_REF] Polonik | Estimation of regression contour clusters: an application of the excess mass approach to regression[END_REF] used estimators based on the maximization of the excess mass which was introduced by Müller and Sawitzki [START_REF] Müller | Excess mass estimates and tests for multimodality[END_REF] and Hartigan [START_REF] Hartigan | Estimation of a convex density contour in two dimensions[END_REF]. Cavalier demonstrated asymptotic minimax rate of convergence for piecewise polynomial estimators using smoothness assumptions on the boundary of the level sets. We used a dierent approach and construct a plug-in estimator using the kernel estimator of the regression. The main advantage of our estimator is the simplicity of his calculation, inherited from the plug-in approach. Moreover, our estimator does not require strong assumptions on the shape of level sets.

All our consistency results are in the sense of the symmetrical dierence (Figure 1), dened by

L n (t)∆L(t) = (L n (t) ∩ L C (t)) ∪ (L C n (t) ∩ L(t)).
Figure 1: Symmetrical dierence (in black) between two sets A (in red) and B (in blue).

Our goal is to establish some consistency results under reasonable assumptions on r and rn . Using a kernel estimator for r, we get a rate of convergence equivalent to the one obtained by Cadre [START_REF] Cadre | Kernel estimation of density level sets[END_REF] for the density function.

This paper is organized as follows. The denition of our estimator and consistency results are given in Section 2. In Section 3 we confront our estimator to simulated data. Finally, proofs are collected in Section 4.

Main results

Construction of the estimator

As announced, we use a plug-in approach. That is, given an estimator r n of r we estimate {x ∈ Λ : r(x) > t} by {x ∈ Λ : r n (x) > t}. To estimate r, we choose to consider a kernel estimator.

Assume that we can write

r(x) = ϕ(x) f (x) ,
where f is the density function of X, and ϕ is dened by ϕ(x) = r(x)f (x).

Let K be a kernel on R d , that is a probability density on R d . We denote h = h n and K h (x) = K(x/h). From an i.i.d. sample (X 1 , Y 1 ), . . . , (X n , Y n ) , we dene, for all x ∈ R d ,

ϕ n (x) = 1 nh d n i=1 Y i K h (x -X i ) and f n (x) = 1 nh d n i=1 K h (x -X i ).
For all x ∈ R d , the kernel estimator of r is then dened by

r n (x) = ϕ n (x)/f n (x) if f n (x) = 0 0 otherwise.
The properties of this estimator are already well studied in the litterature. For instance, the interesting reader can look at Bosq and Lecoutre [START_REF] Bosq | Théorie de l'Estimation Fonctionnelle[END_REF] or Gasser and Müller [START_REF] Gasser | Kernel estimation of regression functions[END_REF].

Under the assumption A0 There exists t -< t such that L(t -) is compact. Besides, λ({r = t}) = 0

(where λ stands for the Lebesgue measure), a rst consistency result can be trivialy obtained from a slight modication of Theorem 3 by Cuevas, González-Manteiga and Rodríguez-Casal [START_REF] Cuevas | Plug-in estimation of general level sets[END_REF] and the consistency properties of the kernel estimator. A1 The functions r and f are twice continuously dierentiable, and, ∀t ∈ Θ , ∃0 < t -< t : inf

L(t -) f > 0; A2 For all t ∈ Θ, inf r -1 ({t}) r > 0,
where, ψ(x) stands for the gradient at x ∈ R d of the dierentiable function ψ : R d → R.

The assumptions A1 on the regularity are inherited from the classical assumptions in kernel estimation [START_REF] Bosq | Théorie de l'Estimation Fonctionnelle[END_REF]. Note that "harder" assumptions on the regularity of r and f will not improve the obtained rate of consistency. Moreover, let us mention that under Assumptions A1 and A2, we have

(Proposition A.2 in [1]) ∀t ∈ Θ : λ(r -1 [t -ε, t + ε]) → 0 as ε → 0.
Let us now introduce the assumptions on the kernel K.

A3 K is a continuously dierentiable with a compact support. Moreover, there exists a decreasing function µ :

R + → R such that K(x) = µ( x ) for all x ∈ R d .
We are now in a position to establish a rate of convergence for E λ(L n (t)∆L(t)).

Theorem 2.1.

Under Assumptions A0 -A3, if nh d /(log n) → ∞ and nh d+4 log n → 0, then for almost all t ∈ Θ E λ(L n (t)∆L(t)) = O(1/ √ nh d ).
Remarks :

• Roughly speaking, the assumptions about the bandwidth impose to take h between ( logn n )

1 d and (n log n) -1 d+4 . Moreover, if we take h = O((n log n) -1 d+4 ), we get √ nh d = O n 2 d+4 (log n) d 2(d+4) = O n 1/3 (log n) 1/6 with d = 2,
that is a rate of the same order as Cadre [START_REF] Cadre | Convergent estimators for the L 1 -median of a Banach valued random variable[END_REF] in the density case.

• A remaining and crucial problem is the research of an optimal bandwidth h for our estimator. Indeed, if they are already results in the literature about an optimal bandwidth for the estimation of r, this bandwidth is not necessarily optimal for estimating L(t). However, in the simulations, we used a cross-validation procedure to choose a bandwidth.

Discussion about the rate

In this section, we provide a short comparison with the estimator proposed by

Cavalier. Indeed, we choose this estimator because it is proven to be optimal [START_REF] Cavalier | Nonparametric estimation of regression level sets[END_REF].

The main idea of this estimator is that the level set L(t) minimises the excess mass

M (G) = G f (x)dx -t * λ(G).
Starting from this, Cavalier proposes to introduce estimators with piecewisepolynomial structure based on the maximization of local empirical excess mass.

Assuming that L(t) can be expressed as

L(t) = {x = (r, ϕ), 0 ≤ r < 2π},
with g a 2π-periodic continuous function on R, one starts by computing a piecewise-polynomial estimator ĝ of g. Then, the estimate of L(t) is given by the closure of

{(r, ϕ) : 0 ≤ r < ĝ(ϕ), 0 ≤ ϕ < 2π}.
Note that this estimate is always star-shaped.

Depending on the used kind of design points, Cavalier obtains optimal rates of consistency.

If our estimator fails to get an optimal rate, its main advantage is its simplicity. Indeed, where getting the estimator ĝ of g could be a little dicult, our estimator is really easy to implement. One only needs to do is compute a kernel estimation of the regression function (with one of the various existing R packages) and use the results to estimate the level set. Moreover, despites the regularity assumptions for f and r inherited from the use of a kernel estimator, our rate of consistency is obtained for general shapes of level sets. For example, we do not require that the level sets are star-shaped.

Study of nite sample behavior

In this section, we illustrate our method on a simple simulated data set. Consider the function r dened on R 2 (Figure 2) by [START_REF] Cadre | Convergent estimators for the L 1 -median of a Banach valued random variable[END_REF][START_REF] Cadre | Convergent estimators for the L 1 -median of a Banach valued random variable[END_REF]. The size of the square is large enough to contain the level sets we will consider. We set Y i = r(X i ) + ε i , where (X 1 , . . . , X n ) is an i.i.d sample distributed as X, and (ε 1 , . . . , ε n ) is an i.i.d. sample with a normal distribution centered on 0 and with standard deviation 0.1 (X ≡ N (0, 0.1)).

r : (u, v) → sin(u) + sin(v) log √ u 2 + v 2 + 1 .

Illustration of the rate

In this section we illustrate our theoretical rate of convergence obtained in The-

Selection of h by cross-validation

Now we use a simple cross-validation to select the bandwidth: For each value of n we use half the dataset to compute the kernel estimator with h (and the level set estimator) on a grid of 20 values between the limits allowed by the assumptions of Theorem 2.1 (see Remark 1 below Theorem 2.1). Then, we use the remaining part of the dataset to evaluate the volume of the symmetrical dierence and select the optimal h. We compare the error obtained to the previous ones in Figure 4. We see that our choice process of h does not improve the estimation of the level sets. If we compare the error multiplied by √ nh 2 (Figure 5), we see that we select a lower bandwidth. However, we cannot generalize about it since we are here in a very simple case.

Moreover, we use a naive cross-validation method. Looking for more ecient methods to derive an optimal bandwidth for the level-set estimation is still an interesting and opened question. For this, we could rst think of the adaptation of method used for density level sets estimation like Rinaldo, Singh, Nugent and Wasserman [START_REF] Rinaldo | Stability of Density-Based Clustering[END_REF] or Samworth and Wand [START_REF] Samworth | Asymptotics and optimal bandwidth selection for highest density region estimation[END_REF] for example.

Proofs

This section is dedicated to the proof of Theorem 2.1. From now on, c is a non-negative constant, which value may change from line to line.

Proof of Theorem 2.1

In this proof, some arguments are classical result from the kernel density (or regression) estimation theory. For more details, we refer the reader to the book by Bosq and Lecoutre [START_REF] Bosq | Théorie de l'Estimation Fonctionnelle[END_REF], chapter 4 and 5.

From now on, we denote by ∂A the boundary of any subset A ⊂ R d . Besides, we introduce H the (d -1)-dimensional Hausdor measure (Evans and Gariepy [START_REF] Evans | Measure Theory and Fine Properties of Functions[END_REF]). Recall that H agrees with ordinary (k -1)-dimensional surface area on nice sets (Proposition A.1 in [START_REF] Cadre | Kernel estimation of density level sets[END_REF]). Finally, we set K = K 2 dλ.

Preliminary results

All the results in this sections are stated under Assumptions A0 -A3. The proof of the theorem relies on the four following lemmas.

Let us dene

Ω n,c = √ nh d sup Ln(t)∪L(t)
|r n -r| ≥ c log n .

Lemma 4.1. If nh d+4 / log n → 0, then there exists Γ > 0 such that

√ nh d P(Ω n,Γ ) → 0.
Note that the condition nh d+4 / log n → 0 is satised under the assumptions of Theorem 2.1.

Proof of Lemma 4.1

As r is continuous, we have sup

L(t -) |r| < c. Assuming that inf L(t -) f > 0, then, since sup L(t -) |f n -f | → 0 a.s.
under the assumptions of Lemma 4.1 (Bosq and Lecoutre [START_REF] Bosq | Théorie de l'Estimation Fonctionnelle[END_REF]), there exists θ > 0 such that inf L(t -) f n > θ a.s. for n large enough. So we can write sup

L(t -) |r n -r| = sup L(t -) ϕ n -ϕ f n + r f n -f f n ≤ c sup L(t -) |ϕ n -ϕ| + sup L(t -) |f n -f | . (1)
We have sup

L(t -) |ϕ n -ϕ| ≤ sup L(t -) |ϕ n -E ϕ n | + sup L(t -) |E ϕ n -ϕ|. We cover L(t -) with n balls B k = B(x k , ρ n ) (k = 1, . . . , n ) of radius ρ n .
Consider x ∈ L(t -), we denote by B k the ball containing x. Then we set, for x, x ∈ L(t -),

A n (x, x ) = 1 n n i=1 Y i [K h (x -X i ) -K h (x -X i )] -E 1 n n i=1 Y i [K h (x -X i ) -K h (x -X i )] ,
which leads us to sup

L(t -) |ϕ n -ϕ| ≤ sup 1≤k≤ n |ϕ n (x k )-Eϕ n (x k )|+ sup x∈L(t -) |A n (x, x k )|+ sup L(t -) |E ϕ n -ϕ|. (2)
Then, since K is Lipschitz, there exists γ > 0 such that

|A n (x, x k )| ≤ ch -d-γ ρ γ n 1 n n i=1 |Y i | + E|Y | ≤ ch -d-γ ρ γ n since Y is bounded.
As a consequence, we have

P sup x∈L(t -) |A n (x, x k )| > c 4 log n nh d ≤ P ch -d-γ ρ γ n > c 4 log n nh d .
One can choose

ρ n = n -a , a > 0 and ρ γ n = o h d+γ log n nh d , such that P sup x∈L(t -) |A n (x, x k )| > log n/ √ nh d = 0. (3)
Then, using the arguments of the proof of Theorem 5.II.3 in [START_REF] Bosq | Théorie de l'Estimation Fonctionnelle[END_REF], we obtain ∀ε > 0, P sup

1≤k≤ln |ϕ n (x k ) -Eϕ n (x k )| > ε < 2 n e -nh d ε 2 c .
If we set ε = ε 0 log n/nh d , we have

P sup 1≤k≤ln |ϕ n (x k ) -Eϕ n (x k )| > ε 0 log n nh d ≤ c n n -2ε0/c
≤ cn -2ε0/c ρ -d n .

Remember that ρ n = n -a , with a > 0, one gets

√ nh d P sup 1≤k≤ln |ϕ n (x k ) -Eϕ n (x k )| > ε 0 log n nh d ≤ cn 1/2+ad-2ε0/c √ h d (4)
which tends to 0 choosing ε 0 > (1/2+ad)c 2 .

Moreover, under A3, K is even which gives us sup

L(t -) |E ϕ n -ϕ| = O log n nh d ,
Then, the Berry-Esseen inequality [START_REF] Berry | The accuracy of the Gaussian approximation to the sum of independent variables[END_REF] gives us

|P(r n (x) < t)-Φ(t n (x))| ≤ c nV n (x, t) 3 E |(Y -t)K h (x -X) -E(Y -t)K h (x -X)| 3 . (6)
Finally, under Assumptions A1 and A3, we have (see for example Bosq and

Lecoutre [START_REF] Bosq | Théorie de l'Estimation Fonctionnelle[END_REF])

sup x∈L(t -) |(Y -t)K h (x -X) -E(Y -t)K h (x -X)| 3 ≤ ch d and inf x∈L(t -) V n (x, t) ≥ ch d .
The lemma can then be deduced from [START_REF] Walther | Granulometric smoothing[END_REF].

Dene now Θ 0 the set of all t in Θ such that lim ε 0

1 ε λ r -1 [t -ε, t] = lim ε 0 1 ε λ r -1 [t, t + ε] = ∂L(t)
r -1 dH.

The following result is proven in Cadre [START_REF] Cadre | Kernel estimation of density level sets[END_REF] (Lemma 3.2).

Lemma 4.3. Θ 0 = Θ almost everywhere.

Note that under Assumptions A1 and A2, we obtain, thanks to Proposition

A.2 in [1], λ r -1 [t -ε, t + ε] = λ r -1 (t -ε, t + ε) ,
for all t ∈ Θ and ε > 0 small enough.

Finally, we set

v(x) = Var(Y |X = x) + r 2 (x),
and, for t ∈ Θ and x ∈ L(t -),

t n (x) = f (x) nh d Kf (x)(v(x) + t 2 ) (t -r(x)).
We are now in a position to prove Lemma 4.4 below.

Lemma 4.4. If nh d /(log n) → ∞ and nh d+4 log n → 0, then for all t ∈ Θ 0 ,

lim n→∞ √ nh d V t n P(r n (x) < t)dx - V t n Φ(t n (x))dx = 0, and lim n→∞ √ nh d V t n P(r n (x) > t)dx - V t n Φ(t n (x))dx = 0.
Proof of Lemma 4.4 We only prove the rst equation, the second one can be obtained with similar arguments.

Dene E n by E n = √ nh d V t n |Φ(t n (x))dx -Φ(t n (x))dx|.
As Φ is Lipschitz we have

E n ≤ c √ nh d λ(V t n ) sup V t n |t n -t n |. (7)
By denition of t n (x) and t n (x), we have, for all x ∈ V t n ,

1 √ nh d |t n (x) -t n (x)| ≤ |t -r(x)| f (x) Kf (x)(v(x) + t 2 ) - E f n (x) V n (x, t)h -d + h d V n (x, t) E f n (x)|r(x) -E r n (x)| ≤ log n nh d |f (x)V n (x, t)h -d -(E f n (x)) 2 K(v(x) + t 2 )| K(v(x) + t 2 )V n (x, t)h -d + h d V n (x, t) E f n (x)|r(x) -E r n (x)| (8)
Remember that

| E r n (x) -r(x)| ≤ 1 f n (x) |Eφ n (x) -φ(x)| + |r(x)||E|f n (x) -f (x)| (9)
Since V t n is included in L(t -), we can deduce (Bosq and Lecoutre [START_REF] Bosq | Théorie de l'Estimation Fonctionnelle[END_REF]) from A1, A3 and (9) that sup

x∈V t n | E r n (x) -r(x)| ≤ ch 2 . ( 10
) Moreover, if we set V 1 n (x) = Var K h (x -X), V 2 n = Var Y K h (x -X), we can write |f (x)V n (x, t)h -d -(E f n (x)) 2 K(v(x) + t 2 )| ≤ |f (x)| V n (x, t)h -d -KE f n (x)(v(x) + t 2 ) + c|f (x) -E f n (x)| ≤ |f (x)| V n (x, t)h -d -Kf (x)(v(x) + t 2 ) + c|f (x) -E f n (x)| ≤ |f (x)| t 2 |V 1 n (x)h -d -Kf (x)| + |V 2 n (x)h -d -Kf (x)v(x)| +2t |Cov (Y K h (x -X), K h (x -X))| + c|f (x) -E f n (x)| ≤ c |V 1 n (x)h -d -Kf (x)| + |V 2 n (x)h -d -Kf (x)v(x)| + |Cov (Y K h (x -X), K h (x -X))| + |f (x) -E f n (x)| .
Again, since V t n ⊂ L(t -), we can deduce (Bosq and Lecoutre [START_REF] Bosq | Théorie de l'Estimation Fonctionnelle[END_REF]) from A1 and A3 that sup

x∈V t n |f (x)V n (x, t)h -d -(E f n (x)) 2 K(v(x) + t 2 )| ≤ ch. (11)
We deduce from ( 8), ( 10) and ( 11) that

sup x∈V t n |t n (x) -t n (x)| ≤ c h log n + √ nh k+4 .
Then, thanks to [START_REF] Müller | Excess mass estimates and tests for multimodality[END_REF] and since t ∈ Θ 0 , we have for n large enough

E n ≤ c log n h log n + √ nh k+4 , (12)
which tends to 0 under the assumptions on h of Theorem 2.

I n = V t n Φ(t n (x))dx, I n = V t n
Φ(t n (x))dx.

We have

I n = 1 2π K V t n bn(x) -∞ exp -u 2
I n ≥ C 1 2π K V t n b n (x) -∞ exp -C 2 1 u 2 2 K
du dx, and

I n ≤ C 2 2π K V t n b n (x) -∞ exp -C 2 2 u 2 2 K du dx.
Using the arguments of the proof of Proposition 3.1 in [START_REF] Cadre | Kernel estimation of density level sets[END_REF], we obtain

C 1 t K √ 2π ∂L(t)

Figure 2 :

 2 Figure 2: Representation of r.

orem 2 . 1 . 1 √ nh 2 in Figure 3 .Figure 3 :

 21133 Figure 3: Illustration of the rate : The curve represent the estimated error multiplied by √ nh 2 .

Figure 4 :

 4 Figure 4: Comparison of the error: the plain line stands for h = (n log n) -1.1/6 , and the dotted line for h selected by a cross-validation.

Figure 5 :

 5 Figure5: Comparison of the rates: the plain line stands for h = (n log n) -1.1/6 , and dotted line for h selected by a cross-validation.

  where b n (x) = f (x)nh d (t -r(x))/ v(x) + t 2 . Besides, b n (x) = |ϕ(x)| v(x) + t 2 b n (x), with b n (x) = √ nh d (t -r(x))/ |r(x)|.Then we can nd two positive constants C 1 and C 2 (whose values will then change from line to line) such thatC 1 b n (x) ≤ b n (x) ≤ C 2 b n (x),which leads us to

  tends to 0 since λ(r -1 [t -ε, t + ε]) → 0. This and (12) ends the proof. -1 [t, t+Γ log n/nh d ]∩L(t -) and Vt n = r -1 [t-Γ log n/nh d , t]∩L(t -).

	and remember that
	V t n = r
			1. Finally, Lemma
	4.2 leads us to
	√	nh d
		V t
	4.1.2. Proof of Theorem 2.1
	We rst note that
	E λ L n (t)∆L(t) =	P(r n (x) < t)dx +	P(r n (x) ≥ t)dx.
		L(t -)∩{r≥t}	L(t -)∩{r<t}

n P(r n (x) < t)dx -V t n Φ(t n (x))dx ≤ cλ(V t n ) which Set P n,t (x) = P(r n (x) < t), P n,t (x) = P(r n (x) ≥ t)

Consider t ∈ Θ 0 and dene

  These inequalities, Lemma 4.4 and Lemma 4.3 concludes the proof.

					1 r	∂H ≤ lim	n→∞	√	nh d I n ≤ C 2	t K √ 2π ∂L(t)	1 r	∂H.
	With similar arguments, we have
	C 1	t 2π	K	∂L(t)	dH r	≤ lim n→∞	√	nh d I n ≤ lim n→∞	√	nh d I n ≤ C 2	t 2π	K	∂L(t)	dH r	.

n→∞ √ nh d I n ≤ lim

Acknowledgements : The authors thank the associate editor and two anonymous referees for relevant remarks and constructive comments on a previous version of the paper.

and, using that nh d+4 / log n → 0 we obtain √ nh d P sup

From (2) and using (3), ( 4) and ( 5) we obtain

From (1) and such as sup L(t -) |f n -f | → 0 a.s., we conclude the proof.

Consider t ∈ Θ. For all x ∈ L(t -), we dene

For all x ∈ L(t -) such that V n (x, t) = 0, we set

Besides, we consider the sets

Finally, we denote by Φ the distribution function of the standard normal N (0, 1), and we dene Φ(x) = 1 -Φ(x).

Lemma 4.2. There exists c > 0 such that for all n ≥ 1, t ∈ R and x ∈ L(t -):

Proof of Lemma 4.2

Set, for i = 1, . . . , n,

By denition, we have V n (x, t) = Var(Z (x, t)), and

Z i (x, t) -E Z(x, t) < t n (x) .