Some endpoint estimates for bilinear Coifman-Meyer multipliers

In this paper we establish mapping properties of bilinear Coifman-Meyer multipliers acting on the product spaces $H^1(\mathbb{R}^n)\times\mathrm{bmo}(\mathbb{R}^n)$ and $L^p(\mathbb{R}^n)\times\mathrm{bmo}(\mathbb{R}^n)$, with $1<p<\infty$. As application of these results, we obtain some related Kato-Ponce-type inequalities involving the endpoint space $\mathrm{bmo}(\mathbb{R}^n)$, and we also study the pointwise product of a function in $\mathrm{bmo}(\mathbb{R}^n)$ with functions in $H^1(\mathbb{R}^n)$, $h^1(\mathbb{R}^n)$ and $L^p(\mathbb{R}^n)$, with $1<p<\infty$.

The literature on mapping properties of bilinear Coifman-Meyer multipliers is vast, so we conceal our discussion to those works more relevant to the current paper. The study of those operators was initiated by R. Coifman and Y. Meyer [7], and can be seen as particular instances of bilinear Calderón-Zygmund operators. As such, it follows from the general theory by L. Grafakos and R. Torres [16], that these operators map L p (R n ) × L q (R n ) continuously into L r (R n ) under the relation 1/r = 1/p + 1/q, where 1 < p, q ≤ ∞ and 1/2 < r < ∞. Moreover, and relevant to this paper, the following endpoint estimates are also known: . Mapping properties of bilinear Coifman-Meyer multipliers acting on bmo(R n ) × bmo(R n ) were established in [23,Theorem 7.1]. Nevertheless, to the best of our knowledge, the boundedness properties of such operators acting on product spaces of the type L p (R m ) × bmo(R n ), for 1 < p < ∞, and H 1 (R m ) × bmo(R n ) is not covered by the results existing in the literature.
The aim of this paper is to study these remaining cases. In this way, our investigation is a continuation of that initiated in [23]. These results are obtained as a consequence (see Corollary 3.5) of a more general theorem (see Theorem 3.2 below), where we establish boundedness properties analogous to (i) and (ii), where L ∞ (R n ) is replaced by a suitable intermediate space X w (R n ), lying in between L ∞ (R n ) and bmo(R n ), associated to an admissible weight w. The adequate choice of weight, allows both to recover the existing boundeness properties (i) and (ii), as well as to obtain those involving bmo(R n ). The precise definition of admissible weights and that of the space X w (R n ) are given in Section 2.2 below.
The range of Coifman-Meyer multipliers acting on bmo(R n ) × bmo(R n ) obtained in [23], lies in a space of generalised smoothness, defined as a potential-type space, involving BMO(R n ), and denoted by J w 1 (BMO(R n )) in this paper (see Section 2.3 for its definition). This space is analogous to those introduced by R. S. Strichartz [26], J s (BMO) with s ∈ R, where J s = (1 − ∆) s/2 denotes the Bessel potential operator. Heuristically, the space J s (BMO) consists of "derivatives of order s" of functions in BMO(R n ), while the space J w 1 (BMO(R n )) consists of "derivatives of logarithmic order" of functions in BMO(R n ). The formal definition and properties of these spaces of generalised smoothness are given in Section 2.3 below.
The literature on the study of spaces of generalised smoothness, and those of logarithmic smoothness in particular, is considerably large. Among those we refer, for their relevance to this paper, to the works of V. Mikhailets and A. A. Murach [20], A. Caetano and S. Moura [5] and S. Moura [21]. Nevertheless, we also refer the interested reader to the recent work of O. Dominguez and S. Tikhonov [8] and the references therein, for a recent overview of the field and related results for function spaces with logarithmic smoothness.
As in [23], to obtain the mapping properties studied in this paper, we introduce a family of spaces of generalised smoothness, of potential-type, associated to either L p (R n ) or H 1 (R n ). We also show (see Proposition 2.26 and Proposition 2.29 below) that, in several cases, these spaces coincide with some spaces already existing in the literature [5,20,21].
The study of bilinear Coifman-Meyer multipliers can be reduced to that of paraproducts of the type where Q t and P t are frequency localisation operators and m is a bounded function (see Section 4). In this way, our main results are obtained by the use of new estimates for bilinear paraproducts (see Theorem 4.3).
One of the applications that follows from the main theorem of this paper is an endpoint inequality of Kato-Ponce-type. These kind of inequalities have been largely studied, and we used as a main reference the works by L. Grafakos and S. Oh [15], by provided s > 0, 1 ≤ p < ∞. We obtain related end-point estimates to this one (see Corollary 6.1 below), with L ∞ (R n ) replaced by bmo(R n ), albeit with the stronger restriction on s to be larger than 4n + 1.
Regarding the endpoint case s = 0, namely estimates for the product of two functions, one in BMO(R n ) and the other in the Hardy space H 1 (R n ), have been investigated by A. Bonami, T. Iwaniec, P. Jones and M. Zinsmeister in [2] and by A. Bonami, S. Grellier and L. D. Ky in [1]. Likewise, J. Cao, L. D. Ky and D. Yang [6] studied the counterpart problem where one of the terms lies in the local Hardy space h p (R n ), for 0 < p ≤ 1, and the other in the local bmo(R n ) space. In these studies, the product is decomposed in two terms, one belonging to L 1 (R n ), and the other in a suitable Musielak-Orlicz-Hardy space.
As a final application of our main results, we obtain some counterparts of these results for the range 1 ≤ p < ∞. In both Corollary 6.2 and Corollary 6.3, the product is realised in one of the potential-type spaces of generalised smoothness introduced in this paper. In addition, Corollary 6.4 allows the product to be decomposed in two terms, one belonging to L 1 (R n ), and the other in a Musielak-Orlicz-Hardy space.
This article is organised as follows. Section 2 is devoted to introduce the notions and tools needed to state and prove the main results. It is divided in three parts. In a first one, we introduce some function spaces and technical results related to them. In a second one, we introduce the notion of admissible weights, and the associated function spaces used throughout the paper. In the last part, we define the potentialtype spaces of generalised smoothness, and study some of their properties.
In Section 3 we state the main theorem of the article. The mapping properties for paraproducts (Theorem 4.3) are presented in Section 4. The proof of Theorem 3.2 is given in Section 5 and some of its consequences are introduced in Section 6.
Finally, we present an Appendix with a self-contained and direct proof of the L 2 -estimates obtained for paraproducts in Theorem 4.3 by interpolation.

Preliminaries
2.1. Generalities. The notation A B will be used to indicate the existence of a constant C > 0 such that A ≤ CB. Similarly, we will write A ≈ B if both A B and B A hold. We will also use the notation ξ = (1 + |ξ| 2 ) 1/2 for ξ ∈ R n .
The space of Schwartz functions will be denoted by S(R n ) and its topological dual, the space of tempered distributions, by S ′ (R n ). For a function f ∈ S(R n ) we define its Fourier transform as and we will write for appropriate symbols a, or simply a(D) when t = 1.
A diverse collection of function spaces will be appearing throughout the paper. Lets recall that the Hardy space H 1 (R n ), is the space of tempered distributions f for which the non-tangential maximal function (2) x → sup .
Here Φ is a Schwartz function with Φ = 1 and the notation Φ t (x) = t −n Φ(x/t) will be used from now on, with t > 0 and x ∈ R n . The local version of the Hardy space, denoted by h 1 (R n ) and introduced by D. Goldberg [11], is the space of tempered distributions f for which the truncated non-tangential maximal function belongs to L 1 (R n ), endowed with the norm given by .
The space of functions of bounded mean oscillation, BMO(R n ), which can be identified with the dual space of H 1 (R n ), is the set of all those locally integrable functions f on R n for which The supremum is taken over all cubes in R n whose sides are parallel to the axis, while |Q| denotes the Lebesgue measure of the cube Q and f Q is the average of f over Q, namely f Q = 1 |Q| Q f (x)dx. The local version of BMO(R n ) was considered by D. Goldberg in [11], and it will be denoted by bmo(R n ). It is defined to be the set of all locally integrable functions f on R n for which (4) f bmo(R n ) := sup Here ℓ(Q) denotes the side length of the cube Q. The function space bmo(R n ) is the dual space of h 1 (R n ) and it is continuously embedded in BMO(R n ). We recall (see e.g. [13, Section 3.3.1]) that a measure dµ(x, t) on R n+1 + is a Carleson measure if there exists a constant C > 0 such that (5) µ(T (Q)) ≤ C |Q| for all cube Q in R n , where T (Q) := Q×(0, ℓ(Q)]. The norm of the Carleson measure, denoted by µ C , is considered to be the infimum of the set of all constants C > 0 satisfying (5).
We shall also recall the relations between Carleson measures and functions in BMO(R n ), given in the the following result, whose proof can be found in [13, Then, for all function b ∈ BMO(R n ), the measure defined by is a Carleson measure with norm bounded by a constant times b 2 BMO(R n ) .
Theorem 2.2. Let K t (x, y), t > 0, be a collection of functions defined on R n × R n for which there exists δ > 0 such that for all t > 0 and x, y ∈ R n . Define for every t > 0 the linear operators Assume that R t 1 ≡ 0 for all t > 0 and that the estimate holds for all f ∈ L 2 (R n ). Then, for all b ∈ BMO(R n ), the measure defined by is a Carleson measure with norm bounded by a constant times b 2 BMO(R n ) . We shall also recall the following classical result of L. Carleson (see [25, p. 236] or [13, Corollary 3.3.6]).
holds, where F * denotes the nontangential maximal function We shall also need the following technical result shown in [24,Proposition 4.11].
Let G(t, x) be a measurable function on R n+1 + , and assume that the measure defined by 2.2. Admissible weights. The endpoint results obtained in this article involve some function spaces introduced in [23]. We shall first recall their definition (see [23,Definition 4.2]), and for convenience, gather together in Proposition 2.6 below, some of their properties discussed in that paper.
Definition 2.5. Let w : (0, ∞) → (0, ∞) be a weight function satisfying the following properties: Let φ be a Schwartz function with frequency support inside a ball centred at the origin, and set P t f := φ(tD)f . Then X w (R n ) is defined to be the set of all locally integrable functions for which Proposition 2.6. Let w be a weight function satisfying I),II) and III), and a function φ as above.
(1) The definition of the space X w (R n ) does not depend on the different choices of function φ, in the sense that different choices induce equivalent norms.
Remark 2.7. Given a weight satisfying I), II) and III) above, and given any positive constant c > 0, the weight cw satisfies the same properties, and X w (R n ) = X cw (R n ) with equivalence of norms. So, without loss of generality, multiplying w by a constant, one can assume that w ≥ 1. This is how the condition III) is stated in [23,Definition 4.2]. Moreover, the class of weights satisfying these conditions were called admissible in that paper. Here, we will reserve that terminology for those weights defined below.
Note also that, as a direct consequence of I), II) and III), w also satisfies that for The following definition is a minor modification of that in the paper of A. Caetano and S. Moura [5, Definition 2.1], that we shall adopt hereafter in this paper.
Definition 2.8. Let w : (0, 1] → (0, ∞) be a monotonic function, and extend it to w : (0, ∞) → (0, ∞) by defining w(t) = w(1) for all t ≥ 1. We say that w is an admissible weight if it satisfies that there exist c, d > 0 such that for all j ≥ 0 Example 2.9. Example of admissible weights are those functions of the form Indeed, it was observed in [5, Example 2.2] that weights defined on (0, 1] by an expression of the form |log cx| b are admissible, provided c ∈ (0, 1]. Note now that for t ∈ (0, 1), we can write Lemma 2.10. Let w be an admissible weight and let Θ : (0, ∞) → (0, ∞) be a monotonic function satisfying (6). Then Θ(w(t)) is also an admissible weight. In particular, w −1 , and cw for all constant c > 0 are also admissible.
The last part of the statement easily follows from the first part by taking Θ(t) equal to t −1 , and ct respectively. Remark 2.11. Using the previous lemma and the example above, one can construct other admissible weights such as Lemma 2.12. Let w be an admissible weight. Then w satisfies I) and II) in Definition 2.5 above. Moreover, condition III) holds for an admissible weight w if, and only if, w is either non-increasing, or satisfying that for all t > 0, w(t) ≈ 1.
Proof. We shall provide a proof of the first part of the statement for w being nonincreasing. The non-decreasing case is treated analogously. We know that, by [5, Lemma 2.3], there exists b ≥ 0 such that for all 0 < t ≤ 1 it holds that Since w is constant on [1, ∞), these inequalities imply from where w satisfies II). Using these two inequalities, and a change of variables in the case that t ≥ 1, we deduce that for all t > 0 These imply that which yields that w satisfies I).
In the non-increasing case, the condition inf t w(t) > 0 is equivalent to say that for all t > 0, w(t) ≥ w(1) > 0, which implies that w satisfies III). If w is non-decreasing, the condition inf t>0 w(t) = w(0 + ) > 0, is equivalent to the property that w(t) ≈ 1 for all t > 0.
Remark 2.13. Observe that if a non-decreasing admissible weight w satisfies inf t>0 w(t) = w(0 + ) > 0 then one has that w(t) ≈ 1. Hence, by Proposition 2.6, the space X w (R n ) in Definition 2.5 coincides with L ∞ (R n ).

2.3.
Generalised smoothness-type spaces. The rest of this section is devoted to the definition and main properties of the spaces of generalised smoothness that appear in the main result of this paper.
The following definition can be implicitly found in [5].
Definition 2.14. Let w be an admissible weight, and let (ϕ j ) j≥0 be a resolution of unity as above. We say that the function is the regularisation of w (associated to the resolution of unity (ϕ j ) j≥0 ).
It was shown in [5, Lemma 3.1], that both w and 1/w are smooth functions on R n such that for all multi-index α ∈ N n , and for all ξ ∈ R n In addition, using (8) and the fact that w(1/ |ξ|) = w(1) for all |ξ| ≤ 1 we obtain the estimates This motivates the terminology of regularisation, as w is smooth and also essentially encodes all the point-wise information of w.
Proof. The boundedness on L p (R n ) for 1 < p < ∞ can be found in [12, Theorem 6.2.7] and that on H 1 (R n ) can be found in [10, Theorem III.7.30]). Duality and self adjointness, yield the boundedness on BMO(R n ).
Definition 2.17. Let w be an admissible weight, and let w denote its regularisation given by (8). We define the linear operators defined initially for f ∈ S(R n ).
Proposition 2.19. Let w be an admissible weight, and let w be its regularisation.
(1) The operators J w and J w −1 are linear and continuous on S(R n ) and S ′ (R n ), being each other inverses. (2) If inf t>0 w(t) > 0, then w −1 ∈ S 0 (R n ), and so, for all 1 ≤ p ≤ ∞, Proof. Both operators are continuous on S(R n ) since w, 1/w are smooth and all their derivatives have at most polynomial growth. This yields that both J w and J w −1 are continuous on S ′ (R n ). In addition, by the commutativity of Fourier multipliers it follows that J w and J w −1 are each other inverses.
The assumption inf t>0 w(t) > 0 and (10) imply that the symbol w −1 ∈ S 0 (R n ), and so the boundedness is a direct consequence of the previous lemma.
Definition 2.20. Let w be an admissible weight and let w be its regularisation given by (8). We define the space Similarly one defines Proposition 2.21. Let 1 ≤ p ≤ ∞, and let w be an admissible weight. The definition of the spaces J w (X p ) and J w −1 (X p ) is independent of the resolution of the unity chosen to regularise w.
Proof. Let (ϕ 1 j ) j and (ϕ 2 j ) j be two resolutions of the unity, and let w 1 and w 2 be the corresponding regularisations of w.
For all multi-index α, the Leibniz rule, (9) and (10) yield and so w 2 w −1 1 ∈ S 0 (R n ). Analogously, one shows that w −1 2 w 1 ∈ S 0 (R n ). Let us prove that J w 1 (X p ) = J w 2 (X p ), by showing that their defining norms are equivalent. By the symmetry of the problem it is indeed enough to prove that J w 2 (X p ) ⊂ J w 1 (X p ). Lemma 2.16 and the commutativity of Fourier multipliers yield The independence of the definition of J w −1 (X p ) on the resolution of the identity is obtained analogously. So we omit the details.
Proposition 2.22. Let w be an admissible weight, and define the admissible weight u = w −1 . Let w and u denote respectively their regularisation given by (8). Then it holds that for all 1 ≤ p ≤ ∞, J w (X p ) = J u −1 (X p ) with equivalent norms. Namely Proof. With an argument similar to the one on the previous proposition, one obtains that both uw and (uw) −1 belong to S 0 (R n ).
The proof runs similarly, and it is a consequence of Lemma 2.16, so the details are left to the reader. Proposition 2.23. Let 1 ≤ p ≤ ∞, and let w be an admissible weight.
(1) For all 1 ≤ p < ∞, the space J w (X p ) endowed with the norm . Jw(X p ) is a Banach space. In the case p = ∞ we have that . Jw(BMO(R n )) defines a norm in J w (BMO(R n )) after identifying functions which differ almost everywhere by a constant, making it a Banach space; the Hölder conjugate exponent of p; (3) If inf t>0 w(t) > 0, then X p is continuously embedded in J w (X p ).
Proof. The fact that . Jw(X p ) defines a norm follows from (X p , . X p ) being a normed space and the linearity of J w −1 , when 1 ≤ p < ∞.
In the endpoint case p = ∞, the space of functions of bounded mean oscillation defines a normed space, provided that functions which differ almost everywhere by a constant are identified. As a consequence, if f Jw(BMO(R n )) = 0 then f = w(0)C in S ′ (R n ) for some constant function C. Hence the same identification is needed on J w (BMO(R n )).
Let 1 ≤ p ≤ ∞. To show the completeness of J w (X p ), let (f n ) n∈N be a Cauchy sequence in J w (X p ). Then (J w −1 f n ) n∈N is a Cauchy sequence in X p and, since X p is complete, we can find g ∈ X p for which J w −1 f n → g in X p as n → ∞. Then J w g belongs to J w (X p ) and the sequence (f n ) n∈N converges to J w g in J w (X p ).
To show the second statement, notice that Λ ∈ (J w (X p )) * is equivalent to Λ • J w ∈ (X p ) * , which is identified with X p ′ . This implies the existence of a unique The other inclusion is obtained analogously.
Finally, the last statement follows from Proposition 2.19.
We notice that the potential-type spaces introduced above coincide with some Triebel-Lizorkin spaces of generalised smoothness for 1 < p < ∞, studied in [5, Section 2.2] and [21]. Let us start by recalling their definition. Following the notation at the beginning of this section, let {ϕ j } j≥0 denote a resolution of the unity.
Definition 2.24. Let s ∈ R, 0 < p < ∞, 0 < q ≤ ∞. Set w for an admissible weight. We define the space F s,w p,q (R n ) to be the set of tempered distributions f for which Remark 2.25. It was shown in [5] that the spaces F s,w p,q (R n ) are independent of the chosen resolution of the unity, in the sense that different resolutions, give rise to the same space with equivalent quasi-norms.
Proposition 2.26. Let w be an admissible weight and 1 < p < ∞. Then the potential-type space J w (L p (R n )) coincides with the Triebel-Lizorkin space F 0,1/w p,2 (R n ) of generalised smoothness, with equivalent norms. Namely, it holds that Proof. The fact that the space L p (R n ) and F 0 p,2 (R n ) coincide with equivalent norms (see e.g. [27]), and Proposition 2.22 yield where u stands for the regularisation of w −1 . Finally, the lifting property [5,Proposition 3.2] gives To finish this section, we shall point out yet another connection of some of the potential-type spaces studied in this paper, with spaces existing in the literature.
and by w b its regularisation given in Definition 2.14, which following the notation there, can be explicitly written as We would like to point out that the spaces J w b (L 2 (R n )) lay within the family of the so-called refined Sobolev scale H ϕ (R n ), which also coincide with the Hörmander spaces B 2,µ (R n ) introduced by L. Hörmander (see [ (ii) It holds that ϕ(t) ≈ ψ(t) for all t ≥ d. Then we define the space H ϕ (R n ) as the set of tempered distributions f for which their Fourier transform f is locally integrable in R n and satisfies More precisely, we have the following identification.
Proposition 2.29. Let b ∈ R. Then the potential-type space J w b (L 2 (R n )) coincides with the refined Sobolev space H ϕ (R n ), with ϕ(t) := w b (1/t), with equivalent norms. More precisely, it holds that Proof. We shall prove first that the function ϕ(t) = w b (1/t) satisfies the conditions required in Definition 2.28, with d = 1 and ψ = ϕ, which would yield (i).
The measurably condition is ensured by the monotonicity of the weights. Furthermore, we have that w b (1/t) and w −1 b (1/t) are both bounded on any interval of the form [1, c] for all c > 1. Indeed, if b ≥ 0 then w b satisfies A similar argument shows the property for b < 0.
To show (i), we notice that, for all λ > 0 To show the equivalence of norms, note that the Pancherel Theorem and (11) yield finishing the proof.
Remark 2.30. Let w be one of the weights considered in Remark 2.11 and denote by w its regularisation. Define ϕ(t) = w(1/t) for t ≥ 1. One can show that this function satisfies the hypothesis in Definition 2.28 since it behaves asymptotically like ψ(t) = log b 1 (t) log(log(t)) b 2 as t >> 1, which satisfies the condition (i) in Definition 2.28 as shown in [20, Example 1.1]. Hence, arguing as in the previous result, one can show that the associated spaces J w (L 2 (R n )) and H ϕ (R n ) coincide.

Main results
It was shown in [23, Theorem 7.1] that bilinear Coifman-Meyer multipliers map X w (R n ) × X w (R n ) continuously into the potential-type space of generalised smoothness J w (BMO(R n )). More specifically, one obtains the following from that result.
Theorem 3.1. Let σ(ξ, η) be a smooth function on R n × R n \ {(0, 0)} satisfying (1) and let T σ be the corresponding bilinear Coifman-Meyer multiplier. Let w be an admissible weight satisfying inf t>0 w(t) > 0 and let w be its regularisation given in Definition 2.14. There exists a constant C such that The aim of the present article is to extend the boundedness range of these bilinear multipliers to the case when one of the two arguments of T σ belongs to the space X w (R n ), while the other one is either in a Lebesgue space L p (R n ), with 1 < p < ∞, or is an element in the Hardy space H 1 (R n ). The obtained results involve, as in [23], potential-type spaces of generalised smoothness. On this occasion, Lebesgue and Hardy potential-type spaces arise. The following is the main result of this paper. Theorem 3.2. Let σ(ξ, η) be a smooth function on R n × R n \ {(0, 0)} satisfying (1) and let T σ be the corresponding bilinear Coifman-Meyer multiplier. Let w be an admissible weight satisfying inf t>0 w(t) > 0 and let w be its regularisation given in Definition 2.14.
(i) Given 1 < p < ∞, there exists a constant C > 0 such that holds for every f ∈ L p (R n ) and g ∈ X w (R n ).
(ii) The symbol σ can be decomposed as the sum of two symbols σ = σ g + σ b , such that we can find constants C ′ , C ′′ > 0 for which hold for every f ∈ H 1 (R n ) and g ∈ X w (R n ). [7,16]).

Remark 3.4.
Analysing the proof of the result above, we note that the conclusions of the theorem, can be achieved by requiring σ to satisfy (1) only for multi-indices α, β such that |α| + |β| ≤ 4n + 1.
If we consider the logarithmic admissible weight w 1 from Example 2.9, and taking into consideration that by Proposition 2.6 we have that bmo(R n ) = X w 1 (R n ), then Theorem 3.2 yields the following endpoint estimates.
Corollary 3.5. Let w 1 (t) = 1 + log + 1/t and let w 1 be its regularisation from Definition 2.27. Let σ and T σ be as in Theorem 3.2.
(1) Given 1 < p < ∞ we can find a constant C > 0 such that the estimate holds for every f ∈ L p (R n ) and g ∈ bmo(R n ).
(2) The symbol σ can be decomposed as the sum of two symbols σ = σ g + σ b such that we can find constants C ′ , C ′′ > 0 for which the estimates and hold for every f ∈ H 1 (R n ) and g ∈ bmo(R n ).

Boundedness of paraproducts
The strategy we follow to prove our main results, relies on obtaining estimates for paraproducts of the type where m(t) is a measurable bounded function on (0, ∞) and Q t , P t are frequency localisation operators, defined as follows. Given ψ ∈ S(R n ), whose Fourier transform is supported in a ring satisfying ∞ 0 ψ(tξ) 2 dt t < ∞, for all ξ = 0, and given φ ∈ S(R n ), whose Fourier transform is supported in a ball centred at the origin, we define the frequency localisation operators where t > 0 and f ∈ S(R n ).
We can then write Π as the sum of two bilinear operators, Here Q (i) t and P (i) t denote the frequency localisation operators associated to ψ (i) and φ (i) respectively, with i = 1, 2.
The boundedness properties of these operators on X w (R n ) × X w (R n ) was studied in [23,Theorem 5.2]. More specifically the following result is consequence of the proof of that theorem: Proposition 4.1. Let w be an admissible weight satisfying inf t>0 w(t) > 0 and let w be its regularisation given in Definition 2.14. We can find constants C ′ , C ′′ > 0 such that holds for all f ∈ BMO(R n ) and g ∈ X w (R n ), while holds for all f, g ∈ BMO(R n ). In consequence, as X w (R n ) ⊂ BMO(R n ), holds for all f ∈ BMO(R n ) and g ∈ X w (R n ).
We are interested in finding estimates for the paraproduct Π when one of the arguments belongs to the space X w (R n ), while the other function belongs to either a Lebesgue space L p (R n ), 1 < p < ∞, or the Hardy space H 1 (R n ).
When the term in X w (R n ) lies in the first argument, these type of boundedness properties are a direct consequence of results already existing in the literature, and we summarise them in the following lemma.
Lemma 4.2. Let w be an admissible weight satisfying inf t>0 w(t) > 0 and let w be its regularisation given in Definition 2.14.
(i) Let 1 < p < ∞. There is a constant C > 0 such that for all f ∈ X w (R n ) and g ∈ L p (R n ). (ii) There is a constant C > 0 such that for all f ∈ X w (R n ) and g ∈ H 1 (R n ).
Proof. If a function f ∈ X w (R n ) ⊂ BMO(R n ) then the linear operator g → Π(f, g) is of Calderón-Zygmund-type (see e.g. [13, Section 4]), and hence it is bounded on any L p (R n ) for 1 < p < ∞, and from H 1 (R n ) to L 1 (R n ) with operator norm at most a multiple of f BMO(R n ) , which is smaller or equal than f Xw(R n ) .
To complete the picture, we need to study the case where the second argument belongs to X w (R n ), while the first one is considered to be in either a Lebesgue space L p (R n ), with 1 < p < ∞, or the Hardy space H 1 (R n ). Theorem 4.3. Let w be an admissible weight satisfying inf t>0 w(t) > 0 and let w be its regularisation given in Definition 2.14.
(i) Let 1 < p < ∞. There is a constant C > 0 such that holds for every f ∈ L p (R n ) and g ∈ X w (R n ). (ii) We can find constants C ′ , C ′′ > 0 such that and hold for every f ∈ H 1 (R n ) and g ∈ X w (R n ).
Proof of Theorem 4.3. Let us begin by proving the last part of the second statement.
To this end lets fix first f ∈ H 1 (R n ) and g ∈ BMO(R n ). We shall prove that Π 2 (f, g) ∈ L 1 (R n ). By duality, it is enough to show that, for H ∈ L ∞ (R n ), we can estimate the expression Note that Theorem 2.1 yields that |(Q (1) t g)(x)| 2 t −1 dtdx is a Carleson measure, whose norm is bounded by a constant times g 2 BMO . Moreover, we have that sup t>0,x∈R n (P Therefore, Proposition 2.4 yields that It follows that Π 2 (f, g) belongs to L 1 (R n ) and To study the stated boundedness for Π 1 , let us fix f ∈ H 1 (R n ) and g ∈ X w (R n ). Consider now a function H of the form H = J w −1 h with h ∈ BMO(R n ). By duality it is enough to estimate the expression where v(t, x) = m(t)(P (1) t g)(x)/w(t). By the definition of the norm on X w (R n ), we have that (15) |v(t, x)| m L ∞ (R n ) g Xw(R n ) , for all t > 0 and x ∈ R n .
Next we can write w(t)(Q where R t is the integral operator defined by Let us now show that the linear operators R t and their kernels K t satisfy the hypotheses of Theorem 2.2. This would imply that the measure defined by is a Carleson measure with norm bounded by a constant times h 2 BMO(R n ) .
To this end, we begin by observing that R t 1 ≡ 0 for all t > 0. To find the kernel estimates, a change of variables and integration by parts yield (17) for any N ≥ 1. Note that the Leibniz rule, the fact that |ξ| ≈ 1 and (10) give Finally, using (17), (18) and the fact that ψ (2) is compactly supported we obtain for any integer N > n/2 that from where the estimates for K t follows.
Finally, the Plancherel theorem and (11) yield the quadratic estimate Then (15), (16) and Proposition 2.4 imply It follows by duality that J w −1 Π 1 (f, g) belongs to the Hardy space H 1 (R n ) and which shows the claimed estimate for Π 1 . Finally, let us show the first part of the statement. To this aim, notice that for a fixed g ∈ X w (R n ), if we consider the linear operator f → J w −1 Π 1 (f, g), by complex interpolation between the results above (see. e.g. [18]), and those in Proposition 4.1, it follows that for all 1 < p < ∞ and for all f ∈ L p (R n ) holds that with constant independent on f or g. Similarly, one obtains that for all 1 < p < ∞ holds for all f ∈ L p (R n ) and g ∈ BMO.
We know from Proposition 2.19 that L p (R n ) ⊂ J w (L p (R n )), from where it follows that L p (R n ) + J w (L p (R n )) = J w (L p (R n )). Using this, and the estimates above one shows the first statement of the theorem.

Proof of Theorem 3.2
Proceeding as in the proof of [7, Proposition 2], let us start by decomposing the symbol σ as the sum of two symbols, where τ 1 and τ 2 still satisfy (1) and τ 1 is supported in {|ξ| ≥ |η| /20}, while τ 2 is supported in {|ξ| ≤ |η| /10}, Next we consider a Schwartz function ψ which is frequency supported in the ring {4/5 ≤ |ξ| ≤ 6/5} and satisfies for all ξ = 0. In addition, we select a Schwartz function φ whose Fourier transform is supported in a ball, and is identically one in the frequency support of ψ. Then τ 1 can be written as For a fixed t > 0, the functionτ 1 is smooth and compactly supported away from the origin. Hence, for N ∈ N, the Fourier inversion formula yields where m 1 (t, u, v) = t −2n τ 1 (u/t, v/t)(1 + |u| 2 + |v| 2 ) N . By (19), the operator T τ 1 can be expressed as where Q u t and P v t are the frequency localisation operator associated to ψ u (x) := ψ(x + u) and φ v (x) := φ(x + v) respectively. An integration by parts argument, jointly with (1) for τ 1 , shows that m 1 (t, u, v) is uniformly bounded in its three variables. Also we can show that for all δ > 0, Next we observe that the bilinear operator dt t is similar to those studied in Section 4. Proceeding as in (12), we can write . Let us show the validity of part (ii) in the statement. To this end, fix two functions f ∈ H 1 (R n ) and g ∈ X w (R n ). Theorem 4.3(ii) shows that (20) Π Here P (1) (u, v) and P (2) (u, v) are polynomials in |u| and |v|, independent of the functions f and g.
Consequently, the operator T τ 1 can be written as τ 1 = τ In particular, by choosing N large enough, the Minkowskii integral inequality, (20) and (21) yield Proceeding in a similar same way as we did for T τ 1 , but interchanging the roles of ξ and η, we have that we can write the operator T τ 2 as Lemma 4.2(ii) yields that the paraproduct where P(u, v) is a polynomial in |u| and |v|, independent of f and g. By choosing N large enough, and using the Minkowskii integral inequality, we conclude that The proof of part (ii) in the statement finishes by taking σ g = τ 1 + τ 2 and σ b = τ 1 . Part (i) of the theorem is proved analogously by combining Lemma 4.2(i) and Theorem 4.3(i).

Applications
In this section we will derive some consequences from Corollary 3.5. Namely, we will give some endpoint inequalities of Kato-Ponce-type missing in the literature, and the reconstruction of the product of functions in the local space version of bmo(R n ) and the local Hardy space h 1 (R n ).
Throughout this section we shall fix the admissible weight w 1 (t) = 1 + log + 1/t and w 1 denotes its regularisation from Definition 2.27.
where B s j are bilinear Fourier multipliers. It was also shown there, that the symbol of both B s 1 and B s 2 satisfy (1) for all multi-indices α, β. Moreover, following the argument in [14,Section 4], one shows that B s 3 satisfies (1) for |α| + |β| ≤ s. Then, in order to apply the results obtained in this paper, as it was pointed out in Remark 3.4, we require s > 4n + 1.
Then, the result is a direct application of Corollary 3.5 to each term.
6.2. Product of functions. The Corollary 3.5 above enables us to study the product of a function in bmo(R n ) with a function in either the Lebesgue space L p (R n ), with 1 < p < ∞, the Hardy space H 1 (R n ) or the local Hardy space h 1 (R n ). Indeed, taking the symbol σ ≡ 1, the following result, that corresponds to Corollary 6.1 for s = 0, follows directly from Corollary 3.5.
Corollary 6.2. Let 1 < p < ∞. Then, for all f ∈ L p (R n ) and g ∈ bmo(R n ) it holds that The case where one of the terms belongs to h 1 (R n ) and the other one to bmo(R n ), is not a direct consequence of our results. Nevertheless, the relation between H 1 (R n ) and its local version, as well as some properties of the latter space and local bmo(R n ) allows to obtain the following. Corollary 6.3. There exist two continuous bilinear operators on the product space Proof. Pick Φ any Schwartz function for which Φ = 1. Let us consider a function f in h 1 (R n ) and a function g in bmo(R n ). So we can decompose f as So, at least formally, we have that we could define the product of f and g as (22) f g := (Lf )g + (Hf )g, provided we could make sense of the right hand side term of this expression. Notice first that [11,Lemma 4] implies that We can interpret the second term in (22) as a bilinear Coifman-Meyer multiplier, where the symbol is identically one, acting on the product space H 1 (R n ) ×bmo(R n ). In this way, Corollary 3.5 provides a decomposition (Hf )g = B (1) (1) 1 and B 2 are two bilinear operators satisfying the estimates and The first term in (22) can we written as B 1 (f, g) := (Lf )g. We shall see that this term is actually a function in L 1 (R n ). Indeed, let {Q i } i≥0 be a countable collection of cubes, all of them with fixed sidelength ℓ ≤ (4n) −1/2 , independent on i, such that it gives a partition of R n . For all i ≥ 0, we denote byQ i the dilation of the cube Q i with sidelength 1. Then we have that where the last inequality follows from (4).
For every i ≥ 0 and for all x,x ∈ Q i , it holds that |x −x| ≤ √ nℓ < 1/2.
where we are using the independence of the h 1 (R n ) norm on the chosen function Ψ. Finally, using this in (24), we obtain that B 1 (f, g) To finish the proof, it is enough to define the bilinear operator B 1 := B 1 +B 1 .
The boundedness of the second term in (22), was obtained by applying the results of Corollary 3.5 above, as it was expressed as a bilinear Coifman-Meyer multiplier whose symbol is identically one. However, since by (23), Hf defines a function in H 1 (R n ) and bmo(R n ) ⊂ BMO(R n ), we could use instead [1, Theorem 1.1] to show that (Hf )g = S(Hf, g) + T (Hf, g), where S : H 1 (R n ) × BMO(R n ) → L 1 (R n ) and T : H 1 (R n ) × BMO(R n ) → H log (R n ).
Using this, one obtains the following result.
Corollary 6.4. There exist two continuous bilinear operators on the product space h 1 (R n ) × bmo(R n ), respectively B 1 : h 1 (R n ) × bmo(R n ) → L 1 (R n ) and B 2 : h 1 (R n ) × bmo(R n ) → H log (R n ), such that f g = B 1 (f, g) + B 2 (f, g). More specifically, a decomposition of the product of functions in h 1 (R n ) and bmo(R n ) into two bilinear operators, one of them mapping continuously h 1 (R n ) × bmo(R n ) into L 1 (R n ), and the other one mapping continuously h 1 (R n ) ×bmo(R n ) into h log (R n ), where h log (R n ) is a local Musielak-Orlicz-Hardy space related to the growth function defined in (25).
In particular, since the inclusion H log (R n ) ⊆ h log (R n ) holds, we notice that Corollary 6.4 above slightly improves [6, Theorem 1.1 and Remark 3.1].
Remark 6.6. It would be an interesting problem to study the relationship between the potential space J w (H 1 (R n )) and the Musielak-Orlicz-Hardy space H log (R n ) appearing in [1], as well as the space h Φ * (R n ) in [6, Theorem 1.1]. The following result can be obtained from either Corollary 6.3, or from Corollary 3.5 by using the embedding H 1 (R n ) ⊆ h 1 (R n ). Corollary 6.7. There exist two continuous bilinear operators on the product space H 1 (R n ) × bmo(R n ), respectively B 1 : H 1 (R n ) × bmo(R n ) → L 1 (R n ) and B 2 : H 1 (R n ) × bmo(R n ) → J w 1 (H 1 (R n )), such that f g = B 1 (f, g) + B 2 (f, g).

Appendix: The L 2 estimates for paraproducts
Although the case p = 2 is covered by interpolation in Theorem 4.3, we shall give here a direct and self-contained proof that relies only on the use of Plancherel's Theorem and quadratic estimates. More precisely, we shall prove the following result.
Proposition 7.1. Let w be an admissible weight satisfying inf t>0 w(t) > 0 and let w be its regularisation given in Definition 2.14. There is a constant C > 0 such that the estimate (26) Π(f, g) Jw(L 2 (R n )) ≤ C f L 2 (R n ) g Xw(R n ) holds for all f ∈ L 2 (R n ) and g ∈ X w (R n ).
Finally, combining (33) and (32) we deduce that for all f, h ∈ L 2 (R n ) and g ∈ X w (R n ), which by duality yields (26), and thus finishing the proof of the result.