Approaching Bilinear Multipliers via a Functional Calculus

We propose a framework for bilinear multiplier operators defined via the (bivariate) spectral theorem. Under this framework, we prove Coifman–Meyer type multiplier theorems and fractional Leibniz rules. Our theory applies to bilinear multipliers associated with the discrete Laplacian on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^d,$$\end{document}Zd, general bi-radial bilinear Dunkl multipliers, and to bilinear multipliers associated with the Jacobi expansions.


Introduction
The theory of spectral multipliers is now a well-established and vast branch of linear harmonic analysis. Its origins lie in trying to extend the Fourier multiplier operators on R given by to other settings. Here m is a bounded function on R whilef (ξ ) = R f (x)e −i xξ dx, ξ ∈ R. For a self-adjoint operator L, its spectral multipliers are the operators m(L) defined by the spectral theorem. In the Fourier case, L is merely i d dx . As in the Fourier to L p , for some 1 < p < ∞, p = 2. The bilinear multipliers for the Fourier transform are the operators x ∈ R, (1.1) with m : R 2 → C being a bounded function. As far as we know, in the bilinear case, there has been no systematic approach to extend the operators F m outside of the Fourier transform setting. The main idea behind the creation of this paper is to provide a theory for bilinear multipliers defined by the (bivariate) spectral theorem that parallels the correspondence between the linear Fourier multipliers and spectral multipliers. Our starting point is the observation that (1.1) may be rephrased as Here, ∂ 1 , ∂ 2 denote the partial derivatives, while m(i∂ 1 , i∂ 2 ) is defined by the bivariate spectral theorem. Note that ∂ 1 = ∂ ⊗ I and ∂ 2 = I ⊗ ∂, where ∂ denotes the derivative on R, while I is the identity operator. We investigate the possibility of replacing i∂ 1 and i∂ 2 by some other operators L 1 = L ⊗ I and L 2 = I ⊗ L. The bilinear multipliers we consider are of the form Here L is a self-adjoint non-negative operator on L 2 (X, ν), and m(L 1 , L 2 ) is defined by the bivariate spectral theorem. We also assume that L is injective on its domain, and that the contractivity condition (CT) (see p. 8) and the well definiteness condition (WD) (see p. 5) are satisfied. These assumptions should be regarded as technical ones. The main assumptions on L that are in force in this paper are the existence of a Mikhlin-Hörmander functional calculus (MH), see p. 4, together with a product formula for the spectral multipliers of L, see (PF) on p. 6. Roughly speaking (PF) states that spectral multipliers of L behave well under pointwise multiplication. There are two main goals of our paper. Firstly, we would like to prove Coifman-Meyer type multiplier theorems outside of the Fourier transform setting. Secondly, we would like to apply these results to obtain fractional Leibniz rules.
The main result of this paper is the following generalized Coifman-Meyer type theorem.
Theorem 2.3 is formally stated and proved in Sect. 2. The main difficulty in obtaining the theorem lies in finding an appropriate proof of the classical Coifman-Meyer multiplier theorem, which is prone to modifications towards our setting. The proof we present in Sect. 2 follows the scheme by Muscalu and Schlag [20, pp. 67-71] and is close to the original proof of Coifman and Meyer [7]. An important ingredient in our proof is a spectrally defined Littlewood-Paley theory. For this method to work, the assumption (PF) (see p. 6) is very useful. Unfortunately, this assumption is violated in some interesting cases. In particular, it fails whenever L has a discrete eigenfunction decomposition with the property that a product of eigenfunctions is not in the linear span of eigenfunctions. This happens for instance when L is the harmonic oscillator on R (in which case the Hermite functions constitute its basis of eigenfunctions). It would be interesting to try to replace (PF) with a less rigid condition. An application of Theorem 2.3 provides Coifman-Meyer type multiplier results for bilinear multipliers given by (1.2) in three cases different from the Fourier transform setting. In Theorem 3.1, we treat bilinear multipliers for L being the discrete Laplacian on Z d . This is close to [5,Theorem 3.4]; however, our results here are of a different kind. In Theorem 4.1, we consider bi-radial bilinear Dunkl multipliers; here, L is the general Dunkl Laplacian. In Corollary 4.2 we also reprove [3,Theorem 4.1]. Finally, in Theorem 5.1, we give a Coifman-Meyer type multiplier result for Jacobi trigonometric polynomials; here, L is the Jacobi operator.
The second main goal of this paper is to obtain fractional Leibniz rules for operators different from the Laplacian. The fractional Leibniz rule states that, if R d is the Laplacian on R d , then for each s ≥ 0 and 1/ p The proof of this inequality can be found in Grafakos and Ou [13], see also Bourgain and Li [6] for the endpoint case. The fractional Leibniz rule is also known as the Kato-Ponce inequality, as Kato and Ponce studied a similar estimate [16] (see also [17]). Generalizations of Kato-Ponce or similar inequalities were considered by many authors. For example, Gulisashvili and Kon [15] developed a fractional Leibniz rule which allowed derivatives of negative orders. Muscalu et al. [19] extended the Kato-Ponce inequality by admitting partial fractional derivatives in R 2 . Bernicot et al. [4] obtained the Kato-Ponce inequality in weighted Lebesgue spaces. Coulhon et al. [9] proved fractional Leibniz rules on Lie groups and Riemannian manifolds. Frey [12] obtained a fractional Leibniz rule for general operators satisfying Davies-Gaffney estimates and p 1 = p = 2, p 2 = ∞.
In the present paper, we obtain fractional Leibniz rules of the form where s > 0 and 1 It is straightforward to extend the result of this paper to the multilinear setting. However, to keep the presentation simple, we decided to limit ourselves to the bilinear case.
Throughout the paper, we use the variable constant convention, where C, C p , C s , etc. may denote different constants that may change even in the same chain of inequalities. We write X Y, whenever X ≤ CY, with C being independent of significant quantities. Similarly, by X ≈ Y we mean that C −1 Y ≤ X ≤ CY. By S(R d ) we denote the space of Schwartz functions. The symbols Z and N denote the sets of integers and non-negative integers, respectively. For a multi-index α ∈ N 2 by |α|, we denote its length α 1 +α 2 . Throughout the paper, for a function ψ : [0, ∞) → C we set

General Bilinear Multipliers
We say that a function μ : (0, ∞) → C satisfies the (one-dimensional) Mikhlin-Hörmander condition of order ρ ∈ N if it is differentiable up to order ρ and Similarly, we say that m : (0, ∞) 2 → C satisfies the (two-dimensional) Mikhlin-Hörmander condition of order s ∈ N, if the partial derivatives ∂ α m exist for multiindices |α| ≤ s and Consider a non-negative self-adjoint operator L on L 2 (X, ν) with domain Dom(L).
Here, (X, ν) is a σ -finite measure space with ν being a Borel measure. Throughout the paper, we assume that L generates a symmetric contraction semigroup, namely for 1 ≤ p ≤ ∞, and that L is injective on Dom(L). Then, for μ : (0, ∞) → C, the spectral theorem allows us to define the multiplier operator μ(L) Here, E is the spectral measure of L , while E f, f is the complex measure defined by We shall need the following assumption on L; L has a Mikhlin-Hörmander functional calculus of a finite order ρ > 0. More precisely, every function μ that satisfies (2.1) gives rise to an operator μ(L) which is bounded on all L p (X, ν), 1 < p < ∞, and Note that if L = (− R ) 1/2 then (MH) follows from the Mikhlin-Hörmander multiplier theorem.
There are two consequences of (MH) which will be needed later. The first of them is well known and follows from Khintchine's inequality.

Proposition 2.2
Let ϕ : [0, ∞) → C be compactly supported, and assume that ϕ ∈ C α ([0, ∞)) for some α > ρ + 2. Then the maximal operator To simplify the proof of our main Theorem 2.3, we will need an auxiliary subspace of L 2 (X, ν). Namely, consider the spaces For the convenience of the reader, we shall justify this statement. Let ψ : [0, ∞) → C be a smooth function which is supported in [1/2, 2] and such that k∈Z ψ k (λ) = 1, λ > 0. Then, for each N ∈ N and f ∈ L 1 (X, ν) ∩ L ∞ (X, ν), the partial sum S N f = N k=−N ψ k (L) f belongs to A by (MH). We claim that S N f → f in L p (X, ν). To see this, we take 1 < r < ∞ if p ≤ 2 or r > p if p > 2. Then we observe that S N f L r (X,ν) is uniformly bounded in N (this follows from (MH)) and that S N f → f in L 2 (X, ν) (this follows from the spectral theorem, since E {0} = 0 by the injectivity of L). Therefore, the log-convexity of L p norms proves the claim. Finally, a density argument together with the fact that S N f L p (X,ν) is uniformly bounded in N shows that A is dense in L p (X, ν) and finishes our task.
Besides being dense in L p (X, ν), the space A has the nice property that each f ∈ A satisfies f = is a fixed integer depending on f and ψ is the function from the previous paragraph. This allows us to deal easily with some rather delicate questions on convergence in the proof of Theorem 2.3.
We proceed to define formally the bilinear multipliers studied in this paper. To do this, we will need the operators L 1 = L ⊗ I and L 2 = I ⊗ L . These may be regarded as non-negative self-adjoint operators on L 2 (X × X, ν ⊗ ν), see [24,Theorem 7.23] and [28,Proposition A.2.2]. Moreover, the spectral measure of L 1 is E L ⊗ I, while the spectral measure of L 2 is I ⊗ E L . Thus, the operators L 1 and L 2 commute strongly and the bivariate spectral theorem, see e.g. [24,Theorem 5.21], allows us to consider multiplier operators Here m : In the most general form, the bilinear multiplier operators studied in the paper are given by is not formal. In order to make it rigorous, we assume that Thus, restricting m(L 1 , For instance, if L = (− R ) 1/2 , then the operator B m coincides with a bilinear multiplier for the Fourier transform. Namely, denoting If m is bounded and f 1 , is well defined (in fact continuous) by the Lebesgue dominated convergence theorem. We need one more assumption to prove the main theorem. Namely, we require that: there is b > 0 with the following property: if ϕ and ψ are bounded smooth functions such that supp whereψ k is a smooth function which is bounded by 1, equals 1 on We remark that, since , so that an application ofψ k (L) to g is legitimate. Note that when L = (− R ) 1/2 the formula (PF) can be easily deduced by using the convolution structure on the frequency space associated with Fourier multipliers.
In what follows we often abbreviate L p := L p (X, ν) and · p := · L p . Let p, p 1 , p 2 > 1. We say that a bilinear operator B is bounded from Note that in this case B has a unique bounded extension from L p 1 × L p 2 to L p .
The main result of this paper is a Coifman-Meyer type general bilinear multiplier theorem.

Theorem 2.3 Let L be a non-negative self-adjoint operator on L 2 (X, ν), which is injective on its domain and satisfies
Proof Let ψ be a smooth function supported in [1/2, 2] and such that k ψ k ≡ 1. We There is no issue of convergence here as for f 1 , f 2 ∈ A each of the sums defining T 1 , T 2 , and T 3 is finite. We estimate separately each of the operators T i , i = 1, 2, 3, starting with T 1 . This is the easiest part, in fact here the assumption (PF) is redundant.
Letψ be another smooth function, which vanishes outside [2 −b−4 , 2 b+4 ] and equals 1 on Then with the Fourier coefficients Now, using integration by parts, together with the assumption (2.2), and the fact that ψ ⊗ φ is compactly supported away from 0, we obtain the uniform in k ∈ Z bound We remark that here, in order to conclude (2.8), it is perfectly enough to assume the Marcinkiewicz 'product' condition Thus, m k can be expressed as By (2.8) and the bivariate spectral theorem, we have that for a.e. x 1 , x 2 ∈ X ; here we have convergence in L 2 (X × X, ν ⊗ ν). Moreover, (2.8) and the assumption (WD) imply that the above sum converges also pointwise (and gives a continuous function on X × X ). Consequently, for x ∈ X , we have where we have used the fact that the sum in k is finite when f 1 , f 2 ∈ A. Now Schwarz's inequality (first inequality below) and Hölder's inequality together with (2.8) (second inequality below) lead to the estimate Thus, taking into account the presence of the modulations e 2πin j 2 −k λ j /a in the definition of ψ n j k , j = 1, 2, and using Proposition 2.1, we obtain However, since we have the rapidly decaying factor in (2.9), if s > 2ρ + 4, we arrive at the desired bound Now we pass to estimating T 2 and T 3 . Since the proofs are mutatis mutandis the same, we treat only the former operator. Setting ϕ = j<−b−2 ψ j , we rewrite T 2 as Recall that in the above decomposition of T 2 all the appearing sums in k, k 1 , and k 2 , are in fact Set m k := ψ k ϕ k m and note that m k is supported in Similarly to the case of T 1 , we expand the function M k = m k (2 k λ) in a Fourier series. Namely, letψ be a smooth function vanishing outside [2 −2 , 2 2 ] and equal to 1 on [2 −1 , 2 1 ], and letφ be a smooth function vanishing outside [0, 2 −b ] and equal to 1 on with the Fourier coefficients As with T 1 , we now use integration by parts, together with the assumption (2.2). Here, it is important that we assume the stronger Mikhlin-Hörmander condition instead of merely the Mikhlin-Marcinkiewicz condition. Indeed, from integration by parts we obtain, for arbitrary β However, as ψ ⊗ ϕ does not vanish for λ 2 close to zero, in order to conclude that the above integral is uniformly bounded, we do need (2.2). In summary, we proved that (2.8) holds also in this case.
Coming back to m k we now write, Thus, m k , k ∈ Z, can be expressed as With the aid of (WD) and (2.8), arguing as on p. 7 we see that where the series on the right converges pointwise to a continuous function on X.
Summarizing the above, we have just decomposed , and the function ψ n 1 k is supported in [2 k−2 , 2 k+2 ], using the assumption (PF) we have Hence, if h is a function in L q , 1/ p + 1/q = 1, then we obtain and, consequently, where we used Proposition 2.1 withψ in the second inequality above. Similarly to the estimate for T 1 , applying Propositions 2.1 and 2.2 leads to Finally, the rapidly decaying factor in (2.10) gives, for s > 2ρ + 4, the desired bound The proof of Theorem 2.3 is thus completed.

Bilinear Multipliers on Z d
In the present section, we formalize Theorem 2.3 for bilinear multiplier operators on Z d . We also prove a fractional Leibniz rule for the discrete Laplacian. Let e j = (0, . . . , 1, . . . , 0) ∈ Z d be the j-th coordinate vector. Consider the discrete Laplacian on Z d , given by Then, since the formula (2.6) takes the form where n ∈ Z d . Note that the space A 2 from (2.5) in this case is given by Throughout this section, we denote by L p the space l p (Z d ) equipped with the counting measure. Using Theorem 2.3, we prove the following Coifman-Meyer multiplier theorem for the discrete Laplacian. Then the bilinear multiplier operator given by (3.1) is bounded from L p 1 × L p 2 to L p , where 1/ p 1 + 1/ p 2 = 1/ p, and p 1 , p 2 , p > 1. Moreover, the bound (2.7) holds.
As a corollary of Theorem 3.1, we prove a fractional Leibniz rule for the discrete Laplacian on Z d . For Re(z) ≥ 0 and h ∈ L 2 , the complex derivative (− Z d ) z h is given by This coincides with taking the n-th composition of (− Z d ) when z = n is a nonnegative integer. Clearly, To see this, we just use the Taylor series expansion of the function . This is legitimate since I + Z d /(2d) is a contraction on all L p spaces. Our fractional Leibniz rule is the following.

Corollary 3.2 Let
Then, for every s > 0, Remark 1 Note that if f, g ∈ A then f g ∈ L 2 , and hence (− Z d ) s ( f g) makes sense.

Remark 2
Since (− Z d ) s is bounded on all L p spaces, 1 ≤ p ≤ ∞, a version of (3.5) without the Laplacians on the right-hand side is obvious. This is in contrast with the fractional Leibniz rule on R d .
In the proof of the corollary, we shall need two lemmata. The first of them is the l p (Z) boundedness of a discrete Hilbert transform.

Lemma 3.3 The one-dimensional linear multiplier operator
is bounded on all l p (Z) spaces, 1 < p < ∞.
The second of the lemmata is the following.
Proof From Theorem 3.1 and the assumptions on ϕ, it follows that B ϕ ( f, g) ∈ 2 (Z). Thus, the left-hand side of (3.6) makes sense as a function on 2 (Z). Moreover, a continuity argument shows that it suffices to demonstrate (3.6) for Re(z) > 0.
We proceed to the proof of the corollary.

Proof of Corollary 3.2
We claim that it is enough to prove the corollary in dimension d = 1. Indeed, fix s > 0 and assume that (3.5) is true in this case. Let Z be the one-dimensional discrete Laplacian on Z. Define L j := − Z ⊗ I ( j) , j = 1, . . . , d, to be the one-dimensional discrete Laplacian acting on the j-th variable, so that, clearly, − Z d = d j=1 L j . Since each L j generates a symmetric contraction semigroup, using e.g., the multivariate multiplier theorem [27,Corollary3.2] we see that the operator is bounded on L p , p > 1. In other words, we have the bound Since the multiplier L s j (− Z d ) −s is bounded on all L p , p > 1, (this again follows from [27,Corollary 3.2]) in order to conclude the proof of our claim it is thus enough to show that L s j ( f g) p L s j f p 1 g p 2 + L s j g p 2 f p 1 , (3.7) for every j = 1, . . . , d.
For notational simplicity, we justify (3.7) only for j = 1, the proofs for other j are analogous. For a sequence h : Clearly, we have ( f g) n (·) = f n (·)g n (·). Then, using (3.5) in the dimension d = 1 (first inequality below), together with the simple fact that (a + b) p ≈ a p + b p (second and last inequalities below), and Hölder's inequality with exponents p 1 / p, p 2 / p > 1 (third inequality below) we obtain Hence, (3.7) is proved.
Having justified the claim, we now focus on proving (3.5) for d = 1. Till the end of the proof of the corollary, we work on Z and write l p and · p for l p (Z) and · l p (Z) , respectively.

In view of Lemma 3.3, to demonstrate (3.8) it suffices to show that
This, however, follows directly from Theorem 3.1, since, for each s > 0, the multipliers m s 1,1 , andm s 1,1 , satisfy Hörmander's condition (2.2) of arbitrary order. Finally, we prove (3.8) for T 1,−1 . For Re(z) ≥ 0, we set Then using (3.1) (in the case d = 1), we rewrite T 1,−1 as Note that A is preserved by (− Z ) s . Thus, by Lemma 3.3, to demonstrate (3.8) it is enough to prove, for f, g ∈ A, the bounds (3.9) We focus only on the first estimate, the reasoning for the second being analogous. We are going to apply Stein's complex interpolation theorem [25] for each fixed f ∈ A.
The argument used here takes ideas from the proof of [15,Theorem 1.4]. For further reference, we note that the formula (3.10) makes sense not only for f, g ∈ A but more generally, for f, g ∈ 2 .

Bilinear Radial Multipliers for the Generic Dunkl Transform
Here, we apply Theorem 2.3 for bilinear multiplier operators associated with the generic Dunkl transform. In the case when the underlying group of reflections is isomorphic to Z 2 , we also prove a fractional Leibniz rule. Let R be a root system in R d and G the associated reflection group (see [22,Chapter 2]). Let σ α (x) denote the reflection of x in the hyperplane orthogonal to α ∈ R d and let κ be a non-negative, G invariant function on R. The differential-difference (rational) Dunkl operators are defined as Here, f is a Schwartz function; R + is a fixed positive subsystem of R; and x, y = d j=1 x j y j is the standard inner product. The fundamental property of the operators δ j is that, similarly to the usual partial derivatives (which appear when we take κ ≡ 0), they commute, i.e., δ l δ j = δ j δ l , l, j = 1, . . . , d. The operators δ j are also symmetric on . Moreover, they leave S(R d ) invariant. Additionally, the Leibniz rule holds under the extra assumption that one of the functions f 1 , f 2 is invariant under G. The easiest case of Dunkl operators arises when G ∼ Z d 2 . In other words, G consists of reflections through the coordinate axes. In this case, where κ j ≥ 0, while σ j (x) denotes the reflection of x in the hyperplane orthogonal to the j-th coordinate vector. In this case, the weight w κ (x) takes the product form In the (general) Dunkl setting, there is an analogue of the Fourier transform, called the Dunkl transform. It is defined by , z) is the so called Dunkl kernel. A defining property of this kernel is the equation The operator D has properties similar to the Fourier transform. Namely, we have the Plancherel formula and the inversion formula, Additionally, the Dunkl transform diagonalizes simultaneously the Dunkl operators δ i , i.e., The Dunkl Laplacian is given by κ = d i=1 δ 2 i . Using the identity the operator − κ may be formally defined as a non-negative self-adjoint operator on L 2 (R d , w). The same is true for L := (− κ ) 1/2 . Then, for a bounded function μ, the spectral multiplier μ(L) is uniquely determined on S(R d ) by Consider now L 1 := L ⊗ I and L 2 = I ⊗ L . Analogously to the case of bilinear Fourier multipliers, the formula (2.6) can given by the Dunkl transform. Namely, for a bounded function m : [0, ∞) 2 → C, we have (4.7) The above formula is valid pointwise e.g., for Schwartz functions f 1 and f 2 on R d . We observe that in this section the space A 2 from (2.5) is (4.8) Thus, by (4.5) the Dunkl derivatives δ j , j = 1, . . . , d, preserve A 2 .
In this section, we will heavily rely on the concepts of Dunkl translation and Dunkl convolution. For x, y ∈ R d , the Dunkl translation is defined by The inversion formula (4.4) and the properties of the Dunkl kernel imply For f, g ∈ A, the Dunkl convolution is . It is known that the Dunkl transform turns this convolution into multiplication, i.e., (4.9) The first result of this section is the following Coifman-Meyer type theorem. In what follows we set λ κ = (d −1)/2+ α∈R + κ(α) and for brevity write L p := L p (R d , w κ ) and · p = · L p . Theorem 4.1 Assume that m satisfies the Mikhlin-Hörmander condition (2.2) of an order s > 2λ κ + 6. Then the bilinear multiplier operator given by (4.7) is bounded from L p 1 × L p 2 to L p , where 1/ p 1 + 1/ p 2 = 1/ p, and p 1 , p 2 , p > 1. Moreover, the bound (2.7) holds.
Proof We are going to apply Theorem 2.3. In order to do so, we need to check that its assumptions are satisfied for the operator L = (− κ ) 1/2 . To see that L is injective on its domain, we merely note that w κ (ξ ) dξ is absolutely continuous with respect to Lebesgue measure. The contractivity condition (CT) follows from [22,Theorem 4.8] and the subordination method. The assumption (WD) is straightforward from (4.7) and the Lebesgue dominated convergence theorem, while (MH) was proved by Dai and Wang [10, Theorem 4.1] (with arbitrary ρ > λ κ + 1).
Thus, we are left with verifying the property (PF), which we prove with b = 2. This will be deduced by using the convolution structure associated with Dunkl operators. Let ϕ k and ψ k be smooth functions such that supp ϕ k ⊆ [0, 2 k−2 ] and supp ψ k ⊆ [2 k−1 , 2 k+1 ]. Letψ k be a smooth function equal 1 on [2 k−5 , 2 k+5 ] and vanishing outside of [2 k−7 , 2 k+7 ]. Taking the Dunkl transform of the both sides of (PF) and using (4.6), we see that our task is equivalent to proving the formula Denote g j = D( f j ), j = 1, 2. By (4.9) and (4.6) the equation above is exactly By definition ofψ to prove the last formula it is enough to show that . We claim that τ ξȟ 2 (y) = 0. This implies (4.10).
Theorem 4.1 is quite far from a general bilinear Dunkl multiplier theorem, i.e., when the multiplier function m is not necessarily radial in each of its variables. However, in the case d = 1 (and G ∼ Z 2 ), Theorem 4.1 implies [3, Theorem 4.1] by Amri, Gasmi, and Sifi. We slightly abuse the notation and, for ϕ : (4.11) This will cause no confusion with (4.7), as till the end of the present section we only use B ϕ given by (4.11). Then the bilinear multiplier operator given by (4.11) is bounded from L p 1 × L p 2 to L p , where 1/ p 1 + 1/ p 2 = 1/ p, and p 1 , p 2 , p > 1.
Remark When κ = 0 we recover the Coifman-Meyer multiplier theorem in the Fourier transform setting.
be the projection onto the positive Dunkl frequencies. The corollary can be deduced from the boundedness of on all L p spaces 1 < p < ∞.
For Re z ≥ 0, let (− κ ) z be the complex Dunkl derivative The natural L 2 domain of this operator is The second main result of this section is the following fractional Leibniz rule for (− κ ) s , in the case G ∼ Z d 2 .

Corollary 4.3
Let G ∼ Z d 2 and take 1/ p = 1/ p 1 + 1/ p 2 , with p, p 1 , p 2 > 1. Then, for any s > 0, we have where f, g ∈ A and at least one of the functions f or g is invariant by G.
Before proving the fractional Leibniz rule, we need a lemma which is an analogue of Lemma 3.4. Its proof is similar, however a bit more technical. Therefore we give more details.

Lemma 4.4
Take d = 1 and let G ∼ Z 2 . Assume that at least one of the functions f, g ∈ A is G-invariant. Take Re(z) ≥ 0 and let ϕ : R 2 → C be a bounded function that satisfies the Mikhlin-Hörmander condition (4.11) of order s > 2λ κ + 6. Then Remark It is not obvious why B ϕ ( f, g) ∈ Dom L 2 ((− κ ) z ). This is explained in the proof of the lemma.
Proof Since the argument is symmetric in f and g, we assume that f is G-invariant. Denote E G (iξ 1 , x) = |G| −1 g∈G E(iξ 1 , gx), and observe that E G is G-invariant in x. Then, since both f and D( f ) are G-invariant our task reduces to proving that for almost all x ∈ R d . For z = n ∈ N, this formula is a direct computation, and follows from the Leibniz rule. Indeed, by (4.1) and (4.2) we have the interchange of differentiation and integration being allowed since f, g ∈ A. Iterating the above equality 2n times, we obtain (4.13) for z = n.
We remark that (4.13) for z ∈ N also explains why does (− κ ) z (B ϕ ( f, g)) make sense for general Re(z) ≥ 0. Indeed, let n be an integer larger than Re(z). Then, to prove that B ϕ ( f, g) ∈ Dom L 2 ((− κ ) z ), it is enough to show that B ϕ ( f, g) ∈ Dom L 2 ((− κ ) n ). Now, using (4.13) for z = n, together with the binomial formula and (4.5), we arrive at with δ being the Dunkl operator on R. Since f, g belong to A 2 the same is true for δ j f and δ 2n− j g. Thus, an application of Corollary 4.2 proves that B ϕ ( f, g) ∈ Dom L 2 ((− κ ) n ), as desired.
We come back to demonstrating (4.13) for general Re(z) ≥ 0. Note first that by a continuity argument it suffices to consider Re(z) > 0. Denoting our task is reduced to proving that (4.14) for h ∈ A 2 ∩S(R) (recall that A 2 is given by (4.8)). This is enough because A 2 ∩S(R) is dense in L 2 . From (4.13) for z ∈ N, we deduce that for any polynomial P it holds  D(B ϕ ( f, g)) D(h) ∈ L 1 , the dominated convergence theorem shows that the left-hand side of (4.16) converges to (− κ ) z (B ϕ ( f, g)), h L 2 as r → ∞. Similarly, since D( f ) and D(g) are supported in [−N , N ], the expression T P r ( f, g)(x) is uniformly bounded in r ∈ N and x ∈ R and converges to T z ( f, g)(x) as r → ∞. As h ∈ S(R) the dominated convergence theorem implies lim r →∞ T P r ( f, g), h L 2 = T z ( f, g), h L 2 . Therefore, (4.14) is justified and hence, also (4.13). This completes the proof of Lemma 4.4.
We now pass to the proof of Corollary 4.3.
Proof By repeating the argument from the beginning of the proof of Corollary 3.2 (with sums replaced by integrals), our task is reduced to d = 1. We devote the present paragraph to a brief justification of this statement. Here we need the fact that for s ≥ 0 and L j = −δ 2 j , the operators (L j ) s (− κ ) −s , as well as (− κ ) s ( d j=1 (L j ) s ) −1 , are bounded on all L p , 1 < p < ∞. This is true by e.g., [27,Corollary 3.2], since in the product setting each L j , j = 1, . . . , d, generates a symmetric contraction semigroup. Then we are left with showing that cf. (3.7). The proof of (4.17) is similar to that of (3.7); thus, we give a sketch when j = 1. For t ∈ R and x ∈ R d−1 , consider the auxiliary functions f x (t) = f ((t, x)) and g x (t) = g ((t, x)). Then, setting w (1) From this point on, we repeat the steps in the proof of (3.7). Namely, we apply the fractional Leibniz rule for d = 1 and Hölder's inequality (for integrals). We omit the details here. From now on, we focus on proving Corollary 4.3 for d = 1.
Let φ be a function in C ∞ ([0, ∞)) supported in [0, 1/4] and such that φ(t) + φ(t −1 ) = 1. Setting and using Lemma 4.4 with ϕ ≡ 1, we rewrite From now on, the proof resembles that of Corollary 3.2 (in fact it is even easier). We need to prove, for f, g ∈ A, the estimate We focus only on the first inequality, as the proof of the second is analogous. For Re(z) ≥ 0, we set m z (ξ 1 , ξ 2 ) = |ξ 1 + ξ 2 | 2z |ξ 1 | 2z φ(|ξ 2 |/|ξ 1 |), ξ ∈ R 2 , so that T 1 ( f, g) = B m s ((− κ ) s f, g). Since A is preserved under (− κ ) s , our task is reduced to showing that, for s > 0 it holds B m s ( f, g) p ≤ C f p 1 g p 2 , f, g ∈ A. (4.18) As in Sect. 3, we are going to apply Stein's complex interpolation theorem. To do this, we need to extend B m z ( f, g) outside of A × A, by allowing g to be a simple function. This may be achieved by a limiting process. Namely, instead of m z , we consider m z ε = m z e −ε|ξ | 2 . Then, converges pointwise to B m s ( f, g) as ε → 0 + , whenever f, g ∈ A. Therefore, by Fatou's Lemma, to prove (4.18) for B m s it is enough to prove it for each B m s ε , ε > 0, as long as where C is independent of ε. The gain is that now (4.19) is well defined for g ∈ L 2 , in particular it is valid for simple functions. Let n > 2λ κ +6. Then the multipliers m n+iv ε , j = 1, 2, v ∈ R, satisfy Hörmander's condition (4.12) of order 2λ κ + 6. Thus, using Corollary 4.2 (with d = 1), we obtain B m n+iv ε ( f, g) p ≤ C n (1 + |v|) 2λ κ +2 f p 1 g p 2 , v ∈ R.
Thus, using [10, Theorem 4.1] followed by Corollary 4.2 (for the multiplier φ(|ξ 2 |/|ξ 1 |)e −ε(|ξ | 2 ) ), we obtain By definition B m z ε ( f, g) Hence, for fixed f ∈ A, the family {B m z ( f, g)} Re(z)>0 consists of analytic operators. This family has admissible growth, more precisely; for each simple function h, we have Consequently, using Stein's complex interpolation theorem is permitted and leads to (4.18). The proof of the corollary is thus finished.

Bilinear Multipliers for Jacobi Trigonometric Polynomials
In this section, we give a bilinear multiplier theorem for expansions in terms of Jacobi trigonometric polynomials. Contrary to the previous sections, we do not prove a fractional Leibniz rule here. The reason for this is that there is no natural first-order operator in the Jacobi setting that satisfies a Leibniz-type rule of integer order.
In this setting, the spectral multipliers of J 1/2 are given by μ(J 1/2 ) f = n∈N μ n + γ f, P k L 2 P k .
The space A from (2.5) coincides with the linear span of {P n } n∈N . We prove the following Coifman-Meyer type multiplier theorem. Remark The theorem implies a Coifman-Meyer type multiplier result for bilinear multipliers associated with the modified Hankel transform. This follows from a transference result of Sato [23].