Estimates for some bilinear wave operators

We consider some bilinear Fourier multiplier operators and give a bilinear version of Seeger, Sogge, and Stein's result for Fourier integral operators. Our results improve, for the case of Fourier multiplier operators, Rodr\'iguez-L\'opez, Rule, and Staubach's result for bilinear Fourier integral operators. The sharpness of the results is also considered.


Introduction
The solution to the wave equation ∂ 2 t u = △u with the initial data u(0, x) = f (x) and u t (0, x) = g(x) is given by where f denotes the Fourier transform of f (for the definition of Fourier transform, see Notation 1.6 below).Several basic properties of the mapping (f, g) → u(t, •) are derived from the estimate of the operator The purpose of this paper is to consider bilinear versions of this operator.
We begin with the definition of linear Fourier multiplier operators.
For θ ∈ L ∞ (R n ), the operator θ(D) is defined by for f in the Schwartz class S(R n ).If X and Y are function spaces on R n equipped with quasi-norms or seminorms • X and • Y , respectively, and if there exists a constant A such that then we say that θ is a Fourier multiplier for X → Y and write θ ∈ M(X → Y ).(Sometimes we write θ(ξ) ∈ M(X → Y ) to mean θ(•) ∈ M(X → Y ).)The minimum of A that satisfies the above inequality is denoted by θ M(X→Y ) .Throughout this paper, H p , 0 < p ≤ ∞, denotes the Hardy space and BMO denotes the space of bounded mean oscillation.We use the convention that H p = L p if 1 < p ≤ ∞.For H p and BMO, see, e.g., [S, Chapters III and IV].
We recall classical results about the operator (1.1) and its generalizations.We use the following notation.
Definition 1.1.We write P = P(R n ) to denote the set of all functions on R n that are real-valued, homogeneous of degree 1, and C ∞ away from the origin.
The following theorem is due to Seeger, Sogge, and Stein [SSS].
Theorem A (Seeger-Sogge-Stein [SSS]).If φ ∈ P(R n ), 1 ≤ p ≤ ∞, and m = −(n − 1)|1/p − 1/2|, then In fact, this theorem is not given in [SSS] in exactly the same form as above; the result given in [SSS] is restricted to local estimate.However, Theorem A can be proved by a slight modification of the argument of [SSS].Or one can appeal to the general results given by Ruzhansky and Sugimoto [RS,Theorems 1.2 and 2.2].
For a proof of this theorem, see [M1,Theorem 1] or [S,Chapter IX,6.13].The purpose of the present paper is to consider bilinear versions of Theorems A and B. We recall the definition of bilinear Fourier multiplier operators.For a bounded measurable function σ = σ(ξ, η) on R n × R n , the bilinear operator T σ is defined by (ξ+η) σ(ξ, η) f(ξ) g(η) dξdη, x ∈ R n , for f, g ∈ S(R n ).If X, Y , and Z are function spaces on R n equipped with quasi-norms or seminorms • X , • Y , and • Z , respectively, and if there exists a constant A such that T σ (f, g) Z ≤ A f X g Y for all f ∈ X ∩ S and all g ∈ Y ∩ S, then we say that σ is a bilinear Fourier multiplier for X ×Y to Z and write σ ∈ M(X ×Y → Z). (Sometimes we write θ(ξ, η) ∈ M(X × Y → Z) to mean θ(•, •) ∈ M(X × Y → Z).)The smallest constant A that satisfies the above inequality is denoted by σ M(X×Y →Z) .
We shall consider the bilinear Fourier multiplier of the form e i(φ 1 (ξ)+φ 2 (η)) σ(ξ, η), φ 1 , φ 2 ∈ P(R n ), σ ∈ S m 1,0 (R 2n ), where the class S m 1,0 (R 2n ) is defined as follows.Definition 1.2.For m ∈ R, the class S m 1,0 (R 2n ) is defined to be the set of all C ∞ functions σ = σ(ξ, η) on R 2n that satisfy the estimate In the theory of bilinear Fourier multipliers, a classical method is known that allows us to write multiplier σ ∈ S m 1,0 (R 2n ) as a sum of multipliers of the product form θ 1 (ξ)θ 2 (η).Using this method, we can deduce the following theorem from Theorem A.
In fact, Rodríguez-López-Rule-Staubach [RRS] considered more general operators, bilinear Fourier integral operators, and proved a theorem that almost covers Theorem 1.3.The statement of the theorem of [RRS] is, however, restricted to local estimate.We shall give a full proof of Theorem 1.3 in a succeeding section, Section 3.
The main purpose of the present paper is to show that the number m = −(n − 1) |1/p − 1/2|+|1/q −1/2| in Theorem 1.3 can be improved and show that the improved m is optimal at least for certain (p, q).
The following is the first main theorem of this paper.
Compare the claims of Theorems 1.3 and 1.4.They are the same in the regions 1 ≤ p, q ≤ 2 and 2 ≤ p, q ≤ ∞, but different outside of these regions.In the typical case (p, q) = (1, ∞), Theorem 1.3 asserts that the multiplier e i(φ In order to show that the number m 1 (p, q) is in fact optimal for some (p, q), we consider the special case φ For p, q ∈ [1, ∞] given, set 1/r = 1/p + 1/q and we consider necessary condition on m ∈ R that allows the assertion ).The following is the second main theorem of this paper.
Theorem 1.5.Let n ≥ 2. (1 This theorem implies that the number m 1 (p, q) of Theorem 1.4 is optimal for p, q in the range given in (1) and (2) of Theorem 1.5.The present authors do not know whether m 1 (p, q) is optimal for other p, q.
The contents of the rest of the paper are as follows.In Section 2, we collect some propositions concerning flag paraproduct, which we will use in the proof of Theorem 1.3.In order not to interrupt the stream of argument, we shall postpone rather long proofs of those propositions to Section 6.In Sections 3, 4, and 5, we prove Theorems 1.3, 1.4, and 1.5, respectively.The last section, Section 6, is devoted to the proofs of the propositions stated in Section 2.
We end this section by introducing some notations used throughout this paper.
Notation 1.6.The Fourier transform and the inverse Fourier transform on R d are defined by

Sometimes we use rude expressions f (x)
∧ or g(ξ) We shall repeatedly use dyadic partition of unity, which is defined as follows.Take a Notice, however, that we will also use the letters ψ, ζ, ϕ in a meaning different from the above.
For a smooth function θ on R d and for a nonnegative integer N, we write θ C N = max |α|≤N sup ξ ∂ α ξ θ(ξ) .The letter n denotes the dimension of the Euclidean space that we consider.Unless further restrictions are explicitly made, n is an arbitrary positive integer.

Some results from bilinear flag paraproducts
In this section, we give some results for the bilinear Fourier multipliers of the form This kind of multipliers with a 0 , a 1 , a 2 being the 0-th order multipliers (i.e. the ones that generalize the homogeneous functions of degree 0) are considered by Muscalu [Mu1,Mu2] and Chapter 8], where their mapping properties between L p spaces are given.In this section, we consider the case where a 0 , a 1 , a 2 are non-zero order multipliers and give estimates including H p and BMO.The results of this section will be used to prove Theorem 1.3.
Proofs of the following two propositions will be given in Section 6.
Then the following hold.

Proof of Theorem 1.3
In order to prove Theorem 1.3, we use the following lemma.
Using the functions ζ and ϕ of Notation 1.6, we decompose τ as We shall prove τ i ∈ M(H p × H q → X r ) for i = 1, 2, 3, 4. Firstly, the multiplier τ 1 is easy to handle.By Lemma 3.1, the inverse Fourier transform of Next, consider τ 2 .We write this as As we have seen above, e iφ 2 (η) ϕ(η) ∈ M(H q → H q ) for 1 ≤ q ≤ ∞.Theorem A implies Combining these results, we see that τ 2 belongs to the same multiplier class as in (3.1), which a fortiori implies τ 2 ∈ M(H p × H q → X r ).
By symmetry, we also have τ 3 ∈ M(H p × H q → X r ).Finally, consider τ 4 .We write this as where ζ is the same as above.Theorem A gives (notice that m 1 < 0 if n ≥ 2 and p = ∞ and that m 2 < 0 if n ≥ 2 and q = ∞).Now combining these results, we see that τ 4 belongs to the same multiplier class as in (3.2), which a fortiori implies τ 4 ∈ M(H p × H q → X r ).This completes the proof of Theorem 1.3.

Proof of Theorem 1.4
In this section, we prove Theorem 1.4.For this, the key is to prove the assertion of Theorem 1.4 in the special case p = 1 and q = ∞, which we shall write here for the sake of reference.
Theorem 1.4 can be deduced from this theorem and from Theorem 1.3.In fact, notice that, by obvious symmetry, we have e i(φ 1 (ξ)+φ 2 (η)) σ(ξ, η) ∈ M(L ∞ × H 1 → L 1 ) under the same assumptions on n and σ.Hence, if Theorem 4.1 is proved, then we can deduce the claims of Theorem 1.4 from the claims of Theorems 1.3 and 4.1 with the aid of complex interpolation.(For the interpolation argument, see, e.g., [BBMNT,Proof of Theorem 2.2] or [MT1, Proof of the 'if' part of Theorem 1.1].)Thus it is sufficient to prove Theorem 4.1.
To prove Theorem 4.1, we use the following lemmas.
Hence the conclusion of the lemma follows.
Then the following hold.
(1) For each positive integer N, there exists a constant c N depending only on n, φ, and N such that (2) There exists a constant c depending only on n and φ such that for all j ∈ N.
Proof.We write To estimate f j (x), we follow the idea given by Seeger-Sogge-Stein [SSS].
where B(x, r) denotes the ball with center x and radius r, and ν runs on an index set of cardinality ≈ (2 j/2 ) n−1 .Take functions {χ ν j } ν such that χ ν j is homogeneous of degree 0 and Using this partition of unity, we decompose f j (x) as The key idea is that the oscillating factor e −iφ(ξ) can be well approximated by e −iξ•∇φ(ξ ν j ) on the support of ψ(2 −j ξ)χ ν j (ξ).We write the phase function ξ • x − φ(ξ) appearing in the last integral as Notice that the support of ψ(2 −j ξ)χ ν j (ξ) is included in the set 2 .The functions appearing in the above integral satisfy the following estimates on E j,ν : By using these estimates and by integration by parts, we obtain the following two estimates: Taking sum over ν's of card ≈ (2 j 2 ) n−1 , we have Proof of (2).Combining (4.2) and (4.1), we have where (x − ∇φ(ξ ν j )) ′ denotes the orthogonal projection of x − ∇φ(ξ ν j ) to the orthogonal complement of the line Rξ ν j .Taking N = 2n − 1 and integrating the above inequality we have Taking sum over ν's of card ≈ (2 j 2 ) n−1 , we obtain the inequality as mentioned in (2).This completes the proof of Lemma 4.3.
Then the following hold.
(1) For each positive integer N > 2n, there exists a constant c N depending only on n, φ, and N such that and for all j ∈ N.
(2) There exists a constant c depending only on n and φ such that Proof.From the definition of ζ and from the assumption on supp θ, we have If |x| > 2R and 1 ≤ k ≤ j + 1, then Lemma 4.3 (1) gives If N > 2n, then taking sum over k, we obtain the inequality mentioned in (1).
For 1 Taking sum over k ≤ j + 1, we obtain the inequality mentioned in (2).Lemma 4.4 is proved.
Consider the multiplier σ I .This is written as Hence by the Fourier series expansion we can write with the coefficient satisfying for any L > 0. Changing variables ξ → 2 −j ξ and η → 2 −j+1 η and multiplying ψ(2 −j ξ)ϕ(2 −j+1 η), we obtain Hence σ I is written as follows: where By a similar argument, σ II and σ III can be written as follows: where the coefficients c III,j satisfy the same estimates as (4.3).Hereafter we shall consider a slightly general multiplier.We assume the multiplier σ is given by where (c j ) j∈N is a sequence of complex numbers satisfying (4.6) with some A ∈ (0, ∞), and θ 1 and θ 2 are functions in For such σ, we shall prove the estimate If this is proved, then by applying it to a) , and , and, thus, taking L sufficiently large and taking sum over a, b ∈ Z n , we obtain In the same way, we obtain Since e i(φ 1 (ξ)+φ 2 (η)) σ(ξ, η)ψ 0 (ξ)ψ 0 (η) is also a multiplier for H 1 ×L ∞ → L 1 by virtue of Lemma 4.2, we will obtain the conclusion of the theorem.Thus the proof is reduced to showing (4.8) for σ given by (4.5), (4.6), and (4.7).
We shall make a further reduction.As in the proof of Theorem 1.3, using the functions ϕ and ζ of Notation 1.6, we decompose the multiplier e i(φ 1 (ξ)+φ 2 (η)) σ(ξ, η) into four parts: The multipliers τ 1 , τ 2 , and τ 3 are easy to handle.For τ 1 , its inverse Fourier transform is given by By Lemma 3.1, we have (4.9) and similar estimate with θ 2 in place of θ 1 .Thus For τ 2 , we use the estimate (4.10) which is given in Lemma 4.4 (2).Using this together with (4.9), we obtain where the last ≈ holds because m < −(n − 1)/2.Similarly, we have Thus the rest of the proof is the estimate for τ 4 .Our purpose is to prove the estimate To prove this, by virtue of the atomic decomposition of H 1 , it is sufficient to prove the uniform estimate of T τ 4 (f, g) L 1 for H 1 -atoms f .By translation, we may assume that the H 1 -atoms are supported on balls centered at the origin.Thus we assume and we shall prove Recall that the bilinear operator T τ 4 is given by Firstly, consider the case r > R. In this case we estimate the L 1 norm as For the L ∞ -norm involving g, we use Lemma 4.4 (2) to obtain (4.11) For the L 1 norm of e iφ 1 (D) ζ(D)θ 1 (2 −j D)f (x) on |x| ≤ 3r, we use the Cauchy-Schwarz inequality to obtain If |x| > 3r and |y| ≤ r, then |x − y| > 2r > 2R.Hence, for |x| > 3r, using Lemma 4.4 (1), we see that which implies Combining the above estimates, we have (4.12) Now from (4.11) and (4.12), we obtain where the last ≈ holds because m < −(n − 1)/2.Secondly, we assume r ≤ R and estimate the L 1 norm of T τ 4 (f, g)(x) on |x| > 3R.We estimate this as For the L ∞ norm involving g, we have (4.11).If |x| > 3R and |y| ≤ r ≤ R, then |x−y| > 2R.Hence, for |x| > 3R, Lemma 4.4 (1) yields This implies Thus we obtain where we used m < −(n − 1)/2 again.Thirdly, we assume r ≤ R and estimate the L 1 norm of T τ 4 (f, g)(x) on |x| ≤ 3R.We set Using Lemma 4.4 (2), we have If |x| ≤ 3R and |y| > 5R, then |x − y| > 2R.Hence, for |x| ≤ 3R, we use Lemma 4.4 (1) to have Thus where we used m < −(n − 1)/2 again.Finally, we estimate the L 1 norm of T τ 4 (f, g1 B )(x) on |x| ≤ 3R.For this, we use the Cauchy-Schwarz inequality to have For the L 2 norm involving g1 B , we have We estimate the L 2 norm of e iφ 1 (D) ζ(D)θ 1 (2 −j D)f in two ways.Firstly, we have On the other hand, using the moment condition of f , we can write Hence . By Plancherel's theorem, Combining (4.13), (4.14), and (4.15), we obtain where we used m = −n/2.This completes the proof of Theorem 4.1.

Necessary conditions on m
In this section, we shall prove Theorem 1.5.
We shall prove that the conditions given in Theorem 1.5 are already necessary for (5.1).We shall prove the following theorem, which asserts that the claims of Theorem 1.5 hold if we replace the condition (1.3) by the condition (5.1).
Lemma 5.2.Let ψ be a C ∞ function on R such that which is the inverse Fourier transform of the radial function e −i|ξ| ψ(2 −j |ξ|) on R n .Then the following hold.
(1) For each L > 0, there exists a constant c L , depending only on n, ψ, and L, such that for all j ∈ N and all x ∈ R n .
Proof.The assertion (3) follows from ( 1) and (2).In fact, the inequality h j H p ≈ h j L p holds because the support of the inverse Fourier transform of h j is included in the annulus 1) and the converse estimate h j L p 2 j( n+1 2 − 1 p ) follows from (2).Thus we only need to prove (1) and ( 2).Since h j (x) is the inverse Fourier transform of a radial function, it is written in terms of Bessel function as is the Bessel function (see e.g.[SW,Theorem 3.3,p. 155]; this formula holds for n = 1 as well since (2π) −1/2 J −1/2 (s) s 1/2 = π −1 cos s).
Proof of (1).Firstly, we estimate of h j (x) for 2 j |x| ≤ 1.For this, we use the power series expansion whose radius of convergence is ∞.Integrating term by term, we have The function arising in the last expression satisfies Hence, for any L ′ ∈ N, we have Thus, for 2 j |x| ≤ 1, we have Since L ′ can be taken arbitrarily large, the above implies the desired estimate of h j (x) for 2 j |x| ≤ 1. Next, we estimate h j (x) for 2 j |x| > 1.For this, we use the asymptotic expansion of the Bessel function, which reads as ) and the remainder terms R ± (s) satisfy Corresponding to the above formula, we write . We shall estimate each of I + j (x), I − j (x), K + j (x), and K − j (x) for 2 j |x| > 1. Estimate of I + j (x) for 2 j |x| > 1.I + j (x) is written as ∧ is a rapidly decreasing function, we have For any given L > 0, the above estimates with a sufficiently large L ′ imply (5.4) Estimate of I − j (x) for 2 j |x| > 1.The function I − j (x) is written as Hence, by the same reason as in the case of I + j (x), for any L ′ > 0. Restricting to the region 2 j |x| > 1, we have which, via Fourier transform, yields for any L ′ > 0. Hence For any L > 0, the above estimates with L ′ sufficiently large implies (5.6) then by the same reasoning as above we obtain Now from (5.4), (5.5), (5.6), and (5.7), we obtain the estimate of h j (x) for 2 j |x| > 1 as claimed in the lemma.Thus the claim (1) is proved.

Proof of (2). The equality (5.3) and the equality b
We set This is a positive number since ψ is nonnegative and not identically equal to 0. Then, from (5.8) and from continuity of the functions, it follows that there exists a number δ > 0 such that 1 On the other hand, the estimates of (5.5), (5.6), and (5.7) imply that there exists a constant . Hence the estimate claimed in (2) of the lemma holds if we take j 0 large enough so that c 1 2 −j 0 ≤ c 0 /20.This completes the proof of Lemma 5.2.
Proof of Theorem 5.1.We define the operator S j by We divide the proof into three cases.
Case 1: 0 < p, q ≤ 2. Assume (5.1) holds, or equivalently (5.9) 2 jm S j f • S j g Xr ≤ A f H p g H q for all j ∈ N.
Take ψ as in Lemma 5.2 and set We shall test (5.9) to f = g = f j .
Since the support of the Fourier transform of f j is included in the annulus and similar estimate holds for f j H q .On the other hand, by the choice of the functions θ and ψ, we have Hence, by Lemma 5.2, there exist δ ∈ (0, ∞) and j 0 ∈ N such that for j > j 0 .Hence, if (5.9) holds, then testing it to f = g = f j we have for j > j 0 , which is possible only when m ≤ −(n − 1) Case 2: 2 ≤ p, q ≤ ∞.Assume (5.9) holds.Using the function ψ of Lemma 5.2, we set Then Lemma 5.2 gives the estimate and similar estimate holds for f j H q .On the other hand, and hence Hence, if (5.9) holds, then by testing it to f = g = f j we have By the symmetry of the situation, it is sufficient to consider the case 1 ≤ p ≤ 2 ≤ q ≤ ∞.Thus we assume 1 ≤ p ≤ 2 ≤ q ≤ ∞ and 1/p + 1/q = 1/r = 1.We assume (5.1) holds, or equivalently, (5.10) and prove that this is possible only when m ≤ −n/p + n/2.We use the same function f j that was used in the proof of Case 1: where ψ is the function given in Lemma 5.2.
As we have seen in Case 1, (5.11) On the other hand, and, hence, Lemma 5.2 (2) gives (5.12) For a sequence of complex numbers α = α ℓ ℓ∈Z n , we define g j,α by where δ ′ is a sufficiently small positive number; for the succeeding argument the choice δ ′ = δ/(2 √ n) will suffice.
Since the operator S j is linear and commutes with translation, we have Now we test (5.10) to f = f j and g = g j,α .Then by (5.11) and (5.13) we have (5.14) 2 jm S j f j (x) (recall that 1/p + 1/q = 1).We take the dual form of this inequality, which reads as We define the cube Then each Q ν is a cube with side length δ ′ 2 −j and all of them constitute a partition of R n .Let ǫ ν ν∈Z n be any sequence of ±1, and apply (5.15) to ϕ(x) = ν∈Z n ǫ ν 1 Qν (x).Then we obtain Notice that this inequality holds uniformly for all choices of ǫ ν = ±1.We take the q ′ -th power of the above inequality, take average over all choices of ǫ ν = ±1, and use Kintchine's inequality; this yields (5.16) Thus (5.16) implies 2 j(nq ′ /2+n) 2 j(n−m)q ′ for j > j 0 , which is possible only when m ≤ −n/2 + n/q = n/2 − n/p.This completes the proof of Theorem 5.1.6. Proofs of Propositions 2.3 and 2.4 6.1.Proof of Proposition 2.3.In order to prove Proposition 2.3, we use the following lemmas.The first two lemmas are given in [MT2].Lemma 6.1 ([MT2, Lemma 2.5]).Let 0 < p, q ≤ ∞ and 1/p + 1/q = 1/r > 0. Assume that ψ and φ are functions on R n such that supp ψ ⊂ {a −1 ≤ |ξ| ≤ a} and , where a, A, B ∈ (0, ∞) and L and L ′ are sufficiently large integers determined by p, q, and n.Then where c = c(n, p, q, a) is a positive constant.Moreover, if p = ∞ then f H p can be replaced by f BM O .Lemma 6.2 ([MT2, Lemma 2.7]).Let 0 < p, q ≤ ∞ and 1/p + 1/q = 1/r > 0. Assume that ψ 1 and ψ 2 are functions on R n such that supp ψ 1 , supp ψ 2 ⊂ {a −1 ≤ |ξ| ≤ a} and , where a, A, B ∈ (0, ∞) and L and L ′ are sufficiently large integers determined by p, q, and n.Then where c = c(n, p, q, a) is a positive constant.Moreover, if p = ∞ (respectively, q = ∞) then f H p (respectively, g H q ) can be replaced by f BM O (respectively, g BM O ).
Lemma 6.3.Let m 2 < 0 and suppose the multiplier τ is given by where (c j,k ) a sequence of complex numbers satisfying Then τ belongs to the following multiplier classes: Moreover, in each case, the multiplier norm of τ is bounded by c ψ 1 C N ψ 2 C N with c = c(n, m 2 , p, q) and N = N(n, p, q).Proof.We divide the proof into several cases.
(4 • ) BMO × H q → L q , 0 < q ≤ 1.By virtue of the atomic decomposition for H q , it is sufficient to show the uniform estimate of T τ (f, g) L q for all H q -atoms g.By translation, it is sufficient to consider the H q -atoms supported on balls centered at the origin.Thus we assume and we shall prove . By the same reason as in (2 • ), the Littlewood-Paley theory for H q reduces the proof to the estimate of We first estimate the L q norm on |x| ≤ 2r.Using Hölder's inequality and using the result proved in (3 • ) (with q = 2), we have Next, we estimate the L q norm on |x| > 2r.Using the inequality ψ 1 (2 ℓ 2 j L q (|x|>2r) To estimate the L q -norm of the functions ψ 2 (2 −k D)g(x) on |x| > 2r, we write Then using the size estimate of g and the moment condition on g, we have (see [MT2,inequalities (2.7) and (2.8)]).Hence From (6.2) and (6.3), we obtain (5 • ) BMO × BMO → BMO.By virtue of the duality between BMO and H 1 , it is sufficient to show the following inequality: then the integral in (6.4) can be written as where the last follows from Lemma 6.2.This completes the proof of Lemma 6.3.Lemma 6.4.Suppose the multiplier τ is defined by Then τ belongs to the following multiplier classes: Moreover, in each case, the multiplier norm of τ is bounded by c ψ 1 C N φ C N with c = c(n, p, q) and N = N(n, p, q).where j−k≥3 , |j−k|≤2 , and j−k≤−3 denote the sums of σ(ξ, η)ψ(2 −j ξ)ψ(2 −k η) over j, k ∈ Z that satisfy the designated restrictions.We shall consider each of σ I , σ II , and σ III .
(1 • ) For the multiplier σ II , we shall prove the following: To prove this, observe that |ξ| ≈ |η| ≈ 2 j on the support of ψ(2 −j ξ)ψ(2 −k η) with |j−k| ≤ 2. From this we see that σ II ∈ Ṡ0 1,0 (R 2n ).Hence Proposition 2.2 implies that σ II is a bilinear Fourier multiplier for the following spaces: We shall prove that the space L ∞ in the above can be replaced by BMO.
(2 • ) For the multiplier σ I , we shall prove the following: Proof of (6.6) in the case m 2 = 0. We write σ (6.10) Since m 2 = 0 and m = m 1 in the present case, we see that b ∈ Ṡ0 1,0 (R 2n ).Thus, Proposition 2.2 implies that b ∈ M(H p × H q → L r ).Also since a 2 ∈ Ṡ0 1,0 (R n ) in the present case, the classical multiplier theorem for linear operators implies a 2 ∈ M(H q → H q ).Hence Proof of (6.6) in the case m 2 < 0. Notice that σ I is supported in |ξ| ≥ 2|η| and satisfies Since m 2 < 0, the theorem of Grafakos and Kalton [GK2,Theorem 7.4] implies σ I ∈ M(H p × H q → L r ).
Another proof of (6.6) in the case m 2 < 0.Here we shall give a direct proof of (6.6) for the case m 2 < 0, which uses only a classical method.
We shall prove each of τ I , τ II , and τ III belongs to the multiplier class as mentioned in the proposition.
We shall prove τ I ∈ M(H p × L ∞ → L p ).By the same argument as in Proof of Proposition 2.3 (see Proof of (6.8)), we can write τ I as (6.17 Taking L sufficiently large and taking sum over a, b ∈ Z n , we obtain τ I ∈ M(H p ×L ∞ → L p ). Thus the part (1) is proved.Proof of (2).Here we assume m 1 < 0. By the results proved in (1 • ) and (3 • ) in Proof of Proposition 2.3, the multipliers τ II and τ III belong to M(BMO × BMO → BMO).Recall that the multiplier τ I is written as (6.17) with c (a,b) j satisfying (6.15) and ψ (a) and ϕ (b) defined by (6.16).Hence we can prove τ I ∈ M(BMO × L ∞ → BMO) by using Lemma 6.4.Thus the part (2) of Proposition 2.4 is proved.This completes the proof of Proposition 2.4.