BILINEAR MULTIPLIERS AND TRANSFERENCE OSCAR BLASCO

We give de Leeuw-type transference theorems for bilinear multipliers. In particular, it is shown that bilinear multipliers arising from regulated functions 
 m ( ξ , η ) in ℝ × ℝ can be transferred to bilinear multipliers acting on 𝕋 × 𝕋 and ℤ × ℤ . The results follow from the description of bilinear multipliers on the discrete real line acting on L p -spaces.

The study of such multipliers was started by Coifman and Meyer (see [3,4,19]) for smooth symbols and new results for nonsmooth symbols, extending the ones given by the bilinear Hilbert transform, have been achieved by Gilbert and Nahmod (see [8,9,10]) and also by Muscalu et al. (see [20]).
We refer the reader also to [7,12,11,15] for new results on bilinear multipliers and related topics.
In a recent paper (see [7]), Fan and Sato have shown certain de Leeuw-type theorems for transferring multilinear operators on Lebesgue and Hardy spaces from R n to T n .Here we will consider bilinear multipliers on Lebesgue spaces L p (R) and get a characterization which allows us to transfer not only to the bilinear multipliers on T but also on Z.Our approach will follow closely the ideas in the original paper by de Leeuw (see [6]) and will provide an alternative proof of some results in [7], whose proof follows, in the multilinear case, the approach used by Stein and Weiss (see [21, page 260]).
We start by setting up natural analogous versions of bilinear multipliers in the periodic and discrete cases.Let m = (m k,k ) be a bounded sequence and let m be a periodic function on for functions f , g defined on T, and for Of course we can see these three cases as instances of the general bilinear multiplier acting on different groups.Let G be a locally compact abelian group and G its dual group with Haar measure µ.Let 1 ≤ p 1 , p 2 ≤ ∞ and let m be a bounded measurable function on G × G.We say that m is a (p 1 , p 2 )-multiplier on G × G if the operator (defined for simple functions f and g) extends to a bounded bilinear operator from , where 1/ p 1 + 1/ p 2 = 1/ p 3 .The reader is referred to [14] for the general theory in the linear case.
The first transference results on linear multipliers were given by de Leeuw (see [6]).He showed, among other things, that if m is regulated (all its points are Lebesgue points) and m is a p-multiplier on R, then (m(εk)) k is a uniformly bounded p-multiplier for all ε > 0 on Z (see [21, page 264] for the converse of this result for continuous multipliers).
Transference results of similar nature are presented in [1].
A general transference method was considered by [5] (see also the generalization given by [13]), but we will not consider these approaches in our bilinear generalization in the paper.
In [7], the multilinear version of the continuous result was shown, namely that for any continuous function m(ξ,η), one has that m is a (p 1 , p 2 )-multiplier on R × R if and only if m(εk,εk ) k,k is a uniformly bounded multiplier on Z × Z for ε > 0. An extension of the result to Lorentz spaces was achieved in [2].
We will first characterize the boundedness of bilinear multipliers on R × R by the existence of a constant K such that for all measures µ, ν, λ of finite supports.This allows us to transfer from the continuous Ꮿ m to the discrete case Ᏸ m recovering some of the Fan-Sato results in [7].
We also obtain the transference from the continuous case Ꮿ m to the periodic case ᏼ m .Our main result establishes that m is a (p 1 , p 2 )-multiplier on R × R if and only if The reader should be aware that the results of the paper can be stated for multilinear multipliers, with the condition 1/ p = n i=1 (1/ p i ), by considering the corresponding multilinear notions, for instance, for m(ξ 1 ,...,ξ n ), one has and similar modifications for ᏼ m and Ᏸ m .We simply do the bilinear case for the sake of simplicity.

Bilinear multipliers on R × R
We start by reformulating the condition of (p 1 , p 2 )-multiplier on R × R using duality.The proof is straightforward and is left to the reader.
Now we present some behavior of multipliers on R × R with respect to convolution and dilation operators to be used later on.
Proof.Let f s (x)= f (x + s) for any s ∈ R and function f .Then for any s,t ∈ R and φ,ψ,ν ∈ with φ p1 = ψ p2 = ν p 3 = 1, we have And the result follows by Lemma 2.1.
Theorem 2.4.Let m(ξ,η) be a bounded continuous function on R × R. The following are equivalent: for all measures µ, ν, λ supported on a finite number of points.
It is easily checked that the Lebesgue convergence theorem implies that where for all measures µ, ν, λ having their supports on finite sets of points.On the other hand, from (i) and Lemma 2.1, we have (2.11) We now choose α = 1/ p 1 , β = 1/ p 2 , and
Finally we have that for φ,ψ,ν ∈ , (2.18) The result now follows from Lemma

Transference theorems
We mention the formulations for (p 1 , p 2 )-multipliers on the groups T and Z which follow directly from duality.
Lemma 3.1.Let m(t,s) be a bounded measurable function on T × T. Then m is a (p 1 , p 2 )multiplier on T × T if and only if there exists a constant K so that for all trigonometric polynomials P, Q, and R.
Theorem 3.3 (see [7,Theorem 1]).Let m(ξ,η) be a regulated bounded function on Proof.According to Lemma 3.2, we have to show that there exists a constant K so that for all trigonometric polynomials P, Q, and R.This follows by selecting the measures µ, ν, λ in Theorem 2.6 such that µ = P, ν = Q, and λ = R. Theorem 3.4.Let m(ξ,η) be a bounded regulated function on R × R. The following are equivalent: Proof.(i)⇒(ii).Using Lemma 3.1, it suffices to show that for any finite sequences (a(n)) n , (b(n)) n , and (c(n)) n with a p1 = b p2 = c p 3 = 1, there exists a constant K > 0 such that where Using now the assumption and Shanon's sampling theorem, one gets ψ a L p (R) ≤ C 1 φ a L p (R) ≤ C 2 a p ≤ C 3 ψ a L p (R) for some constants C i for i = 1,2,3.Hence the desired inequality follows.Now we apply Lemma 2.3 to get the result for each ε.(ii)⇒(i).We take φ and ψ such that supp φ and suppψ are contained in [−1/4,1/4].For a fixed u ∈ [−1/2,1/2], consider the periodic extensions of the functions φ(ξ)e 2πiuξ , ψ(η)e 2πiuη to be denoted P u and Q u , respectively.