Bilinear multipliers of weighted Wiener amalgam spaces and variable exponent Wiener amalgam spaces

R E S E A R C H

Open Access

Bilinear multipliers of weighted Wiener

amalgam spaces and variable exponent

Wiener amalgam spaces

Öznur Kulak

1*

_{and Ahmet Turan Gürkanlı}

2

Dedicated to Professor Ravi P Agarwal.

*_{Correspondence:} oznurn@omu.edu.tr 1_{Department of Mathematics,} Faculty of Arts and Sciences, Ondokuz Mayıs University, Kurupelit, Samsun, 55139, Turkey Full list of author information is available at the end of the article

Abstract

Let

ω

1,

ω

2be slowly increasing weight functions, and let

ω

3be any weight function

onRn. Assume that m(ξ,

η) is a bounded, measurable function on

Rn_{× R}n_{. We define}

Bm(f , g)(x) =  Rn  Rn ˆf(ξ)ˆg(

η)m(ξ

,

η)e

2πiξ+η,x_d

_ξ

_d

_η

for all f , g∈ C_c∞(Rn_{). We say that m(ξ,η) is a bilinear multiplier onR}n_{of type}

(W(p1, q1,ω1; p2, q2,

ω

2; p3, q3,

ω

3)) if Bmis a bounded operator from

W(Lp1_{, L}q1 ω1)× W(L p2_{, L}q2 ω2) to W(L p3_{, L}q3 ω3), where 1≤ p1≤ q1<∞, 1 ≤ p2≤ q2<∞,

1 < p3, q3≤ ∞. We denote by BM(W(p1, q1,

ω

1; p2, q2,

ω

2; p3, q3,ω3)) the vector space

of bilinear multipliers of type (W(p1, q1,

ω

1; p2, q2,

ω

2; p3, q3,

ω

3)). In the first section of

this work, we investigate some properties of this space and we give some examples of these bilinear multipliers. In the second section, by using variable exponent Wiener amalgam spaces, we define the bilinear multipliers of type

(W(p1(x), q1,

ω

1; p2(x), q2,ω2; p3(x), q3,

ω

3)) from W(Lp1(x), Lqω11)× W(L

p2(x)_{, L}q2

ω2) to

W(Lp3(x)_{, L}q3

ω3), where p∗1, p∗2, p∗3<∞, p1(x)≤ q1, p2(x)≤ q2, 1≤ q3≤ ∞ for all

p1(x), p2(x), p3(x)∈ P(Rn). We denote by BM(W(p1(x), q1,

ω

1; p2(x), q2,ω2; p3(x), q3,ω3))

the vector space of bilinear multipliers of type

(W(p1(x), q1,

ω

1; p2(x), q2,ω2; p3(x), q3,

ω

3)). Similarly, we discuss some properties of this

space.

MSC: 42A45; 42B15; 42B35

Keywords: bilinear multipliers; weighted Wiener amalgam space; variable exponent

Wiener amalgam space

1 Introduction

Throughout this paper we will work onRn _{with Lebesgue measure dx. We denote by}

C_c∞(Rn_{), C}

c(Rn) and S(Rn) the space of infinitely differentiable complex-valued functions

with compact support onRn_{, the space of all continuous, complex-valued functions with}

compact support onRn and the space of infinitely differentiable complex-valued func-tions onRnthat rapidly decrease at infinity, respectively. Let f be a complex-valued mea-surable function onRn_{. The translation, character and dilation operators T}

x, Mxand Ds

are defined by Txf(y) = f (y – x), Mxf(y) = eπ ix,yf(y) and Dptf(y) = t–

n p_f(y

t), respectively,

©2014Kulak and Gürkanlı; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

for x, y∈ Rn_{,  < p, t <∞. With this notation out of the way, one has, for  ≤ p ≤ ∞ and}  p+  p = , (Txf)ˆ(ξ) = M–xˆf(ξ), (Mxf)ˆ(ξ) = Txˆf(ξ),  Dp_tfˆ(ξ) = Dp_t– ˆf(ξ).

For ≤ p ≤ ∞, Lp_(Rn_{) denotes the usual Lebesgue space. A continuous function ω}

satis-fying ≤ ω(x) and ω(x + y) ≤ ω(x)ω(y) for x, y ∈ Rn_{will be called a weight function onR}n_.

If ω(x)≤ ω(x) for all x∈ Rn, we say that ω≤ ω. For ≤ p ≤ ∞, we set

Lp_ωRn=f : f ω∈ LpRn.

It is known that Lpω(Rn) is a Banach space under the norm

f p,ω= f ω p=  Rn f(x)ω(x)pdx  p , ≤ p < ∞ or f ∞,ω= f ω ∞= ess sup x∈Rn _f(x)ω(x)_{, p=∞.}

The dual of the space Lpω(Rn) is the space Lq_ω–(Rn), where_p +_q=  and ω–(x) =_ω_(x). We

say that a weight function υsis of polynomial type if υs(x) = ( +|x|)sfor s≥ . Let f be a

measurable function onRn_{. If there exist C >  and N∈ N such that}

f(x) ≤C + xN

for all x∈ Rn_{, then f is said to be a slowly increasing function []. It is easy to see that}

polynomial-type weight functions are slowly increasing. For f ∈ L(Rn), the Fourier trans-form of f is denoted by ˆf. We know that ˆf is a continuous function onRn_{which vanishes at}

infinity and it has the inequality ˆf ∞≤ f . We denote by M(Rn) the space of bounded

regular Borel measures, by M(ω) the space of μ in M(Rn_{) such that}

μ ω=



Rn

ωd|μ| < ∞.

If μ∈ M(Rn), the Fourier-Stieltjes transform of μ is denoted by ˆμ []. The space (Lp_(Rn₎₎

loc consists of classes of measurable functions f onRn such that

f χK∈ Lp(Rn) for any compact subset K⊂ Rn, where χK is the characteristic function of

K. Let us fix an open set Q⊂ Rn_{with compact closure and ≤ p, q ≤ ∞. The weighted}

Wiener amalgam space W (Lp, Lqω) consists of all elements f ∈ (Lp(Rn))locsuch that Ff(z) =

f χz+Q pbelongs to Lqω(Rn); the norm of W (Lp, Lqω) is f _W(Lp_,Lq

ω)= Ff q,ω[–]. In this paper, P(Rn_{) denotes the family of all measurable functions p :R}n_{→ [, ∞). We}

put

p∗= ess inf

x∈Rn p(x), p

∗_{= ess sup}

We shall also use the notation ∞=x∈ Rn: p(x) =∞.

The variable exponent Lebesgue spaces (or generalized Lebesgue spaces) Lp(x)_(Rn_{) are}

defined as the set of all (equivalence classes) measurable functions f onRn _{such that}

p(λf ) <∞ for some λ > , equipped with the Luxemburg norm

f p(x)= inf  λ>  : p f λ  ≤  . If p∗<∞, then f ∈ Lp(x)_(Rn_{) if}

p(f ) <∞. The set Lp(x)(Rn) is a Banach space with the norm

· p(x). If p(x) = p is a constant function, then the norm · p(x)coincides with the usual

Lebesgue norm · p[]. The spaces Lp(x)(Rn) and Lp(Rn) have many common properties.

A crucial difference between Lp(x)_(Rn_{) and the classical Lebesgue spaces L}p_(Rn_{) is that}

Lp(x)_(Rn_{) is not invariant under translation in general. If p}∗_{<∞, then C}∞

c (Rn) is dense in

Lp(x)_(Rn_{). The space L}p(x)_(Rn_{) is a solid space, that is, if f∈ L}p(x)_(Rn_{) is given and g∈ L} loc(Rn)

satisfies|g(x)| ≤ |f (x)| a.e., then g ∈ Lp(x)_(Rn_{) and g}

p(x)≤ f p(x)by []. In this paper we

will assume that p∗<∞. The space (Lp(x)_(Rn₎₎

loc consists of classes of measurable functions f onRnsuch that

f χK ∈ Lp(x)(Rn) for any compact subset K ⊂ Rn. Let us fix an open set Q⊂ Rn with

compact closure, p(x)∈ P(Rn) and ≤ q ≤ ∞. The variable exponent amalgam space W(Lp(x)_{, L}q

ω) consists of all elements f ∈ (Lp(x)(Rn))locsuch that Ff(z) = f χz+Q p(x)belongs

to Lqω(Rn); the norm of W (Lp(x), Lqω) is f _W(Lp(x)_,Lq

ω)= Ff q,ω[].

2 The bilinear multipliers space BM[W(p1, q1,ω1; p2, q2,

ω

2; p3, q3,

ω

3)]

Lemma . Let≤ p ≤ q < ∞ and ω be a slowly increasing weight function. Then C_c∞(Rn₎

is dense in the Wiener amalgam space W(Lp_{, L}q ω).

Proof Since Cc(Rn) = Lqω(Rn) [], we have Cc(Rn) = W (Lp, Lqω) by a lemma in []. Also we

have the inclusion C_c∞Rn⊂ Cc  Rn_{⊂ W}_Lp_{, L}q ω  .

For the proof that C∞_c (Rn_{) is dense in W (L}p_{, L}q

ω), take any f ∈ W(Lp, Lqω). For given ε > ,

there exists g∈ Cc(Rn) such that

f – g _W(Lp_,Lq ω)<

ε

. (.)

Also, since g∈ Cc(Rn)⊂ Lqω(Rn) and C_c∞(Rn) is dense in Lqω(Rn), by Lemma . in [],

there exists h∈ C_c∞(Rn_{) such that}

g – h q,ω<

ε .

Furthermore, by using the inequality p≤ q, we write g – h W(Lp_,Lq

ω)≤ g – h q,ω< ε

(see [] and []). Combining (.) and (.), we obtain f – h W(Lp_,Lq

ω)≤ f – g W(Lp,Lqω)+ h – g W(Lp,Lqω)< ε.

This completes the proof. 

Definition . Let ≤ p≤ q<∞,  ≤ p≤ q<∞,  < p, q≤ ∞ and ω, ω, ωbe

weight functions onRn_{. Assume that ω}

, ωare slowly increasing functions and m(ξ , η) is

a bounded, measurable function onRn_{× R}n_{. Define}

Bm(f , g)(x) =  Rn  Rnˆf(ξ)ˆg(η)m(ξ,η)e π iξ+η,x_{dξ dη} for all f , g∈ C_c∞(Rn_).

mis said to be a bilinear multiplier onRn_{of type (W (p}

, q, ω; p, q, ω; p, q, ω)) if

there exists C >  such that

Bm(f , g)_W(Lp,Lq_ω)≤ C f W(Lp,Lq_ω) g W(Lp,Lq_ω)

for all f , g∈ C_c∞(Rn_{). That means that B}

m extends to a bounded bilinear operator from

W(Lp_{, L}q ω)× W(Lp, L q ω) to W (Lp, L q ω).

We denote by BM[W (p, q, ω; p, q, ω; p, q, ω)] the space of all bilinear multipliers

of type (W (p, q, ω; p, q, ω; p, q, ω)) and m (W (p,q,ω;p,q,ω;p,q,ω))= Bm .

The following theorem is an example to a bilinear multiplier onRnof type (W (p, q, ω;

p, q, ω; p, q, ω)). Theorem . Let _p  +  p =  p,  q +  q =  q and ω≤ ω. If K∈ L  ω(R n_{), then m(ξ , η) =}

ˆK(ξ – η) defines a bilinear multiplier and m (W (p,q,ω;p,q,ω;p,q,ω))≤ K ,ω.

Proof We know by Theorem . in [] that for f , g∈ C∞_c (Rn), Bm(f , g)(t) =



Rn

f(t – y)g(t + y)K (y) dy. (.)

Also by Proposition .. in [], Tyf ∈ W(Lp, Lqω), T–yg∈ W(Lp, Lqω). So, we write FTyf ∈ Lq

ω(Rn), FT–yg∈ L

q

ω(Rn).

Using the Minkowski inequality and the generalized Hölder inequality, we have Bm(f , g)_W(Lp,Lq_ω) =Bm(f , g)χQ+xpq,ω

= 

Rnf(t – y)g(t + y)K (y) dy χQ+x   p   q,ω ≤ Rn f(t – y)g(t + y)χQ+x(t)_p q,ωK(y)dy ≤  Rn

f(t – y)χQ+x(t)_pg(t + y)χQ+x(t)_{pq,ω}K(y)dy

= 

Rn

Again, by using Proposition .. in [] and the assumption ω≤ ω, we write

FTyfω q≤ ω(y) f W(Lp,Lq_ω)<∞. (.)

From this result, we find FTyf ∈ L

q

ω(Rn). Hence by (.), (.) and the generalized Hölder

inequality, we obtain Bm(f , g)_W(Lp,Lq_ω)≤  Rn FTyf(x)ω(x)_qFT–yg(x)_qK(y)dy ≤  Rn f W(Lp,Lqω) g W(Lp,Lqω) K(y)ω(y) dy = f _W(Lp,Lq_ω) g W(Lp,Lq_ω) K ,ω = C f _W(Lp,Lq_ω) g W(Lp,Lq_ω), (.)

where C = K ,ω. Then m(ξ , η) = ˆK(ξ – η) defines a bilinear multiplier. Finally, using

(.), we obtain m (W (p,q,ω;p,q,ω;p,q,ω))= sup f _{W(Lp ,L}q ω )≤, g W(Lp ,Lqω )≤ Bm(f , g) _W(Lp,Lq_ω) f _W(Lp,Lq_ω) g W(Lp,Lq_ω) ≤ K ,ω. 

Definition . Let ≤ p≤ p<∞,  ≤ q≤ q<∞,  < p, q≤ ∞ and ω, ω, ωbe

weight functions onRn_{. Suppose that ω}

, ωare slowly increasing functions. We denote

by ˜M[(p, q, ω; p, q, ω; p, q, ω)] the space of measurable functions M :Rn→ C such

that m(ξ , η) = M(ξ – η)∈ BM[W(p, q, ω; p, q, ω; p, q, ω)], that is to say,

BM(f , g)(x) =  Rn  Rnˆf(ξ)ˆg(η)M(ξ – η)e π iξ+η,x_{dξ dη}

extends to a bounded bilinear map from W (Lp_{, L}q

ω)× W(Lp, L q ω) to W (Lp, L q ω). We denote M (W (p,q,ω;p,q,ω;p,q,ω))= BM .

Let ω be a weight function. The continuous function ω– _{cannot be a weight}

func-tion. But the following lemma can be proved easily by using the technique of the proof of Lemma ..

Lemma . Let≤ p ≤ q < ∞ and ω be a slowly increasing continuous weight function. Then C_c∞(Rn_{) is dense in W (L}p_{, L}q

ω–) Wiener amalgam space. Theorem . Let_p  +  p_ = ,  q + 

q_ = , q ≥ p≥  and ωbe a continuous, symmetric

slowly increasing weight function. Then m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)] if and

only if there exists C>  such that  _Rn  Rnˆf(ξ)ˆg(η)ˆh(ξ + η)m(ξ,η)dξ dη   ≤ C f W(Lp,Lq_ω) g W(Lp,Lq_ω) h _W(Lp,Lq ω–) for all f, g, h∈ C_c∞(Rn_).

Proof We assume that m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)]. By Theorem . in [],

we write, for all f , g, h∈ C_c∞(Rn_),

 _Rn  Rnˆf(ξ)ˆg(η)ˆh(ξ + η)m(ξ,η)dξ dη   =_Rnh(y) ˜Bm(f , g)(y) dy   ≤ Rn h(y)˜Bm(f , g)(y)dy, (.)

where ˜Bm(f , g)(y) = Bm(f , g)(–y). If we set –t = u, we have

˜Bm(f , g)_W(Lp,Lq_ω)=F Q ˜Bm(f ,g)  q,ω=Bm(f , g)(u)χQ+x(–u)pq,ω =Bm(f , g)(u)χ–Q–x(u)_{pq,ω}=FB–Qm(f ,g)(–x)q,ω. (.)

Since ωis a symmetric weight function, if we set –x = y, we have

˜Bm(f , g)_W(Lp,Lq_ω)=F –Q

Bm(f ,g)(y)q,ω. (.)

We know from [] and [] that the definition of W (Lp_{, L}q

ω) is independent of the choice

of Q. Then there exists C >  such that F_B–Q_{m(f ,g)}(y)_q

,ω≤ CF

Q

Bm(f ,g)(y)q,ω. (.)

Hence, by (.) and (.), we have ˜Bm(f , g)_W(Lp,Lq_ω)≤ CF

Q

Bm(f ,g)(y)q,ω= CBm(f , g)W(Lp,Lqω). (.) Since from the assumption m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)] the right-hand side

of (.) is finite, thus ˜Bm(f , g)∈ W(Lp, Lωq). On the other hand, since m∈ BM[W(p, q,

ω; p, q, ω; p, q, ω)], there exists C>  such that

Bm(f , g)_W(Lp,Lq_ω)≤ C f W(Lp,Lq_ω) g W(Lp,Lq_ω). (.)

Combining (.) and (.), we have

˜Bm(f , g)_W(Lp,Lq_ω)≤ CC f W(Lp,Lq_ω) g W(Lp,Lq_ω). (.)

If we apply the Hölder inequality to the right-hand side of inequality (.) and use inequal-ity (.), we obtain  _Rn  Rnˆf(ξ)ˆg(η)ˆh(ξ + η)m(ξ,η)dξ dη   ≤˜Bm(f , g)_W(Lp,Lq_ω) h _W(Lp_,Lq ω–) ≤ CC f _W(Lp,L_ωq) g W(Lp,Lq_ω) h W(Lp,Lq ω–_ ) .

For the proof of converse, assume that there exists a constant C >  such that  _Rn  Rnˆf(ξ)ˆg(η)ˆh(ξ + η)m(ξ,η)dξ dη   ≤ C f _W(Lp,Lq_ω) g W(Lp,Lq_ω) h _W(Lp,Lq ω–_ ) (.)

for all f , g, h∈ C∞_c (Rn_{). From the assumption and (.), we write}

 _Rnh(y) ˜Bm(f , g)(y) dy   ≤ C f W(Lp,Lq_ω) g W(Lp,Lq_ω) h _W(Lp,Lq ω–_ ) . (.)

Define a function  from C_c∞(Rn_{)⊂ W(L}p_{, L}q   ω–) toC such that (h) =  Rn

h(y) ˜Bm(f , g)(y) dy.

is linear and bounded by (.). Also, since q_≥ p_≥ , we have C_c∞(Rn_{) = W (L}p_{, L}q

ω_–)

by Lemma .. Thus  extends to a bounded function from W (Lp_{, L}q

ω–_ ) toC. Then  ∈ (W (Lp_{, L}q   ω–_ )) ∗_{= W (L}p_{, L}q

ω). Again, since the definition of W (Lp, L

q

ω) is independent of

the choice of Q, there exists C>  such that

Bm(f , g)_W(Lp,Lq_ω)≤ C˜Bm(f , g)W(Lp,Lq_ω). (.)

Combining (.) and (.), we obtain

Bm(f , g)_W(Lp,Lq_ω)≤ C˜Bm(f , g)W(Lp,Lq_ω) = C  = C sup h W(Lp ,Lq ω–_ ) ≤ |(h)| h W(Lp,Lq ω–_ ) ≤ CC f _W(Lp,Lq_ω) g W(Lp,Lq_ω).

This completes proof. 

The following theorem is a generalization of Theorem ..

Theorem . Let  p +  p =  p,  q +  q =  q, ω ≤ ω and υ(x) = C( + x ₎N_{, C≥ ,}

N ∈ N be a weight function. If μ ∈ M(υ) and m(ξ, η) = ˆμ(αξ + βη) for α, β ∈ R, then m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)]. Moreover,

m (W (p,q,ω;p,q,ω;p,q,ω))≤ μ υ if|α| ≤ ,

m (W (p,q,ω;p,q,ω;p,q,ω))≤ |α| N _μ

υ if|α| > .

Proof Let f , g∈ C∞_c (Rn_{). By Theorem . in [], we have}

Bm(f , g)(t) =



Also by [] we write the inequalities

Tαyf _W(Lp,Lq_ω)≤ ω(αy) f W(Lp,Lq_ω) (.)

and

Tβyg _W(Lp,Lq_ω)≤ ω(αy) g W(Lp,Lq_ω). (.)

From these inequalities, we have FTαyf∈ L

q

ω(R

n_{) and F} Tβyg∈ L

q

ω(Rn). If we use the

in-equality ω≤ ωand set x – αt = u, we obtain

FTαyfω q≤ FTαyfω q≤ ω(αy) f W(Lp,Lq_ω), (.)

and hence FTαyfω∈ Lq(Rn). Then by (.), (.), (.), (.) and the Hölder inequality,

we have _{Bm(f , g)} W(Lp,Lq_ω)≤  _Rn _{f(t – αy)g(t – βy)χQ+x(t)} pd|μ|(y)   q,ω ≤  Rn f(t – αy)χQ+x(t) pg(t – βy)χQ+x(t)pd|μ|(y)   q,ω ≤  Rn

FTαyf(x)FTβyg(x)_q,ωd|μ|(y) ≤  Rn _{FTαyf(x)ω(x)} qFTβyg(x)qd|μ|(y) ≤  Rn ω(αy) f _W(Lp,Lq_ω) g W(Lp,Lq)d|μ|(y) ≤ f _W(Lp,Lq_ω) g W(Lp,Lq_ω)  Rn ω(αy) d|μ|(y). (.)

Now, suppose that α≤ . Since ω is a slowly increasing weight function, there exist

C≥  and N ∈ N such that ω(x)≤ C   + xN_{= υ(x).} Then  Rn ω(αy) d|μ|(y) ≤  Rn C + αyNd|μ|(y) ≤  Rn C + yNd|μ|(y) = μ υ. Hence by (.) Bm(f , g)_W(Lp,Lq_ω)≤ f W(Lp,Lq_ω) g W(Lp,Lq_ω) μ υ. (.)

Thus m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)], and by (.) we obtain

m (W (p,q,ω;p,q,ω;p,q,ω))= sup f _{W(Lp ,L}q ω )≤, g W(Lp ,Lqω)≤ Bm(f , g) _W(Lp,Lq_ω) f _W(Lp,Lq_ω) g W(Lp,Lq_ω) ≤ μ υ.

Similarly, if α > , then  Rn ω(αy) d|μ|(y) ≤  RnC  α+ αyNd|μ|(y) = αN  Rn υ(y) d|μ|(y) = αN μ υ. Therefore by (.) we have Bm(f , g)_W(Lp,Lq_ω)≤ α N _f W(Lp,Lq_ω) g W(Lp,Lq_ω) μ υ. (.)

Hence, we obtain m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)], and by (.)

m (W (p,q,ω;p,q,ω;p,q,ω))= sup f W(Lp ,Lq_{ω )}≤, g W(Lp ,Lq_ω)≤ Bm(f , g) _W(Lp,Lq_ω) f _W(Lp,Lq_ω) g W(Lp,Lq_ω) ≤ αN _μ υ. 

Now, we will give some properties of the space BM[W (p, q, ω; p, q, ω; p, q, ω)]. Theorem . Let m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)].

(a) T(ξ,η)m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)]for each (ξ, η)∈ Rnand

T(ξ,η)m (W (p,q,ω;p,q,ω;p,q,ω))= m (W (p,q,ω;p,q,ω;p,q,ω)).

(b) M(ξ,η)m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)]for each (ξ, η)∈ R n_and

M(ξ,η)m (W (p,q,ω;p,q,ω;p,q,ω))

≤ ω(–ξ)ω(–η) m (W (p,q,ω;p,q,ω;p,q,ω)).

Proof (a) Let f , g∈ C_c∞(Rn_{). From Theorem . in [], we write the equality}

BT_{(ξ,η)}m(f , g)(x) = eπ iξ+η,xBm(M–ξf, M–ηg)(x). (.)

Also the equalities M–ξf W(Lp,L_ωq)= f W(Lp,Lq_ω) and M–ηg W(Lp,Lq_ω)= g W(Lp,Lq_ω)

are satisfied. Then, using equality (.) and the assumption m∈ BM[W(p, q, ω; p, q,

ω; p, q, ω)], we have BT_{(ξ,η)}m(f , g)_W(Lp,Lq_ω) =e π iξ+η,x_B m(M–ξf, M–ηg)_W(Lp,Lq_ω) =Bm(M–ξf, M–ηg)_W(Lp,Lq_ω) ≤ C f _W(Lp,Lq_ω) g W(Lp,Lq_ω)

for some C > . Thus T(ξ,η)m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)]. Moreover, we

ob-tain T(ξ,η)m (W (p,q,ω;p,q,ω;p,q,ω)) = BT_{(ξ,η)}m = sup f _{W(Lp ,L}q ω )≤, g W(Lp ,Lqω)≤ BT_{(ξ,η)}m(f , g) _W(Lp,Lq_ω) f W(Lp,Lq_ω) g W(Lp,Lq_ω)

= sup M_–ξf W(Lp ,Lq_{ω )}≤, M–ηg W(Lp ,Lq_ω)≤ BT_{(ξ,η)}m(M–ξf, M–ηg) W(Lp,Lq_ω) M–ξf W(Lp,Lq_ω) M–ηg W(Lp,Lq_ω) = Bm = m (W (p,q,ω;p,q,ω;p,q,ω)).

(b) For any f , g∈ C_c∞(Rn), we write

BM_{(ξ,η)}m(f , g)(x) = Bm(T–ξf, T–ηg)(x) (.)

by Theorem . in []. Also, the inequalities T–ξf W(Lp,Lq_ω)≤ ω(–ξ) f W(Lp,Lq_ω)and

T–ηg W(Lp,Lq_ω)≤ ω(–η) g W(Lp,Lq_ω) are satisfied []. Since m∈ BM[W(p, q, ω; p,

q, ω; p, q, ω)], by (.) we have

BM_{(ξ,η)}m(f , g)_W(Lp,Lq_ω) =Bm(T–ξf, T–ηg)_W(Lp,Lq_ω)

≤ Bm T–ξf W(Lp,Lq_ω) T–ηg W(Lp,Lq_ω)

≤ ω(–ξ)ω(–η) Bm f _W(Lp,Lq_ω) g W(Lp,Lq_ω) (.)

and hence M(ξ,η)m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)]. So, by (.) we obtain

M(ξ,η)m (W (p,q,ω;p,q,ω;p,q,ω)) = sup f _{W(Lp ,L}q ω )≤, g W(Lp ,Lqω)≤ BM_{(ξ,η)}m(f , g) _W(Lp,Lq_ω) f W(Lp,Lq_ω) g W(Lp,Lq_ω) ≤ ω(–ξ)ω(–η) Bm = ω(–ξ)ω(–η) m (W (p,q,ω;p,q,ω;p,q,ω)).  Lemma . If ω is a slowly increasing weight function such that ω(x)≤ C( + x)N= υ(x)

and f ∈ W(Lp_{, L}q ω), then Dpyf ∈ W(Lp, Lqυ). Moreover, Dp_yf_W(Lp_,Lq ω)≤ C f W(Lp,L q υ) if y≤ , Dp_yf_W(Lp_,Lq ω)< Cy n q+N f W(Lp_,Lq υ) if y>  for some C> .

Proof Take any f ∈ W(Lp_{, L}q

ω). If we get t_y= u, we obtain Dp_yf_W(Lp_,Lq ω)=  _Rn Dp_yf(t)χQ+x(t)pdt  p  q,ω =  Rn  y–n_p f t y  pχQ+x(t) dt  p  q,ω =  Rn f(u)pχ_y–_Q+y–_x(u) du  p  q,ω =F_fy–Qy–x_q,ω. (.)

Again, if we say y–_{x= s and use ω to be slowly increasing, then there exist C} >  and N∈ N such that Dp_yf_W(Lp_,Lq ω) =  Rn F_fy–Qy–xqω(x)qdx  q = ynq  Rn F_fy–Q(s)qω(ys)qds  q ≤ ynq  Rn F_fy–Q(s)qC   + ysNqds  q (.) by equation (.).

Let y≤ . Using inequality (.), we have Dp_yf_W(Lp_,Lq ω)≤  Rn F_fy–Q(s)qC   + sNqds  q =F_fy–Q_q,υ. (.) Since y–_{Qis a compact set and the definition of W (L}p_{, L}q

υ) is independent of the choice

of a compact set Q, then there exists C >  such that

F_fy–Q_q,υ≤ CF_fQ_q,υ (.) by [, ]. Then by (.) we write Dp_yf_W(Lp_,Lq ω)≤F y–Q f q,υ≤ CF Q f q,υ= C f W(Lp_,Lq υ). Thus we have Dpyf ∈ W(Lp, Lqυ).

Now, assume that y > . Similarly, by (.) and (.), we get Dp_yf_W(Lp_,Lq ω) < y n q  Rn F_fy–Q(s)qC  y+ ysNqds  q = ynq+N  Rn F_fy–Q(s)qυ(s)qds  q ≤ ynq+N f W(Lp_,Lq υ). Hence Dpyf ∈ W(Lp, Lqυ). 

Theorem . Let υi(x) = Ci( + x)Ni, Ci> , Ni>  for i = , , , and let ωbe a slowly

in-creasing weight function. If _q=_p

+ 

p– 

p,  < y <∞ and m ∈ BM[W(p, q, ω; p, q, ω;

p, q, υ)], then Dqym∈ BM[W(p, q, υ; p, q, υ; p, q, ω)]. Moreover, then

Dq_ym_{(W (p} ,q,υ;p,q,υ;p,q,ω)) ≤ Cy–_qn–N _m (W (p,q,ω;p,q,ω;p,q,υ)) if y≤ , Dq_ym_{(W (p} ,q,υ;p,q,υ;p,q,ω)) < Cyqn+qn+N+N _m (W (p,q,ω;p,q,ω;p,q,υ)) if y> .

Proof Let f ∈ W(Lp_{, L}q_ω

) and g∈ W(Lp, L

q

ω) be given. From Lemma ., we have D

p

y f ∈

W(Lp_{, L}q_ω ) and D

p

y g∈ W(Lp, Lqω). Also it is known by Theorem . in [] that

B_Dq ym(f , g)(y) = D p y–Bm  Dp yf, Dpyg  (y). If we use this equality, we write

_B Dqym(f , g)W(Lp,Lq_ω)=  _Rn _Dp y–Bm  Dp yf, Dpyg  (t)χQ+x(t) p dt  p  q,ω =  Rn  yn p_Bm_Dp yf, Dpyg  t y–  χQ+x(t)  pdt  p  q,ω =  Rny n_B m  Dp yf, Dpyg  (ty)χQ+x(t)pdt  p  q,ω .

If we say yt = u in the last equality, we have B_Dq ym(f , g)W(Lp,Lq_ω)=  _Rn Bm  Dp y f, Dpyg 

(u)χyQ+yx(u)pdt

 p  q,ω =FyQ Bm(Dpy f,Dpy g)(yx)  q,ω. (.)

On the other hand, since ωis a slowly increasing weight function, there exist C> ,

N>  such that ω(x)≤ C( + x)N= υ(x). If we make the substitution yx = s in equality

(.), we obtain B_Dq ym(f , g)W(Lp,Lq_ω) =F yQ Bm(Dpy f,Dpy g)(yx)  q,ω =  Rn FyQ Bm(Dpy f,Dpy g)(s) q ω  y–sqy–nds  q = y–qn  Rn FyQ Bm(Dpy f,Dpy g)(s) q_ω   y–sq ds  q ≤ y–_qn Rn _FyQ Bm(Dpy f,Dpy g)(s) q C   + y–sNqds  q . We assume that y≤ . Then

B_Dq ym(f , g)W(Lp,Lq_ω)≤ y –_qn Rn FyQ Bm(Dpy f,Dpy g)(s) q C  y–+ y–sNqds  q = y–qn–N FyQ Bm(Dpy f,Dpy g)  q,υ.

Also, since yQ is a compact set, we have B_Dq ym(f , g)W(Lp,Lq_ω)≤ Cy –_qn–N_FQ Bm(Dpyf,Dpy g)  q,υ = Cy–qn–N Bm  Dp y f, DpygW(Lp,Lq_υ). (.)

Since m∈ BM[W(p, q, ω; p, q, ω; p, q, υ)], by Lemma . and inequality (.), we obtain B_Dq ym(f , g)W(Lp,Lq_ω) ≤ Cy–_qn–N _m (W (p,q,ω;p,q,ω;p,q,υ)) f W(Lp,Lq_υ) g W(Lp,Lq_υ). (.)

Then Dqym∈ BM[W(p, q, υ; p, q, υ; p, q, ω)], and by (.) we have

Dq_ym_{(W (p}

,q,υ;p,q,υ;p,q,ω))≤ Cy

–_qn–N _m

(W (p,q,ω;p,q,ω;p,q,υ)).

Now let y > . Again, since m∈ BM[W(p, q, ω; p, q, ω; p, q, υ)], by Lemma . and

inequality (.), we obtain B_Dq ym(f , g)W(Lp,Lq_ω) < CBm  Dp yf, DpygW(Lp,Lq_υ) < Cyqn+qn+N+N _m (W (p,q,ω;p,q,ω;p,q,υ)) f W(Lp,Lq_υ) g W(Lp,Lq_υ). (.)

Thus Dqym∈ BM[W(p, q, υ; p, q, υ; p, q, ω)], and by (.) we have

Dq_ym_{(W (p} ,q,υ;p,q,υ;p,q,ω))< Cy n q+qn+N+N _m (W (p,q,ω;p,q,ω;p,q,υ)).  Theorem . Let m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)].

(a) If ∈ L(Rn), then ∗ m ∈ BM[W(p, q, ω; p, q, ω; p, q, ω)]and

 ∗ m (W (p,q,ω;p,q,ω;p,q,ω))≤   m (W (p,q,ω;p,q,ω;p,q,ω)).

(b) If ∈ L

ω(Rn)such that ω(u, υ) = ω(u)ω(υ), then

ˆm ∈ BM[W(p, q, ω; p, q, ω; p, q, ω)]and

ˆm (W (p,q,ω;p,q,ω;p,q,ω))≤  ,ω m (W (p,q,ω;p,q,ω;p,q,ω)).

Proof (a) Let f , g∈ C_c∞(Rn_{) be given. By Proposition . in []}

B∗m(f , g)(y) =



Rn 

Rn

(u, v)BT(u,v)m(f , g)(y) du dv.

If we use Theorem . and the assumption m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)], we

have B∗m(f , g)_W(Lp,Lq_ω) ≤  Rn  Rn (u, v)BT(u,v)m(f , g)W(Lp,Lq_ω)du dv ≤  Rn  Rn (u, v) T(u,v)m (W (p,q,ω;p,q,ω;p,q,ω)) f W(Lp,L_ωq) g W(Lp,Lq_ω)du dv = m (W (p,q,ω;p,q,ω;p,q,ω))   f W(Lp,Lq_ω) g W(Lp,Lq_ω)<∞. (.)

Hence ∗ m ∈ BM[W(p, q, ω; p, q, ω; p, q, ω)], and by (.) we obtain

 ∗ m (W (p,q,ω;p,q,ω;p,q,ω))≤   m (W (p,q,ω;p,q,ω;p,q,ω)).

(b) Let ∈ Lω(Rn). Take any f , g∈ C∞c (Rn). It is known by Proposition . in [] that

B_ˆm(f , g)(x) =  Rn  Rn (u, v)BM(–u,–v)m(f , g)(x) du dv.

Since m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)], we have M(–u,–v)m∈ BM[W(p, q, ω; p,

q, ω; p, q, ω)] and M(–u,–v)m (W (p,q,ω;p,q,ω;p,q,ω))≤ ω(u)ω(v) m (W (p,q,ω;p,q,ω;p,q,ω)) by Theorem .. Then _Bˆm(f , g)_W(Lp,Lq ω) ≤  Rn  Rn (u, v)BM(–u,–v)m(f , g)W(Lp,Lq_ω)du dv ≤  Rn  Rn (u, v) M(–u,–v)m (W (p,q,ω;p,q,ω;p,q,ω)) f p,ω g p,ωdu dv ≤  Rn  Rn (u, v)ω(u)ω(υ) m (W (p,q,ω;p,q,ω;p,q,ω)) × f W(Lp,Lq_ω) g W(Lp,Lq_ω)du dv = m (W (p,q,ω;p,q,ω;p,q,ω)) f W(Lp,Lq_ω) g W(Lp,Lq_ω)  ,ω. (.)

Thus from (.) we obtain ˆm∈ BM[W(p, q, ω; p, q, ω; p, q, ω)] and

ˆm (W (p,q,ω;p,q,ω;p,q,ω))≤  ,ω m (W (p,q,ω;p,q,ω;p,q,ω)).  Theorem . Let m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)]. If Q, Qare bounded

mea-surable sets inRn_{, then}

h(ξ , η) =  μ(Q× Q)  Q×Q m(ξ + u, η + v) du dv ∈ BM W(p, q, ω; p, q, ω; p, q, ω)  .

Proof Let f , g∈ C∞_c (Rn). We know by Theorem . in [] that Bh(f , g)(x) =  μ(Q× Q)  Q×Q BT(–u,–v)m(f , g)(x) du dv.

From Theorem ., we have Bh(f , g)_W(Lp,Lq_ω) ≤  μ(Q× Q)  Q×Q BT(–u,–v)m(f , g)W(Lp,Lq_ω)du dv

≤  μ(Q× Q) Q×Q T(–u,–v)m (W (p,q,ω;p,q,ω;p,q,ω)) × f W(Lp,Lq_ω) g W(Lp,Lq_ω)du dv =  μ(Q× Q) m (W (p,q,ω;p,q,ω;p,q,ω)) × f _W(Lp,Lq_ω) g W(Lp,Lq_ω)μ(Q× Q) = m (W (p,q,ω;p,q,ω;p,q,ω)) f W(Lp,Lq_ω) g W(Lp,Lq_ω). Hence, we obtain h(ξ , η)∈ BM[W(p, q, ω; p, q, ω; p, q, ω)].  Theorem . Let ω be a continuous, symmetric, slowly increasing weight function,

ω(u, v) = ω(u)ω(v), ω≤ ω,_q_+_q_ =_q_,_p_+_p_ =_p_,_p_+_p  = ,  q+  q_ =  and q ≥ p. Assume that ∈ L ω(Rn), ∈ Lω(R n_{) and } ∈ Lω(R n_{). If m(ξ , η) = ˆ} (ξ ) ˆ(ξ , η) ˆ(η), then m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)].

Proof Take any f , g, h∈ C∞_c (Rn_{). Then, by Theorem . in [], we write}

 _Rn  Rnˆf(ξ)ˆg(η)ˆh(ξ + η)m(ξ,η)dξ dη   ≤_Rn h(y) ˜B_ˆ(f∗ , g∗ )(y)dy.

On the other hand, we know that the inequalities

f ∗  _W(Lp,Lq_ω)≤ C f W(Lp,Lq_ω)  ,ω (.)

and

g ∗  _W(Lp,Lq_ω)≤ C g W(Lp,Lq_ω)  ,ω (.)

hold, where C> , C>  by []. That means f∗∈ W(Lp, Lωq) and g∗∈ W(Lp, L

q

ω).

Also, every constant function is bilinear multiplier of type (W (p, q, ω; p, q, ω; p, q,

ω)) under the given conditions. So, by Theorem ., we have ˆ∈ BM[W(p, q, ω; p, q,

ω; p, q, ω)]. Now, if we say that –y = u, we have

˜B_ˆ(f∗ , g∗ )(y)_W(Lp,Lq_ω) =  Rn B_ˆ(f ∗ , g∗ )(u) p χ–Q–x(u) du  p  q,ω =F_B–Q ˆ(f∗,g∗)(–x)q,ω.

In this here, we set –x = u and use ωto be symmetric. Then we have

˜B_ˆ(f∗ , g∗ )(y)_W(Lp,Lq_ω)≤ CF

Q

B_ˆ(f∗,g∗)q,ω

by []. Using the Hölder inequality, inequalities (.), (.), (.) and ˆ∈ BM[W(p, q, ω; p, q, ω; p, q, ω)], we find  _Rn  Rnˆf(ξ)ˆg(η)ˆh(ξ + η)m(ξ,η)dξ dη   ≤ h W(Lp,Lq ω–_ ) B_ˆ(f∗ , g∗ )_W(Lp,Lq_ω) ≤ CCC h W(Lp,Lq ω–_ ) Bˆ  ,ω  ,ω f W(Lp,Lq_ω) g W(Lp,Lq_ω).

If we say C = CCC Bˆ  ,ω  ,ω, then we obtain

 _Rn  Rnˆf(ξ)ˆg(η)ˆh(ξ + η)m(ξ,η)dξ dη   ≤ C f W(Lp,Lq_ω) g W(Lp,Lq_ω) h _W(Lp,Lq ω–_ ) .

From Theorem ., we have m∈ BM[W(p, q, ω; p, q, ω; p, q, ω)].  Theorem . Let ≤ p, p≤ p ≤ ,  ≤ q, q≤ r ≤ , p ≥ p, q ≥ r and q ≥ p

such that _p  =  p +  p –  p and  q =  q +  q – 

r. Assume that ω is a continuous,

bounded, symmetric weight function. If m∈ W(Lp_(Rn_{), L}r_(Rn_{))∩ L}∞_(Rn_{), then m∈}

BM[W (p, q, ω; p, q, ω; p, q, ω)].

Proof Firstly, we show that m∈ BM[W(p, r, ω; p, r, ω;∞, ∞, ω)]. Take any f , g, h ∈

C_c∞(Rn_{). Let A× B ⊂ R}n _{be a closed and bounded rectangle. Since the definition of}

W(Lp_(Rn_{), L}r_(Rn_{)) is independent of the choice of a compact set Q, then, by using}

Fubini’s theorem, we get ˆf(ξ )ˆg(η)_W(Lp_(Rn_),Lr_(Rn₎₎

=FQ_ˆfˆgLr_(Rn₎≤ CFA_ˆfˆg×B_Lr_(Rn₎

= C ˆf χx+A p ˆgχy+B p_Lr_(Rn₎= CFA_ˆf(x)FˆgB(y)Lr_(Rn₎

= CFA_ˆfrFB_ˆgr= C ˆf _W(Lp_,Lr₎ ˆg _W(Lp_,Lr₎ (.) for some C> . So, we have ˆf(ξ )ˆg(η) ∈ W(Lp



(Rn_{), L}r_(Rn_{)). By using the Hölder}

inequal-ity, the Hausdorff-Young inequality and equality (.), we obtain  _Rn  Rnˆf(ξ)ˆg(η)ˆh(ξ + η)m(ξ,η)dξ dη   ≤ C h ˆf(ξ )ˆg(η)_W(Lp_(Rn_),Lr_(Rn₎₎ m W(Lp_(Rn_),Lr_(Rn₎₎ ≤ C h  ˆf _W(Lp_,Lr₎ ˆg _W(Lp_,Lr₎ m W(Lp_(Rn_),Lr_(Rn₎₎ ≤ CC h W(L_,L₎ f _W(Lp_,Lr₎ g _W(Lp_,Lr₎ m _W(Lp_(Rn_),Lr_(Rn₎₎ ≤ CC ω ∞ m W(Lp_(Rn_),Lr_(Rn₎₎ f _W(Lp_,Lr ω) g W(Lp,Lrω) h W(L_,L ω–_ )

Now, we show that m∈ BM[W(p, r, ω; , , ω; p, r, ω)]. Again, using Fubini’s theorem,

we have

ˆf(–v) ˆh(u)_W(Lp_(Rn_),Lr_(Rn₎₎

≤ Cˆf(–v) ˆh(u)χ(x,y)+A×B_Lp_(Rn_)Lr_(Rn₎

= C ˆf χ–x–A p ˆhχy+B p_Lr_(Rn₎= CF–A_ˆf (–x)F_ˆhB(y)_Lr_(Rn₎

= CF–A_{ˆf r}F_ˆhB_r≤ CC ˆf W(Lp_,Lr₎ ˆh _W(Lp_,Lr₎ (.) for some C> , C> . So, ˆf(–v) ˆh(u)∈ W(Lp



(Rn_{), L}r_(Rn_{)). Similarly, we have}

m(–v, u + v)_W(Lp_(Rn_),Lr_(Rn₎₎≤ CFm(–A)×(A+B)Lr_(Rn₎

≤ CC m W(Lp_(Rn_),Lr_(Rn₎₎ (.)

for some C> , C> . That means m(–v, u + v)∈ W(Lp(Rn), Lr(Rn)). We set ξ + η = u

and ξ = –v. Then, by using the Hölder inequality, the Hausdorff-Young inequality, (.) and (.), we get  _Rn  Rnˆf(ξ)ˆg(η)ˆh(ξ + η)m(ξ,η)dξ dη   =  Rn 

Rnˆf(–v)ˆg(u + v)ˆh(u)m(–v,u + v)dudv   ≤ g ˆf(–v)ˆg(u)_W(Lp_(Rn_),Lr_(Rn₎₎m(–v, u + v)_W(Lp_(Rn_),Lr_(Rn₎₎ ≤ CCCC g  ˆf _W(Lp_,Lr₎ ˆh _W(Lp_,Lr₎ m W(Lp_(Rn_),Lr_(Rn₎₎ ≤ CCCCC ω ∞ m W(Lp_(Rn_),Lr_(Rn₎₎ f _W(Lp_,Lr ω) g W(L_,L ω) h W(Lp,Lrω–_ ). Thus, by Theorem ., we obtain m∈ BM[W(p, r, ω; , , ω; p, r, ω)]. Similarly, if we

change the variables ξ + η = u and η = –v, then  _Rn  Rnˆf(ξ)ˆg(η)ˆh(ξ + η)m(ξ,η)dξ dη   = f   Rn  Rnˆg(–v) ˆh(u) m(u + v, –v)du dv ≤ C ω ∞ m W(Lp(Rn_),Lr_(Rn₎₎ f _W(L_,L ω) g W(Lp,Lωr ) h W(Lp,Lr_ω–  ). Hence m∈ BM[W(, , ω; p, r, ω; p, r, ω)].

We take ˜p, ˜q,˜pand˜qsuch that ≤ ˜p≤ p,  ≤ ˜q≤ r, p≤ ˜p≤ ∞ and r≤ ˜q≤ ∞.

Since m∈ BM[W(p, r, ω; p, r, ω;∞, ∞, ω)] and m∈ BM[W(, , ω; p, r, ω; p, r, ω)], we

have m∈ BM[W(˜p,˜q, ω; p, r, ω;˜p,˜q, ω)] by the interpolation theorem in [, ] such

that  ˜p = – θ p + θ ,  ˜p = – θ ∞ + θ p, (.)  ˜q = – θ r + θ ,  ˜q = – θ ∞ + θ r (.)

for all  ≤ θ ≤ . On the other hand, from equalities (.) and (.), we obtain the equalities _˜p  –  ˜p =  p and  ˜q –  ˜q = 

r. Similarly, we take ˜p, ˜q, ˜r and ˜s such

that  ≤ ˜p ≤ p,  ≤ ˜q ≤ r, p ≤ ˜r ≤ ∞ and r ≤ ˜s ≤ ∞. Again, if we use m ∈

BM[W (p, r, ω; p, r, ω;∞, ∞, ω)] and m∈ BM[W(p, r, ω; , , ω; p, r, ω)], we have m∈

BM[W (p, r, ω;˜p,˜q, ω;˜r,˜s, ω)] by the interpolation theorem in [, ] such that

 ˜p = – θ p + θ ,  ˜r = – θ ∞ + θ p, (.)  ˜q = – θ r + θ ,  ˜s = – θ ∞ + θ r (.)

for all ≤ θ ≤ . So, from equalities (.) and (.), we have _˜p

 –  ˜r =  pand  ˜q –  ˜s =  r.

Now, we choose˜p,˜psuch that ≤ ˜p≤ p< p and ≤ ˜p≤ p< p. Let these numbers

have the following conditions:  p –  p= ( – θ )  ˜p – p  , (.)  p –  p= ( – θ )  ˜p –  p  (.) for  < θ < . Again, we choose ˜q, ˜q such that ≤ ˜q≤ q< r and ≤ ˜q≤ q< r. Let

these numbers have the following conditions:  q – r= ( – θ )  ˜q – r  , (.)  q – r= ( – θ )  ˜q – r  (.) for  < θ < . If we use equalities (.), (.), (.), (.), (.) and (.), we write

 p = – θ ˜p +θ p,  p = – θ p + θ ˜p , (.)  q = – θ ˜q +θ r,  q = – θ r + θ ˜q . (.)

Moreover, using the equalities _˜p

–  ˜p =  p,  ˜p –  ˜r = 

p, (.) and the assumption

 p =  p+  p –  p, we obtain  p = – θ ˜p + θ ˜r . (.)

Similarly, using the equalities _˜q

–  ˜q =  r,  ˜q –  ˜s = 

r, (.) and the assumption

 q =  q+  q –  r, we obtain  q = – θ ˜q + θ ˜s . (.) Since m∈ BM[W(˜p,˜q, ω; p, r, ω;˜p,˜q, ω)], m∈ BM[W(p, r, ω;˜p,˜q, ω;˜r,˜s, ω)],

then the bilinear multipliers Bm : W (L˜p, Lω˜q) × W(Lp, Lrω)→ W(L ˜p_{, L}˜q