The spaces of bilinear multipliers of weighted Lorentz type modulation spaces
Authors: Gürkanlı Ahmet Turan , Kulak Öznur , Sandıkçı Ayşe 
Published: March 2016

Abstract
Fix a nonzero window

g
∈

𝒮
⁢

(

ℝ
n

)

${g\in\mathcal{S}(\mathbb{R}^{n})}$

, a weight function w on

ℝ

2
⁢
n

${\mathbb{R}^{2n}}$

and

1
≤
p

,

q
≤
∞

${1\leq p,q\leq\infty}$

. The weighted Lorentz type modulation
space

M
⁢

(
p
,
q
,
w
)

⁢

(

ℝ
n

)

${M(p,q,w)(\mathbb{R}^{n})}$

consists of all tempered distributions

f
∈

𝒮
′

⁢

(

ℝ
n

)

${f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})}$

such that the short time Fourier transform

V
g

⁢
f

${V_{g}f}$

is in the
weighted Lorentz space

L
⁢

(
p
,
q
,

w

⁢
d
⁢
μ

)

⁢

(

ℝ

2
⁢
n

)

${L(p,q,w\,d\mu)(\mathbb{R}^{2n})}$

.
The norm on

M
⁢

(
p
,
q
,
w
)

⁢

(

ℝ
n

)

${M(p,q,w)(\mathbb{R}^{n})}$

is

∥
f
∥

M
⁢

(
p
,
q
,
w
)

=

∥

V
g

⁢
f

∥

p
⁢
q

,
w

${\|f\/\|_{M(p,q,w)}=\|V_{g}f\/\|_{pq,w}}$

. This space was firstly defined and some of its properties were investigated for the unweighted case by Gürkanlı in [9] and generalized to the weighted case by Sandıkçı and Gürkanlı in [16]. Let

1
<

p
1

,

p
2

<
∞

${1<p_{1},p_{2}<\infty}$

,

1
≤

q
1

,

q
2

<
∞

${1\leq q_{1},q_{2}<\infty}$

,

1
≤

p
3

,

q
3

≤
∞

${1\leq p_{3},q_{3}\leq\infty}$

,

ω
1

,

ω
2

${\omega_{1},\omega_{2}}$

be polynomial weights and

ω
3

${\omega_{3}}$

be a weight function on

ℝ

2
⁢
n

${\mathbb{R}^{2n}}$

.
In the present paper, we define the bilinear multiplier operator from

M
⁢

(

p
1

,

q
1

,

ω
1

)

⁢

(

ℝ
n

)

×
M

⁢

(

p
2

,

q
2

,

ω
2

)

⁢

(

ℝ
n

)

${M(p_{1},q_{1},\omega_{1})(\mathbb{R}^{n})\times M(p_{2},q_{2},\omega_{2})(%
\mathbb{R}^{n})}$

to

M
⁢

(

p
3

,

q
3

,

ω
3

)

⁢

(

ℝ
n

)

${M(p_{3},q_{3},\omega_{3})(\mathbb{R}^{n})}$

in the following way.
Assume that

m
⁢

(
ξ
,
η
)

${m(\xi,\eta)}$

is a bounded function on

ℝ

2
⁢
n

${\mathbb{R}^{2n}}$

, and define

B
m

⁢

(
f
,
g
)

⁢

(
x
)

=

∫

ℝ
n

∫

ℝ
n

f
^

⁢

(
ξ
)

⁢

g
^

⁢

(
η
)

⁢
m
⁢

(
ξ
,
η
)

⁢

e

2
⁢
π
⁢
i
⁢

〈

ξ
+
η

,
x
〉

⁢

𝑑

ξ

⁢

𝑑
η

for all

f
,
g

∈

𝒮
⁢

(

ℝ
n

)

.

$B_{m}(f,g)(x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\hat{f}(\xi)\hat{g}(%
\eta)m(\xi,\eta)e^{2\pi i\langle\xi+\eta,x\rangle}\,d\xi\,d\eta\quad\text{for %
all ${f,g\in\mathcal{S}(\mathbb{R}^{n})}$. }$

The function m is said to be a bilinear multiplier on

ℝ
n

${\mathbb{R}^{n}}$

of type

(

p
1

,

q
1

,

ω
1

;

p
2

,

q
2

,

ω
2

;

p
3

,

q
3

,

ω
3

)

${(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})}$

if

B
m

${B_{m}}$

is the bounded bilinear operator from

M
⁢

(

p
1

,

q
1

,

ω
1

)

⁢

(

ℝ
n

)

×
M

⁢

(

p
2

,

q
2

,

ω
2

)

⁢

(

ℝ
n

)

${M(p_{1},q_{1},\omega_{1})(\mathbb{R}^{n})\times M(p_{2},q_{2},\omega_{2})(%
\mathbb{R}^{n})}$

to

M
⁢

(

p
3

,

q
3

,

ω
3

)

⁢

(

ℝ
n

)

${M(p_{3},q_{3},\omega_{3})(\mathbb{R}^{n})}$

.
We denote by

BM
⁢

(

p
1

,

q
1

,

ω
1

;

p
2

,

q
2

,

ω
2

)

⁢

(

ℝ
n

)

${\mathrm{BM}(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2})(\mathbb{R}^{n})}$

the space of all bilinear multipliers of type

(

p
1

,

q
1

,

ω
1

;

p
2

,

q
2

,

ω
2

;

p
3

,

q
3

,

ω
3

)

${(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})}$

, and define

∥
m
∥

(

p
1

,

q
1

,

ω
1

;

p
2

,

q
2

,

ω
2

;

p
3

,

q
3

,

ω
3

)

=

∥

B
m

∥

${\|m\|_{(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})%
}=\|B_{m}\|}$

.
We discuss the necessary and sufficient conditions for

B
m

${B_{m}}$

to be bounded.
We investigate the properties of this space and we give some examples.