Affiliation:
1. Department of Mathematics , Faculty of Science , İstanbul University , İstanbul , Turkey
Abstract
Abstract
Let
Φ
i
{\Phi_{i}}
be Young functions and
ω
i
{\omega_{i}}
be weights on
ℝ
d
{\mathbb{R}^{d}}
,
i
=
1
,
2
,
3
{i=1,2,3}
. A locally integrable function
m
(
ξ
,
η
)
{m(\xi,\eta)}
on
ℝ
d
×
ℝ
d
{\mathbb{R}^{d}\times\mathbb{R}^{d}}
is said to be a bilinear multiplier on
ℝ
d
{\mathbb{R}^{d}}
of type
(
Φ
1
,
ω
1
;
Φ
2
,
ω
2
;
Φ
3
,
ω
3
)
{(\Phi_{1},\omega_{1};\Phi_{2},\omega_{2};\Phi_{3},\omega_{3})}
if
B
m
(
f
1
,
f
2
)
(
x
)
=
∫
ℝ
d
∫
ℝ
d
f
1
^
(
ξ
)
f
2
^
(
η
)
m
(
ξ
,
η
)
e
2
π
i
〈
ξ
+
η
,
x
〉
𝑑
ξ
𝑑
η
B_{m}(f_{1},f_{2})(x)=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\hat{f_{1}}(%
\xi)\hat{f_{2}}(\eta)m(\xi,\eta)e^{2\pi i\langle\xi+\eta,x\rangle}\,d\xi\,d\eta
defines a bounded bilinear operator from
L
ω
1
Φ
1
(
ℝ
d
)
×
L
ω
2
Φ
2
(
ℝ
d
)
{L^{\Phi_{1}}_{\omega_{1}}(\mathbb{R}^{d})\times L^{\Phi_{2}}_{\omega_{2}}(%
\mathbb{R}^{d})}
to
L
ω
3
Φ
3
(
ℝ
d
)
{L^{\Phi_{3}}_{\omega_{3}}(\mathbb{R}^{d})}
.
We deduce some properties of this class of operators. Moreover,
we give the methods to generate bilinear multipliers between weighted Orlicz spaces.