Affiliation:
1. College of Mathematics and System Science , 47907 Xinjiang University , Xinjiang 830046 , P. R. China
Abstract
Abstract
In this paper, we deal with the
L
p
L^{p}
boundedness of rough Fourier integral operators
T
a
,
φ
T_{a,\varphi}
with amplitude
a
(
x
,
ξ
)
∈
L
∞
S
ρ
m
a(x,\xi)\in L^{\infty}S_{\rho}^{m}
and phase function
φ
(
x
,
ξ
)
∈
L
∞
Φ
2
\varphi(x,\xi)\in{L^{\infty}}{\Phi^{2}}
which satisfies a measure condition.
We show that
T
a
,
φ
T_{a,\varphi}
is bounded on
L
p
L^{p}
for
1
≤
p
≤
∞
1\leq p\leq\infty
if
m
<
n
(
ρ
−
1
)
p
−
ρ
(
n
−
1
)
2
p
m<\frac{n(\rho-1)}{p}-\frac{\rho(n-1)}{2p}
when
1
≤
p
≤
2
1\leq p\leq 2
or
m
<
n
(
ρ
−
1
)
2
−
ρ
(
n
−
1
)
2
(
1
−
1
p
)
m<\frac{n(\rho-1)}{2}-\frac{\rho(n-1)}{2}(1-\frac{1}{p})
when
2
≤
p
≤
∞
2\leq p\leq\infty
.
Our main results extend and improve some known results about
L
p
L^{p}
boundedness of Fourier integral operators.
Funder
Natural Science Foundation of Xinjiang Province
National Natural Science Foundation of China