Abstract
We study the local maximal oscillatory integral operator
T
α
,
β
∗
(
f
)
(
x
)
=
sup
0
<
t
<
1
|
∫
ℝ
n
e
i
|
t
ξ
|
α
|
t
ξ
|
β
Ψ
(
|
t
ξ
|
)
f
^
(
ξ
)
e
2
π
i
〈
x
,
ξ
〉
𝑑
ξ
|
,
\displaystyle T_{\alpha,\beta}^{\ast}(f)(x)=\sup_{0<t<1}\Bigg{|}\int_{\mathbb{%
R}^{n}}\frac{e^{i|t\xi|^{\alpha}}}{|t\xi|^{\beta}}\Psi(|t\xi|)\widehat{f}(\xi)%
e^{2\pi i\langle x,\xi\rangle}\,d\xi\Bigg{|},
where
α
∈
(
0
,
1
)
{\alpha\in(0,1)}
,
β
>
0
{\beta>0}
, and Ψ is a cutoff function that vanishes in a neighborhood of the origin. First, in the case
0
<
p
<
1
{0<p<1}
, we obtain the
H
p
(
ℝ
n
)
→
L
p
(
ℝ
n
)
{{{H^{p}}({{\mathbb{R}^{n}}})}\rightarrow{{L^{p}({{\mathbb{R}^{n}}})}}}
boundedness of
T
α
,
β
∗
{T_{\alpha,\beta}^{\ast}}
with the sharp relation among
α
,
β
{\alpha,\beta}
and p. Then, using interpolation, we obtain the
L
p
(
ℝ
n
)
{{{L^{p}({{\mathbb{R}^{n}}})}}}
boundedness on
T
α
,
β
∗
{T_{\alpha,\beta}^{\ast}}
when
p
>
1
{p>1}
, which is an improvement of the recent result by Kenig and Staubach. At the critical case
p
=
1
{p=1}
and
β
=
n
α
2
{\beta=\frac{n\alpha}{2}}
, we show
T
α
,
β
∗
:
B
q
(
ℝ
n
)
→
L
1
,
∞
(
ℝ
n
)
{T_{\alpha,\beta}^{\ast}:B_{q}({\mathbb{R}^{n}})\rightarrow L^{1,\infty}({%
\mathbb{R}^{n}})}
, where
B
q
(
ℝ
n
)
{B_{q}({\mathbb{R}^{n}})}
is the block space introduced by Lu, Taibleson and Weiss in order to study the almost every convergence of the Bochner–Riesz means at the critical index. As a further application, we obtain the convergence speed of a combination to the fractional Schrödinger operators
{
e
i
t
k
|
△
|
α
}
{\{e^{itk|\triangle|^{\alpha}}\}}
.