A note on maximal estimate for an oscillatory operator

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}}\}} .