Abstract:

We consider maximal estimates for solution to the generalized dispersive equation

$\left\{ \begin{align} & \text{i}{{\partial }_{t}}u+\phi (\sqrt{-\Delta })u=0,\ \ \ \ \ (x,t)\in {{\mathbb{R}}^{n}}\times \mathbb{R}, \\ & u(x,0)=f(x),\ \ \ \ \ \ \ \ \ \ \ \ \ f\in {\cal S}({{\mathbb{R}}^{n}}), \\ \end{align} \right.\ \ \ \ \ \ \ (*)$

where $\phi(\sqrt{-\Delta})$ is a pseudo-differential operator with symbol $\phi(|\xi|)$. When $\phi$ satisfies suitable growth conditions and the initial data $f$ belong to the Sobolev space $H^{s}({\Bbb R}^{n})$, we obtain the global estimate for the maximal operator $S_{\phi}^*$ generated by the operators family $\{S_{t, \phi}\}_{0 <t <1}, $ where $S_{\phi}^*$ is defined by $S^{\ast}_{\phi}f(x)=\displaystyle\sup_{0 <t <1}|S_{t, \phi}f(x)|, $ and $S_{t, \phi}f$ is a formal solution of the equation $(\ast)$. These estimates are apparently good extensions to the current results for the fractional Schrödinger equation and these estimates were obtained in a general unified way.

Key words: Dispersive equation, Local maximal operator, Global estimate