Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local $L^{2}$ estimate of the maximal operator $S^\ast_{\phi,\gamma}$ of the operator family $\{S_{t,\phi,\gamma}\}$ defined initially by $$ S_{t,\phi,\gamma}f(x):=\rme^{\rmi t\phi(\sqrt{-\Delta})} f(\gamma(x,t))=(2\pi)^{-1} \int_{\mathbb{R}} \rme^{\rmi \gamma(x,t)\cdot\xi+\rmi t\phi(|\xi|)}\wh{f}(\xi)\rmd\xi,\quad f\in\mathcal{S}(\mathbb{R}), $$ which is the solution (when $n=1$) of the following dispersive equations ($\ast$) along a curve $\gamma$: $$ \bigg\{\!\!\!\begin{array}{ll} \rmi\partial_{t}u+\phi(\sqrt{-\Delta})u=0, & (x,t)\in \mathbb{R}^n\times \mathbb{R},\u(x,0)=f(x), & f\in \mathcal{S}(\mathbb{R}^{n}), \end{array}\eqno(\ast) $$ where $\phi: \mathbb{R}^{+}\rightarrow\mathbb{R}$ satisfies some suitable conditions and $\phi(\sqrt{-\Delta})$ is a pseudo-dif\/ferential operator with symbol $\phi(|\xi|)$. As a consequence of the above result, the authors give the pointwise convergence of the solution (when $n=1$) of the equation ($\ast$) along curve $\gamma$. Moreover, a global $L^{2}$ estimate of the maximal operator $S^\ast_{\phi,\gamma}$ is also given in this paper.