Journal of the
Korean Mathematical Society JKMS

In the present paper, we give some characterization of the $L^{2}$ maximal estimate for the operator $P_{a,\gamma}^{t}f\big(\Gamma(x,t)\big)$ along curve with complex time, which is defined by $$P_{a,\gamma}^{t}f\big(\Gamma(x,t)\big) =\int_{\mathbb{R}} e^{i\Gamma(x,t)\xi}e^{it|\xi|^{a}} e^{-t^{\gamma}|\xi|^{a}} \hat{f}(\xi)d\xi,$$ where $t,\gamma>0$ and $a\geq2,$ curve $\Gamma$ is a function such that $\Gamma:\mathbb{R}\times[0,1]\rightarrow\mathbb{R},$ and satisfies H\"{o}lder's condition of order $\sigma$ and bilipschitz conditions. The authors extend the results of the Schr\"{o}dinger type with complex time of Bailey \cite{Bailey} and Cho, Lee and Vargas \cite{CLV} to Schr\"{o}dinger operators along the curves.

Keywords: Schr\"{o}dinger equation, curve, maximal operator, global estimate, local estimate

MSC numbers: Primary 42B25; Secondary 35Q55

Supported by: The work is supported by NSFC (No.11661061, No.11561062, No.11761054), Inner Mongolia University scientific research projects (No. NJZY17289, NJZY19186), and the natural science foundation of Inner Mongolia (No.2019MS01003).