Maximal Schrödinger operators with complex time

Abstract

For $\gamma >0$ and $a>0,$ the operator $P_{a,\gamma }^{t}f$ of Schrödinger type with complex time is defined by

$$\begin{aligned} P_{a,\gamma }^{t}f(x)=S_{a}^{t+it^{\gamma }}f(x) =\int _{{\mathbb {R}}} e^{ix\xi }e^{it|\xi |^{a}}e^{-t^{\gamma }|\xi |^{a}} {\hat{f}}(\xi )d\xi , \end{aligned}$$

and the corresponding maximal operator $P_{a,\gamma }^{*}$ is defined by

$$\begin{aligned} P_{a,\gamma }^{*}f(x) =\displaystyle \sup _{0<t<1}|P_{a,\gamma }^{t}f(x)|,\quad x\in {\mathbb {R}}. \end{aligned}$$

When $0<a<1$ and $\gamma >1,$ some characterization of the global $L^{2}$ estimate for the maximal operator $P_{a,\gamma }^{*}$ is obtained. The authors extend the results of the maximal operator $P_{a,\gamma }^{*}$ for $a>1$ and $\gamma >1$ in Bailey (Rev. Mat. Iberoam 29: 531-546, 2013).