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Let $H=(-\Delta)^m+V $ be a higher-order elliptic operator on $L^2(R^n)$ witha general bounded decaying potential $V$. This paper is devoted to consider theglobal Kato smoothing and Strichartz estimates for the solution to theSchr\"odinger type equation associated with $H$. Among these results, the sharpglobal Kato smoothing estimates for $e^{itH}$ are first established based onthe uniform resolvent estimates of Kato-Yajima type for the absolutelycontinuous part of $H$. As a consequence, the optimal local decay estimates arealso obtained. Then, with the aid of such local decay estimates,the full set ofStrichartz estimates (including the endpoint case) have been proved. Inparticular, we have obtained Strichartz estimates with sharp smoothing effectsfor higher-order cases with rough potentials, which will be useful to the studyof nonlinear higher-order Schr\"odinger equations. Finally, we point out thatthe new uniform Sobolev estimates of Kenig-Ruiz-Sogge type with extraderivatives term are deduced to establish the sharp Kato smoothing estimates.

Global Kato smoothing and Strichartz estimates for higher-order Schr\"odinger operators with rough decay potentials