Global Kato smoothing and Strichartz estimates for higher-order Schrödinger operators with rough decay potentials (2004.10115v2)

Abstract: Let ( H = (-\Delta)^m + V ) be a higher-order elliptic operator on ( L^2(\mathbb{R}^n) ), where ( V ) is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to Schr\"odinger-type equation associated with ( H ). In particular, we first establish sharp global Kato smoothing estimates for ( e^{itH} ), based on uniform resolvent estimates of Kato-Yajima type for the absolutely continuous part of ( H ). As a consequence, we also obtain optimal local decay estimates. Using these local decay estimates, we then prove the full set of Strichartz estimates, including the endpoint case. Notably, we derive Strichartz estimates with sharp smoothing effects for higher-order cases with rough potentials, which are applicable to the study of nonlinear higher-order Schr\"odinger equations. Finally, we introduce new uniform Sobolev estimates of the Kenig-Ruiz-Sogge type, incorporating an additional derivative term, which are crucial for establishing the sharp Kato smoothing estimates.