Convex Hypersurfaces and $L^p$ Estimates for Schrödinger Equations

Quan Zheng, Xiaohua Yao, Da Fan

Published 2004-03-19, updated 2004-04-27Version 2

This paper is concerned with Schr\"odinger equations whose principal operators are homogeneous elliptic. When the corresponding level hypersurface is convex, we show the $L^p$-$L^q$ estimate of solution operator in free case. This estimate, combining with the results of fractionally integrated groups, allows us to further obtain the $L^p$ estimate of solutions for the initial data belonging to a dense subset of $L^p$ in the case of integrable potentials.