Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications

Abstract

We obtain weighted \begin{document}$L^2$\end{document} Strichartz estimates for Schrodinger equations \begin{document}$i\partial_tu+(-\Delta)^{a/2}u=F(x, t)$\end{document} , \begin{document}$u(x, 0)=f(x)$\end{document} , of general orders \begin{document}$a>1$\end{document} with radial data \begin{document}$f, F$\end{document} with respect to the spatial variable \begin{document}$x$\end{document} , whenever the weight is in a Morrey-Campanato type class. This is done by making use of a useful property of maximal functions of the weights together with frequency-localized estimates which follow from using bilinear interpolation and some estimates of Bessel functions. As consequences, we give an affirmative answer to a question posed in [ 1 ] concerning weighted homogeneous Strichartz estimates, and improve previously known Morawetz estimates. We also apply the weighted \begin{document}$L^2$\end{document} estimates to the well-posedness theory for the Schrodinger equations with time-dependent potentials in the class.