Strichartz estimates for Schr\"odinger equations in weighted $L^2$ spaces and their applications

TL;DR

This paper establishes weighted $L^2$ Strichartz estimates for Schrödinger equations of general order with radial data, using Morrey-Campanato weights, and applies these results to improve well-posedness and Morawetz estimates.

Contribution

It provides new weighted $L^2$ Strichartz estimates for general order Schrödinger equations with radial data, answering an open question and enhancing existing estimates.

Findings

Affirmative answer to a previously open question on weighted homogeneous Strichartz estimates.

Improved Morawetz estimates for Schrödinger equations.

Application of estimates to well-posedness with time-dependent potentials.

Abstract

We obtain weighted $L^2$ Strichartz estimates for Schr\"odinger equations $i\partial_tu+(-\Delta)^{a/2}u=F(x,t)$, $u(x,0)=f(x)$, of general orders $a>1$ with radial data $f,F$ with respect to the spatial variable $x$, 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 \cite{BBCRV} concerning weighted homogeneous Strichartz estimates, and improve previously known Morawetz estimates. We also apply the weighted $L^2$ estimates to the well-posedness theory for the Schr\"odinger equations with time-dependent potentials in the class.