On inhomogeneous Strichartz estimates for fractional Schr\"odinger equations and their applications

TL;DR

This paper establishes new inhomogeneous Strichartz estimates for the radial fractional Schrödinger equation and applies them to prove well-posedness results for equations with radial initial data and potentials below L^2.

Contribution

It introduces novel inhomogeneous Strichartz estimates for radial fractional Schrödinger equations and applies these to analyze well-posedness with less regular initial data and potentials.

Findings

New inhomogeneous Strichartz estimates for radial fractional Schrödinger equations

Well-posedness results for equations with radial initial data below L^2

Applicability to equations with radial potentials in critical scaling range

Abstract

In this paper we obtain some new inhomogeneous Strichartz estimates for the fractional Schr\"odinger equation in the radial case. Then we apply them to the well-posedness theory for the equation $i\partial_{t}u+|\nabla|^{\alpha}u=V(x,t)u$, $1<\alpha<2$, with radial $\dot{H}^\gamma$ initial data below $L^2$ and radial potentials $V\in L_t^rL_x^w$ under the scaling-critical range $\alpha/r+n/w=\alpha$.