Inhomogeneous Strichartz estimates for Schr\"odinger's equation

TL;DR

This paper extends the known range of inhomogeneous Strichartz estimates for Schrödinger's equation by establishing the estimates at the previously excluded corner points of the parameter domain.

Contribution

The authors prove the inhomogeneous Strichartz estimates at the corner points P and P', completing the range of validity for these estimates.

Findings

Established estimates at corner points P and P'

Extended the range of inhomogeneous Strichartz estimates

Confirmed the estimates hold at previously excluded boundary points

Abstract

Foschi and Vilela in their independent works (\cite{F},\cite{V}) showed that the range of $(1/r,1/\widetilde{r})$ for which the inhomogeneous Strichartz estimate $ \big\|\int_{0}^{t}e^{i(t-s)\Delta}F(\cdot,s)ds\big\|_{L^{q}_tL^{r}_x} \lesssim \|F\|_{L^{\widetilde{q}'}_tL^{\widetilde{r}'}_x} $ holds for some $q,\widetilde{q}$ is contained in the closed pentagon with vertices $A,B,B',P,P'$ except the points $P,P'$ (see Figure 1). We obtain the estimate for the corner points $P,P'$.