Inhomogeneous Strichartz estimates for Schrödinger's equation

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 $P,P'$.