Restrictions on local inhomogeneous Strichartz estimates for the Schrödinger equation

We find new necessary conditions for the estimate ||u||Ltq(R;Lxr(Rn))≲||F||Ltq~′(R;Lxr~′(Rn))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${||u||_{L^{q}_{t} (\mathbb{R}; L^{r}_{x} (\mathbb{R}^{n}))} \lesssim\,||F||_{L^{{\tilde{q}}^{\prime}}_{t}(\mathbb{R};L^{{\tilde{r}}^{\prime}}_{x}(\mathbb{R}^{n}))}}$$\end{document}, where u =  u(t, x) is the solution to the Cauchy problem associated with the free inhomogeneous Schrödinger equation with identically zero initial data and inhomogeneity F =  F(t, x).