Abstract
We find new necessary conditions for the estimate \({||u||_{L^{q}_{t} (\mathbb{R}; L^{r}_{x} (\mathbb{R}^{n}))} \lesssim\,||F||_{L^{{\tilde{q}}^{\prime}}_{t}(\mathbb{R};L^{{\tilde{r}}^{\prime}}_{x}(\mathbb{R}^{n}))}}\), 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).
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Abdelhakim, A.A. Restrictions on local inhomogeneous Strichartz estimates for the Schrödinger equation. Arch. Math. 102, 165–169 (2014). https://doi.org/10.1007/s00013-014-0608-6
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DOI: https://doi.org/10.1007/s00013-014-0608-6