Strichartz Estimates for the Liouville Equation on Euclidean Tori and Applications to Kakeya

Abstract

We prove Strichartz estimates for the space-time density $\rho$ of solutions to the free Liouville equation on flat tori. In dimension one, we obtain the optimal range of estimates for the density $\rho \in L_{t,x}^p$ in terms of $f_0\in L_v^a{L}_x^b$. In higher dimensions, we prove that such estimates cannot hold and that a weight has to be added: $\rho$ can be bounded in terms of the norm of $|v{|}^{\gamma }{f}_0$. We conjecture a range of optimal estimates, and partially prove them. Finally, these results have natural applications to the $X$-ray transform and Kakeya problems on Euclidean cylinders.

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