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

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 $\rho \in L^p_{t,x}$ in terms of $f_0 \in L^{a}_vL^{b}_x$.

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.