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arXiv · 2606.29993

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

Pierre Germain·Mickaël Latocca
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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 $\rho \in L^p_{t,x}$ 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.

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Pierre Germain, Mickaël Latocca. 2026-06-29. Strichartz Estimates for the Liouville Equation on Euclidean Tori and Applications to Kakeya. https://arxiv.org/abs/2606.29993

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