{"paper":{"title":"Strichartz Estimates for the Liouville Equation on Euclidean Tori and Applications to Kakeya","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Micka\\\"el Latocca, Pierre Germain","submitted_at":"2026-06-29T09:04:33Z","abstract_excerpt":"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 cylin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29993","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.29993/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}