{"paper":{"title":"Nonlinear Bounds in H\\\"older Spaces for the Monge-Amp\\`ere Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessio Figalli, Connor Mooney, Yash Jhaveri","submitted_at":"2015-08-11T19:10:48Z","abstract_excerpt":"We demonstrate that $C^{2,\\alpha}$ estimates for the Monge-Amp\\`{e}re equation depend in a highly nonlinear way both on the $C^{\\alpha}$ norm of the right-hand side and $1/\\alpha$. First, we show that if a solution is strictly convex, then the $C^{2,\\alpha}$ norm of the solution depends polynomially on the $C^{\\alpha}$ norm of the right-hand side. Second, we show that the $C^{2,\\alpha}$ norm of the solution is controlled by $\\exp((C/\\alpha)\\log(1/\\alpha))$ as $\\alpha \\to 0$. Finally, we construct a family of solutions in two dimensions to show the sharpness of our results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02692","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}