{"paper":{"title":"On Kato-Ponce and fractional Leibniz","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dong Li","submitted_at":"2016-09-06T22:38:49Z","abstract_excerpt":"We show that in the Kato-Ponce inequality $\\|J^s(fg)-fJ^s g\\|_p \\lesssim \\| \\partial f \\|_{\\infty} \\| J^{s-1} g \\|_p + \\| J^s f \\|_p \\|g\\|_{\\infty}$, the $J^s f$ term on the RHS can be replaced by $J^{s-1} \\partial f$. This solves a question raised in Kato-Ponce \\cite{KP88}. We propose and prove a new fractional Leibniz rule for $D^s=(-\\Delta)^{s/2}$ and similar operators, generalizing the Kenig-Ponce-Vega estimate \\cite{KPV93} to all $s>0$. We also prove a family of generalized and refined Kato-Ponce type inequalities which include many commutator estimates as special cases. To showcase the s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.01780","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"}