{"paper":{"title":"On the Kato problem and extensions for degenerate elliptic operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Cristian Rios, David Cruz-Uribe, Jos\\'e Mar\\'ia Martell","submitted_at":"2015-10-23T00:01:17Z","abstract_excerpt":"We study the Kato problem for degenerate divergence form operators. This was begun by Cruz-Uribe and Rios who proved that given an operator $L_w=-w^{-1}{\\rm div}(A\\nabla)$, where $w\\in A_2$ and $A$ is a $w$-degenerate elliptic measure (i.e, $A=w\\,B$ with $B$ an $n\\times n$ bounded, complex-valued, uniformly elliptic matrix), then $L_w$ satisfies the weighted estimate $\\|\\sqrt{L_w}f\\|_{L^2(w)}\\approx\\|\\nabla f\\|_{L^2(w)}$. Here we solve the $L^2$-Kato problem: under some additional conditions on the weight $w$, the following unweighted $L^2$-Kato estimates hold $$ \\|L_w^{1/2}f\\|_{L^2(\\mathbb{R}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.06790","kind":"arxiv","version":7},"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"}