{"paper":{"title":"Corona-type theorems and division in some function algebras on planar domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Raymond Mortini, Rudolf Rupp","submitted_at":"2013-01-31T16:23:10Z","abstract_excerpt":"Let $A$ be an algebra of bounded smooth functions on the interior of a compact set in the plane. We study the following problem: if $f,f_1,\\dots,f_n\\in A$ satisfy $|f|\\leq \\sum_{j=1}^n |f_j|$, does there exist $g_j\\in A$ and a constant $N\\in\\N$ such that $f^N=\\sum_{j=1}^n g_j f_j$? A prominent role in our proofs is played by a new space, $C_{\\dbar, 1}(K)$, which we call the algebra of $\\dbar$-smooth functions.\n  In the case $n=1$, a complete solution is given for the algebras $A^m(K)$ of functions holomorphic in $K^\\circ$ and whose first $m$-derivatives extend continuously to $\\ov{K^\\circ}$. T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.7668","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":""},"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"}