{"paper":{"title":"Level sets of certain classes of $\\alpha$-analytic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Abtin Daghighi, Frank Wikstr\\\"om","submitted_at":"2016-12-21T06:53:45Z","abstract_excerpt":"For an open set $V\\subset\\mathbb{C}^n$, denote by $\\mathscr{M}_{\\alpha}(V)$ the family of $\\alpha$-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded domain $\\Omega\\subset \\mathbb{C}^n$, with continuous boundary (that in each variable separately allows a solution to the Dirichlet problem), a function $f \\in \\mathscr{M}_{\\alpha}(\\Omega\\setminus f^{-1}(0))$ automatically satisfies $f\\in \\mathscr{M}_{\\alpha}(\\Omega)$, if it is $C^{\\alpha_j-1}$-smooth, in the $z_j$ variable, $\\alpha\\in \\mathbb{Z}^n_+$, up to the boundary. For a submanifold $U\\subset \\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.06990","kind":"arxiv","version":4},"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"}