{"paper":{"title":"Extremal Problems in Bergman Spaces and an Extension of Ryabykh's Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Timothy Ferguson","submitted_at":"2013-01-31T15:56:49Z","abstract_excerpt":"We study linear extremal problems in the Bergman space $A^p$ of the unit disc for $p$ an even integer. Given a functional on the dual space of $A^p$ with representing kernel $k \\in A^q$, where $1/p + 1/q = 1$, we show that if the Taylor coefficients of $k$ are sufficiently small, then the extremal function $F \\in H^{\\infty}$. We also show that if $q \\le q_1 < \\infty$, then $F \\in H^{(p-1)q_1}$ if and only if $k \\in H^{q_1}$. These results extend and provide a partial converse to a theorem of Ryabykh."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.7659","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"}