{"paper":{"title":"Characterizations of Besov and Triebel-Lizorkin Spaces via Averages on Balls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Amiran Gogatishvili, Dachun Yang, Feng Dai, Wen Yuan","submitted_at":"2015-07-29T01:38:46Z","abstract_excerpt":"Let $\\ell\\in\\mathbb{N}$ and $p\\in(1,\\infty]$. In this article, the authors prove that the sequence $\\{f-B_{\\ell,2^{-k}}f\\}_{k\\in\\mathbb{Z}}$ consisting of the differences between $f$ and the ball average $B_{\\ell,2^{-k}}f$ characterizes the Besov space $\\dot B^\\alpha_{p,q}(\\rn)$ with $q\\in (0, \\infty]$ and the Triebel-Lizorkin space $\\dot F^\\alpha_{p,q}(\\rn)$ with $q\\in (1,\\infty]$ when the smoothness order $\\alpha\\in(0,2\\ell)$. More precisely, it is proved that $f-B_{\\ell,2^{-k}}f$ plays the same role as the approximation to the identity $\\varphi_{2^{-k}}\\ast f$ appearing in the definitions o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08004","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"}