{"paper":{"title":"Littlewood-Paley Characterizations of Haj{\\l}asz-Sobolev 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":"Dachun Yang, Der-Chen Chang, Jun Liu, Wen Yuan","submitted_at":"2016-01-14T02:30:34Z","abstract_excerpt":"Let $p\\in(1,\\infty)$ and $q\\in[1,\\infty)$. In this article, the authors characterize the Triebel-Lizorkin space ${F}^\\alpha_{p,q}(\\mathbb{R}^n)$ with smoothness order $\\alpha\\in(0,2)$ via the Lusin-area function and the $g_\\lambda^*$-function in terms of difference between $f(x)$ and its average $B_tf(x):=\\frac1{|B(x,t)|}\\int_{B(x,t)}f(y)\\,dy$ over a ball $B(x,t)$ centered at $x\\in\\mathbb{R}^n$ with radius $t\\in(0,1)$. As an application, the authors obtain a series of characterizations of $F^\\alpha_{p,\\infty}(\\mathbb{R}^n)$ via pointwise inequalities, involving ball averages, in spirit close t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03467","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"}