{"paper":{"title":"Littlewood-Paley Characterizations of Fractional Sobolev 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, Feng Dai, Jun Liu, Wen Yuan","submitted_at":"2015-11-24T07:31:13Z","abstract_excerpt":"In this paper, the authors characterize Sobolev spaces $W^{\\alpha,p}({\\mathbb R}^n)$ with the smoothness order $\\alpha\\in(0,2]$ and $p\\in(\\max\\{1, \\frac{2n}{2\\alpha+n}\\},\\infty)$, via the Lusin area function and the Littlewood-Paley $g_\\lambda^\\ast$-function in terms of centered ball averages. The authors also show that the condition $p\\in(\\max\\{1, \\frac{2n}{2\\alpha+n}\\},\\infty)$ is nearly sharp in the sense that these characterizations are no longer true when $p\\in (1,\\max\\{1, \\frac{2n}{2\\alpha+n}\\})$. These characterizations provide a new possible way to introduce fractional Sobolev spaces w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07598","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"}