{"paper":{"title":"Characterization of Sobolev spaces on the sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"J. A. Barcel\\'o, S. P\\'erez-Esteva, T. Luque","submitted_at":"2019-07-02T18:13:49Z","abstract_excerpt":"We prove a characterization of the Sobolev spaces $H^\\alpha$ on the unit sphere $\\mathbb{S}^{d-1}$, where the smoothness index $\\alpha$ is any positive real number and $d\\geq 2$. This characterization does not use differentiation and it is given in terms of $([\\alpha/2]+1)$-multidimensional square functions $S_\\alpha$. For $[\\alpha/2]=0,$ a function $f\\in L^2(\\mathbb{S}^{d-1})$ belongs to $H^\\alpha(\\mathbb{S}^{d-1})$ if and only if $S_\\alpha (f)\\in L^2(\\mathbb{S}^{d-1})$. If $n=[\\alpha/2]>0$, the membership of $f$ is equivalent to the existence of $g_1,\\cdots,g_n$ in $L^2(\\mathbb{S}^{d-1})$ su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.01571","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1907.01571/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}