{"paper":{"title":"On functions of bounded $\\beta$-dimensional mean oscillation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Daniel Spector, You-Wei Benson Chen","submitted_at":"2022-07-14T15:02:37Z","abstract_excerpt":"In this paper, we define a notion of $\\beta$-dimensional mean oscillation of functions $u: Q_0 \\subset \\mathbb{R}^d \\to \\mathbb{R}$ which are integrable on $\\beta$-dimensional subsets of the cube $Q_0$: \\begin{align*} \\|u\\|_{BMO^{\\beta}(Q_0)}:= \\sup_{Q \\subset Q_0} \\inf_{c \\in \\mathbb{R}} \\frac{1}{l(Q)^\\beta} \\int_{Q} |u-c| \\;d\\mathcal{H}^{\\beta}_\\infty, \\end{align*} where the supremum is taken over all finite subcubes $Q$ parallel to $Q_0$, $l(Q)$ is the length of the side of the cube $Q$, and $\\mathcal{H}^{\\beta}_\\infty$ is the Hausdorff content. In the case $\\beta=d$ we show this definition"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.06979","kind":"arxiv","version":2},"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/2207.06979/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"}