{"paper":{"title":"Weak integral conditions for BMO","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alexander A. Logunov, Dmitriy M. Stolyarov, Leonid Slavin, Pavel B. Zatitskiy, Vasily Vasyunin","submitted_at":"2013-09-26T10:05:42Z","abstract_excerpt":"We study the question of how much one can weaken the defining condition of BMO. Specifically, we show that if $Q$ is a cube in $\\mathbb{R}^n$ and $h:[0,\\infty)\\to[0,\\infty)$ is such that $h(t)\\underset{t\\to\\infty}{\\longrightarrow}\\infty,$ then\n  $$ \\sup_{J \\text{subcube} Q} \\frac1{|J|}\\int_J h(|\\varphi-\\frac1{|J|} \\int_J\\varphi |)<\\infty \\Longrightarrow \\varphi\\in BMO(Q). $$\n  Under some additional assumptions on $h$ we obtain estimates on $\\|\\varphi\\|_{BMO}$ in terms of the supremum above. We also show that even though the condition $h(t)\\underset{t\\to\\infty}{\\longrightarrow}\\infty$ is not ne"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6780","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":""},"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"}