{"paper":{"title":"Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Dachun Yang, Haibo Lin","submitted_at":"2013-08-27T13:41:19Z","abstract_excerpt":"Let $({\\mathcal X},\\,d,\\,\\mu)$ be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of T. Hyt\\\"onen. In this paper, the authors prove that the $L^p(\\mu)$ boundedness with $p\\in(1,\\,\\infty)$ of the Marcinkiewicz integral is equivalent to either of its boundedness from $L^1(\\mu)$ into $L^{1,\\infty}(\\mu)$ or from the atomic Hardy space $H^1(\\mu)$ into $L^1(\\mu)$. Moreover, the authors show that, if the Marcinkiewicz integral is bounded from $H^1(\\mu)$ into $L^1(\\mu)$, then it is also bounded from $L^\\infty(\\mu)$ into the space ${\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5869","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"}