{"paper":{"title":"On weak$^*$-convergence in the localized Hardy spaces $H^1_\\rho(\\mathcal X)$ and its application","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Dinh Thanh Duc, Ha Duy Hung, Luong Dang Ky","submitted_at":"2015-11-25T14:43:02Z","abstract_excerpt":"Let $(\\mathcal X, d, \\mu)$ be a complete RD-space. Let $\\rho$ be an admissible function on $\\mathcal X$, which means that $\\rho$ is a positive function on $\\mathcal X$ and there exist positive constants $C_0$ and $k_0$ such that, for any $x,y\\in \\mathcal X$, $$\\rho(y)\\leq C_0 [\\rho(x)]^{1/(1+k_0)} [\\rho(x)+d(x,y)]^{k_0/(1+k_0)}.$$\n  In this paper, we define a space $VMO_\\rho(\\mathcal X)$ and show that it is the predual of the localized Hardy space $H^1_\\rho(\\mathcal X)$ introduced by Yang and Zhou \\cite{YZ}. Then we prove a version of the classical theorem of Jones and Journ\\'e \\cite{JJ} on we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08075","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"}