{"paper":{"title":"Remark on atomic decompositions for Hardy space $H^1$ in the rational Dunkl setting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Agnieszka Hejna, Jacek Dziuba\\'nski","submitted_at":"2018-03-27T20:22:48Z","abstract_excerpt":"Let $\\Delta$ be the Dunkl Laplacian on $\\mathbb R^N$ associated with a normalized root system $R$ and a multiplicity function $k(\\alpha)\\geq 0$. We say that a function $f$ belongs to the Hardy space $H^1_{\\Delta}$ if the nontangential maximal function $\\mathcal M_H f(\\mathbf x)=\\sup_{\\| \\mathbf x-\\mathbf y\\|<t} |\\exp(t^2\\Delta )f(\\mathbf x)|$ belongs to $L^1(w(\\mathbf x)\\, d\\mathbf x)$, where $w(\\mathbf x)=\\prod_{\\alpha\\in R} |\\langle \\alpha,\\mathbf x\\rangle|^{k(\\alpha)}$. We prove that $H^1_\\Delta$ coincides with the space $H^1_{\\rm atom}(\\mathbb R^N, \\| \\mathbf x-\\mathbf y\\|, w(\\mathbf x)d\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.10302","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":""},"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"}