{"paper":{"title":"Intrinsic nature of the Stein-Weiss $H^1$-inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jie Xiao, Liguang Liu","submitted_at":"2019-04-08T12:28:35Z","abstract_excerpt":"This paper explores the intrinsic nature of the celebrated Stein-Weiss $H^1$-inequality\n  $$\n  \\|I_s u\\|_{L^\\frac{n}{n-s}}\\lesssim \\|u\\|_{L^1}+\\|\\vec{R}u\\|_{L^{1}}=\\|u\\|_{H^1}\n  $$ through the tracing and duality laws based on Riesz's singular integral operator $I_s$. We discover that $f\\in I_s\\big([\\mathring{H}^{s,1}_{-}]^\\ast\\big)$ if and only if $\\exists\\ \\vec{g}=(g_1,...,g_n)\\in \\big(L^\\infty\\big)^n$ such that $f=\\vec{R}\\cdot\\vec{g}=\\sum_{j=1}^n R_jg_j$ in $\\mathrm{BMO}$ (the John-Nirenberg space introduced in their 1961 {\\it Comm. Pure Appl. Math.} paper \\cite{JN}) where $\\vec{R}=(R_1,..."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.03994","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"}