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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,..."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1904.03994","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-04-08T12:28:35Z","cross_cats_sorted":[],"title_canon_sha256":"766567ee2dcd2956eda65e4b0fd003824cedd0de4f07552b2bdb3063adbf9154","abstract_canon_sha256":"3137a2cd440b3171dbbf8ac59c1da9a65912a6362682a720f02eb60403a77ddb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:17.883517Z","signature_b64":"G9BusEdC+kiwHZ3ctMU0DJf3lq+uPrBZFi3eMZkRFsbnInfqPXJcZgywMR98mpR9ciMvEX6u/KBdvbb8ymw5Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d34e6f71b5afad23a74beeda51cade6d025e26534eb66838f29416633b8c9bb1","last_reissued_at":"2026-05-17T23:46:17.882942Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:17.882942Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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