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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,...","authors_text":"Jie Xiao, Liguang Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-04-08T12:28:35Z","title":"Intrinsic nature of the Stein-Weiss $H^1$-inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.03994","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6d80063a244060b0d4dd4145d4270fe83ea37e6733e871e545d87ea5c44f7152","target":"record","created_at":"2026-05-17T23:46:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3137a2cd440b3171dbbf8ac59c1da9a65912a6362682a720f02eb60403a77ddb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-04-08T12:28:35Z","title_canon_sha256":"766567ee2dcd2956eda65e4b0fd003824cedd0de4f07552b2bdb3063adbf9154"},"schema_version":"1.0","source":{"id":"1904.03994","kind":"arxiv","version":2}},"canonical_sha256":"d34e6f71b5afad23a74beeda51cade6d025e26534eb66838f29416633b8c9bb1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d34e6f71b5afad23a74beeda51cade6d025e26534eb66838f29416633b8c9bb1","first_computed_at":"2026-05-17T23:46:17.882942Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:46:17.882942Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"G9BusEdC+kiwHZ3ctMU0DJf3lq+uPrBZFi3eMZkRFsbnInfqPXJcZgywMR98mpR9ciMvEX6u/KBdvbb8ymw5Bw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:46:17.883517Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.03994","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6d80063a244060b0d4dd4145d4270fe83ea37e6733e871e545d87ea5c44f7152","sha256:05b78eed78d59a91412b43267bc7263e54a5fdf062e854a07e1778bdecce08a7"],"state_sha256":"e2815c175531c33043d9b8fcfe22730e51c46c3b280329d408c99236d73bb244"}