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In this article, the authors establish the Riesz transform characterization of the Hardy space $H_{L_w}^p(\\mathbb{R}^n)$ associated with $L_w$, for $w\\in A_{q}(\\mathbb{R}^n)$ and $w^{-1}\\in A_{2-\\frac{2}{n}}(\\mathbb{R}^n)$ with $n\\geq 3$, $q\\in[1,2]$ and $p\\in(q(\\frac{1}{r}+\\frac{q-1}{2}+\\frac{1}{n})^{-1},1]$ if, for some $r\\in[1,\\,2)$, $\\{tL_w e^{-tL_w}\\}_{t\\geq 0}$ satisfies the weighted $L^r-L^2$ full off-diagonal estimate"},"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":"1509.05479","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-09-18T00:26:09Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"e89cb2e039c4bd4232e10e052474e485eacfda659d3353fb6d52f852555676e0","abstract_canon_sha256":"263fe5584151dd63184d38e60906d0579b02c4ef7777141cc6d3f22c744936fc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:43.063272Z","signature_b64":"oNchBbxIpRy3LVHU+yXPiJCpyBrgBtqFaCnNsijRk2GsC7VM749WM60C3/WoTe4RvOvv4ric1stgsWRpKDlVCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"621bc5d4a98d6f3c2d8b1c58b5d93674bb8dd7d2c16ea035578e87f46e6a43ee","last_reissued_at":"2026-05-18T01:32:43.062873Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:43.062873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Dachun Yang, Junqiang Zhang","submitted_at":"2015-09-18T00:26:09Z","abstract_excerpt":"Let $w$ be a Muckenhoupt $A_2(\\mathbb{R}^n)$ weight and $L_w:=-w^{-1}\\mathop\\mathrm{div}(A\\nabla)$ the degenerate elliptic operator on the Euclidean space $\\mathbb{R}^n$. 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