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We provide the two-weight inequality for the Poisson operator $\\mathsf{P}^{[\\lambda]}_t=e^{-t\\sqrt{\\Delta_\\lambda}}$ in this Bessel setting. In particular, we prove that for a measure $\\mu$ on $\\mathbb{R}^2_{+,+}:=(0,\\infty)\\times (0,\\infty)$ and $\\sigma$ on $\\mathbb{R}_+$: $$ \\|\\mathsf{P}^{[\\lambda]}_\\sigma(f)\\|_{L^2(\\mathbb{R}^2_{+,+};\\mu)} \\lesssim \\|f\\|_{L^2(\\mathbb{R}_+;\\sigma)}, $$ if "},"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":"1707.07492","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-07-24T11:25:20Z","cross_cats_sorted":[],"title_canon_sha256":"66ad23bb1f26815dcd2f012bd8eab8ed6a04a0dd5e88956b88f61047e20976c3","abstract_canon_sha256":"ea4c06e74a95fcea5b698a93fae44eb1bb7652a0009e275e08866dc58e566cec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:44.517972Z","signature_b64":"KmKCExmFCxBxxWEfSfQjhcvTbKf5Ismcs2yZHeWIUTiAaLTwsNkCU5i8aO8AH1xZzMeltJuVTk96v/O1e85+CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"edf7a74f4498e8006640bfa8bc17ed6623ff1e3c2509750d37e4c28e9d5e5a7a","last_reissued_at":"2026-05-17T23:52:44.517500Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:44.517500Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Two-Weight Inequality for the Poisson Operator in the Bessel Setting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Brett D. Wick, Ji Li","submitted_at":"2017-07-24T11:25:20Z","abstract_excerpt":"Fix $\\lambda>0$. Consider the Bessel operator $\\Delta_\\lambda:=-\\frac{d^2}{dx^2}-\\frac{2\\lambda}{x}\\frac d{dx}$ on $\\mathbb{R}_+:=(0,\\infty)$ and the harmonic conjugacy introduced by Muckenhoupt and Stein. We provide the two-weight inequality for the Poisson operator $\\mathsf{P}^{[\\lambda]}_t=e^{-t\\sqrt{\\Delta_\\lambda}}$ in this Bessel setting. In particular, we prove that for a measure $\\mu$ on $\\mathbb{R}^2_{+,+}:=(0,\\infty)\\times (0,\\infty)$ and $\\sigma$ on $\\mathbb{R}_+$: $$ \\|\\mathsf{P}^{[\\lambda]}_\\sigma(f)\\|_{L^2(\\mathbb{R}^2_{+,+};\\mu)} \\lesssim \\|f\\|_{L^2(\\mathbb{R}_+;\\sigma)}, $$ if "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.07492","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1707.07492","created_at":"2026-05-17T23:52:44.517570+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.07492v2","created_at":"2026-05-17T23:52:44.517570+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.07492","created_at":"2026-05-17T23:52:44.517570+00:00"},{"alias_kind":"pith_short_12","alias_value":"5X32OT2ETDUA","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_16","alias_value":"5X32OT2ETDUAAZSA","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_8","alias_value":"5X32OT2E","created_at":"2026-05-18T12:31:03.183658+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5X32OT2ETDUAAZSAX6ULYF7NMY","json":"https://pith.science/pith/5X32OT2ETDUAAZSAX6ULYF7NMY.json","graph_json":"https://pith.science/api/pith-number/5X32OT2ETDUAAZSAX6ULYF7NMY/graph.json","events_json":"https://pith.science/api/pith-number/5X32OT2ETDUAAZSAX6ULYF7NMY/events.json","paper":"https://pith.science/paper/5X32OT2E"},"agent_actions":{"view_html":"https://pith.science/pith/5X32OT2ETDUAAZSAX6ULYF7NMY","download_json":"https://pith.science/pith/5X32OT2ETDUAAZSAX6ULYF7NMY.json","view_paper":"https://pith.science/paper/5X32OT2E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.07492&json=true","fetch_graph":"https://pith.science/api/pith-number/5X32OT2ETDUAAZSAX6ULYF7NMY/graph.json","fetch_events":"https://pith.science/api/pith-number/5X32OT2ETDUAAZSAX6ULYF7NMY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5X32OT2ETDUAAZSAX6ULYF7NMY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5X32OT2ETDUAAZSAX6ULYF7NMY/action/storage_attestation","attest_author":"https://pith.science/pith/5X32OT2ETDUAAZSAX6ULYF7NMY/action/author_attestation","sign_citation":"https://pith.science/pith/5X32OT2ETDUAAZSAX6ULYF7NMY/action/citation_signature","submit_replication":"https://pith.science/pith/5X32OT2ETDUAAZSAX6ULYF7NMY/action/replication_record"}},"created_at":"2026-05-17T23:52:44.517570+00:00","updated_at":"2026-05-17T23:52:44.517570+00:00"}