{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:MEAV5LD5AC5J7LII43U6GSGZTU","short_pith_number":"pith:MEAV5LD5","canonical_record":{"source":{"id":"1406.5914","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-06-23T14:13:12Z","cross_cats_sorted":[],"title_canon_sha256":"33c361b322c88de9e137d8af7e285a097b7a9963f99f4ca953730735b1afdae6","abstract_canon_sha256":"260008a3da4278f4b227c42572d9e3d1e117c21bf6dd2a2c9c992a835215ffda"},"schema_version":"1.0"},"canonical_sha256":"61015eac7d00ba9fad08e6e9e348d99d25d4ce453b691dc5e49b5f46e3068325","source":{"kind":"arxiv","id":"1406.5914","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.5914","created_at":"2026-05-18T02:49:10Z"},{"alias_kind":"arxiv_version","alias_value":"1406.5914v1","created_at":"2026-05-18T02:49:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.5914","created_at":"2026-05-18T02:49:10Z"},{"alias_kind":"pith_short_12","alias_value":"MEAV5LD5AC5J","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MEAV5LD5AC5J7LII","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MEAV5LD5","created_at":"2026-05-18T12:28:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:MEAV5LD5AC5J7LII43U6GSGZTU","target":"record","payload":{"canonical_record":{"source":{"id":"1406.5914","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-06-23T14:13:12Z","cross_cats_sorted":[],"title_canon_sha256":"33c361b322c88de9e137d8af7e285a097b7a9963f99f4ca953730735b1afdae6","abstract_canon_sha256":"260008a3da4278f4b227c42572d9e3d1e117c21bf6dd2a2c9c992a835215ffda"},"schema_version":"1.0"},"canonical_sha256":"61015eac7d00ba9fad08e6e9e348d99d25d4ce453b691dc5e49b5f46e3068325","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:10.510967Z","signature_b64":"RqAvMpwNBFR+t2bsa8pRBN25bib2v8DOIfuZtneACvocT4svsniRg443MQ7DiVY6ygD/ChjiLar4kBpsNytzCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"61015eac7d00ba9fad08e6e9e348d99d25d4ce453b691dc5e49b5f46e3068325","last_reissued_at":"2026-05-18T02:49:10.510516Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:10.510516Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1406.5914","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:49:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ado6kCvBunqb8RSseq2J7O9oO/Y1F3FBQFypm1SNoAQu3WerDXhMf6tQIH13VRJ2pNUEq+IGq8Nmiuf2nPsNBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:59:07.707454Z"},"content_sha256":"15561031f884ff4dd6fdd62a9f687eb2f15736cc9c93c98cc1f542b4937ca1f5","schema_version":"1.0","event_id":"sha256:15561031f884ff4dd6fdd62a9f687eb2f15736cc9c93c98cc1f542b4937ca1f5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:MEAV5LD5AC5J7LII43U6GSGZTU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Characterization of the Two-weight Inequality for Riesz Potentials on Cones of Radially Decreasing Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Alexander Meskhi, Ghulam Murtaza, Muhammad Sarwar","submitted_at":"2014-06-23T14:13:12Z","abstract_excerpt":"We establish necessary and sufficient conditions on a weight pair $(v,w)$ governing the boundedness of the Riesz potential operator $I_{\\alpha}$ defined on a homogeneous group $G$ from $L^p_{dec,r}(w, G)$ to $L^q(v, G)$, where $L^p_{dec,r}(w, G)$ is the Lebesgue space defined for non-negative radially decreasing functions on $G$. The same problem is also studied for the potential operator with product kernels $I_{\\alpha_1, \\alpha_2}$ defined on a product of two homogeneous groups $G_1\\times G_2$. In the latter case weights, in general, are not of product type. The derived results are new even "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.5914","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:49:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hQVhL3Lv5Rxrm+ROJ4WYUCZ3T38P35wNfCwooRoYjxLDkRUn9dpvh2iFM1/dG+F8HrV37NoHqBSzhB5+69HXAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:59:07.707812Z"},"content_sha256":"50fd81878c596b5da2e36da2250bcd7f2285e59ce2ef011935ef0628a5589a4d","schema_version":"1.0","event_id":"sha256:50fd81878c596b5da2e36da2250bcd7f2285e59ce2ef011935ef0628a5589a4d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MEAV5LD5AC5J7LII43U6GSGZTU/bundle.json","state_url":"https://pith.science/pith/MEAV5LD5AC5J7LII43U6GSGZTU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MEAV5LD5AC5J7LII43U6GSGZTU/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-25T23:59:07Z","links":{"resolver":"https://pith.science/pith/MEAV5LD5AC5J7LII43U6GSGZTU","bundle":"https://pith.science/pith/MEAV5LD5AC5J7LII43U6GSGZTU/bundle.json","state":"https://pith.science/pith/MEAV5LD5AC5J7LII43U6GSGZTU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MEAV5LD5AC5J7LII43U6GSGZTU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:MEAV5LD5AC5J7LII43U6GSGZTU","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"260008a3da4278f4b227c42572d9e3d1e117c21bf6dd2a2c9c992a835215ffda","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-06-23T14:13:12Z","title_canon_sha256":"33c361b322c88de9e137d8af7e285a097b7a9963f99f4ca953730735b1afdae6"},"schema_version":"1.0","source":{"id":"1406.5914","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.5914","created_at":"2026-05-18T02:49:10Z"},{"alias_kind":"arxiv_version","alias_value":"1406.5914v1","created_at":"2026-05-18T02:49:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.5914","created_at":"2026-05-18T02:49:10Z"},{"alias_kind":"pith_short_12","alias_value":"MEAV5LD5AC5J","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MEAV5LD5AC5J7LII","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MEAV5LD5","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:50fd81878c596b5da2e36da2250bcd7f2285e59ce2ef011935ef0628a5589a4d","target":"graph","created_at":"2026-05-18T02:49:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We establish necessary and sufficient conditions on a weight pair $(v,w)$ governing the boundedness of the Riesz potential operator $I_{\\alpha}$ defined on a homogeneous group $G$ from $L^p_{dec,r}(w, G)$ to $L^q(v, G)$, where $L^p_{dec,r}(w, G)$ is the Lebesgue space defined for non-negative radially decreasing functions on $G$. The same problem is also studied for the potential operator with product kernels $I_{\\alpha_1, \\alpha_2}$ defined on a product of two homogeneous groups $G_1\\times G_2$. In the latter case weights, in general, are not of product type. The derived results are new even ","authors_text":"Alexander Meskhi, Ghulam Murtaza, Muhammad Sarwar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-06-23T14:13:12Z","title":"A Characterization of the Two-weight Inequality for Riesz Potentials on Cones of Radially Decreasing Functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.5914","kind":"arxiv","version":1},"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":"sha256:15561031f884ff4dd6fdd62a9f687eb2f15736cc9c93c98cc1f542b4937ca1f5","target":"record","created_at":"2026-05-18T02:49:10Z","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":"260008a3da4278f4b227c42572d9e3d1e117c21bf6dd2a2c9c992a835215ffda","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-06-23T14:13:12Z","title_canon_sha256":"33c361b322c88de9e137d8af7e285a097b7a9963f99f4ca953730735b1afdae6"},"schema_version":"1.0","source":{"id":"1406.5914","kind":"arxiv","version":1}},"canonical_sha256":"61015eac7d00ba9fad08e6e9e348d99d25d4ce453b691dc5e49b5f46e3068325","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"61015eac7d00ba9fad08e6e9e348d99d25d4ce453b691dc5e49b5f46e3068325","first_computed_at":"2026-05-18T02:49:10.510516Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:49:10.510516Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RqAvMpwNBFR+t2bsa8pRBN25bib2v8DOIfuZtneACvocT4svsniRg443MQ7DiVY6ygD/ChjiLar4kBpsNytzCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:49:10.510967Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.5914","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:15561031f884ff4dd6fdd62a9f687eb2f15736cc9c93c98cc1f542b4937ca1f5","sha256:50fd81878c596b5da2e36da2250bcd7f2285e59ce2ef011935ef0628a5589a4d"],"state_sha256":"72d857f35549072710ba173c87cf276fd889d729482079553297761f2f37c896"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yEwmo1iah28ouvXrKGV73RQ3M0cZXD8LcB6u7VwxM4T/52brDmeSQlW8kAXsarT8sV7I52DeBUmhqgRQXFzRBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T23:59:07.710150Z","bundle_sha256":"6d50ad259558e40310a3ff25c81fc99089997202f4bd537c968ef60e5986d8b5"}}