{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:BKWNYISD353BUYFZVYGEKGENHD","short_pith_number":"pith:BKWNYISD","canonical_record":{"source":{"id":"0903.4720","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2009-03-27T01:09:31Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"ec8fbfc4cc42a65bf09c965028865baf24612feac2c6cd89319ba922cfb10610","abstract_canon_sha256":"a7a1fb10f4e999f84ded973c44da251862374dc976b137458f67ac7f89fd0412"},"schema_version":"1.0"},"canonical_sha256":"0aacdc2243df761a60b9ae0c45188d38d362c3e3efcdda8eb57687877e86221e","source":{"kind":"arxiv","id":"0903.4720","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.4720","created_at":"2026-05-18T02:14:19Z"},{"alias_kind":"arxiv_version","alias_value":"0903.4720v2","created_at":"2026-05-18T02:14:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.4720","created_at":"2026-05-18T02:14:19Z"},{"alias_kind":"pith_short_12","alias_value":"BKWNYISD353B","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"BKWNYISD353BUYFZ","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"BKWNYISD","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:BKWNYISD353BUYFZVYGEKGENHD","target":"record","payload":{"canonical_record":{"source":{"id":"0903.4720","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2009-03-27T01:09:31Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"ec8fbfc4cc42a65bf09c965028865baf24612feac2c6cd89319ba922cfb10610","abstract_canon_sha256":"a7a1fb10f4e999f84ded973c44da251862374dc976b137458f67ac7f89fd0412"},"schema_version":"1.0"},"canonical_sha256":"0aacdc2243df761a60b9ae0c45188d38d362c3e3efcdda8eb57687877e86221e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:14:19.194624Z","signature_b64":"cmK8Rb3VqYJUXI9Mk77gFvl3BnWua7pWK7++UGctnIe05y6ZFnAXoFPhMIVbs37ruPmG7rtuLVAS8S6wJo1UDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0aacdc2243df761a60b9ae0c45188d38d362c3e3efcdda8eb57687877e86221e","last_reissued_at":"2026-05-18T02:14:19.193964Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:14:19.193964Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0903.4720","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:14:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kfZJvzZR9m3L7mnGCOk69wFTnm9ITBT3oyvi2+SeH3xEJZsMfICkwSVVtzugTg7E6uWO0vZRbBS51EBjrrx/Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T08:37:23.354551Z"},"content_sha256":"445df2d4bbae7cc475bd76849cbcde79a9bc2ae2af716552b602b36fb8d4f34a","schema_version":"1.0","event_id":"sha256:445df2d4bbae7cc475bd76849cbcde79a9bc2ae2af716552b602b36fb8d4f34a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:BKWNYISD353BUYFZVYGEKGENHD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Anisotropic Singular Integrals in Product Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Baode Li, Dachun Yang, Marcin Bownik, Yuan Zhou","submitted_at":"2009-03-27T01:09:31Z","abstract_excerpt":"Let $A_i$ for $i=1, 2$ be an expansive dilation, respectively, on ${\\mathbb R}^n$ and ${\\mathbb R}^m$ and $\\vec A\\equiv(A_1, A_2)$. Denote by ${\\mathcal A}_\\infty(\\rnm; \\vec A)$ the class of Muckenhoupt weights associated with $\\vec A$. The authors introduce a class of anisotropic singular integrals on $\\mathbb R^n\\times\\mathbb R^m$, whose kernels are adapted to $\\vec A$ in the sense of Bownik and have vanishing moments defined via bump functions in the sense of Stein. Then the authors establish the boundedness of these anisotropic singular integrals on $L^q_w(\\mathbb R^n\\times\\mathbb R^m)$ wi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.4720","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:14:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nsi3kKWOjev7I2pHAuu7nPDgpvPqcm1rBgplD4J/+Rwh9HVw6dbBO0y9qcrERGRtzmF1Vj6b19zY4LKykQaAAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T08:37:23.355226Z"},"content_sha256":"8c15da0ef531d4bf98a31de92892c12b39e579ddbbca6a8d834f4edd39578bea","schema_version":"1.0","event_id":"sha256:8c15da0ef531d4bf98a31de92892c12b39e579ddbbca6a8d834f4edd39578bea"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BKWNYISD353BUYFZVYGEKGENHD/bundle.json","state_url":"https://pith.science/pith/BKWNYISD353BUYFZVYGEKGENHD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BKWNYISD353BUYFZVYGEKGENHD/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":"2026-05-27T08:37:23Z","links":{"resolver":"https://pith.science/pith/BKWNYISD353BUYFZVYGEKGENHD","bundle":"https://pith.science/pith/BKWNYISD353BUYFZVYGEKGENHD/bundle.json","state":"https://pith.science/pith/BKWNYISD353BUYFZVYGEKGENHD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BKWNYISD353BUYFZVYGEKGENHD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:BKWNYISD353BUYFZVYGEKGENHD","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":"a7a1fb10f4e999f84ded973c44da251862374dc976b137458f67ac7f89fd0412","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2009-03-27T01:09:31Z","title_canon_sha256":"ec8fbfc4cc42a65bf09c965028865baf24612feac2c6cd89319ba922cfb10610"},"schema_version":"1.0","source":{"id":"0903.4720","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.4720","created_at":"2026-05-18T02:14:19Z"},{"alias_kind":"arxiv_version","alias_value":"0903.4720v2","created_at":"2026-05-18T02:14:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.4720","created_at":"2026-05-18T02:14:19Z"},{"alias_kind":"pith_short_12","alias_value":"BKWNYISD353B","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"BKWNYISD353BUYFZ","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"BKWNYISD","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:8c15da0ef531d4bf98a31de92892c12b39e579ddbbca6a8d834f4edd39578bea","target":"graph","created_at":"2026-05-18T02:14:19Z","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":"Let $A_i$ for $i=1, 2$ be an expansive dilation, respectively, on ${\\mathbb R}^n$ and ${\\mathbb R}^m$ and $\\vec A\\equiv(A_1, A_2)$. Denote by ${\\mathcal A}_\\infty(\\rnm; \\vec A)$ the class of Muckenhoupt weights associated with $\\vec A$. The authors introduce a class of anisotropic singular integrals on $\\mathbb R^n\\times\\mathbb R^m$, whose kernels are adapted to $\\vec A$ in the sense of Bownik and have vanishing moments defined via bump functions in the sense of Stein. Then the authors establish the boundedness of these anisotropic singular integrals on $L^q_w(\\mathbb R^n\\times\\mathbb R^m)$ wi","authors_text":"Baode Li, Dachun Yang, Marcin Bownik, Yuan Zhou","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2009-03-27T01:09:31Z","title":"Anisotropic Singular Integrals in Product Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.4720","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":"sha256:445df2d4bbae7cc475bd76849cbcde79a9bc2ae2af716552b602b36fb8d4f34a","target":"record","created_at":"2026-05-18T02:14:19Z","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":"a7a1fb10f4e999f84ded973c44da251862374dc976b137458f67ac7f89fd0412","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2009-03-27T01:09:31Z","title_canon_sha256":"ec8fbfc4cc42a65bf09c965028865baf24612feac2c6cd89319ba922cfb10610"},"schema_version":"1.0","source":{"id":"0903.4720","kind":"arxiv","version":2}},"canonical_sha256":"0aacdc2243df761a60b9ae0c45188d38d362c3e3efcdda8eb57687877e86221e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0aacdc2243df761a60b9ae0c45188d38d362c3e3efcdda8eb57687877e86221e","first_computed_at":"2026-05-18T02:14:19.193964Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:14:19.193964Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cmK8Rb3VqYJUXI9Mk77gFvl3BnWua7pWK7++UGctnIe05y6ZFnAXoFPhMIVbs37ruPmG7rtuLVAS8S6wJo1UDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:14:19.194624Z","signed_message":"canonical_sha256_bytes"},"source_id":"0903.4720","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:445df2d4bbae7cc475bd76849cbcde79a9bc2ae2af716552b602b36fb8d4f34a","sha256:8c15da0ef531d4bf98a31de92892c12b39e579ddbbca6a8d834f4edd39578bea"],"state_sha256":"74a40301519c238bc2bf1107072aaf1e3a7f0fcdd5d47b1597d740ccdbdc707b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kLl3cIft/Zs6CjR6aRVDmZkmPeHkNblP0RsDwdMlv9zXG2BWAi+aNagnelNXsSAvT6Dm0XvglCy6SMUYAamGBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T08:37:23.358742Z","bundle_sha256":"9bb61cd5ef6a8ba87b3b0088cf6a8c48e5b8cbdac3454812be13fbd9adbc66e7"}}