{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:XG7FWWR4GFQA23HQWXEAKE45IF","short_pith_number":"pith:XG7FWWR4","canonical_record":{"source":{"id":"1507.05291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-19T14:22:42Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"da816b07a48c538e042fce76c7ea5cd69cccbd2bebaaf766589bc4d3cc5f0d05","abstract_canon_sha256":"b7881138216b5a9453f358b65a1e5336ef2f0bb58738396eb714b17ab9cb2f85"},"schema_version":"1.0"},"canonical_sha256":"b9be5b5a3c31600d6cf0b5c805139d416853fbbacfe690da1c748257f16bbb74","source":{"kind":"arxiv","id":"1507.05291","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.05291","created_at":"2026-05-18T01:36:37Z"},{"alias_kind":"arxiv_version","alias_value":"1507.05291v1","created_at":"2026-05-18T01:36:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.05291","created_at":"2026-05-18T01:36:37Z"},{"alias_kind":"pith_short_12","alias_value":"XG7FWWR4GFQA","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"XG7FWWR4GFQA23HQ","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"XG7FWWR4","created_at":"2026-05-18T12:29:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:XG7FWWR4GFQA23HQWXEAKE45IF","target":"record","payload":{"canonical_record":{"source":{"id":"1507.05291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-19T14:22:42Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"da816b07a48c538e042fce76c7ea5cd69cccbd2bebaaf766589bc4d3cc5f0d05","abstract_canon_sha256":"b7881138216b5a9453f358b65a1e5336ef2f0bb58738396eb714b17ab9cb2f85"},"schema_version":"1.0"},"canonical_sha256":"b9be5b5a3c31600d6cf0b5c805139d416853fbbacfe690da1c748257f16bbb74","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:37.286802Z","signature_b64":"xSKTEgbsB/ncj3P4sm8fhyBCZTPXUQp/biNac9SMNMU7UPf1v+91aSiIPd5bK63Wd3KQBgTtJ8BGV7RROMt1CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9be5b5a3c31600d6cf0b5c805139d416853fbbacfe690da1c748257f16bbb74","last_reissued_at":"2026-05-18T01:36:37.285372Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:37.285372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.05291","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":"2026-05-18T01:36:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zL5kKBP8WDZZASjoK+mX78uEJ3BZRP/JPMtD5qDeiq/z8IputW6o8Kz1VjvDoe36wq0+YrMxH6XfHYVpIA8FAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T15:33:31.430487Z"},"content_sha256":"e735bfa15be8d56998608289328b3012c1a68b67329510663202d97cc7320a54","schema_version":"1.0","event_id":"sha256:e735bfa15be8d56998608289328b3012c1a68b67329510663202d97cc7320a54"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:XG7FWWR4GFQA23HQWXEAKE45IF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A non-homogeneous local $Tb$ theorem for Littlewood-Paley $g_{\\lambda}^{*}$-function with $L^p$-testing condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Mingming Cao, Qingying Xue","submitted_at":"2015-07-19T14:22:42Z","abstract_excerpt":"In this paper, we present a local $Tb$ theorem for the non-homogeneous Littlewood-Paley $g_{\\lambda}^{*}$-function with non-convolution type kernels and upper power bound measure $\\mu$. We show that, under the assumptions $\\supp b_Q \\subset Q$, $|\\int_Q b_Q d\\mu| \\gtrsim \\mu(Q)$ and $||b_Q||^p_{L^p(\\mu)} \\lesssim \\mu(Q)$, the norm inequality $\\big\\| g_{\\lambda}^{*}(f) \\big\\|_{L^p(\\mu)} \\lesssim \\big\\| f \\big\\|_{L^p(\\mu)}$ holds if and only if the following testing condition holds : $$\\sup_{Q : cubes \\ in \\ \\Rn} \\frac{1}{\\mu(Q)}\\int_Q \\bigg(\\int_{0}^{\\ell(Q)} \\int_{\\Rn} \\Big(\\frac{t}{t+|x-y|}\\B"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05291","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":"2026-05-18T01:36:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NoLfaFwLQxaCBf5VkovlCzTLp4KGz75K0RlHBcKCYv7Gexo7yroH2IhwjAilIAN6Ju5eZnmpZS+wwBQeIl9zAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T15:33:31.430839Z"},"content_sha256":"3c04d0802b7ffeeff1ccacf93ff59e7223959bbf62c4fa86eb6c515e845b8edc","schema_version":"1.0","event_id":"sha256:3c04d0802b7ffeeff1ccacf93ff59e7223959bbf62c4fa86eb6c515e845b8edc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XG7FWWR4GFQA23HQWXEAKE45IF/bundle.json","state_url":"https://pith.science/pith/XG7FWWR4GFQA23HQWXEAKE45IF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XG7FWWR4GFQA23HQWXEAKE45IF/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":"2026-06-22T15:33:31Z","links":{"resolver":"https://pith.science/pith/XG7FWWR4GFQA23HQWXEAKE45IF","bundle":"https://pith.science/pith/XG7FWWR4GFQA23HQWXEAKE45IF/bundle.json","state":"https://pith.science/pith/XG7FWWR4GFQA23HQWXEAKE45IF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XG7FWWR4GFQA23HQWXEAKE45IF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:XG7FWWR4GFQA23HQWXEAKE45IF","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":"b7881138216b5a9453f358b65a1e5336ef2f0bb58738396eb714b17ab9cb2f85","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-19T14:22:42Z","title_canon_sha256":"da816b07a48c538e042fce76c7ea5cd69cccbd2bebaaf766589bc4d3cc5f0d05"},"schema_version":"1.0","source":{"id":"1507.05291","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.05291","created_at":"2026-05-18T01:36:37Z"},{"alias_kind":"arxiv_version","alias_value":"1507.05291v1","created_at":"2026-05-18T01:36:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.05291","created_at":"2026-05-18T01:36:37Z"},{"alias_kind":"pith_short_12","alias_value":"XG7FWWR4GFQA","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"XG7FWWR4GFQA23HQ","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"XG7FWWR4","created_at":"2026-05-18T12:29:50Z"}],"graph_snapshots":[{"event_id":"sha256:3c04d0802b7ffeeff1ccacf93ff59e7223959bbf62c4fa86eb6c515e845b8edc","target":"graph","created_at":"2026-05-18T01:36:37Z","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":"In this paper, we present a local $Tb$ theorem for the non-homogeneous Littlewood-Paley $g_{\\lambda}^{*}$-function with non-convolution type kernels and upper power bound measure $\\mu$. We show that, under the assumptions $\\supp b_Q \\subset Q$, $|\\int_Q b_Q d\\mu| \\gtrsim \\mu(Q)$ and $||b_Q||^p_{L^p(\\mu)} \\lesssim \\mu(Q)$, the norm inequality $\\big\\| g_{\\lambda}^{*}(f) \\big\\|_{L^p(\\mu)} \\lesssim \\big\\| f \\big\\|_{L^p(\\mu)}$ holds if and only if the following testing condition holds : $$\\sup_{Q : cubes \\ in \\ \\Rn} \\frac{1}{\\mu(Q)}\\int_Q \\bigg(\\int_{0}^{\\ell(Q)} \\int_{\\Rn} \\Big(\\frac{t}{t+|x-y|}\\B","authors_text":"Mingming Cao, Qingying Xue","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-19T14:22:42Z","title":"A non-homogeneous local $Tb$ theorem for Littlewood-Paley $g_{\\lambda}^{*}$-function with $L^p$-testing condition"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05291","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:e735bfa15be8d56998608289328b3012c1a68b67329510663202d97cc7320a54","target":"record","created_at":"2026-05-18T01:36:37Z","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":"b7881138216b5a9453f358b65a1e5336ef2f0bb58738396eb714b17ab9cb2f85","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-19T14:22:42Z","title_canon_sha256":"da816b07a48c538e042fce76c7ea5cd69cccbd2bebaaf766589bc4d3cc5f0d05"},"schema_version":"1.0","source":{"id":"1507.05291","kind":"arxiv","version":1}},"canonical_sha256":"b9be5b5a3c31600d6cf0b5c805139d416853fbbacfe690da1c748257f16bbb74","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b9be5b5a3c31600d6cf0b5c805139d416853fbbacfe690da1c748257f16bbb74","first_computed_at":"2026-05-18T01:36:37.285372Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:37.285372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xSKTEgbsB/ncj3P4sm8fhyBCZTPXUQp/biNac9SMNMU7UPf1v+91aSiIPd5bK63Wd3KQBgTtJ8BGV7RROMt1CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:37.286802Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.05291","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e735bfa15be8d56998608289328b3012c1a68b67329510663202d97cc7320a54","sha256:3c04d0802b7ffeeff1ccacf93ff59e7223959bbf62c4fa86eb6c515e845b8edc"],"state_sha256":"62d2cd4c9ff3b501436635ff2a81fda18cb18ca1ea92b9dc65cc1a546b0d66c6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9uz5uQgpPVv8gixk/nC7TEoSq/AQBGIwQuyUe7GOCOkUQlamlytLWhhHWCedgCkZLZ6H/urczJPJCSf9wfcoAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T15:33:31.432833Z","bundle_sha256":"802e0898656328b0e1aff39c1bf695d3bcc4926ef8e41c29991068a3f0efc67a"}}