{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:BE5ZFCUCVTH6K3BOMPGGNXL772","short_pith_number":"pith:BE5ZFCUC","canonical_record":{"source":{"id":"1201.3809","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-18T15:03:04Z","cross_cats_sorted":[],"title_canon_sha256":"fd960a2a8f3a6bb517cd85dea2e45b010e3bbd0bac6cac2fd6679b1983414ccd","abstract_canon_sha256":"0660a5f2ba9927b271bf6406f1bee91e9e08b1c790ab66bda2dcf04f0f457732"},"schema_version":"1.0"},"canonical_sha256":"093b928a82accfe56c2e63cc66dd7ffe8240c870c52c63067e659e92671ecf98","source":{"kind":"arxiv","id":"1201.3809","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3809","created_at":"2026-05-18T04:04:17Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3809v1","created_at":"2026-05-18T04:04:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3809","created_at":"2026-05-18T04:04:17Z"},{"alias_kind":"pith_short_12","alias_value":"BE5ZFCUCVTH6","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"BE5ZFCUCVTH6K3BO","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"BE5ZFCUC","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:BE5ZFCUCVTH6K3BOMPGGNXL772","target":"record","payload":{"canonical_record":{"source":{"id":"1201.3809","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-18T15:03:04Z","cross_cats_sorted":[],"title_canon_sha256":"fd960a2a8f3a6bb517cd85dea2e45b010e3bbd0bac6cac2fd6679b1983414ccd","abstract_canon_sha256":"0660a5f2ba9927b271bf6406f1bee91e9e08b1c790ab66bda2dcf04f0f457732"},"schema_version":"1.0"},"canonical_sha256":"093b928a82accfe56c2e63cc66dd7ffe8240c870c52c63067e659e92671ecf98","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:04:17.945744Z","signature_b64":"749ovWKVuJpvhuw9Hmibg9COFrNk2xXgoyF4KrpsoLjKfmQERydYyg7gVtXKkTs7ezOLaa31sMN58wgg0TghCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"093b928a82accfe56c2e63cc66dd7ffe8240c870c52c63067e659e92671ecf98","last_reissued_at":"2026-05-18T04:04:17.945155Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:04:17.945155Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1201.3809","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":"2026-05-18T04:04:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8R0sGcw+I0rDq93xZPNcFrzN3Qq4gNn5i+hRoslepQowGnYRVCERcqKgqIXRcTkb9W546S3G/0mcMp/8VzNSBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T04:08:18.959056Z"},"content_sha256":"ffe08c4f0609559e91bc57454d9fcf81295c90db180468ea42e8e60b9c01d4d6","schema_version":"1.0","event_id":"sha256:ffe08c4f0609559e91bc57454d9fcf81295c90db180468ea42e8e60b9c01d4d6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:BE5ZFCUCVTH6K3BOMPGGNXL772","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Maximal $L^2$ regularity for Dirichlet problems in Hilbert spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessandra Lunardi, Giuseppe Da Prato","submitted_at":"2012-01-18T15:03:04Z","abstract_excerpt":"We consider the Dirichlet problem $\\lambda U - {\\mathcal{L}}U= F$ in \\mathcal{O}, U=0 on $\\partial \\mathcal{O}$. Here $F\\in L^2(\\mathcal{O}, \\mu)$ where $\\mu$ is a nondegenerate centered Gaussian measure in a Hilbert space $X$, $\\mathcal{L}$ is an Ornstein-Uhlenbeck operator, and $\\mathcal{O}$ is an open set in $X$ with good boundary. We address the problem whether the weak solution $U$ belongs to the Sobolev space $W^{2,2}(\\mathcal{O}, \\mu)$. It is well known that the question has positive answer if $\\mathcal{O} = X$; if $\\mathcal{O} \\neq X$ we give a sufficient condition in terms of geometri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3809","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":"2026-05-18T04:04:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yvc/m5aTPh64fhixG0UYO9YYmKsMjz3bhmEqQFal3NS11c+9lF7mIT4+o1y6anMlR/F2eL+A+HnDPX4DcIZ0DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T04:08:18.959416Z"},"content_sha256":"6c46f252acffe195c4f80a70c2fd495000c4a75d79be7ac5e7c2623b8a6b0604","schema_version":"1.0","event_id":"sha256:6c46f252acffe195c4f80a70c2fd495000c4a75d79be7ac5e7c2623b8a6b0604"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BE5ZFCUCVTH6K3BOMPGGNXL772/bundle.json","state_url":"https://pith.science/pith/BE5ZFCUCVTH6K3BOMPGGNXL772/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BE5ZFCUCVTH6K3BOMPGGNXL772/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":"2026-07-02T04:08:18Z","links":{"resolver":"https://pith.science/pith/BE5ZFCUCVTH6K3BOMPGGNXL772","bundle":"https://pith.science/pith/BE5ZFCUCVTH6K3BOMPGGNXL772/bundle.json","state":"https://pith.science/pith/BE5ZFCUCVTH6K3BOMPGGNXL772/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BE5ZFCUCVTH6K3BOMPGGNXL772/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:BE5ZFCUCVTH6K3BOMPGGNXL772","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":"0660a5f2ba9927b271bf6406f1bee91e9e08b1c790ab66bda2dcf04f0f457732","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-18T15:03:04Z","title_canon_sha256":"fd960a2a8f3a6bb517cd85dea2e45b010e3bbd0bac6cac2fd6679b1983414ccd"},"schema_version":"1.0","source":{"id":"1201.3809","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3809","created_at":"2026-05-18T04:04:17Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3809v1","created_at":"2026-05-18T04:04:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3809","created_at":"2026-05-18T04:04:17Z"},{"alias_kind":"pith_short_12","alias_value":"BE5ZFCUCVTH6","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"BE5ZFCUCVTH6K3BO","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"BE5ZFCUC","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:6c46f252acffe195c4f80a70c2fd495000c4a75d79be7ac5e7c2623b8a6b0604","target":"graph","created_at":"2026-05-18T04:04:17Z","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 consider the Dirichlet problem $\\lambda U - {\\mathcal{L}}U= F$ in \\mathcal{O}, U=0 on $\\partial \\mathcal{O}$. Here $F\\in L^2(\\mathcal{O}, \\mu)$ where $\\mu$ is a nondegenerate centered Gaussian measure in a Hilbert space $X$, $\\mathcal{L}$ is an Ornstein-Uhlenbeck operator, and $\\mathcal{O}$ is an open set in $X$ with good boundary. We address the problem whether the weak solution $U$ belongs to the Sobolev space $W^{2,2}(\\mathcal{O}, \\mu)$. It is well known that the question has positive answer if $\\mathcal{O} = X$; if $\\mathcal{O} \\neq X$ we give a sufficient condition in terms of geometri","authors_text":"Alessandra Lunardi, Giuseppe Da Prato","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-18T15:03:04Z","title":"Maximal $L^2$ regularity for Dirichlet problems in Hilbert spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3809","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:ffe08c4f0609559e91bc57454d9fcf81295c90db180468ea42e8e60b9c01d4d6","target":"record","created_at":"2026-05-18T04:04:17Z","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":"0660a5f2ba9927b271bf6406f1bee91e9e08b1c790ab66bda2dcf04f0f457732","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-18T15:03:04Z","title_canon_sha256":"fd960a2a8f3a6bb517cd85dea2e45b010e3bbd0bac6cac2fd6679b1983414ccd"},"schema_version":"1.0","source":{"id":"1201.3809","kind":"arxiv","version":1}},"canonical_sha256":"093b928a82accfe56c2e63cc66dd7ffe8240c870c52c63067e659e92671ecf98","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"093b928a82accfe56c2e63cc66dd7ffe8240c870c52c63067e659e92671ecf98","first_computed_at":"2026-05-18T04:04:17.945155Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:04:17.945155Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"749ovWKVuJpvhuw9Hmibg9COFrNk2xXgoyF4KrpsoLjKfmQERydYyg7gVtXKkTs7ezOLaa31sMN58wgg0TghCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:04:17.945744Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.3809","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ffe08c4f0609559e91bc57454d9fcf81295c90db180468ea42e8e60b9c01d4d6","sha256:6c46f252acffe195c4f80a70c2fd495000c4a75d79be7ac5e7c2623b8a6b0604"],"state_sha256":"abebb704c5f3ca93cd8305c293feec1ba9d9fb0f5209fe7a2584c346ecc0bb28"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H/Ph7I3kG1yv2IMOuHJ6cJXP0uVoChMALyWhWKO07AZbmJjA+6v1dBWmDX1TP/iKCi+rpfNd2UoFQKgtxA6dAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T04:08:18.961376Z","bundle_sha256":"8310328e2995f0fa941de07fc1180b58ce645a0a9a7264be5d01a0702d0f968c"}}