{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:VE64EYGKSCGGPJPUO7YLZH6XVS","short_pith_number":"pith:VE64EYGK","canonical_record":{"source":{"id":"2607.02312","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T15:24:53Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"026d92d799c13bf0ba38ff8dfcb492ccb4ddcc8a8334038240d1747b46386963","abstract_canon_sha256":"6594c4c2c328179680097833ef53cb1adc7a2b9b44651b5dc78143654abbf8fa"},"schema_version":"1.0"},"canonical_sha256":"a93dc260ca908c67a5f477f0bc9fd7ac97c1fcfa4fccd55458a10104bcee6256","source":{"kind":"arxiv","id":"2607.02312","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.02312","created_at":"2026-07-03T01:17:48Z"},{"alias_kind":"arxiv_version","alias_value":"2607.02312v1","created_at":"2026-07-03T01:17:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.02312","created_at":"2026-07-03T01:17:48Z"},{"alias_kind":"pith_short_12","alias_value":"VE64EYGKSCGG","created_at":"2026-07-03T01:17:48Z"},{"alias_kind":"pith_short_16","alias_value":"VE64EYGKSCGGPJPU","created_at":"2026-07-03T01:17:48Z"},{"alias_kind":"pith_short_8","alias_value":"VE64EYGK","created_at":"2026-07-03T01:17:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:VE64EYGKSCGGPJPUO7YLZH6XVS","target":"record","payload":{"canonical_record":{"source":{"id":"2607.02312","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T15:24:53Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"026d92d799c13bf0ba38ff8dfcb492ccb4ddcc8a8334038240d1747b46386963","abstract_canon_sha256":"6594c4c2c328179680097833ef53cb1adc7a2b9b44651b5dc78143654abbf8fa"},"schema_version":"1.0"},"canonical_sha256":"a93dc260ca908c67a5f477f0bc9fd7ac97c1fcfa4fccd55458a10104bcee6256","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:48.158439Z","signature_b64":"r8J37O7AucpZm8JDRB6538Mutovo9Wtct/BTpjq+TrYwimFOHact04zbDxHvPoEBwytQP0YTvukUcIvBcrAKBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a93dc260ca908c67a5f477f0bc9fd7ac97c1fcfa4fccd55458a10104bcee6256","last_reissued_at":"2026-07-03T01:17:48.158053Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:48.158053Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.02312","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":"2026-07-03T01:17:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ltOsEEdRsfdRSeUr6jP5K0lr8ztFMWnZLmW+wU/0idNbLLLzY9V3UZxMdvCvWb8+KYNHoarqg4u7CTOPIRPHDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T13:44:38.248423Z"},"content_sha256":"4dead3eee377f9451eb1ae93a519822e3051cc970caad57efd7ec9928cc2bcbe","schema_version":"1.0","event_id":"sha256:4dead3eee377f9451eb1ae93a519822e3051cc970caad57efd7ec9928cc2bcbe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:VE64EYGKSCGGPJPUO7YLZH6XVS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The structure of solution spaces for fractional-order operators, with gradient estimates","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Gerd Grubb","submitted_at":"2026-07-02T15:24:53Z","abstract_excerpt":"The solution space of the homogeneous Dirichlet problem for the fractional Laplacian $(-\\Delta )^{a}$ ($0<a<1$) or a pseudodifferential generalization $P$, on a bounded open set $\\Omega \\subset R^n$ with $C^{1+\\tau }$-boundary, $$ Pu=f \\text{ on }\\Omega ,\\quad u=0 \\text{ on }R^n\\setminus \\Omega , $$ is analysed in detail. It is shown, both for solutions in Sobolev spaces of Bessel-potential type $H_q^t$ and in H\\\"older-Zygmund spaces $C_*^t$, that the solution space for $f$ of regularity $s\\in [0,\\tau -2a)$ is the direct sum of a component $\\dot H_q^{2a+s}(\\bar\\Omega)$ resp. $\\dot C_*^{2a+s}(\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02312","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.02312/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-03T01:17:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H2bl59ppNFpcdezFNrVzn74oH82pqsKWQQ31VZ2Pcg8lXYdpJdUANFy3ywRTjRUY+KfA4+V2mI2cgiHzc7sOCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T13:44:38.248803Z"},"content_sha256":"12a76e1fabdbde14e88526e967cfada1c8ea3b45450792c80267b424078ac51f","schema_version":"1.0","event_id":"sha256:12a76e1fabdbde14e88526e967cfada1c8ea3b45450792c80267b424078ac51f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VE64EYGKSCGGPJPUO7YLZH6XVS/bundle.json","state_url":"https://pith.science/pith/VE64EYGKSCGGPJPUO7YLZH6XVS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VE64EYGKSCGGPJPUO7YLZH6XVS/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":"2026-07-07T13:44:38Z","links":{"resolver":"https://pith.science/pith/VE64EYGKSCGGPJPUO7YLZH6XVS","bundle":"https://pith.science/pith/VE64EYGKSCGGPJPUO7YLZH6XVS/bundle.json","state":"https://pith.science/pith/VE64EYGKSCGGPJPUO7YLZH6XVS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VE64EYGKSCGGPJPUO7YLZH6XVS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VE64EYGKSCGGPJPUO7YLZH6XVS","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":"6594c4c2c328179680097833ef53cb1adc7a2b9b44651b5dc78143654abbf8fa","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T15:24:53Z","title_canon_sha256":"026d92d799c13bf0ba38ff8dfcb492ccb4ddcc8a8334038240d1747b46386963"},"schema_version":"1.0","source":{"id":"2607.02312","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.02312","created_at":"2026-07-03T01:17:48Z"},{"alias_kind":"arxiv_version","alias_value":"2607.02312v1","created_at":"2026-07-03T01:17:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.02312","created_at":"2026-07-03T01:17:48Z"},{"alias_kind":"pith_short_12","alias_value":"VE64EYGKSCGG","created_at":"2026-07-03T01:17:48Z"},{"alias_kind":"pith_short_16","alias_value":"VE64EYGKSCGGPJPU","created_at":"2026-07-03T01:17:48Z"},{"alias_kind":"pith_short_8","alias_value":"VE64EYGK","created_at":"2026-07-03T01:17:48Z"}],"graph_snapshots":[{"event_id":"sha256:12a76e1fabdbde14e88526e967cfada1c8ea3b45450792c80267b424078ac51f","target":"graph","created_at":"2026-07-03T01:17:48Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.02312/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The solution space of the homogeneous Dirichlet problem for the fractional Laplacian $(-\\Delta )^{a}$ ($0<a<1$) or a pseudodifferential generalization $P$, on a bounded open set $\\Omega \\subset R^n$ with $C^{1+\\tau }$-boundary, $$ Pu=f \\text{ on }\\Omega ,\\quad u=0 \\text{ on }R^n\\setminus \\Omega , $$ is analysed in detail. It is shown, both for solutions in Sobolev spaces of Bessel-potential type $H_q^t$ and in H\\\"older-Zygmund spaces $C_*^t$, that the solution space for $f$ of regularity $s\\in [0,\\tau -2a)$ is the direct sum of a component $\\dot H_q^{2a+s}(\\bar\\Omega)$ resp. $\\dot C_*^{2a+s}(\\","authors_text":"Gerd Grubb","cross_cats":["math.FA"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T15:24:53Z","title":"The structure of solution spaces for fractional-order operators, with gradient estimates"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02312","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":"sha256:4dead3eee377f9451eb1ae93a519822e3051cc970caad57efd7ec9928cc2bcbe","target":"record","created_at":"2026-07-03T01:17:48Z","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":"6594c4c2c328179680097833ef53cb1adc7a2b9b44651b5dc78143654abbf8fa","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T15:24:53Z","title_canon_sha256":"026d92d799c13bf0ba38ff8dfcb492ccb4ddcc8a8334038240d1747b46386963"},"schema_version":"1.0","source":{"id":"2607.02312","kind":"arxiv","version":1}},"canonical_sha256":"a93dc260ca908c67a5f477f0bc9fd7ac97c1fcfa4fccd55458a10104bcee6256","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a93dc260ca908c67a5f477f0bc9fd7ac97c1fcfa4fccd55458a10104bcee6256","first_computed_at":"2026-07-03T01:17:48.158053Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-03T01:17:48.158053Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r8J37O7AucpZm8JDRB6538Mutovo9Wtct/BTpjq+TrYwimFOHact04zbDxHvPoEBwytQP0YTvukUcIvBcrAKBw==","signature_status":"signed_v1","signed_at":"2026-07-03T01:17:48.158439Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.02312","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4dead3eee377f9451eb1ae93a519822e3051cc970caad57efd7ec9928cc2bcbe","sha256:12a76e1fabdbde14e88526e967cfada1c8ea3b45450792c80267b424078ac51f"],"state_sha256":"b103e876c16b1472fe6a5103cd770455e4d6e146c2497db701a7067d1c978169"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E2GE3XAexRrXDbaGqb4Qy5RupC4z3elPtRaoMmXZZ7DiUM6NzprssecpY0VTiA0jyIk1gJtmCYP8FsZjtwvNCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T13:44:38.250779Z","bundle_sha256":"2e3c6ccaad9aa1aa5d5ccf5631d1b9a24ff1eadb08a6520d46f691ffceddd18a"}}