{"as_of":"2026-08-21T14:16:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:4eba465f69494661914e0497b3fdb03901869664940eec12626c056a0af364a6","coverage":[{"denominator":50,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":50,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-14T04:21:18.242546Z","state":"measured"},{"denominator":50,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":50,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-21T06:32:19.484+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2608.09034/citation-record","integrity":"/paper/2608.09034/integrity","json":"/paper/2608.09034/citation-record.json","paper":"/paper/2608.09034"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:19.085162Z","title":"Aubin, Probl` emes isop´ erim´ etriques et espaces de Sobolev,J","venue":null,"work_id":"1517f358-557c-43e9-8b79-c5583b2dacd8","year":1976},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.046401Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:aaa270ff21a21ae084758104ef1f0b3cd092ba16a4030e80b3e8675e4daa783e","observation_id":"4e07f02b-1e5d-4d6e-a115-b81b0f25b9f6","resolution":{"observed_at":"2026-08-14T04:21:19.089520Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:19.072367Z","title":"Bianchi, H","venue":null,"work_id":"e9137e60-08d8-457e-ad00-3e0ebf366d22","year":1991},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.050222Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:4e60ab29a2b92539c67d15295c0e54de38e02d78e0e21c6c242e6118dfb16355","observation_id":"2f5a6cf9-47dc-4f07-b039-591770ce5def","resolution":{"observed_at":"2026-08-14T04:21:19.076088Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:19.060206Z","title":"Brezis, E","venue":null,"work_id":"eb99e926-3f12-405a-8045-a9463c27e94e","year":1983},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.054012Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:f9e3c5f76bcf6be106a6ed4c450969a424005c411a9733a614efd727f6663362","observation_id":"63c09b8b-328c-4a50-bd5c-fd8a13400d5f","resolution":{"observed_at":"2026-08-14T04:21:19.063877Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:19.049741Z","title":"Brezis, E.H","venue":null,"work_id":"7ebe6e60-4f80-44ca-9b5f-c6aaaa16696e","year":1985},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.058175Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:bcf1f26b7068262e8e75ea65278252d069a410f9995dc12ec66f02c3df713ac5","observation_id":"62d0e8d7-2e3b-4d51-8038-1d24cdc0fada","resolution":{"observed_at":"2026-08-14T04:21:19.053359Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:19.038592Z","title":"Carlen, Duality and stability for functional inequalities,Ann","venue":null,"work_id":"ea3215bc-cbcf-47b9-8705-681000cabf0a","year":2017},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.062633Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:29dfc9622a56976156918bebd5985d3715651a4c0655569e63b24ca7770cae6d","observation_id":"5737cab4-0758-4c5b-a803-d8409e018829","resolution":{"observed_at":"2026-08-14T04:21:19.042399Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2505.07039","last_updated":"2025-07-16T04:35:07Z","snapshot_observed_at":"2026-08-19T12:24:28.803201Z","submitted_at":"2025-05-11T16:21:29Z","title":"Existence of Bianchi-Egnell stability extremizer for the Hardy-Sobolev inequality","version":2},"cited_work":{"arxiv_id":"2505.07039","doi":null,"metadata_source":"pith","pith_arxiv_id":"2505.07039","snapshot_observed_at":"2026-08-14T04:21:18.577081Z","title":"Existence of Bianchi-Egnell stability extremizer for the Hardy-Sobolev inequality","venue":"math.AP","work_id":"3b5af486-3ca5-4968-9d4a-ed634eca7c77","year":2025},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.066701Z"},"links":{"cited_paper":"/paper/2505.07039","citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:9f0d131c5fc779fad57814b46d137aaba5022954cbbb67410710b347b8a65852","observation_id":"4d54f38f-87f9-4146-ac04-d4d759f3eafb","resolution":{"observed_at":"2026-08-14T04:21:18.581677Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:19.028123Z","title":"Chen, R.L","venue":null,"work_id":"aadd1e0b-34a6-4bdb-983d-0f022e53d4cb","year":2013},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.072319Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:b36793324d3c2cd81098297131cda5ec1f3b9fd8af439d3605cfeb6a7147eb2d","observation_id":"9fca40ba-4a03-484e-910f-42f58210d9b7","resolution":{"observed_at":"2026-08-14T04:21:19.031615Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:19.016996Z","title":"Chen, G.Z","venue":null,"work_id":"173d0785-2604-4873-bad2-9b6e7c8a827c","year":2024},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.076709Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:6dea0ef0756b77fe70ef5e7ba596e4cf1a4c0b440256e1198e8d03bbaadea4ea","observation_id":"911a28e8-ff33-4074-8fde-fee50ebb924c","resolution":{"observed_at":"2026-08-14T04:21:19.020707Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:19.006934Z","title":"Chen, G.Z","venue":null,"work_id":"5b6cee8f-4a96-44ae-99ec-060cfac86111","year":2026},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.080621Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:16914b27f22302ee10df9f704a54ee388850de385289fdb85ac42f899f0c8f70","observation_id":"1eba439c-17fc-4fbc-bcce-7d38ff2b4ff8","resolution":{"observed_at":"2026-08-14T04:21:19.010087Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.986052Z","title":"Chen, G.Z","venue":null,"work_id":"a9dafa92-129d-457c-9d7f-0bd41353d4b5","year":2025},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.084516Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:eecc1c60a0ea418c002d0ef15e642466183e3843a15df0527d7a0f8e2f1e4475","observation_id":"7aedeab9-a737-47b2-9727-6b580d47dbc5","resolution":{"observed_at":"2026-08-14T04:21:18.994746Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.973859Z","title":"Chen, G.Z","venue":null,"work_id":"cb98e2cd-8287-4538-87fd-09691bac75d3","year":2026},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.088308Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:316e2915c5a1ebff735d35f36056874036c3d3fb3a2fb2fcd2893a64217e673e","observation_id":"0a266f1b-2c37-49b8-8050-6875ea8cad53","resolution":{"observed_at":"2026-08-14T04:21:18.977826Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":"2602.08375","doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.556590Z","title":"Chen, Z.X","venue":null,"work_id":"5b901a64-6d78-4d10-8465-6a13dc347da6","year":2026},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.093871Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:deaba243fd39705b5a69305608f731cf81ad9d62df5302823540dcafb238dd30","observation_id":"95bc659c-7d96-47fd-8c5a-89ccdd56d924","resolution":{"observed_at":"2026-08-14T04:21:18.562982Z","resolver_source":"raw_fallback","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.962248Z","title":"De Nitti, T","venue":null,"work_id":"3a119c6c-ffa2-4535-a107-65fa9ba226e5","year":2023},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.097875Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:52bd8b5096ada0a324704948646260d05194c561ee06ed648532732fdcbd9483","observation_id":"90f5685d-8f1d-40be-b0f5-4c543d62d095","resolution":{"observed_at":"2026-08-14T04:21:18.966357Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.952095Z","title":"Deng, W.S","venue":null,"work_id":"d56b9b9b-8689-43b4-b791-d75045e67aa5","year":2025},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.101347Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:bd45bca3983d550ff114ca9518f7b4e3eb987e8a2ae7ce1f63b3a3b293248246","observation_id":"c616d627-6cb3-4fd0-9d0e-cf46b26d34ad","resolution":{"observed_at":"2026-08-14T04:21:18.955522Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2308.04111","last_updated":"2024-12-28T02:40:44Z","snapshot_observed_at":"2026-07-06T16:03:46.860343Z","submitted_at":"2023-08-08T07:53:21Z","title":"On the stability constant of Caffarelli-Kohn-Nirenberg inequality","version":4},"cited_work":{"arxiv_id":"2308.04111","doi":null,"metadata_source":"pith","pith_arxiv_id":"2308.04111","snapshot_observed_at":"2026-08-14T04:21:18.491332Z","title":"On the stability constant of Caffarelli-Kohn-Nirenberg inequality","venue":"math.AP","work_id":"2632d541-e0a1-4728-a1e8-a969ed71fb59","year":2023},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.104748Z"},"links":{"cited_paper":"/paper/2308.04111","citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:57bcb885254058a80ea9f785c26588787e1965c194b77a3cc3bc3ce9f5a0d66e","observation_id":"60125815-9bad-4639-9d34-041bd1e00261","resolution":{"observed_at":"2026-08-14T04:21:18.496039Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.940936Z","title":"Deng, X.L","venue":null,"work_id":"28c9c21c-dea2-4db9-b800-c041b5b531d7","year":2025},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.109120Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:cfcc8bb474c0e337884e586c4bd33b8d44bcbe210a23d0282005374e7e272cd1","observation_id":"e0d589bf-27e8-44d3-bb67-4e6ffda9e200","resolution":{"observed_at":"2026-08-14T04:21:18.944776Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.114153Z","title":"Deng, X.L","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.114153Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:de8b35c5353f95e5ff0c7c4a536a5830f99f4359efe9216fe5e57133bfa93570","observation_id":"7a5b4dbd-efef-45ee-8e0a-d979927fa645","resolution":{"observed_at":"2026-08-14T04:21:18.114153Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.929870Z","title":"Deng, X.L","venue":null,"work_id":"8331e54c-5916-479c-a370-a6c5cb7f7c53","year":2024},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.117968Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:c3c6ae18d23cac5ebb8328b0d1784ccf5a137954a8cc590369eb78a9455ee77f","observation_id":"d0a220a5-9f90-4b66-abc9-2a938be5eb1e","resolution":{"observed_at":"2026-08-14T04:21:18.933710Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.917636Z","title":"Dolbeault, M.J","venue":null,"work_id":"297a1da2-1a1a-4ab4-9713-9ea054c6c2ee","year":2025},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.121751Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:2101cf2bb3b084b5a84a3ec6ccdd52c2618a9f643eccf576da921553ce34c182","observation_id":"77a435f0-8b0f-44a9-88f1-ad313961e4b0","resolution":{"observed_at":"2026-08-14T04:21:18.921710Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.906750Z","title":null,"venue":null,"work_id":"b4343d83-98f8-4af0-81f5-04e09fd5a7d7","year":2019},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.125802Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:2c4f8d33e469f77ba3254405a6793292859462c86bd1cc7be1a572f3a3c92afe","observation_id":"2db68118-1b6e-4bd4-a1a0-20bb15c718f9","resolution":{"observed_at":"2026-08-14T04:21:18.910289Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.897089Z","title":"Figalli, R","venue":null,"work_id":"baf8241d-11f8-4df5-91c8-3a94b8eb5ce9","year":2019},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.129553Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:e2c073e8368a5b61b00af143e2ca50a5b8d21029ac8daaaf58bff2deb18453cd","observation_id":"2172d112-3731-4875-870a-c02710472b15","resolution":{"observed_at":"2026-08-14T04:21:18.900422Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.880585Z","title":"Figalli, Y.R.Y","venue":null,"work_id":"784e2cd5-427f-429f-b63c-568823e1f961","year":2022},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.132953Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:fd9d47b5ec4dc3e0f9bce309841c7e67a4d48663d7ba4424d464d312a15cce11","observation_id":"b7276a56-1e46-4fd7-8e98-050b3e0e8846","resolution":{"observed_at":"2026-08-14T04:21:18.884302Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.869555Z","title":"Folland, E.M","venue":null,"work_id":"0b4c6706-2231-4f07-adbf-eee16ac8aedd","year":1974},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.136484Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:29b68d516c1ce3f171cbd13777a2e2e34564543c6665625bd0dd4a77054dc63a","observation_id":"e080daa0-7e87-471e-bb08-62a2f0878800","resolution":{"observed_at":"2026-08-14T04:21:18.873609Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.857763Z","title":"Frank, E.H","venue":null,"work_id":"37d6fab8-ea1a-4418-b287-5d0b16878b8b","year":2012},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.139675Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:75ca6120344cf59e122affcf0a8cbc227be25e6b6f60aabe94310393020f2d66","observation_id":"90f4bd44-36e0-4e7c-b6a8-474a70b6643c","resolution":{"observed_at":"2026-08-14T04:21:18.861717Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.846039Z","title":null,"venue":null,"work_id":"e2ea10ee-b1d8-441a-81f3-b8b186607f3a","year":2020},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.143728Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:f398ac6932ff4f74118053b2bd9341923b2e2ae904dbcf4fee7d3a28105d0f6c","observation_id":"f22c1439-971e-4cbb-acd1-5864f5ad79d0","resolution":{"observed_at":"2026-08-14T04:21:18.849626Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.828580Z","title":"Guo, T.X","venue":null,"work_id":"d6a0bffd-661b-4a9c-8ef5-d06fedeef37d","year":2019},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.148308Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:e27136c4e5cf79f161ffe04b7da9d2996072d396ddce8c3559edad1a020cf7ec","observation_id":"245ee950-eda4-4543-a547-0635c7a89767","resolution":{"observed_at":"2026-08-14T04:21:18.838244Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.814293Z","title":"Hardy, J.E","venue":null,"work_id":"57a702a9-e988-4217-8a27-0cdb9f13b6c3","year":1928},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.152470Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:e40c2a458993e02965043d0b48a1bd5654019f797e8ee11fa648afdeabd13ac1","observation_id":"b70d09b8-fabe-407d-ba06-a19e91b5f00a","resolution":{"observed_at":"2026-08-14T04:21:18.818408Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.803436Z","title":"Jerison, J.M","venue":null,"work_id":"4cd2e46f-f964-4927-a6d0-966db3a57d68","year":1988},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.156483Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:5a7b866d3984df54a12018b10b829888de3c4ca6aaee16a01c625699ade3a0b0","observation_id":"19afbda6-3d5f-469a-bf95-65a1d94b363c","resolution":{"observed_at":"2026-08-14T04:21:18.806773Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.792304Z","title":"K¨ onig, On the sharp constant in the Bianchi-Egnell stability inequality,Bull","venue":null,"work_id":"300c9635-a3cc-447e-b0db-40fb542d93e4","year":2023},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.161077Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:7665c400a552e8d1a67655501e56adc4146a9a9929ce84f4df4c02a5b7984fa0","observation_id":"e4cf74d9-c2c1-4fac-9ca0-1b5196a5201a","resolution":{"observed_at":"2026-08-14T04:21:18.796335Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.164613Z","title":"K¨ onig, Stability for the Sobolev inequality: existence of a minimizer,J","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.164613Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:303c429a05ec7168f8736968527664e9195d9c6bf6f78c0ba006e7c3ead31eb1","observation_id":"3c0827f0-1c14-4a9a-afd5-c72e0fb2b29d","resolution":{"observed_at":"2026-08-14T04:21:18.164613Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.169615Z","title":null,"venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.169615Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:474d0d110182dbb7a16afce7699ae305644fac7fde31db30d4034d570de164af","observation_id":"3609a40f-ec45-4e5c-b824-e1f466b4e0d1","resolution":{"observed_at":"2026-08-14T04:21:18.169615Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.778737Z","title":"Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,Ann","venue":null,"work_id":"253e58b9-7c14-4c06-8cb2-66439dd41780","year":1983},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.173305Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:e6db0a5666c617bb55da308cffe1d520c99b2f89574db95ed7863ee93681c25f","observation_id":"e4c0011d-2af1-4738-9722-1ce7e4ac4f29","resolution":{"observed_at":"2026-08-14T04:21:18.782670Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.765933Z","title":null,"venue":null,"work_id":"fa7f1b50-23ad-4686-8a2d-90fe0775e9d6","year":2001},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.176742Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:2715204b0ec96b79d082cb4d8d00413c1a94f7ec12e2d022fda57c3c6a5baa8f","observation_id":"d58f861b-19ff-43aa-83d9-711e5414c0ae","resolution":{"observed_at":"2026-08-14T04:21:18.769807Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.754590Z","title":"Lions, The concentration-compactness principle in the calculus of variations","venue":null,"work_id":"4a901de1-eb59-47a5-8b6b-5175847eab50","year":1985},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.179917Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:78cec032954d36699fc9641f931fe73720717e4eaba260840a7994f70a992c1f","observation_id":"21057174-f8cf-4fc5-84f3-bdd1bac5da32","resolution":{"observed_at":"2026-08-14T04:21:18.758256Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.742702Z","title":null,"venue":null,"work_id":"a5bf8214-57c5-4dad-bcc6-5fe001af8160","year":2015},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.183449Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:7c1f4d98a83facfa1a096fbd4ac37c5f2e6a628a5de9f9f8094e057214cc5cdf","observation_id":"5e4253ed-fcd1-454f-99f4-d0005c25b992","resolution":{"observed_at":"2026-08-14T04:21:18.746717Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.732968Z","title":"Loiudice, Improved Sobolev inequalities on the Heisenberg group,Nonlinear Anal.,62(2005): 953-962","venue":null,"work_id":"c2a0edcb-b79c-4d0a-9f4b-484033bb02af","year":2005},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.187616Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:cb29832696dd4020084192765eda5305c2190c479f72102719e9d58aadda93be","observation_id":"2ee83dff-06ba-4d50-9d2a-da9feb7ca820","resolution":{"observed_at":"2026-08-14T04:21:18.736435Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.721513Z","title":null,"venue":null,"work_id":"4df379bc-d5a9-49a8-b7b3-21c02d6a2c31","year":2000},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.191443Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:2c807b824ccaa30c0875163f4b6611614305535ad2b72a40c164ed7691dfbcaa","observation_id":"c846677a-f9dc-489b-9d2e-0e29a14f21d3","resolution":{"observed_at":"2026-08-14T04:21:18.725887Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2503.06636","last_updated":"2025-03-09T14:31:17Z","snapshot_observed_at":"2026-08-16T12:51:46.734076Z","submitted_at":"2025-03-09T14:31:17Z","title":"Remainder terms, profile decomposition and sharp quantitative stability in the fractional nonlocal Sobolev-type inequality with $n>2s$","version":1},"cited_work":{"arxiv_id":"2503.06636","doi":null,"metadata_source":"pith","pith_arxiv_id":"2503.06636","snapshot_observed_at":"2026-08-14T04:21:18.412613Z","title":"Remainder terms, profile decomposition and sharp quantitative stability in the fractional nonlocal Sobolev-type inequality with $n>2s$","venue":"math.AP","work_id":"e537d2cb-0314-4c53-8735-3c796416797a","year":2025},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.195200Z"},"links":{"cited_paper":"/paper/2503.06636","citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:30b8a1037c74cf12f4b006ee41ec141b86f2c3e3ee9c3992e92b3af93a9ddd19","observation_id":"811c604f-dc4a-4668-8bd8-dc1be1e56b08","resolution":{"observed_at":"2026-08-14T04:21:18.417408Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.710318Z","title":"Malchiodi, F","venue":null,"work_id":"2dab2f2c-d4a6-40b2-97c0-8c095322fda8","year":2002},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.200130Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:59e48ac1132ec227d3b793e48e9a961d9d56d37c5d2941f1d003e1174aa20425","observation_id":"1b93346a-38f9-4198-9a0a-a0f7ec899047","resolution":{"observed_at":"2026-08-14T04:21:18.713946Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.690337Z","title":"Neumayer, A note on strong-form stability for the Sobolev inequality,Calc","venue":null,"work_id":"12150211-bec4-43c2-8483-83eb19fa74e2","year":2020},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.204083Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:d2ad0e9d0b4bcca8f784a0d148d83b2231ef99f447e8b8b11f32bf0980fc8a37","observation_id":"6ab77e6c-90c1-4f29-adbb-57d26152fa3d","resolution":{"observed_at":"2026-08-14T04:21:18.694437Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.669877Z","title":"Platis, Quasiconformal mappings on the Heisenberg group: an overview, IRMA Lect","venue":null,"work_id":"ec5ae106-c1ac-4578-82f3-96e4cd12c4c6","year":2016},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.207611Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:43c81cbc281e99c715260a56c17ca0c7c6d89738bf7105d42ef79a493af25ebb","observation_id":"8a150052-800b-4115-a5a2-bbff27ab2523","resolution":{"observed_at":"2026-08-14T04:21:18.673322Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.657393Z","title":"Talenti, Best constant in Sobolev inequality,Ann","venue":null,"work_id":"65261ebd-ce27-4d54-8fbe-b6ef7127fe6d","year":1976},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.211346Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:3726ec95c1c8084cd1f21b4180aa6b6c62fe247131edc8e2556aeb5b25f748ef","observation_id":"ff27bd1e-6dd4-4e1e-a142-4db64fc61bed","resolution":{"observed_at":"2026-08-14T04:21:18.661802Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.645450Z","title":"Tang, B.W","venue":null,"work_id":"fccf5dde-5c9f-468c-9f39-2911053ec6e4","year":2024},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.215249Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:815159c1c25545f0315ee6bafe790bdc2ec3e0303639f98a541b5e58db5a39aa","observation_id":"17001a04-cc35-43d0-9ef9-05208a374186","resolution":{"observed_at":"2026-08-14T04:21:18.649989Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.631973Z","title":"Wei, Y.Z","venue":null,"work_id":"11ae6a33-bed3-443e-ad59-958bc19c83d4","year":2024},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.219190Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:992a0727c1f29b5a1e4c7cdb76210c0089448d70a9f0fe05a53a080559d928a3","observation_id":"e94fdd89-d521-4b88-8564-d6280458c43f","resolution":{"observed_at":"2026-08-14T04:21:18.635993Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2508.07719","last_updated":"2026-08-18T00:46:14Z","snapshot_observed_at":"2026-08-21T14:09:14.104535Z","submitted_at":"2025-08-11T07:46:35Z","title":"Nondegeneracy of positive solutions for critical Hartree equation on Heisenberg group and it's applications","version":2},"cited_work":{"arxiv_id":"2508.07719","doi":null,"metadata_source":"pith","pith_arxiv_id":"2508.07719","snapshot_observed_at":"2026-08-14T04:21:18.395607Z","title":"Nondegeneracy of positive solutions for critical Hartree equation on Heisenberg group and it's applications","venue":"math.AP","work_id":"2588015f-5141-4a3e-b9aa-c0c6fdb3455d","year":2025},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.223114Z"},"links":{"cited_paper":"/paper/2508.07719","citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:f754c66ae079c4a6ab7850e6b97af579affe8c3adf2fe0ecaad0ed9b3602f0d8","observation_id":"a51b2f03-9029-4d07-a4a4-57532151f664","resolution":{"observed_at":"2026-08-14T04:21:18.399937Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.618000Z","title":null,"venue":null,"work_id":"ce16d5d7-f8aa-4c28-87fd-7d86c3389237","year":2025},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.228113Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:24bcc98349a8497cf50d9fc3829bc7e2dd07fcceb0bf5e0adf5c6948db0fa856","observation_id":"22427220-787d-4d97-9ca2-300c11e02cf5","resolution":{"observed_at":"2026-08-14T04:21:18.624262Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.606634Z","title":"Zhang, Existence of extremals for stability of nonlocal Sobolev inequality,Adv","venue":null,"work_id":"4d3727e5-2a69-4f91-b465-203493befc8e","year":2026},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":47,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.231251Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:eb1469d56bbea107a9543194d6dcfd596e848125c6dc714c82203d24dfa99c6b","observation_id":"8a225803-22f7-49db-89e6-42c516c66d3f","resolution":{"observed_at":"2026-08-14T04:21:18.610735Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2511.20264","last_updated":"2026-08-18T00:45:04Z","snapshot_observed_at":"2026-08-21T14:09:22.438015Z","submitted_at":"2025-11-25T12:49:28Z","title":"Symmetry and uniqueness of the positive solution for the critical Hartree equation on the Heisenberg group","version":4},"cited_work":{"arxiv_id":"2511.20264","doi":null,"metadata_source":"raw_reference","pith_arxiv_id":"2511.20264","snapshot_observed_at":"2026-08-19T01:25:33.586599Z","title":"Zhang, J.L","venue":null,"work_id":"5e31178a-4b00-46bd-876b-8ea979080d13","year":2025},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.234929Z"},"links":{"cited_paper":"/paper/2511.20264","citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:a84d7b685ffa2a96645ffdbfd82241e13a49ed03632588c358211a3e52411456","observation_id":"9ad50ff2-7af5-4ec3-adaf-c5919c434e49","resolution":{"observed_at":"2026-08-19T01:25:33.586599Z","resolver_source":"raw_fallback","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2508.08614","last_updated":"2026-08-18T00:45:46Z","snapshot_observed_at":"2026-08-21T14:09:14.168539Z","submitted_at":"2025-08-12T03:54:23Z","title":"Quantitative stability of critical points for the nonlocal-Sobolev inequality in Heisenberg group","version":2},"cited_work":{"arxiv_id":"2508.08614","doi":null,"metadata_source":"pith","pith_arxiv_id":"2508.08614","snapshot_observed_at":"2026-08-14T04:21:18.290515Z","title":"Quantitative stability of critical points for the nonlocal-Sobolev inequality in Heisenberg group","venue":"math.AP","work_id":"0b681eb2-f307-452b-beb5-51642618bbc2","year":2025},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":49,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.238562Z"},"links":{"cited_paper":"/paper/2508.08614","citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:dd7b29000c24ef43518aa067f906501d1266235b052b98a153ee1ea7de839f12","observation_id":"1e829ec6-95f3-4e15-aaf9-8ff8b70c8a85","resolution":{"observed_at":"2026-08-14T04:21:18.296476Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T04:21:18.590081Z","title":"Zhang, Y.X","venue":null,"work_id":"8023332d-4ac0-4d92-8b77-d3aeafb3ce45","year":2025},"citing_paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","version":1},"reference_index":50,"source":"pdf_text","source_observed_at":"2026-08-14T04:21:18.242546Z"},"links":{"citing_paper":"/paper/2608.09034"},"observation_digest":"sha256:690114e13e08709e1a0e6725ab6eb7e7a916c698073036ab63c8ec99ef0f86e6","observation_id":"87681062-f65a-497e-80fd-c25445daaea3","resolution":{"observed_at":"2026-08-14T04:21:18.594306Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2608.09034","last_updated":"2026-08-10T02:36:40Z","latest_version":1,"primary_category":"math.AP","snapshot_observed_at":"2026-08-17T00:42:01.714400Z","submitted_at":"2026-08-10T02:36:40Z","title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group"},"reference_resolution":{"displayed":50,"state_counts":{"malformed_identifier":1,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":8,"verified_exact":7,"verified_fuzzy":34},"total_outbound_references":50},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"thesis":"As of 21 August 2026, this Paper Citation Record lists 50 of 50 outbound references and 0 inbound Pith citation observations for arXiv:2608.09034."}