{"paper":{"title":"A unified geometric perspective on Zygmund's conjecture for maximal functions associated with vector fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Refining Bourgain's argument identifies a weaker condition for boundedness of maximal functions associated with planar vector fields.","cross_cats":[],"primary_cat":"math.CA","authors_text":"Lingxiao Zhang","submitted_at":"2026-05-06T01:34:44Z","abstract_excerpt":"The Zygmund vector field maximal function conjecture is a long-standing open problem. This paper establishes a new boundedness criterion that significantly weakens the existing conditions in the literature. Specifically, the required decay condition is relaxed from the power-type decay of Bourgain for Zygmund's conjecture and the exponential-logarithmic decay of Lacey and Li for Stein's conjecture, to a logarithmic polynomial decay. Unlike the traditional framework that separates finite-type and non-finite-type operators, this paper offers a unified geometric view of both settings. The new cri"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"By refining Bourgain's argument for maximal functions associated with planar vector fields, we identify a condition ensuring boundedness that is weaker than previously known.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The refinement assumes the vector fields satisfy the (unspecified in abstract) weaker condition identified by the refined argument; without the full paper the exact form and verification of this condition cannot be assessed.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Refines Bourgain's method to establish boundedness of maximal functions under a weaker condition than previously known, strengthening an implicit Lacey-Li result.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Refining Bourgain's argument identifies a weaker condition for boundedness of maximal functions associated with planar vector fields.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3b6268bbcb06efe9d1b814fac6f8c4f8779dd00fc774fa45b896ef3b6fe8b604"},"source":{"id":"2605.04394","kind":"arxiv","version":2},"verdict":{"id":"2be98dcf-f7cb-4ebe-b50f-8425709b8264","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T16:55:35.178926Z","strongest_claim":"By refining Bourgain's argument for maximal functions associated with planar vector fields, we identify a condition ensuring boundedness that is weaker than previously known.","one_line_summary":"Refines Bourgain's method to establish boundedness of maximal functions under a weaker condition than previously known, strengthening an implicit Lacey-Li result.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The refinement assumes the vector fields satisfy the (unspecified in abstract) weaker condition identified by the refined argument; without the full paper the exact form and verification of this condition cannot be assessed.","pith_extraction_headline":"Refining Bourgain's argument identifies a weaker condition for boundedness of maximal functions associated with planar vector fields."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04394/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T11:44:00.960619Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:20.011448Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:29:10.181808Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"e015d598fff024546c44347ba1d8006dbc948019434d7e4cf89b307d11297c04"},"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"}