What was already known
Kinnunen established W¹,p control for p>1, while the strong maximal inequality used there fails at p=1; Hajłasz and Onninen asked whether the endpoint gradient estimate nevertheless holds.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Resolves the planar centered-disk case of a longstanding Sobolev endpoint question.
Limitations
The theorem concerns real W¹,¹ inputs in the plane and centered Euclidean disks; it does not assert global integrability of Mf or a BV extension.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note covers local W¹,¹ regularity and the global L¹ weak-gradient bound for centered disk averages.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- For every real f in W¹,¹(R²), the centered-disk maximal function has a globally integrable weak gradient and satisfies ||∇Mf||₁ ≤ C||∇f||₁ for an absolute constant C.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.
Read this published revision