OpenAI · 085-01 · 2026-09-26 · Real and complex analysis
An Endpoint Gradient Bound for the Centered Disk Maximal Operator
preprints/An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026
Family 085. Endpoint regularity of the planar centered maximal function.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real , the centered disk maximal function satisfies with an absolute constant. It belongs locally to and has a globally integrable weak gradient.
We prove the endpoint gradient bound for every real-valued , where is the supremum of the averages of over disks centered at x, and C is an absolute constant. The maximal function belongs to and has a globally integrable weak gradient. This gives a positive resolution of the planar centered-disk case of the endpoint question of Hajłasz and Onninen.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 085: Endpoint Sobolev regularity of centered disk averages. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.