Remark: Family 085. Endpoint regularity of the planar centered maximal function

Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real $f\in W^{1,1}({\mathbb{R}}^2)$, the centered disk maximal function satisfies $\Vert \nabla Mf{\Vert }_1\le C\Vert \nabla f{\Vert }_1$ with an absolute constant. It belongs locally to $W^{1,1}$ and has a globally integrable weak gradient.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.