Endpoint Sobolev regularity of centered disk averages

What OpenAI claims

Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real f ∈ W^1,1(ℝ²), the centered disk maximal function satisfies ‖∇ Mf‖₁ ≤ C‖∇ f‖₁ with an absolute constant. It belongs locally to W^1,1 and has a globally integrable weak gradient. (Lean)

Paper

1. An Endpoint Gradient Bound for the Centered Disk Maximal Operator

   26 September 2026 · Preprint, not peer reviewed · PDF

   We prove the endpoint gradient bound ‖∇ Mf‖₁ ≤ C‖∇ f‖₁ for every real-valued f ∈ W^1,1(ℝ²), where Mf(x) is the supremum of the averages of |f| over disks centered at x, and C is an absolute constant. The maximal function belongs to W^1,1_loc(ℝ²) 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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.