Endpoint Sobolev regularity of centered disk averages

1 paperLean

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 \(\|\nabla Mf\|_1\le C\|\nabla f\|_1\) with an absolute constant. It belongs locally to \(W^{1,1}\) and has a globally integrable weak gradient.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 1

An Endpoint Gradient Bound for the Centered Disk Maximal Operator

OpenAI · September 26, 2026

We prove the endpoint gradient bound \(\|\nabla Mf\|_1\le C\|\nabla f\|_1\) for every real-valued \(f\in W^{1,1}(\mathbb R^2)\), 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 \(W^{1,1}_{\mathrm{loc}}(\mathbb R^2)\) 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.

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@misc{OAI:An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026,
  author = {{OpenAI}},
  title = {{An Endpoint Gradient Bound for the Centered Disk Maximal Operator}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026/article.pdf}{OAI:An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

An Endpoint Gradient Bound for the Centered Disk Maximal Operator

Scope

The centered disk maximal operator takes the supremum of averages of \(|f|\) over disks centered at each point. The formalization proves that every real \(f\in W^{1,1}(\mathbb R^2)\) has a maximal function that is finite almost everywhere, belongs to \(W^{1,1}_{\mathrm{loc}}\), and has a globally integrable weak gradient satisfying \(\|\nabla Mf\|_1\le C\|\nabla f\|_1\) for one absolute constant \(C\).

The earlier signed finite-band estimate for smooth compactly supported functions is also retained. The general endpoint statement covers the Sobolev setting of the paper; a BV extension is outside these statements.

Comparator links

Result	Comparator statement
Signed finite-band disk-maximal gradient bound	SignedFiniteBand.lean
Endpoint gradient bound for the centered disk maximal operator	DiskMaximal.lean

Scope note on GitHub →