Result 085, Real and complex analysis

Endpoint Sobolev regularity of centered disk averages

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.

Lean formalization Proof

The bigger picture

Why it matters

Taking the largest average of a function's magnitude over all disks centered at a point can emphasize features at many scales. The reported result says this operation still controls total variation measured by an integrable gradient.

What changes?

The unreviewed manuscript considers every real-valued function on the plane that is integrable and has integrable weak first derivatives. Its centered disk maximal function takes, at each point, the supremum of averages of the function's absolute value over disks centered there. The claim is that this maximal function has locally integrable values and weak first derivatives. Its gradient is globally integrable, with its integrated magnitude bounded by an absolute constant times the original function's integrated gradient magnitude.

What does that help mathematicians do?

This rules out a specific loss of regularity: selecting the largest disk average cannot turn a finite integrated gradient into an infinite one under these assumptions. The conclusion concerns derivatives, not global integrability of the maximal function itself. It settles the planar centered-disk case of the Hajłasz–Onninen question, without claiming the same result for other averaging shapes or higher dimensions.

Are there practical applications?

The immediate value is foundational. In arguments that replace a planar function by its centered disk maximal function, researchers gain a bound on the resulting weak derivatives using only the original gradient. This supports reasoning about such averages within Sobolev analysis; the supplied material does not establish a practical computational or physical application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

An Endpoint Gradient Bound for the Centered Disk Maximal Operator

September 26, 2026 45 pages Main result formalized in Lean

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/085.md.

Endpoint Sobolev regularity of centered disk averages

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

At a glance

Subject: Real and complex analysis
Manuscripts: 1
Pages: 45
Dated: Sep 26, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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