The Hajlasz-Onninen endpoint question for the centered maximal function, planar disk case: is $\|\nabla Mf\|_1\le C\|\nabla f\|_1$ on $W^{1,1}(\mathbb R^2)$ ?

For $f\in L^1_{loc}(\mathbb R^n)$ let $Mf(x)=\sup_{r>0}|B(x,r)|^{-1}\int_{B(x,r)}|f|$ be the centered Hardy-Littlewood maximal function. Kinnunen proved that $M$ is bounded on $W^{1,p}$ for $1<p<\infty$ , using $|\nabla Mf|\le M|\nabla f|$ , but this argument fails at $p=1$ because $M$ is not bounded on $L^1$ . Hajlasz and Onninen (2004, Question 1) asked whether the endpoint gradient bound nevertheless holds: is $f\mapsto|\nabla Mf|$ bounded from $W^{1,1}(\mathbb R^n)$ to $L^1(\mathbb R^n)$ ? Known before this work: one dimension (Tanaka and Aldaz-Perez Lazaro for the uncentered operator, Kurka for the centered one), radial or block-decreasing inputs and the uncentered operator in higher dimensions (Luiro, Aldaz-Perez Lazaro), dyadic and uncentered-cube variants (Weigt), and fractional centered operators (Weigt). The centered operator in dimension at least two was open in general. In the plane, with disks: is there an absolute $C$ such that $\|\nabla Mf\|_{L^1(\mathbb R^2)}\le C\|\nabla f\|_{L^1(\mathbb R^2)}$ for every $f\in W^{1,1}(\mathbb R^2)$ , with $Mf$ weakly differentiable?

Result: Proved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Argument
Field: Harmonic analysis: regularity of maximal functions
Posed by: P. Hajlasz and J. Onninen, On boundedness of maximal functions in Sobolev spaces, Ann. Acad. Sci. Fenn. Math. 29 (2004), Question 1
Year posed: 2004
Years open: 22y
Solved: 2026-09-26
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 24 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: there is an absolute $C<\infty$ such that for every real $f\in W^{1,1}(\mathbb R^2)$ the centered disk maximal function $Mf$ is locally integrable, lies in $W^{1,1}_{loc}(\mathbb R^2)$ and satisfies $\int|\nabla Mf|\le C\int|\nabla f|$ . The key step is a signed finite-band estimate for $\sup_{a\le t\le b}A_tg$ uniform in the number of dyadic scales, combined with Lahti-Weigt's conditional regularity theorem. Not shown: dimensions $n\ge3$ , balls in other norms, the BV (variation) version, the optimal constant, or continuity of $f\mapsto\nabla Mf$ .

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The release also supplies a Lean formalization of the main theorem, produced as part of the same release.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the TeX source against Hajlasz-Onninen Question 1, and the Lean statement lean/ComparatorChallenges/DiskMaximal.lean (theorem OAI.CenteredDiskEndpoint.main_endpoint), whose solution module OAI/Analysis/DiskMaximal/Main.lean exists at the pinned commit. This challenge is not in the formalization catalogue (formalization.yaml lists only the signed finite-band lemma, SignedFiniteBand); its statement was read here and was not rebuilt. It states the headline: an absolute $C$ such that for every integrable $f$ on the Euclidean plane with integrable weak gradient, $Mf$ is finite a.e., locally integrable, in $W^{1,1}_{loc}$ , with an integrable weak gradient of $L^1$ norm at most $C\|\nabla f\|_1$ . The paper itself limits the result to disks in the plane; higher dimensions and a BV version are not covered.

The Hajlasz-Onninen endpoint question for the centered maximal function, planar disk case: is $\|\nabla Mf\|_1\le C\|\nabla f\|_1$ on $W^{1,1}(\mathbb R^2)$ ?

What was actually shown

What the AI did

Verification

Sources

Discussion

The Hajlasz-Onninen endpoint question for the centered maximal function, planar disk case: is ∥∇Mf∥1≤C∥∇f∥1\|\nabla Mf\|_1\le C\|\nabla f\|_1∥∇Mf∥1​≤C∥∇f∥1​ on W1,1(R2)W^{1,1}(\mathbb R^2)W1,1(R2)?

What was actually shown

What the AI did

Verification

Sources

Discussion

The Hajlasz-Onninen endpoint question for the centered maximal function, planar disk case: is $\|\nabla Mf\|_1\le C\|\nabla f\|_1$ on $W^{1,1}(\mathbb R^2)$ ?