Endpoint Sobolev regularity of centered disk averages

1 · The essentials

Imagine measuring the average size of a function over a disk, then trying every disk size while keeping its center fixed. The largest average is the value of a new function at that center. The manuscript studies how much this new function can change across the plane. It starts with real-valued functions whose total size and total amount of change are both finite, in a precise mathematical sense. Its main result bounds the new function’s total amount of change by a fixed constant times that of the starting function. The new function also has finite total size on every bounded region, though the stated conclusion does not require finite total size over the whole plane. This result has a Lean formalization and resolves the planar centered-disk case of a question attributed to Hajłasz and Onninen.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a map of brightness: at each point, average the brightness inside circles of every possible size and retain the largest average. The result says that this largest-average map has a controlled total amount of change if the original map meets the stated finiteness conditions; the image does not capture the precise mathematical meaning of change.

Why it matters

Taking the largest average over all disk sizes raises the question of whether the resulting function retains control over its changes. The manuscript supplies that control in the plane under the stated assumptions. It gives a positive resolution of the centered-disk case of the endpoint regularity question of Hajłasz and Onninen, rather than a resolution for every possible setting.

2 · Going further

For a real-valued function $f \in W^{1, 1} (R^{2})$ , define the centered disk maximal function by $M f (x) = sup_{r > 0} \frac{1}{∣ B ( x , r ) ∣} \int_{B (x, r)} ∣ f (y) ∣ d y$ , where $B (x, r)$ is the disk centered at $x$ with radius $r$ , and $∣ B (x, r) ∣$ is its area. The problem is to control the weak gradient of this supremum using the weak gradient of $f$ . The main result is $∥\nabla M f ∥_{1} \leq C ∥\nabla f ∥_{1}$ , with an absolute constant $C$ . It also states that $M f \in W_{loc}^{1, 1} (R^{2})$ and that its weak gradient is globally integrable.

The source identifies this as a positive resolution of the planar centered-disk case of the endpoint question of Hajłasz and Onninen. Here the endpoint setting is the stated $W^{1, 1}$ assumption and the corresponding $L^{1}$ gradient estimate. The supplied abstract gives the conclusion but does not describe the argument, so no proof mechanism can be reported from it. The main result has a Lean formalization; the source provides no further breakdown of its coverage. Only the main manuscript is supplied, with no companion manuscripts described.

Useful notions

Averages over disks
Suprema, or least upper bounds
Integrable functions and the L¹ norm
Weak derivatives
Local versus global integrability

What was known before / what is new

Before

The source identifies an endpoint maximal-function regularity question posed by Hajłasz and Onninen. It does not supply earlier estimates, partial resolutions, or a historical account of that question.

New

The manuscript establishes the absolute-constant bound $∥\nabla M f ∥_{1} \leq C ∥\nabla f ∥_{1}$ for every real-valued $f \in W^{1, 1} (R^{2})$ . It also states local Sobolev membership of $M f$ and global integrability of its weak gradient. The main result has a Lean formalization.

Limits and assumptions

The hypotheses concern real-valued functions in $W^{1, 1} (R^{2})$ and averages of their absolute values over centered disks. The conclusion gives local $W^{1, 1}$ membership and a globally integrable weak gradient, not an assertion that $M f$ itself is globally integrable. The main result is Lean-formalized, but the source does not detail the formalization’s coverage of individual assertions. No conclusions for other dimensions or averaging shapes are supplied, and the source does not specify which other cases remain open.

Mini glossary

Centered disk maximal function: The function assigning to each point the supremum of the averages of the original function’s absolute value over disks centered there.
Weak gradient: A collection of derivatives understood through integration rather than requiring ordinary derivatives at every point.
W^{1,1}: The space of integrable functions whose first weak derivatives are also integrable.
Local Sobolev membership: Membership in a Sobolev space on bounded regions rather than necessarily over the entire plane.
L¹ norm: The integral of the absolute value of a function, or of the magnitude of a vector-valued function such as a gradient.
Absolute constant: A fixed constant in the estimate that does not depend on the chosen function.

Keywords : Centered disk maximal function · Endpoint Sobolev regularity · Weak gradient bound

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

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lean/docs/085.md

Manuscripts in this family

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An Endpoint Gradient Bound for the Centered Disk Maximal Operator

26/09/2026

Verified by Lean

We prove the endpoint gradient bound $∥\nabla M f ∥_{1} \leq C ∥\nabla f ∥_{1}$ for every real-valued $f \in W^{1, 1} (R^{2})$ , where $M f (x)$ is the supremum of the averages of $∣ f ∣$ over disks centered at x, and C is an absolute constant. The maximal function belongs to $W_{loc}^{1, 1} (R^{2})$ 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 GitHub folder Lean formalization