Endpoint gradients of the centered disk maximal function

Source and verification

This article explains family 085 of the October 6, 2026 OpenAI collection. The research conclusions are attributed to its manuscripts. The source review covers statements, introductions, and the proof mechanisms described below; it is not an independent verification of the complete proofs.

Overview

The Hardy–Littlewood maximal operator is not bounded on $L^1$, but its gradient may behave better than the function itself. Family 085 claims a global $L^1$ gradient estimate for the planar centered-disk maximal function on $W^{1,1}$ inputs.

🏷️ Preliminaries: maximal averages and weak derivatives

For locally integrable $f$ on ${\mathbb{R}}^2$, define

$$Mf(x)=\underset{r>0}{\sup }\frac{1}{\pi r^2}{\int }_{B(x,r)}|f(y)|dy.$$

Every disk is centered at the evaluation point. Changing to uncentered disks or to cubes changes the geometry of the operator.

The space $W^{1,1}({\mathbb{R}}^2)$ consists of integrable functions whose distributional first derivatives are integrable functions. By contrast, a function of bounded variation may have a derivative that is a finite measure with a singular part. Controlling total variation does not by itself prove Sobolev regularity.

For $p>1$, a pointwise gradient comparison with $M(|\nabla f|)$ leads to an $L^p$ bound. At $p=1$, the maximal operator has only weak-type control, so that last step cannot produce a strong $L^1$ gradient estimate.

🏷️ The claimed endpoint theorem

Family 085, as stated in the manuscript

For every real $f\in W^{1,1}({\mathbb{R}}^2)$, the centered disk maximal function belongs to $W_{\mathrm{loc}}^{1,1}$ and satisfies

$${\int }_{{\mathbb{R}}^2}|\nabla Mf|dx\le C{\int }_{{\mathbb{R}}^2}|\nabla f|dx$$

with an absolute constant. The input need not be radial or compactly supported. (OpenAI, 2026)

The conclusion does not say $Mf\in W^{1,1}({\mathbb{R}}^2)$ globally, since $Mf$ itself need not be integrable.

For example, take a nonzero nonnegative smooth function supported in a fixed disk, with mass $m>0$. At a distant point $x$, a centered disk of radius comparable to $|x|$ captures that mass, so $Mf(x)\gtrsim m|x{|}^{-2}$. Its integral in two dimensions diverges like ${\int }^{i}nftydr/r$. There is therefore no conflict between a global gradient bound and failure of global integrability of the maximal function.

🏷️ The signed finite-band estimate

For a smooth real $g$, let

$$\begin{array}{c}{A}_{t}g(x)=\frac{1}{\pi t^2}{\int }_{B(x,t)}g(y)dy,\\ S_{a,b}g=\underset{a\le t\le b}{\sup }{A}_{t}g.\end{array}$$

The source claims a gradient bound for $S_{a,b}g$ independent of the number of dyadic radii between $a$ and $b$. The averages are signed: they contain $g$, not $|g|$.

This signed formulation permits local subtraction of a constant. On a square $Q$, use $g_Q={\chi }_Q(g-c_Q)$, where $c_Q$ is a local average and ${\chi }_Q$ is a cutoff equal to one on the disks under consideration. There, $A_t{g}_Q=A_{t}g-c_Q$, so the envelopes have the same gradient. Poincaré’s inequality controls the cutoff error by the local gradient of $g$. Requiring $g_Q$ to be nonnegative would obstruct this localization.

🏷️ Geometry of optimizing disks

At an interior maximizing radius, the derivative of $A_{t}g$ with respect to $t$ vanishes. A rotational integration identity supplies another vanishing moment. The source combines them into a first-moment cancellation for suitable mixtures of nearby disks.

The practical effect is that a balanced kernel can be modified by a factor vanishing at a selected boundary point. Mass concentrated near that point then makes a small contribution. A geometric sweep by tangent disks controls the exceptional centers where too much shell mass concentrates. Boundary radii and changing maximizing radii need separate estimates; the interior critical-point identity does not cover them automatically.

🏷️ Paying once for cancellation across scales

The input gradient is a vector measure. The manuscript decomposes it using positive grid partitions and a finite collection of direction cones. Within each scalar coordinate, opposite signs are paired in equal masses, while the constant-sign remainder is passed to a larger scale. Each cancellation retains its support and the scale at which it first occurs.

An error born at scale $s$ and observed at scale $r\ge s$ is charged with a decay factor $(s/r{)}^{\alpha }$, for a fixed $\alpha >0$. Summing this geometric decay and the paired masses gives cell costs $D_Q$ satisfying

$$\sum _Q{D}_Q\le C\int |\nabla g|.$$

The same costs must cover changes of direction as well as scalar cancellations. This is what prevents a loss proportional to the number of radius scales. The final localization argument removes a possible singular derivative measure and passes from smooth finite-band envelopes to $Mf$.

🏷️ Key steps and proof outlines

The optimizing-radius condition creates a balanced disk mixture

For fixed $x$, differentiate the signed disk average $A_{t}g(x)$ with respect to radius. At an interior local maximum, the inward and outward difference quotients have opposite signs. The source averages their corresponding moment identities over small intervals below and above the maximizing radius, using probability densities proportional to $t^{-3}$. This weight matches the radius derivative of the disk average and is needed to preserve the signs. Let the resulting real moments be $M_{-}\ge 0$ and $M_{+}\le 0$.

Choose weights

$$a=\frac{-M_{+}}{M_{-}-M_{+}},1-a=\frac{M_{-}}{M_{-}-M_{+}}$$

when the denominator is positive. The convex mixture has zero real moment. If both moments vanish, either mixture already works. A rotational identity supplies the other moment cancellation. Thus nearby radii produce a balanced kernel even when differentiating the supremum directly is awkward. See “Disk moments and an affine cancellation.” (OpenAI, 2026)

The resulting zero first moment permits subtraction of an affine factor chosen to vanish at a boundary point. This reduces the contribution of mass concentrated near that point. A geometric sweep by tangent disks then estimates the exceptional centers where this local cancellation is ineffective. Endpoints of the allowed radius band are treated separately, because there is no two-sided maximizing condition there.

Exact scalar pairing records cancellation once

For two positive measures $U,V$ with total masses $P,N$, set $d=\min (P,N)$. If $P,N>0$, remove

$${\mu }_{+}=(d/P)U,{\mu }_{-}=(d/N)V.$$

Both have mass $d$, and the remainder $(U-{\mu }_{+})-(V-{\mu }_{-})$ has a constant sign. If one mass is zero, the corresponding removed measure is zero. This simple identity lets the proof pair opposite signs without changing their supports or inventing cancellation at a larger scale.

Apply the pairing from the finest dyadic cells upward. A paired portion is recorded at the scale where it first cancels; only the constant-sign remainder passes to its parent. Therefore the same mass is not independently charged as new cancellation at every ancestor. For vector gradients, a finite collection of direction cones reduces the accounting to scalar coordinates while retaining costs for direction changes.

Geometric decay makes the multiscale price finite

A cancellation piece born at scale $s$ contributes less when observed through a much larger disk of radius $r$. The local estimates give a factor $(s/r{)}^{\alpha }$ for some fixed $\alpha >0$. At dyadic radii $r=2^{k}s$, its total charge is bounded by

$$\sum _{k\ge 0}{2}^{-k\alpha }=\frac{1}{1-2^{-\alpha }}.$$

Multiplying by the paired mass and summing all pieces produces the common cell prices $D_Q$. The same prices must also dominate errors from changing direction labels; assigning a separate unrestricted budget to those changes would reintroduce a scale loss. The finite-state label construction arranges that directions change slowly enough in both position and scale for this common accounting to work.

Runs assemble the signed finite-band envelope

For a fixed label history, a run is a maximal consecutive group of scales carrying one label. The envelope over the whole radius band is the maximum of the run envelopes, whose adjacent radius intervals meet at endpoints. At almost every point where one run is active, the gradient of the maximum agrees with a gradient of an active run. The local estimates and common prices control the total contribution of those gradients, including changes of the active run.

This yields a bound independent of the number of dyadic radii. The signed formulation is then used in localization: subtract a local average $c_Q$, multiply by a cutoff, and apply Poincaré to control the cutoff derivative. The averages change by the constant $c_Q$ on the relevant disks, so their envelope gradients are unchanged.

The limit must exclude singular derivatives

For a fixed lower radius $a>0$, translation of the input gives

$$|S_{a,b}g(x+z)-S_{a,b}g(x)|\le \frac{|z|}{\pi a^2}\Vert \nabla g{\Vert }_1.$$

Thus finite-band envelopes are Lipschitz. Uniform total variation bounds permit passage to a limit as the band expands, but would initially yield only a bounded-variation function. The source uses the localized estimates to establish absolute continuity of the limiting derivative, eliminating a possible singular measure. Smooth approximation and cutoff approximation then extend the argument to all $W^{1,1}$ inputs. Finally the Sobolev chain rule applies to $|f|$, returning from signed averages to the usual maximal operator. The separate absolute-continuity step is what supports the stated $W_{\mathrm{loc}}^{1,1}$ conclusion.

See Also

• Family 081: Riesz transforms also requires a multiscale packing argument beyond a local geometric improvement.
• Family 083: Lipschitz directions illustrates another setting where scale-independent estimates depend on carefully summing directional changes.

🏷️ Manuscripts and verification scope

• An Endpoint Gradient Bound for the Centered Disk Maximal Operator — the pinned manuscript; its directory supplies the source and citation metadata.

The repository provides a formalization-scope page. Its scope description lists the signed finite-band estimate and the general Sobolev-input gradient bound; a BV-input extension is outside those statements. These are statements about the repository’s documentation: no Lean build, axiom audit, or comparator-equivalence check was performed for this article.

OpenAI collection gives the reading sequence and source policy; coverage register records which families have articles and which remain to be written.

🐻  OpenAI 2026. An Endpoint Gradient Bound for the Centered Disk Maximal Operator.