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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.

Manuscripts in this family · 1

An Endpoint Gradient Bound for the Centered Disk Maximal Operator

Sep 26, 2026·article.pdfLean

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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