Analysis · Oct 6, 2026

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.

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