Endpoint Sobolev regularity of centered disk averages
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real f∈ W^{1,1}(\mathbb R^2), the centered disk maximal function satisfies \|\nabla Mf\|_1≤ 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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MANUSCRIPT 01
An Endpoint Gradient Bound for the Centered Disk Maximal Operator
We prove the endpoint gradient bound \|\nabla Mf\|_1≤ C\|\nabla f\|_1 for every real-valued f∈ 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}_{loc}(\mathbb R^2) and has a globally integrable weak gradient.…
Full source abstract
We prove the endpoint gradient bound \|\nabla Mf\|_1≤ C\|\nabla f\|_1 for every real-valued f∈ 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}_{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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