Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 085-01. An Endpoint Gradient Bound for the Centered Disk Maximal Operator
Manuscript map · lines 4125–4134
OpenAI@2026-10-07#rem:oai-p-085-01
Authors: OpenAI. Family 085.
Title: An Endpoint Gradient Bound for the Centered Disk Maximal Operator.
Abstract: 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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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