{"id":"openai-math/An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator","pdf":"https://github.com/openai/math/blob/main/preprints/An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026/article.pdf","url":"https://github.com/openai/math/tree/main/preprints/An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026","tags":["harmonic-analysis","sobolev-spaces"],"score":0,"title":"An Endpoint Gradient Bound for the Centered Disk Maximal Operator","domain":"Real and complex analysis","family":"085","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CA","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-26","reactions":{},"formalized":true,"family_title":"Endpoint Sobolev regularity of centered disk averages"}