An Endpoint Gradient Bound for the Centered Disk Maximal Operator
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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