An Endpoint Gradient Bound for the Centered Disk Maximal Operator

We prove the endpoint gradient bound $\Vert \nabla Mf{\Vert }_1\le C\Vert \nabla f{\Vert }_1$ 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 $W_{\mathrm{loc}}^{1,1}({\mathbb{R}}^2)$ 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.