The Hajlasz-Onninen endpoint question for the centered maximal function, planar disk case: is $\|\nabla Mf\|_1\le C\|\nabla f\|_1$ on $W^{1,1}(\mathbb R^2)$?

Theorem 1.1: there is an absolute $C<\infty$ such that for every real $f\in W^{1,1}(\mathbb R^2)$ the centered disk maximal function $Mf$ is locally integrable, lies in $W^{1,1}_{loc}(\mathbb R^2)$ and satisfies $\int|\nabla Mf|\le C\int|\nabla f|$. The key step is a signed finite-band estimate for $\sup_{a\le t\le b}A_tg$ uniform in the number of dyadic scales, combined with Lahti-Weigt's conditional regularity theorem. Not shown: dimensions $n\ge3$, balls in other norms, the BV (variation) version, the optimal constant, or continuity of $f\mapsto\nabla Mf$.

Theorem 1.1: there is an absolute $C<\infty$ such that for every real $f\in W^{1,1}(\mathbb R^2)$ the centered disk maximal function $Mf$ is locally integrable, lies in $W^{1,1}_{loc}(\mathbb R^2)$ and satisfies $\int|\nabla Mf|\le C\int|\nabla f|$. The key step is a signed finite-band estimate for $\sup_{a\le t\le b}A_tg$ uniform in the number of dyadic scales, combined with Lahti-Weigt's conditional regularity theorem. Not shown: dimensions $n\ge3$, balls in other norms, the BV (variation) version, the optimal constant, or continuity of $f\mapsto\nabla Mf$.