Is the local maximal function bounded from $W^{1, 1}$ to $L^1$?

Question

Let $f \in W^{1, 1} (\mathbb R^d)$. For every $\varepsilon > 0$, we consider the local maximal function $M_\varepsilon f: \mathbb R^d \to \mathbb R$, defined by

$$M f_{\varepsilon} (x) = \sup_{r \leq \varepsilon} \frac{1}{|B_r (x)|} \int_{B_r (x)} |f(y) - f(x)| \, dy.$$

Question: Is it true that there exists some $C > 0$ such that for every $K > 0$, there exists some $\delta> 0$ such that for all $f \in W^{1, 1}$ with $\|f\|_{W^{1, 1}} \leq K$ and $\varepsilon < \delta$, we have that $M_\varepsilon f \in L^1$, and

$$\|M_\varepsilon f\|_{L^1} \leq C\|f\|_{W^{1,1}}?$$

Remark: It is known (see, eg here) that the usual global maximal function is bounded from $W^{1, p}$ to itself for all $p > 1$, but not for $p = 1$. I was unable to find results on whether it is even bounded from $W^{1, 1}$ to $L^1$.

Answer 1

Yes I think this is true.

Let $\mathcal{G} := \{ \times_{i=1}^n[k_i,k_i+1] : k_i\in \mathbb{Z} \} $ be the grid of cubes of side length $1$ and vertices in $ \mathbb{Z}^n$. For $Q\in \mathcal{G} $ let $2Q$ be the cube with same center but double the side length. Notice that for $\varepsilon < \frac 12$ if $f$ is supported in $2Q$ then $M_\varepsilon f$ is supported in $3Q$. Let also $\phi \in C^\infty(\mathbb{R}^n), 0\leq \phi \leq 1$ supported in $ [-1,1]^n $ and $ \phi(x) = 1, \forall x \in [-\frac 12, \frac 12]^n $ and $ \phi_Q(x) = \phi(x-c_Q)$, where $c_Q$ is the center of the cube $Q$. Then for $f\in W^{1,1}(\mathbb{R^n})$ we have \begin{align*} \Vert M_\varepsilon f \Vert_{L^1(\mathbb{R}^n)} & \leq \sum_{Q\in \mathcal{G}} \Vert M_\varepsilon (f \phi_Q ) \Vert_{L^1(\mathbb{R}^n)} \\ & = \sum_{Q\in \mathcal{G}} \Vert M_\varepsilon (f \phi_Q ) \Vert_{L^1(3Q)} \\&\leq C_n \sum_{Q\in \mathcal{G}} \Vert M_\varepsilon (f \phi_Q ) \Vert_{L^{\frac{n}{n-1}}(3Q)} \\ & \leq C_n \sum_{Q\in \mathcal{G}} \Vert f \phi_Q \Vert_{L^{\frac{n}{n-1}}(\mathbb{R}^n)} \\ & \leq C_n \sum_{Q\in \mathcal{G}} \Vert f \phi_Q \Vert_{W^{1,1}(\mathbb{R}^n)} \leq C_n \Vert f \Vert_{W^{1,1}(\mathbb{R}^n).} \end{align*} The constant $C_n$ is a dimensional constant which is different in each occurance. Passing from the third to the fourth line you use the boundedness of the maximal function and the second to last inequality is Sobolev's embedding theorem.