  We characterize the space $BV(I)$ of functions of bounded variation on an
arbitrary interval $I\subset \mathbb{R}$, in terms of a uniform boundedness
condition satisfied by the local uncentered maximal operator $M_R$ from $BV(I)$
into the Sobolev space $W^{1,1}(I)$.
  By restriction, the corresponding characterization holds for $W^{1,1}(I)$. We
also show that if $U$ is open in $\mathbb{R}^d, d >1$, then boundedness from
$BV(U)$ into $W^{1,1}(U)$ fails for the local directional maximal operator
$M_T^{v}$, the local strong maximal operator $M_T^S$, and the iterated local
directional maximal operator $M_T^{d}\circ ...\circ M_T^{1}$. Nevertheless, if
$U$ satisfies a cone condition, then $M_T^S:BV(U)\to L^1(U)$ boundedly, and the
same happens with $M_T^{v}$, $M_T^{d} \circ ...\circ M_T^{1}$, and $M_R$.
