Boundedness and unboundedness results for some maximal operators on functions of bounded variation

Abstract

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 $M_R$.