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 $M_R$.

15 pages

Cited by in corpus (7)

• Regularity of the Hardy-Littlewood maximal operator on block decreasing functions
• On high dimensional maximal operators
• Regularity and continuity of the multilinear strong maximal operators
• Optimal bounds on the modulus of continuity of the uncentered Hardy-Littlewood maximal function
• The best constant for the centered maximal operator on radial decreasing functions
• Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities
• Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures