Boundedness and Unboundedness Results for Some Maximal Operators on Functions of Bounded Variation

We characterize the space BV (I) of functions of bounded variation on an arbitrary interval I ⊂ R, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator MR from BV (I) into the Sobolev space W (I). By restriction, the corresponding characterization holds for W (I). We also show that if U is open in R, d > 1, then boundedness from BV (U) into W (U) fails for the local directional maximal operator M T , the local strong maximal operator M S T , and the iterated local directional maximal operator M T ◦ · · · ◦ M 1 T . Nevertheless, if U satisfies a cone condition, then M T : BV (U) → L (U) boundedly, and the same happens with M T , M d T ◦ · · · ◦M 1 T , and MR.