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 W1,1(I ). By restriction, the corresponding characterization holds for W1,1(I ). We also show that if U is open in Rd , d > 1, then boundedness from BV(U) into W1,1(U) fails for the local directional maximal operator Mv T , the local strong maximal operator MS T , and the iterated local directional maximal operator Md T ◦ ··· ◦M1 T . Nevertheless, if U satisfies a cone condition, then MS T :BV(U)→L1(U) boundedly, and the same happens with Mv T , Md T ◦ ··· ◦M1 T , and MR. © 2007 Elsevier Inc. All rights reserved.