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 subset of 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,W-1(I). By restriction, the corresponding characterization holds for W-1,W-1(I). We also show that if U is open in R-d, d > 1, then boundedness from BV(U) into W-1,W-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) o...o M-T(1). Nevertheless, if U satisfies a cone condition, then M-T(S):BV(U) -> L-1(U) boundedly, and the same happens with M-T(v), M-T(d) o...o M-T(1), and M-R. (C) 2007 Elsevier Inc. All rights reserved.