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

摘要

We characterize the space BV(I) of functions of bounded variation on an arbitrary interval I C 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 MTv, 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.