A NOTE ON THE ENDPOINT REGULARITY OF THE DISCRETE MAXIMAL OPERATOR

In this note we study the regularity properties of the discrete maximal operators at endpoint. Precisely, we show that the general discrete centered and non-centered maximal operators are bounded and continuous from l(1)(Z) to BV(Z), as well as the non-centered discrete maximal operator maps BV(Z) -> BV(Z) boundedly under a more restrictive condition, where BV(Z) denotes the set of functions of bounded variation defined on Z. As an immediate consequence, we obtain somewhat unexpected endpoint regularities of the discrete fractional maximal functions.