ON HARDY-LITTLEWOOD MAXIMAL FUNCTIONS IN ORLICZ SPACES

Let phi(t) and psi(t) be the functions having the following representa tions phi(t)=integral(0)(t) a(s) ds and psi(t)=integral(0)(t) b(s) ds, where a(s) is a positive continuous function such that integral(1)(in finity) a(s)/s ds=+infinity and b(s) is an increasing function such th at lim(s-->infinity) b(s)=+infinity. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (i) there exist positive constants c(1) and s(0) such that integral(0)(s)a(t)/t greater than or equal to c(1)b(c(1)s) for all s greater than or equal to s(0)>1; (ii) there exist positive constant c(2) and c(3) such that integral(IRn)psi(c(2)/f(x)/)dx less than or equal to c(3)//f//L(1)+c( 3) integral(IRn)phi(Mf(x))dx for all f is an element of L(1)(IR(n)).