Weighted inequalities for iterated maximal functions in Orlicz spaces

Abstract

AbstractLet M be the classical Hardy‐Littlewood maximal operator. The object of our investigation in this paper is the iterated maximal function Mkf(x) = M(Mk−1f) (x) (k ≥ 2). Let Φ be a φ‐function which is not necessarily convex and Ψ be a Young function. Suppose that w is an A′∞ weight and that k is a positive integer. If there exist positive constants C1 and C2 such that equation image then there exist positive constants C3 and C4 such that equation image where the functions a(t) and b(t) are the right derivatives of Φ(t) and Ψ(t), respectively. Conversely, if w is an A1 weight, then (II) implies (I). Another necessary and sufficient condition will be given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)