Weighted inequalities for iterated maximal functions in Orlicz spaces

✍ Scribed by Hiro-o Kita

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 156 KB
• Volume: 278
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310301

✦ Synopsis

Abstract

Let M be the classical Hardy‐Littlewood maximal operator. The object of our investigation in this paper is the iterated maximal function M^k^f(x) = M(M^k−1^f) (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 C~1~ and C~2~ such that

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then there exist positive constants C~3~ and C~4~ such that

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where the functions a(t) and b(t) are the right derivatives of Φ(t) and Ψ(t), respectively. Conversely, if w is an A~1~ weight, then (II) implies (I). Another necessary and sufficient condition will be given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)