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On maximal functions in Orlicz spaces

Author: Hiro-o Kita
Journal: Proc. Amer. Math. Soc. 124 (1996), 3019-3025
MSC (1991): Primary 42B25, 46E30
MathSciNet review: 1376993
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Abstract: Let $\Phi (t)$ and $\Psi (t)$ be the functions having the representations $\Phi (t)=\int _{0}^{t} a(s)ds$ and $\Psi (t)=\int _{0}^{t} b(s)ds$, where $a(s)$ is a positive continuous function such that $\int _{1}^{\infty }\frac {a(s)}{s}ds=+\infty $ and $b(s)$ is quasi-increasing. Then the maximal function $Mf$ is a function in Orlicz space $L^{\Phi }$ for all $f\in L^{\Psi }$ if and only if there exists a positive constant $c_{1}$ such that $\int _{1}^{s} \frac {a(t)}{t}dt\leq c_{1}b(c_{1}s)$ for all $s\geq 1$.

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Additional Information

Hiro-o Kita
Affiliation: Department of Mathematics, Faculty of Education, Oita University, 700 Dannoharu Oita 870-11, Japan

Keywords: Hardy-Littlewood maximal function, Orlicz space
Received by editor(s): December 6, 1993
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society

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