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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On the Hardy-Littlewood maximal function and some applications


Author: C. J. Neugebauer
Journal: Trans. Amer. Math. Soc. 259 (1980), 99-105
MSC: Primary 42B25; Secondary 28A15
MathSciNet review: 561825
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Abstract: With a monotone family $ F\, = \,\{ {S_\alpha }\} ,\,{S_\alpha }\, \subset \,{{\textbf{R}}^n}$, we associate the Hardy-Littlewood maximal function $ {M_F}f(x)\, = \,{\sup _\alpha }(1/\left\vert {{S_\alpha }} \right\vert)\int_{{S_\alpha }\, + \,x} {\left\vert f \right\vert} $. In general, $ {M_F}$ is not weak type (1.1). However, if we replace in the denominator $ {S_\alpha }$ by $ S_F^ {\ast} \, = \,\{ x\, - \,y:\,x,\,y\, \in \,{S_\alpha }\} $, and denote the resulting maximal function by $ M_F^ {\ast} $, then $ M_F^ {\ast} $ is weak type (1, 1) with weak type constant 1.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0561825-7
PII: S 0002-9947(1980)0561825-7
Keywords: Hardy-Littlewood maximal function, monotone family, weak type
Article copyright: © Copyright 1980 American Mathematical Society