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On certain maximal functions and approach regions. (English) Zbl 0546.42017
To each set $$\Omega\subset {\mathbb{R}}_+^{n+1}$$ a version of the Hardy- Littlewood maximal function is associated as follows: $M_{\Omega}f(x_ 0)=\sup_{(x,y)\in\Omega }(1/| B(0,y)|)\int_{B(0,y)}| f(x_ 0+x+t)| dt$ where $$B(0,y)=\{t\in {\mathbb{R}}^ n; | t| <y\}$$. A necessary and sufficient condition in order that the operator $$M_{\Omega}$$ is weak type (1,1) and strong type (p,p) for $$1<p\leq\infty$$ is established. Some generalization is given and is applied to the study of certain tangential maximal functions of Poisson integrals of potentials.
Reviewer: H.Tanabe

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 42B30 $$H^p$$-spaces
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