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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

42B25 Maximal functions, Littlewood-Paley theory
42B30 \(H^p\)-spaces
Full Text: DOI
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