an:03869945
Zbl 0546.42017
Nagel, Alexander; Stein, Elias M.
On certain maximal functions and approach regions
EN
Adv. Math. 54, 83-106 (1984).
00149730
1984
j
42B25 42B30
maximal function; weak type (1,1); strong type (p,p); Poisson integrals
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.
H.Tanabe