×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] \scR. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, to appear. · Zbl 0569.42016
[2] Fatou, P, Séries trigonométriques et séries de Taylor, Acta math., 30, 335-400, (1906) · JFM 37.0283.01
[3] Fefferman, C; Stein, E.M, Hp spaces of several variables, Acta math., 129, 137-193, (1972) · Zbl 0257.46078
[4] Folland, G.B; Stein, E.M, Hardy spaces on homogeneous groups, () · Zbl 0508.42025
[5] Littlewood, J.E, On a theorem of Fatou, J. London math. soc., 2, 172-176, (1927) · JFM 53.0283.07
[6] Nagel, A; Rudin, W; Shapiro, J, Tangential boundary behavior of functions in Dirichlet type spaces, Ann. math., 116, 331-360, (1982) · Zbl 0531.31007
[7] Rudin, W, Inner function images of radii, (), 357-360 · Zbl 0406.30022
[8] Stein, E.M, Singular integrals and differentiability properties of functions, (1970), Princeton Univ. Press Princeton, N. J · Zbl 0207.13501
[9] Zygmund, A, On a theorem of Littlewood, Summa brasil. math., 2, 1-7, (1949)
[10] Watson, G.N, Theory of Bessel functions, (1944), Cambridge Univ. Press London/New York · Zbl 0063.08184
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.