Author:
H. Kita Kagoshima University Department of Mathematics, Faculty of Education 20-6 Korimoto 1-Chome Kagoshima 890-0065 Japan 20-6 Korimoto 1-Chome Kagoshima 890-0065 Japan

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## Abstract

Let Φ(t)= ∫_0^t a(s) ds and Ψ(t)= ∫_0^t b(s) ds, where a(s) is a positive continuous function such that ∫_0^1 \frac{a(s)}{s} ds < ∞and ∫_1^{\∞}\frac{a(s)}{s} ds= +\∞, and b(s) is an increasing function such that \lim_{s\to\∞}b(s)= +\∞. Letw be a weight function and suppose that w∈A1\∩ A'. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent:(I) there exist positive constants C1 and C2 such that

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int_0^s {\frac{{a\left( t \right)}}{t}dt \geqq } C_1 b\left( {C_2 s} \right)foralls > 0;$$ \end{document}
(II) there exist positive constants C3 and C4 such that
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int {_{R^n } } \Psi \left( {C_3 \left| {f\left( x \right)} \right|} \right)w\left( x \right)dx \leqq C_4 \int {_{R^n } } \Phi \left( {Mf\left( x \right)} \right)w\left( x \right)dxforallf \in {\mathcal{R}}_0 \left( w \right)$$ \end{document}

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation
1950
Volumes
per Year
3
Issues
per Year
6
Founder's
H-1051 Budapest, Hungary, Széchenyi István tér 9.
Springer Nature Switzerland AG
Publisher's
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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