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Acta Mathematica Hungarica
Volume/Issue:
Volume 72: Issue 4
DOI:
https://doi.org/10.1007/bf00114542
Author:
H. Kita
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Acta Mathematica Hungarica
Volume/Issue:
Volume 77: Issue 1-2
DOI:
https://doi.org/10.1023/a:1006581920044
Author:
H. Kita
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Acta Mathematica Hungarica
Volume/Issue:
Volume 57: Issue 3-4
DOI:
https://doi.org/10.1007/bf01903673
Author:
H. Kita
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Acta Mathematica Hungarica
Volume/Issue:
Volume 81: Issue 3
DOI:
https://doi.org/10.1023/a:1006577116061
Author:
H. Kita
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Acta Mathematica Hungarica
Volume/Issue:
Volume 67: Issue 1-2
DOI:
https://doi.org/10.1007/bf01874524
Author:
H. Kita
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Acta Mathematica Hungarica
Volume/Issue:
Volume 46: Issue 1-2
DOI:
https://doi.org/10.1007/bf01961008
Author:
H. Kita
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Acta Mathematica Hungarica
Volume/Issue:
Volume 98: Issue 1-2
DOI:
https://doi.org/10.1023/a:1022809522201
Author:
H. Kita

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 C 1 and C 2 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 C 3 and C 4 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
Volume/Issue:
Volume 56: Issue 3-4
DOI:
https://doi.org/10.1007/bf01903837
Authors:
H. Kita
and
K. Yoneda
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Acta Mathematica Hungarica
Volume/Issue:
Volume 65: Issue 4
DOI:
https://doi.org/10.1007/bf01876036
Authors:
H. Kita
and
K. Yoneda
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