On the equivalence of boundedness for multiple Hardy-Littlewood averages and related operators
Keyword(s):
Necessary and sufficient conditions for the weight function $u$ are obtained, which provide the boundedness for a class of averaging operators from $L_p^+$ to $L_{q,u}^+$. These operators include the multiple Hardy-Littlewood averages and related maximal operators, geometric mean operators, and geometric maximal operators. We show that, under suitable conditions, the boundedness of these operators are equivalent. Our theorems extend several one-dimensional results to multi-dimensional cases and to operators with multiple kernels. We also show that in the case $p<q$, some one-dimensional results do not carry over to the multi-dimensional cases, and the boundedness of $T$ from $L_p^+$ to $L_{q,u}^+$ holds only if $u=0$ almost everywhere.
1990 ◽
Vol 10
(1)
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pp. 43-62
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2002 ◽
Vol 12
(04)
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pp. 709-737
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1988 ◽
Vol 93
(2)
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pp. 279-293
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2002 ◽
Vol 131
(1)
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pp. 219-229
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1995 ◽
Vol 129
(3)
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pp. 225-244
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2009 ◽
Vol 29
(2)
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pp. 715-731
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1989 ◽
Vol 105
(1)
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pp. 177-184
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