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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Weighted inequalities for the one-sided Hardy-Littlewood maximal functions


Author: E. Sawyer
Journal: Trans. Amer. Math. Soc. 297 (1986), 53-61
MSC: Primary 42B25
MathSciNet review: 849466
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Abstract: Let $ {M^ + }f(x) = {\sup _{h > 0}}(1/h)\int_x^{x + h} {\vert f(t)\vert\,dt} $ denote the one-sided maximal function of Hardy and Littlewood. For $ w(x) \geqslant 0$ on $ R$ and $ 1 < p < \infty $, we show that $ {M^ + }$ is bounded on $ {L^p}(w)$ if and only if $ w$ satisfies the one-sided $ {A_p}$ condition:

$\displaystyle \left( {A_p^ + } \right)\qquad \left[ {\frac{1} {h}\int_{a - h}^a... ...1} {h}\int_a^{a + h} {w{{(x)}^{ - 1/(p - 1)}}dx} } \right]^{p - 1}} \leqslant C$

for all real $ a$ and positive $ h$. If in addition $ v(x) \geqslant 0$ and $ \sigma = {v^{ - 1/(p - 1)}}$,then $ {M^ + }$ is bounded from $ {L^p}(v)$ to $ {L^p}(w)$ if and only if

$\displaystyle \int_I {{{[{M^ + }({\chi _I}\sigma )]}^p}w \leqslant C\int_I {\sigma < \infty } } $

for all intervals $ I = (a,b)$ such that $ \int_{ - \infty }^a {w > 0} $. The corresponding weak type inequality is also characterized. Further properties of $ A_p^ + $ weights, such as $ A_p^ + \Rightarrow A_{p - \varepsilon }^ + $ and $ A_p^ + = (A_1^ + ){(A_1^ - )^{1 - p}}$, are established.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0849466-0
PII: S 0002-9947(1986)0849466-0
Article copyright: © Copyright 1986 American Mathematical Society