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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Weighted inequalities for one-sided maximal functions


Authors: F. J. Martín-Reyes, P. Ortega Salvador and A. de la Torre
Journal: Trans. Amer. Math. Soc. 319 (1990), 517-534
MSC: Primary 42B25
DOI: https://doi.org/10.1090/S0002-9947-1990-0986694-9
MathSciNet review: 986694
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Abstract: Let $ M_g^ + $ be the maximal operator defined by

$\displaystyle M_g^ + f(x) = \mathop {\sup }\limits_{h > 0} \left( {\int_x^{x + ... ... f(t)\vert g(t)dt} } \right){\left( {\int_x^{x + h} {g(t)dt} } \right)^{ - 1}},$

where $ g$ is a positive locally integrable function on $ {\mathbf{R}}$. We characterize the pairs of nonnegative functions $ (u,v)$ for which $ M_g^ + $ applies $ {L^p}(v)$ in $ {L^p}(u)$ or in weak- $ {L^p}(u)$. Our results generalize Sawyer's (case $ g = 1$) but our proofs are different and we do not use Hardy's inequalities, which makes the proofs of the inequalities self-contained.

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DOI: https://doi.org/10.1090/S0002-9947-1990-0986694-9
Keywords: One-sided maximal functions, weighted inequalities, weights
Article copyright: © Copyright 1990 American Mathematical Society

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