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Weighted Inequalities for One-Sided Maximal Functions

F. J. Martin-Reyes, P. Ortega Salvador and A. De La Torre
Transactions of the American Mathematical Society
Vol. 319, No. 2 (Jun., 1990), pp. 517-534
DOI: 10.2307/2001252
Stable URL: http://www.jstor.org/stable/2001252
Page Count: 18
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Abstract

Let M+ g be the maximal operator defined by $M^+_g f(x) = \sup_{h > 0}\Bigg(\int^{x+h}_x |f(t)|g(t)dt\Bigg) \Bigg(\int^{x+h}_x g(t)dt\Bigg)^{-1}$, where g is a positive locally integrable function on R. We characterize the pairs of nonnegative functions (u, v) for which M+ g applies Lp(v) in Lp(u) or in weak-Lp(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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