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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On weighted norm inequalities for positive linear operators


Authors: R. Kerman and E. Sawyer
Journal: Proc. Amer. Math. Soc. 105 (1989), 589-593
MSC: Primary 26D15; Secondary 26A33, 44A10, 47B38
MathSciNet review: 947314
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Abstract: Let $ T$ be a positive linear operator defined for nonnegative functions on a $ \sigma $-finite measure space $ \left( {X,m,\mu } \right)$. Given $ 1 < p < \infty $ and a nonnegative weight function $ w$ on $ X$, it is shown that there exists a nonnegative weight function $ v$, finite $ \mu $-almost everywhere on $ X$, such that (1)

$\displaystyle \int_X {{{\left( {Tf} \right)}^p}wd\mu \leq \int _X {{f^p}vd\mu } } ,\quad {\text{for all }}f\leq 0$

, if and only if there exists $ \phi $ positive $ \mu $-almost everywhere on $ X$ with (2)

$\displaystyle \int\limits_X {{{\left( {T\phi } \right)}^p}wd\mu < \infty .} $

In case (2) holds, we may take $ v = {\phi ^{1 - p}}{T^*}\left[ {{{\left( {T\phi } \right)}^{p - 1}}w} \right]$ in (1). This partially answers a question of B. Muckenhoupt in [5]. Applications to some specific operators are also given.

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DOI: https://doi.org/10.1090/S0002-9939-1989-0947314-X
Article copyright: © Copyright 1989 American Mathematical Society