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Transactions of the American Mathematical Society

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

 

 

Boundedness of fractional operators on $ L\sp{p}$ spaces with different weights


Authors: Eleonor Harboure, Roberto A. Macías and Carlos Segovia
Journal: Trans. Amer. Math. Soc. 285 (1984), 629-647
MSC: Primary 26D10; Secondary 42B25, 47B38
MathSciNet review: 752495
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Abstract: Let $ {T_\alpha }$ be either the fractional integral operator $ \smallint f(y)\vert x - y{\vert^{\alpha - n}}\; dy$, or the fractional maximal operator $ \sup \left\{ {{r^{\alpha - n}}{\smallint_{\vert x - y\vert < r}}\vert f(y)\vert dy:\,r > 0} \right\}$. Given a weight $ w$ (resp. $ \upsilon $), necessary and sufficient conditions are given for the existence of a nontrivial weight $ \upsilon $ (resp. $ w$) such that $ {(\smallint \vert{T_\alpha }f{\vert^q}\upsilon \;dx)^{1/q}} \leqslant {(\smallint\vert f{\vert^p}w\;dx)^{1/p}}$ holds. Weak type substitutes in limiting cases are considered.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0752495-7
Article copyright: © Copyright 1984 American Mathematical Society