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



A two weight inequality for the fractional integral when $ p=n/\alpha $

Authors: Eleonor Harboure, Roberto A. Macías and Carlos Segovia
Journal: Proc. Amer. Math. Soc. 90 (1984), 555-562
MSC: Primary 26D10
MathSciNet review: 733405
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Abstract: Let $ {I_\alpha }$ be the fractional integral operator defined as

$\displaystyle {I_\alpha }f(x) = \int {f(y){{\left\vert {x - y} \right\vert}^{\alpha - n}}dy.} $

Given a weight $ w$ (resp. $ \upsilon $), necessary and sufficient conditions are given for the existence of a nontrivial weight $ \upsilon $ (resp. $ w$) such that

$\displaystyle {\left\Vert {\upsilon {\chi _B}} \right\Vert _\infty }\frac{1}{{\... ...\left( {\int {{{\left\vert f \right\vert}^{n/\alpha }}w} } \right)^{\alpha /n}}$

holds for any ball $ B$ such that $ {\left\Vert {v{\chi _B}} \right\Vert _\infty } > 0$.

References [Enhancements On Off] (What's this?)

  • [1] E. Harboure, R. A. Macías and C. Segovia, Boundedness of fractional operators on $ {L^p}$ spaces with different weights, preprint.
  • [2] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-274. MR 0340523 (49:5275)

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