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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Weighted norm inequalities for fractional integrals


Author: G. V. Welland
Journal: Proc. Amer. Math. Soc. 51 (1975), 143-148
MSC: Primary 26A86; Secondary 26A33
MathSciNet review: 0369641
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Abstract: A simpler proof of an inequality of Muckenhoupt and Wheeden is given. Let $ {T_\alpha }f(x) = \smallint f(y)\vert x - y{\vert^{\alpha - d}}dy$ be given for functions defined in $ {{\mathbf{R}}^d}$. Let $ \upsilon $ be a weight function which satisfies

$\displaystyle (\vert Q{\vert^{ - 1}}\int_Q {{{[\upsilon (x)]}^q}dx{)^{1/q}}(\vert Q{\vert^{ - 1}}\int_Q {{{[\upsilon (x)]}^{ - p'}}dx{)^{1/p'}} \leq K} } $

for each cube, $ Q$, with sides parallel to a standard system of axes and $ \vert Q\vert$ is the measure of such a cube. Suppose $ 1/q = 1/p - \alpha /d$ and $ 0 < \alpha < d,1 < p < d/\alpha $. Then there exists a constant such that $ \vert\vert({T_\alpha }f)\upsilon \vert{\vert _q} \leq C\vert\vert f\upsilon \vert{\vert _p}$. Certain results for $ p = 1$ and $ q = \infty $ are also given.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0369641-X
PII: S 0002-9939(1975)0369641-X
Article copyright: © Copyright 1975 American Mathematical Society