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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Weighted norm inequalities for fractional integrals


Authors: Benjamin Muckenhoupt and Richard Wheeden
Journal: Trans. Amer. Math. Soc. 192 (1974), 261-274
MSC: Primary 26A33
MathSciNet review: 0340523
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Abstract: The principal problem considered is the determination of all nonnegative functions, $ V(x)$, such that $ \left\Vert{T_\gamma }f(x)V(x)\right\Vert _q \leq C\left\Vert f(x)V(x)\right\Vert _p$ where the functions are defined on $ {R^n},0 < \gamma < n,1 < p < n/\gamma ,1/q = 1/p - \gamma /n$, C is a constant independent of f and $ {T_\gamma }f(x) = \smallint f(x - y)\vert y{\vert^{\gamma - n}}dy$. The main result is that $ V(x)$ is such a function if and only if

$\displaystyle {\left( {\frac{1}{{\vert Q\vert}}\int_Q {{{[V(x)]}^q}dx} } \right... ...{\frac{1}{{\vert Q\vert}}\int_Q {{{[V(x)]}^{ - p'}}dx} } \right)^{1/p'}} \leq K$

where Q is any n dimensional cube, $ \vert Q\vert$ denotes the measure of Q, $ p' = p/(p - 1)$ and K is a constant independent of Q. Substitute results for the cases $ p = 1$ and $ q = \infty $ and a weighted version of the Sobolev imbedding theorem are also proved.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0340523-6
PII: S 0002-9947(1974)0340523-6
Article copyright: © Copyright 1974 American Mathematical Society