# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Weighted norm inequalities for fractional integralsHTML articles powered by AMS MathViewer

by Benjamin Muckenhoupt and Richard Wheeden
Trans. Amer. Math. Soc. 192 (1974), 261-274 Request permission

## Abstract:

The principal problem considered is the determination of all nonnegative functions, $V(x)$, such that $\left \|{T_\gamma }f(x)V(x)\right \|_q \leq C\left \|f(x)V(x)\right \|_p$ 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 ${T_\gamma }f(x) = \smallint f(x - y)|y{|^{\gamma - n}}dy$. The main result is that $V(x)$ is such a function if and only if ${\left ( {\frac {1}{{|Q|}}\int _Q {{{[V(x)]}^q}dx} } \right )^{1/q}}{\left ( {\frac {1}{{|Q|}}\int _Q {{{[V(x)]}^{ - p’}}dx} } \right )^{1/p’}} \leq K$ where Q is any n dimensional cube, $|Q|$ 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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