# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## A two weight inequality for the fractional integral when $p=n/\alpha$HTML articles powered by AMS MathViewer

by Eleonor Harboure, Roberto A. Macías and Carlos Segovia
Proc. Amer. Math. Soc. 90 (1984), 555-562 Request permission

## Abstract:

Let ${I_\alpha }$ be the fractional integral operator defined as ${I_\alpha }f(x) = \int {f(y){{\left | {x - y} \right |}^{\alpha - n}}dy.}$ Given a weight $w$ (resp. $\upsilon$), necessary and sufficient conditions are given for the existence of a nontrivial weight $\upsilon$ (resp. $w$) such that ${\left \| {\upsilon {\chi _B}} \right \|_\infty }\frac {1}{{\left | B \right |}}\int _B {\left | {{I_\alpha }f(x) - {m_B}({I_\alpha }f)} \right |} dx \leqslant C{\left ( {\int {{{\left | f \right |}^{n/\alpha }}w} } \right )^{\alpha /n}}$ holds for any ball $B$ such that ${\left \| {v{\chi _B}} \right \|_\infty } > 0$.
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