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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 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A two weight inequality for the fractional integral when $p=n/\alpha$
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by Eleonor Harboure, Roberto A. Macías and Carlos Segovia
Proc. Amer. Math. Soc. 90 (1984), 555-562
DOI: https://doi.org/10.1090/S0002-9939-1984-0733405-0

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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Bibliographic Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 90 (1984), 555-562
  • MSC: Primary 26D10
  • DOI: https://doi.org/10.1090/S0002-9939-1984-0733405-0
  • MathSciNet review: 733405