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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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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Weighted norm inequalities for fractional integrals
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by G. V. Welland PDF
Proc. Amer. Math. Soc. 51 (1975), 143-148 Request permission

Abstract:

A simpler proof of an inequality of Muckenhoupt and Wheeden is given. Let ${T_\alpha }f(x) = \smallint f(y)|x - y{|^{\alpha - d}}dy$ be given for functions defined in ${{\mathbf {R}}^d}$. Let $\upsilon$ be a weight function which satisfies \[ (|Q{|^{ - 1}}\int _Q {{{[\upsilon (x)]}^q}dx{)^{1/q}}(|Q{|^{ - 1}}\int _Q {{{[\upsilon (x)]}^{ - p’}}dx{)^{1/p’}} \leq K} } \] for each cube, $Q$, with sides parallel to a standard system of axes and $|Q|$ 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 $||({T_\alpha }f)\upsilon |{|_q} \leq C||f\upsilon |{|_p}$. Certain results for $p = 1$ and $q = \infty$ are also given.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 51 (1975), 143-148
  • MSC: Primary 26A86; Secondary 26A33
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0369641-X
  • MathSciNet review: 0369641