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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 weak-type $(1,1)$ inequalities for rough operators
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by Steve Hofmann PDF
Proc. Amer. Math. Soc. 107 (1989), 423-435 Request permission

Abstract:

Let $\Omega$ be homogeneous of degree 0, have mean value 0 on the circle, and belong to ${L^q}\left ( {{S^1}} \right ),1 < q \leq \infty$. Then the two-dimensional operator defined by \[ Tf\left ( x \right ) = ''{\text {pv''}}\int {\Omega \left ( y \right ){{\left | y \right |}^{ - 2}}} f\left ( {x - y} \right )dy\] is shown to be of weak-type $(1,1)$ with respect to the weighted measures ${\left | x \right |^\alpha }dx$, if $- 2 + 1/q < \alpha < 0$. Under the weaker assumption that $\Omega$ belongs to $L {\log ^ + }L\left ( {{S^1}} \right )$, the same result holds if $- 1 < \alpha < 0$. Similar results are also obtained for the related maximal operator \[ {M_\Omega }f\left ( x \right ) = \sup \limits _{r > 0} {r^{ - 2}}\int _{|y| < r} {\left | {\Omega \left ( y \right )f\left ( {x - y} \right )} \right |dy.} \]
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 107 (1989), 423-435
  • MSC: Primary 42B20
  • DOI: https://doi.org/10.1090/S0002-9939-1989-0984795-X
  • MathSciNet review: 984795