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Proceedings of the American Mathematical Society

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

 

 

Weighted weak-type $ (1,1)$ inequalities for rough operators


Author: Steve Hofmann
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
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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

$\displaystyle Tf\left( x \right) = ''{\text{pv''}}\int {\Omega \left( y \right){{\left\vert y \right\vert}^{ - 2}}} f\left( {x - y} \right)dy$

is shown to be of weak-type $ (1,1)$ with respect to the weighted measures $ {\left\vert x \right\vert^\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

$\displaystyle {M_\Omega }f\left( x \right) = \mathop {\sup }\limits_{r > 0} {r^... ... {\left\vert {\Omega \left( y \right)f\left( {x - y} \right)} \right\vert dy.} $


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DOI: https://doi.org/10.1090/S0002-9939-1989-0984795-X
Article copyright: © Copyright 1989 American Mathematical Society