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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Averages over starlike sets, starlike maximal functions, and homogeneous singular integrals


Authors: David K. Watson and Richard L. Wheeden
Journal: Trans. Amer. Math. Soc. 363 (2011), 5179-5206
MSC (2000): Primary 42B20, 42B25
Published electronically: May 18, 2011
MathSciNet review: 2813412
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Abstract: We improve some of the results in our 1999 paper concerning weighted norm estimates for homogeneous singular integrals with rough kernels. Using a representation of such integrals in terms of averages over starlike sets, we prove a two-weight $ L^{p}$ inequality for $ 1 < p < 2$ which we were previously able to obtain only for $ p \geq 2$. We also construct examples of weights that satisfy conditions which were shown in our earlier paper to be sufficient for one-weight inequalities when $ 1<p<\infty $.


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Additional Information

David K. Watson
Affiliation: Department of Mathematics, The College of New Jersey, 2000 Pennington Road, Ewing, New Jersey 08628
Email: davidkirkwatson@gmail.com

Richard L. Wheeden
Affiliation: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854
Email: wheeden@math.rutgers.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05135-4
PII: S 0002-9947(2011)05135-4
Keywords: Calderón–Zygmund singular integrals, weighted norm inequalities
Received by editor(s): July 14, 2008
Received by editor(s) in revised form: June 10, 2009
Published electronically: May 18, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.