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

DOI:
https://doi.org/10.1090/S0002-9947-2011-05135-4

Published electronically:
May 18, 2011

MathSciNet review:
2813412

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Abstract | References | Similar Articles | Additional Information

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 inequality for which we were previously able to obtain only for . We also construct examples of weights that satisfy conditions which were shown in our earlier paper to be sufficient for one-weight inequalities when .

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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:
https://doi.org/10.1090/S0002-9947-2011-05135-4

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.