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Maximal potentials, maximal singular integrals, and the spherical maximal function

Authors: Piotr Hajłasz and Zhuomin Liu
Journal: Proc. Amer. Math. Soc. 142 (2014), 3965-3974
MSC (2010): Primary 46E35; Secondary 42B20, 42B25
Published electronically: July 31, 2014
MathSciNet review: 3251736
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Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a notion of maximal potentials and we prove that they form bounded operators from $ L^p$ to the homogeneous Sobolev space $ \dot {W}^{1,p}$ for all $ n/(n-1)<p<n$. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.

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Additional Information

Piotr Hajłasz
Affiliation: Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

Zhuomin Liu
Affiliation: Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Address at time of publication: Department of Mathematics and Statistics, P. O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland

Keywords: Sobolev spaces, potentials, singular integrals, spherical maximal function
Received by editor(s): November 3, 2012
Received by editor(s) in revised form: December 31, 2012
Published electronically: July 31, 2014
Additional Notes: The first author was supported by NSF grant DMS-0900871
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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