Maximal potentials, maximal singular integrals, and the spherical maximal function
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- Proc. Amer. Math. Soc. 142 (2014), 3965-3974 Request permission
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.References
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Additional Information
- Piotr Hajłasz
- Affiliation: Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
- MR Author ID: 332316
- Email: hajlasz@pitt.edu
- 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
- Email: liuzhuomin@hotmail.com
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 3965-3974
- MSC (2010): Primary 46E35; Secondary 42B20, 42B25
- DOI: https://doi.org/10.1090/S0002-9939-2014-12129-0
- MathSciNet review: 3251736