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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Pitt's inequality with sharp convolution estimates


Author: William Beckner
Journal: Proc. Amer. Math. Soc. 136 (2008), 1871-1885
MSC (2000): Primary 58J70, 42B10, 35A15
Published electronically: November 30, 2007
MathSciNet review: 2373619
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Abstract: Sharp $ L^p$ extensions of Pitt's inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and Stein-Weiss potentials. Optimal constants are obtained for the full Stein-Weiss potential as a map from $ L^p$ to itself which in turn yield semi-classical Rellich inequalities on $ \mathbb{R}^n$. Additional results are obtained for Stein-Weiss potentials with gradient estimates and with mixed homogeneity. New proofs are given for the classical Pitt and Stein-Weiss estimates.


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Additional Information

William Beckner
Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712-0257
Email: beckner@math.utexas.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09216-7
PII: S 0002-9939(07)09216-7
Received by editor(s): December 19, 2006
Published electronically: November 30, 2007
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2007 American Mathematical Society