Pitt’s inequality with sharp convolution estimates
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- by William Beckner
- Proc. Amer. Math. Soc. 136 (2008), 1871-1885
- DOI: https://doi.org/10.1090/S0002-9939-07-09216-7
- Published electronically: November 30, 2007
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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.References
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Bibliographic Information
- William Beckner
- Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712-0257
- MR Author ID: 33405
- ORCID: 0000-0002-5667-3920
- Email: beckner@math.utexas.edu
- Received by editor(s): December 19, 2006
- Published electronically: November 30, 2007
- Communicated by: Michael T. Lacey
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 1871-1885
- MSC (2000): Primary 58J70, 42B10, 35A15
- DOI: https://doi.org/10.1090/S0002-9939-07-09216-7
- MathSciNet review: 2373619