Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Pitt's inequality with sharp convolution estimates

Author(s): William Beckner
Journal: Proc. Amer. Math. Soc. 136 (2008), 1871-1885.
MSC (2000): Primary 58J70, 42B10, 35A15
Posted: November 30, 2007
MathSciNet review: 2373619
Retrieve article in: PDF

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Abstract: Sharp 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 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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