Pitt's inequality with sharp convolution estimates

Abstract

Sharp $L^p$ extensions of Pitt's inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and Stein-Weiss potentials. More generally, optimal constants are obtained for the full Stein-Weiss potential as a map from $L^p$ to itself, and new proofs are given for the classical Pitt and Stein-Weiss estimates.

v.2 Added referenes, new results on logarithmic uncertainty, gradient estimates and mixed homogeneity, and corrected misprints. 12 pages, amslatex