Pitt’s inequality with sharp convolution estimates

Sharp L p 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 L^p to itself which in turn yield semi-classical Rellich inequalities on R n 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.