  Sharp error estimates in terms of the fractional Laplacian and a weighted
Besov norm are obtained for Pitt's inequality by using the spectral
representation with weights for the fractional Laplacian due to Frank, Lieb and
Seiringer and the sharp Stein-Weiss inequality. Dilation invariance, group
symmetry on a non-unimodular group and a nonlinear Stein-Weiss lemma are used
to provide short proofs of the Frank-Seiringer "Hardy inequalities" where
fractional smoothness is measured by a Besov norm.
