Hardy inequalities with optimal constants and reminder terms

We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces W1,p 0 and in higher-order Sobolev spaces on a bounded domain ! ! Rn can be refined by adding remainder terms which involve Lp norms. In the higher-order case further Lp norms with lower-order singular weights arise. The case 1 < p < 2 being more involved requires a different technique and is developed only in the space W1,p 0