Sharp Hardy inequalities in the half space with trace remainder term

Abstract

In this paper, we deal with a class of inequalities which interpolate the Kato inequality and the Hardy inequality in the half space. Starting from the classical Hardy’s inequality in the half space R+n=Rn−1×(0,∞), we show that, if we replace the optimal constant (n−2)24 with a smaller one (β−2)24, 2≤β<n, then we can add an extra trace-term similar to that one that appears in the Kato inequality. The constant in the trace remainder term is optimal and it tends to zero when β goes to n, while it is equal to the optimal constant in Kato’s inequality when β=2. The approach is based on a very classical method of Calculus of Variation due to Weierstrass (and developed by Hilbert) that usually is considered to prove that the solutions of the Euler–Lagrange equation associated to a functional are, in fact, extremals. In this paper, we will show how this method is well suited also to functionals that have no extremals.