A unified approach to improved Lᵖ Hardy inequalities with best constants

We present a unified approach to improved Lᵖ Hardy inequalities in RN. We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where the distance is taken from a surface of codimension $1<k<N$. In our main result, we add to the right hand side of the classical Hardy inequality a weighted Lᵖ norm with optimal weight and best constant. We also prove nonhomogeneous improved Hardy inequalities, where the right hand side involves weighted Lq norms, q ≠ p.