Equivalent and attained version of Hardy's inequality in Rn

We investigate connections between Hardy's inequality in the whole space Rnand embedding inequalities for Sobolev–Lorentz spaces. In particular, we complete previous results due to [1, Alvino] and [28, Talenti] by establishing optimal embedding inequalities for the Sobolev–Lorentz quasinorm ‖∇⋅‖p,qalso in the range p