Lᵖ Hardy inequalities with homogeneous weights

For p∈ (1,∞) and α, we consider measurable functions g on SN-1 that satisfy the following weighted Hardy inequality: {equation}{abs} ∫R^N{ g (x/|x|)}{|x|p+α}|u(x)|^p dx ≤ C∫R^N|∇ u(x)|^p/|x|^α dx, ∀\,u∈ C_c^∞(R^N), {equation} for some constant $C>0$. Depending on N, p, and $α$, we identify suitable function spaces for g so that {abs} holds. The constant obtained is sharp, in the sense that it is sharp when g ≡ 1. Furthermore, we establish the sharp fractional Hardy inequality with homogeneous weights.