$L^p$ Hardy inequalities with homogeneous weights

For $p\in (1,\mathrm{\infty })$ and $\alpha \in \mathbb{R}$, we consider measurable functions $g$ on ${\mathbb{S}}^{N-1}$ that satisfy the following weighted Hardy inequality: \begin{equation}\label{abs} \int_{$\mathbb{R}$^N}\frac{ g (x/|x|)}{|x|^{p+α}}|u(x)|^p dx \leq C\int_{$\mathbb{R}$^N}\frac{|\nabla u(x)|^p}{|x|^α} dx, \quad\forall,u\in $\mathcal{C}$_c^\infty($\mathbb{R}$^N), \end{equation} for some constant $C>0$. Depending on $N$, $p$, and $\alpha$, we identify suitable function spaces for $g$ so that \eqref{abs} holds. The constant obtained is sharp, in the sense that it is sharp when $g\equiv 1$. Furthermore, we establish the sharp fractional Hardy inequality with homogeneous weights.