$L^p$ Hardy inequalities with homogeneous weights

For $p\in (1,\infty)$ 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+\alpha}}|u(x)|^p dx \leq C\int_{\mathbb{R}^N}\frac{|\nabla u(x)|^p}{|x|^\alpha} 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.