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.
Categories: math.AP