Sharp constants in the Hardy-Rellich inequalities
Optimal constants are found in the inequalities \[ \int_{\mathbb R^d}|x|^{-2l}|u(x)|^2dx\leq C_l\int_{\mathbb R^d}|\xi|^{2l}|\hat u(\xi)|^2d\xi \] with \(l<d/2\) and \[ \int_{\mathbb R^d}|x|^{-2l}\Bigl|u(x)-\sum_{|\alpha|\leq [l-d/2]}(\alpha!)^{-1}(\partial^\alpha u)(0)x^\alpha\Bigr|^2dx \leq C_l\int_{\mathbb R^d}|\xi|^{2l}|\hat u(\xi)|^2d\xi \] with \(l>d/2\), \(l-d/2\notin\mathbb Z\), to hold for all \(u\) such that \(\hat u\in C^\infty_0(\mathbb R^d\setminus\{0\})\). The result is then extended to larger classes of functions and the case \(l-d/2\in\mathbb Z\) is discussed.
scientific article; zbMATH DE number 1374984