\(L^{\infty }\)-estimates for Sobolev functions in terms of \(\nabla \) and \(\Delta \) (Q1022323)

The author presents several inequalities of the form \[ \|u\|_{L^\infty}^p \leq C \|\Delta^k u\|^{p-\alpha}_p \|\nabla(\Delta^l u)\|^{\alpha}_p \text{ for all }u\in C_c^m({\mathbb R}^n), \] where \(k,l \in {\mathbb N}\cup \{0\}\) are determined by \(m\). If \(p = 2\), the estimates are given with explicit constants; for \(p \neq 2\), explicit constants are obtained in two simple cases.