Factorizations and Hardy-Rellich-Type Inequalities

The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the two-parameter $n$-dimensional homogeneous scalar differential expressions $T_{\alpha,\beta} := - \Delta + \alpha |x|^{-2} x \cdot \nabla + \beta |x|^{-2}$,$\alpha, \beta \in $\mathbb{R}$$,$x \in $\mathbb{R}$^n \backslash {0}$,$n \in $\mathbb{N}$$,$n \geq 2$,anditsformaladjoint,denotedby$T_{\alpha,\beta}^+$,weshowthatnonnegativityof$T_{\alpha,\beta}^+ T_{\alpha,\beta}$on$C_0^{\infty}($\mathbb{R}$^n \backslash {0})$ implies the fundamental inequality, \begin{align} \int_{$\mathbb{R}$^n} [(\Delta f)(x)]^2 , d^n x &\geq [(n - 4) \alpha - 2 \beta] \int_{$\mathbb{R}$^n} |x|^{-2} |(\nabla f)(x)|^2 , d^n x \notag \ & \quad - \alpha (\alpha - 4) \int_{$\mathbb{R}$^n} |x|^{-4} |x \cdot (\nabla f)(x)|^2 , d^n x \notag \ & \quad + \beta [(n - 4) (\alpha - 2) - \beta] \int_{$\mathbb{R}$^n} |x|^{-4} |f(x)|^2 , d^n x, \notag \end{align} for $f\in C_0^{\mathrm{\infty }}({\mathbb{R}}^n\mathrm{\backslash }\{0\})$. A particular choice of values for $\alpha$ and $\beta$ yields known Hardy-Rellich-type inequalities, including the classical Rellich inequality and an inequality due to Schmincke. By locality, these inequalities extend to the situation where ${\mathbb{R}}^n$ is replaced by an arbitrary open set $\Omega \subseteq $\mathbb{R}$^n$forfunctions$f \in C^{\infty}_0(\Omega \backslash {0})$. Perhaps more importantly, we will indicate that our method, in addition to being elementary, is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order situations.