Factorizations and Hardy–Rellich-type inequalities

Abstract

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 precisely, introducing the two-parameter $n$-dimensional homogeneous scalar differential expressions $T_{\alpha ,\beta }:=-\Delta +\alpha |x{|}^{-2}x\cdot \nabla +\beta |x{|}^{-2}$, $\alpha ,\beta \in \mathbb{R}$, $x\in {\mathbb{R}}^n\setminus \{0\}$, $n\in \mathbb{N}$, $n\ge 2$, and its formal adjoint, denoted by $T_{\alpha ,\beta }^{+}$, we show that nonnegativity of $T_{\alpha ,\beta }^{+}{T}_{\alpha ,\beta }$ on $C_0^{\infty }({\mathbb{R}}^n\setminus \{0\})$ implies the fundamental inequality (*)

\[ \tag{$*$}\label{0.1} \begin{aligned} \int_{\mathbb R^n} [(\Delta f)(x)]^2 \, d^n x & \geq [(n - 4) \alpha - 2 \beta] \int_{\mathbb R^n} |x|^{-2} |(\nabla f)(x)|^2 \, d^n x \\ & \quad - \alpha (\alpha - 4) \int_{\mathbb R^n} |x|^{-4} |x \cdot (\nabla f)(x)|^2 \, d^n x \\ & \quad + \beta [(n - 4) (\alpha - 2) - \beta] \int_{\mathbb R^n} |x|^{-4} |f(x)|^2 \, d^n x,\\ &&\llap {f \in C^{\infty}_0(\mathbb R^n \setminus \{0\}).} \end{aligned} \]

A particular choice of values for $\alpha$ and $\beta$ in (*) 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 $\Omega \subseteq {\mathbb{R}}^n$ for functions $f\in C_0^{\infty }(\Omega \setminus \{0\})$.

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 operators.