# Factorizations and Hardy–Rellich-type inequalities

### Fritz Gesztesy

Baylor University, Waco, USA### Lance Littlejohn

Baylor University, Waco, USA

A subscription is required to access this book chapter.

## 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 $T_{α,β}:=−Δ+α∣x∣_{−2}x⋅∇+β∣x∣_{−2}$, $α,β∈R$, $x∈R_{n}∖{0}$, $n∈N$, $n≥2$, and its formal adjoint, denoted by $T_{α,β}$, we show that nonnegativity of $T_{α,β}T_{α,β}$ on $C_{0}(R_{n}∖{0})$ 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 $α$ and $β$ 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 $R_{n}$ is replaced by an arbitrary open set $Ω⊆R_{n}$ for functions $f∈C_{0}(Ω∖{0})$.

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.