Factorizations and Hardy–Rellich-type inequalities

  • Fritz Gesztesy

    Baylor University, Waco, USA
  • Lance Littlejohn

    Baylor University, Waco, USA
Factorizations and Hardy–Rellich-type inequalities cover

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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 nn-dimensional homogeneous scalar differential expressions Tα,β:=Δ+αx2x+βx2T_{\alpha,\beta} := - \Delta + \alpha |x|^{-2} x \cdot \nabla + \beta |x|^{-2}, α,βR\alpha, \beta \in \mathbb R, xRn{0}x \in \mathbb R^n \setminus \{0\}, nNn \in \mathbb N, n2n \geq 2, and its formal adjoint, denoted by Tα,β+T_{\alpha,\beta}^+, we show that nonnegativity of Tα,β+Tα,βT_{\alpha,\beta}^+ T_{\alpha,\beta} on C0(Rn{0})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 α\alpha and β\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 Rn\mathbb R^n is replaced by an arbitrary open set ΩRn\Omega \subseteq \mathbb R^n for functions fC0(Ω{0})f \in C^{\infty}_0(\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.