Hardy and Rellich inequalities with Bessel pairs

Abstract

In this paper, we establish suitable characterisations for a pair of functions $(W(x),H(x))$ on a bounded, connected domain $\Omega \subset {\mathbb{R}}^n$ in order to have the following Hardy inequality \begin{equation*} \int_Ω W(x) |\nabla u|_A^2 dx \geq \int_Ω |\nabla d|^2_AH(x)|u|^2 dx, \,\,\, u \in C^{1}_0(Ω), \end{equation*} where $d(x)$ is a suitable quasi-norm (gauge), $|\xi {|}_A^2=⟨A(x)\xi ,\xi ⟩$ for $\xi \in {\mathbb{R}}^n$ and $A(x)$ is an $n\times n$ symmetric, uniformly positive definite matrix defined on a bounded domain $\Omega \subset {\mathbb{R}}^n$. We also give its $L^p$ analogue. As a consequence, we present examples for a standard Laplacian on ${\mathbb{R}}^n$, Baouendi-Grushin operator, and sub-Laplacians on the Heisenberg group, the Engel group and the Cartan group. Those kind of characterisations for a pair of functions $(W(x),H(x))$ are obtained also for the Rellich inequality. These results answer the open problems of Ghoussoub-Moradifam \cite{GM_book}.

References in corpus (1)

• Principal frequency of p-sub-Laplacians for general vector fields