Hardy and Rellich inequalities with Bessel pairs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866911102470717440 |
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| author | Ruzhansky, Michael Sabitbek, Bolys |
| author_facet | Ruzhansky, Michael Sabitbek, Bolys |
| contents | In this paper, we establish suitable characterisations for a pair of functions $(W(x),H(x))$ on a bounded, connected domain $Ω\subset \mathbb{R}^n$ 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), $|ξ|^2_A = \langle A(x)ξ, ξ\rangle$ for $ξ\in \mathbb{R}^n$ and $A(x)$ is an $n\times n$ symmetric, uniformly positive definite matrix defined on a bounded domain $Ω\subset \mathbb{R}^n$. 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_07008 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Hardy and Rellich inequalities with Bessel pairs Ruzhansky, Michael Sabitbek, Bolys Analysis of PDEs 35A23, 35R45, 35B09, 34A40 In this paper, we establish suitable characterisations for a pair of functions $(W(x),H(x))$ on a bounded, connected domain $Ω\subset \mathbb{R}^n$ 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), $|ξ|^2_A = \langle A(x)ξ, ξ\rangle$ for $ξ\in \mathbb{R}^n$ and $A(x)$ is an $n\times n$ symmetric, uniformly positive definite matrix defined on a bounded domain $Ω\subset \mathbb{R}^n$. 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}. |
| title | Hardy and Rellich inequalities with Bessel pairs |
| topic | Analysis of PDEs 35A23, 35R45, 35B09, 34A40 |
| url | https://arxiv.org/abs/2101.07008 |