$L^p$ Hardy inequalities with homogeneous weights
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911542116614144 |
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| author | Roy, Subhajit |
| author_facet | Roy, Subhajit |
| contents | For $p\in (1,\infty)$ and $α\in\mathbb{R}$, we consider measurable functions $g$ on $\mathbb{S}^{N-1}$ that satisfy the following weighted Hardy inequality: \begin{equation}\label{abs}
\int_{\mathbb{R}^N}\frac{ g (x/|x|)}{|x|^{p+α}}|u(x)|^p dx \leq C\int_{\mathbb{R}^N}\frac{|\nabla u(x)|^p}{|x|^α} dx, \quad\forall\,u\in \mathcal{C}_c^\infty(\mathbb{R}^N), \end{equation} for some constant $C>0$. Depending on $N$, $p$, and $α$, we identify suitable function spaces for $g$ so that \eqref{abs} holds. The constant obtained is sharp, in the sense that it is sharp when $g \equiv 1$. Furthermore, we establish the sharp fractional Hardy inequality with homogeneous weights. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05674 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L^p$ Hardy inequalities with homogeneous weights Roy, Subhajit Analysis of PDEs 35A23, 46E35, 49J40 For $p\in (1,\infty)$ and $α\in\mathbb{R}$, we consider measurable functions $g$ on $\mathbb{S}^{N-1}$ that satisfy the following weighted Hardy inequality: \begin{equation}\label{abs} \int_{\mathbb{R}^N}\frac{ g (x/|x|)}{|x|^{p+α}}|u(x)|^p dx \leq C\int_{\mathbb{R}^N}\frac{|\nabla u(x)|^p}{|x|^α} dx, \quad\forall\,u\in \mathcal{C}_c^\infty(\mathbb{R}^N), \end{equation} for some constant $C>0$. Depending on $N$, $p$, and $α$, we identify suitable function spaces for $g$ so that \eqref{abs} holds. The constant obtained is sharp, in the sense that it is sharp when $g \equiv 1$. Furthermore, we establish the sharp fractional Hardy inequality with homogeneous weights. |
| title | $L^p$ Hardy inequalities with homogeneous weights |
| topic | Analysis of PDEs 35A23, 46E35, 49J40 |
| url | https://arxiv.org/abs/2509.05674 |