Admissible function spaces for weighted Sobolev inequalities
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arXiv
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| Format: | Preprint |
| Published: |
2020
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| _version_ | 1866911006670716928 |
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| author | Anoop, T. V. Biswas, Nirjan Das, Ujjal |
| author_facet | Anoop, T. V. Biswas, Nirjan Das, Ujjal |
| contents | Let $k,N \in \mathbb{N}$ with $1\le k\le N$ and let $Ω=Ω_1 \times Ω_2$ be an open set in $\mathbb{R}^k \times \mathbb{R}^{N-k}$. For $p\in (1,\infty)$ and $q \in (0,\infty),$ we consider the following Hardy-Sobolev type inequality:
\begin{align}
\int_Ω |g_1(y)g_2(z)| |u(y,z)|^q \, dy \, dz \leq C \left( \int_Ω | \nabla u(y,z) |^p \, dy \, dz \right)^{\frac{q}{p}}, \quad \forall \, u \in \mathcal{C}^1_c(Ω),
\end{align}
for some $C>0$. Depending on the values of $N,k,p,q,$ we have identified various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue spaces for $(g_1, g_2)$ so that the above inequality holds. Furthermore, we give a sufficient condition on $g_1,g_2$ so that the best constant in the above inequality is attained in the Beppo-Levi space $\mathcal{D}^{1,p}_0(Ω)$-the completion of $\mathcal{C}^1_c(Ω)$ with respect to $\|\nabla u\|_{L^p(Ω)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_04622 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Admissible function spaces for weighted Sobolev inequalities Anoop, T. V. Biswas, Nirjan Das, Ujjal Analysis of PDEs Functional Analysis 35A23, 46E30, 46E35, 47J30 Let $k,N \in \mathbb{N}$ with $1\le k\le N$ and let $Ω=Ω_1 \times Ω_2$ be an open set in $\mathbb{R}^k \times \mathbb{R}^{N-k}$. For $p\in (1,\infty)$ and $q \in (0,\infty),$ we consider the following Hardy-Sobolev type inequality: \begin{align} \int_Ω |g_1(y)g_2(z)| |u(y,z)|^q \, dy \, dz \leq C \left( \int_Ω | \nabla u(y,z) |^p \, dy \, dz \right)^{\frac{q}{p}}, \quad \forall \, u \in \mathcal{C}^1_c(Ω), \end{align} for some $C>0$. Depending on the values of $N,k,p,q,$ we have identified various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue spaces for $(g_1, g_2)$ so that the above inequality holds. Furthermore, we give a sufficient condition on $g_1,g_2$ so that the best constant in the above inequality is attained in the Beppo-Levi space $\mathcal{D}^{1,p}_0(Ω)$-the completion of $\mathcal{C}^1_c(Ω)$ with respect to $\|\nabla u\|_{L^p(Ω)}$. |
| title | Admissible function spaces for weighted Sobolev inequalities |
| topic | Analysis of PDEs Functional Analysis 35A23, 46E30, 46E35, 47J30 |
| url | https://arxiv.org/abs/2012.04622 |