Higher order fractional weighted homogeneous spaces: characterization and finer embeddings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909365972238336 |
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| author | Biswas, Nirjan Kumar, Rohit |
| author_facet | Biswas, Nirjan Kumar, Rohit |
| contents | In this article, for $N \geq 2, s \in (1,2), p\in (1, \frac{N}{s}), σ=s-1 $ and $a \in [0, \frac{N-sp}{2})$, we establish an isometric isomorphism between the higher order fractional weighted Beppo-Levi space \begin{align*}
{\mathcal D}^{s,p}_a(\mathbb{R}^N) := \overline{\mathcal{C}_c^{\infty}(\mathbb{R}^N)}^{[\cdot]_{s,p,a}} \text{ where } [u]_{s,p,a} := \left( \iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{\left| \nabla u(x) -\nabla u(y) \right|^p}{\left|x-y \right|^{N+σp}} \, \frac{\mathrm{d}x}{|x|^a} \frac{\mathrm{d}y}{|y|^a} \right)^{\frac{1}{p}}, \end{align*} and higher order fractional weighted homogeneous space \begin{align*}
\mathring{W}^{s,p}_a(\mathbb{R}^N):= \left\{u \in L_a^{p^*_s}(\mathbb{R}^N): \| \nabla u \|_{L_a^{p^*_σ}(\mathbb{R}^N)} + [u]_{s,p,a} < \infty \right\} \end{align*} with the weighted Lebesgue norm \begin{align*}
\| u \|_{L_a^{p^*_α}(\mathbb{R}^N)}:= \left( \int_{\mathbb{R}^N} \frac{ |u(x)|^{p^*_α}}{|x|^{\frac{2ap^*_α}{p}}} \, {\mathrm{d}x} \right)^{\frac{1}{p^*_α}}, \text{ where } p^*_α=\frac{Np}{N-αp} \text{ for } α= s,σ. \end{align*} To achieve this, we prove that $\mathcal{C}_c^{\infty}(\mathbb{R}^N)$ is dense in $\mathring{W}^{s,p}_a(\mathbb{R}^N)$ with respect to $[\cdot]_{s,p,a}$, and $[\cdot]_{s,p,a}$ is an equivalent norm on $\mathring{W}^{s,p}_a(\mathbb{R}^N)$. Further, we obtain a finer embedding of ${\mathcal D}^{s,p}_a(\mathbb{R}^N)$ into the Lorentz space $L^{\frac{Np}{N-sp-2a}, p}(\mathbb{R}^N)$, where $L^{\frac{Np}{N-sp-2a}, p}(\mathbb{R}^N) \subsetneq L_a^{p^*_s}(\mathbb{R}^N)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05788 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher order fractional weighted homogeneous spaces: characterization and finer embeddings Biswas, Nirjan Kumar, Rohit Analysis of PDEs 46E35, 46E30 In this article, for $N \geq 2, s \in (1,2), p\in (1, \frac{N}{s}), σ=s-1 $ and $a \in [0, \frac{N-sp}{2})$, we establish an isometric isomorphism between the higher order fractional weighted Beppo-Levi space \begin{align*} {\mathcal D}^{s,p}_a(\mathbb{R}^N) := \overline{\mathcal{C}_c^{\infty}(\mathbb{R}^N)}^{[\cdot]_{s,p,a}} \text{ where } [u]_{s,p,a} := \left( \iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{\left| \nabla u(x) -\nabla u(y) \right|^p}{\left|x-y \right|^{N+σp}} \, \frac{\mathrm{d}x}{|x|^a} \frac{\mathrm{d}y}{|y|^a} \right)^{\frac{1}{p}}, \end{align*} and higher order fractional weighted homogeneous space \begin{align*} \mathring{W}^{s,p}_a(\mathbb{R}^N):= \left\{u \in L_a^{p^*_s}(\mathbb{R}^N): \| \nabla u \|_{L_a^{p^*_σ}(\mathbb{R}^N)} + [u]_{s,p,a} < \infty \right\} \end{align*} with the weighted Lebesgue norm \begin{align*} \| u \|_{L_a^{p^*_α}(\mathbb{R}^N)}:= \left( \int_{\mathbb{R}^N} \frac{ |u(x)|^{p^*_α}}{|x|^{\frac{2ap^*_α}{p}}} \, {\mathrm{d}x} \right)^{\frac{1}{p^*_α}}, \text{ where } p^*_α=\frac{Np}{N-αp} \text{ for } α= s,σ. \end{align*} To achieve this, we prove that $\mathcal{C}_c^{\infty}(\mathbb{R}^N)$ is dense in $\mathring{W}^{s,p}_a(\mathbb{R}^N)$ with respect to $[\cdot]_{s,p,a}$, and $[\cdot]_{s,p,a}$ is an equivalent norm on $\mathring{W}^{s,p}_a(\mathbb{R}^N)$. Further, we obtain a finer embedding of ${\mathcal D}^{s,p}_a(\mathbb{R}^N)$ into the Lorentz space $L^{\frac{Np}{N-sp-2a}, p}(\mathbb{R}^N)$, where $L^{\frac{Np}{N-sp-2a}, p}(\mathbb{R}^N) \subsetneq L_a^{p^*_s}(\mathbb{R}^N)$. |
| title | Higher order fractional weighted homogeneous spaces: characterization and finer embeddings |
| topic | Analysis of PDEs 46E35, 46E30 |
| url | https://arxiv.org/abs/2406.05788 |