preprint

Inserted: 11 jun 2024
Last Updated: 12 jun 2024

Year: 2024

Abstract:

In this article, for $N \geq 2, s \in (1,2), p\in (1, \frac{N}{s}), \sigma=s-1 $ and $a \in [0, \frac{N-sp}{2})$, we establish an isometric isomorphism between the higher order fractional weighted Beppo-Levi space $ {\mathcal D}^{s,p}_a(\mathbb{R}^N) := \overline{\mathcal{C}_c^{\infty}(\mathbb{R}^N)}^{[\cdot]_{s,p,a}} $ where

$[u]_{s,p,a} := \left( \iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{| \nabla u(x) -\nabla u(y) |^p}{|x-y |^{N+\sigma p}} \, \frac{\mathrm{d}x}{|x|^a} \frac{\mathrm{d}y}{|y|^a} \right)^{\frac{1}{p}},$

and higher order fractional weighted homogeneous space

$ \mathring{W}^{s,p}_a(\mathbb{R}^N):= \left\{u \in L_a^{p^*_s}(\mathbb{R}^N): || \nabla u ||_{L_a^{p^*_{\sigma}}(\mathbb{R}^N)} + [u]_{s,p,a} < \infty \right\}$

with the weighted Lebesgue norm

$ ||u ||_{L_a^{p^*_{\alpha}}(\mathbb{R}^N)}:= \left( \int_{\mathbb{R}^N} \frac{ |u(x)|^{p^*_{\alpha}}}{|x|^{\frac{2ap^*_{\alpha}}{p}}} \, {\mathrm{d}x} \right)^{\frac{1}{p^*_{\alpha}}},$

where $p^*_{\alpha}=\frac{Np}{N-\alpha p} \text{ for } \alpha= s,\sigma. $ 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 $L^{\frac{Np}{N-sp-2a}, p}(\mathbb{R}^N) \subsetneq L_a^{p^*_s}(\mathbb{R}^N)$.