Higher order fractional weighted homogeneous spaces: characterization and finer embeddings

In this article, for $N\ge 2,s\in (1,2),p\in (1,\frac{N}{s}),\sigma =s-1$ and $a\in [0,\frac{N-sp}{2})$, we establish an isometric isomorphism between the higher order fractional weighted Beppo-Levi space \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 ${\mathcal{C}}_c^{\mathrm{\infty }}({\mathbb{R}}^N)$ is dense in ${\mathring{W}}_a^{s,p}({\mathbb{R}}^N)$ with respect to $[\cdot {]}_{s,p,a}$, and $[\cdot {]}_{s,p,a}$ is an equivalent norm on ${\mathring{W}}_a^{s,p}({\mathbb{R}}^N)$. Further, we obtain a finer embedding of ${\mathcal{D}}_a^{s,p}({\mathbb{R}}^N)$ into the Lorentz space $L^{\frac{Np}{N-sp-2a},p}({\mathbb{R}}^N)$, where $L^{\frac{Np}{N-sp-2a},p}({\mathbb{R}}^N)⊊L_a^{p_s^{\ast }}({\mathbb{R}}^N)$.