  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
\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+\sigma p}} \,
\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^*_{\sigma}}(\mathbb{R}^N)} + [u]_{s,p,a} < \infty
\right\} \end{align*} with the weighted Lebesgue norm \begin{align*}
  \| 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}}}, \text{ where }
p^*_{\alpha}=\frac{Np}{N-\alpha p} \text{ for } \alpha= s,\sigma. \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 $L^{\frac{Np}{N-sp-2a},
p}(\mathbb{R}^N) \subsetneq L_a^{p^*_s}(\mathbb{R}^N)$.
