About the space of function used in the harmonic extensions for elliptic problems involving the fractional laplacian

recently i am studying the following paper:

A concave–convex elliptic problem involving the fractional Laplacian - C. Brandle, E. Colorado, A. de Pablo and U. Sánchez.

At the Pgs 41, 42, the authors cited the space $X ^\alpha,$ which is defined as

the completation of $C ^\infty _0 (\overline {\mathbb{R}_+^{N+1} })$ with respect to the norm $$\| \varphi \| _{X ^\alpha}= \sqrt{\int _{\mathbb{R} _+^{N+1}} y ^{1-\alpha} |\nabla \varphi| ^2 dxdy},$$ for $0<\alpha<2.$

My question:

is $X ^\alpha$ is a function space? I.e, is it true that for any cauchy sequence $(\varphi _n)$ in $( C ^\infty _0 (\overline {\mathbb{R}_+^{N+1} }), \| \cdot \| _{X ^\alpha}),$ there exists $\phi : \mathbb{R} _+^{N+1} \rightarrow \mathbb{R}$ such that

(i) $\varphi _n (x) \rightarrow \phi (x)$ a.e in $\mathbb{R}_+^{N+1};$

(ii) There exists $\phi _{z_j}$ in the weak sense and $( \varphi _n )_{z_j} \rightarrow \phi _{z_j}$ in $L ^2 (\mathbb{R}_+^{N+1} , y ^{1- \alpha}),$ for $j = 1, \ldots, N+1$ ?

Observe that the function $\eta (x,y) = y ^{1-\alpha}$ belongs to the Muckenhoupt class $A_2,$ also if there exists a inequality such as

$$\| \varphi \| _{L^p (\mathbb{R} _+^{N+1})} \leq C \| \varphi \| _{X ^\alpha}, \quad\forall \varphi \in C ^\infty _0 (\overline {\mathbb{R}_+^{N+1} })$$ for some $C>0$ and $p>1,$ (i) and (ii) follows immediatly.

Thank you.