Split the $N-$dimensional Euclidian space as $\mathbb{R}^N = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$. A vector in $\mathbb{R}^N$ will be denoted by $z = (x,y)$. Let $\alpha > 0$ and consider the space $$ S^{1,2}(\Omega) = \left\{u \in L^2(\Omega) : u_{x_i} \in L^2(\Omega), |x|^{\alpha}u_{y_j}\in L^2(\Omega), i=1,...,N_1, j=1,...,N_2 \right\}. $$ This space is endowed with the inner product $$ (u,v) = \int_{\Omega} [\nabla_x u(z) \cdot \nabla_x v(z) + |x|^{2\alpha} \nabla_y u(z) \cdot \nabla_y v(z)] + \int_{\Omega} uv. $$ The respective norm is $$ ||u||^2 = \int_{\Omega} |\nabla_x u(z)|^2 + |x|^{2\alpha}|\nabla_y u(z)|^2 + \int_{\Omega} u^2. $$ This space is related to the Grushin operator. Notice when $\alpha = 0$, $S^{1,2}(\Omega) = W^{1,2}(\Omega)$.
In the paper "Compact Embedding of a Degenerate Sobolev Space and Existence of Entire Solutions to a Semilinear Equation for a Grushin-type Operator" is said the $S^{1,2}(\Omega)$ is a Hilbert space. I'm trying to prove it, but I got stuck. Any sugestion is welcome.