Let $\Gamma \subset \mathbb{R}^{n+1}$ be a $n$-dimensional (closed) $C^k$-hypersurface. Consider the problem $$\nabla_\Gamma \cdot (y^{1-2s}\nabla_\Gamma u)(x,y) =0 \quad \text{for $(x,y) \in \Gamma \times \mathbb{R}^+$}$$ $$u(x,0) = f(x) \quad \text{for $x \in \Gamma$.}$$ Is it true that one can define the fractional Laplace-Beltrami operator as $$(-\Delta_\Gamma)^{s}f = -d_s\lim_{y \to 0}y^{1-2s}\partial_y u$$ where $d_s$ is a constant involving the Gamma function? Here, $\nabla_\Gamma$ is the surface or tangential gradient.
I have seen something like this for non-compact manifolds (paper of Banica, Gonzalez, Saez) and I must confess my knowledge of manifolds is not good so I am doubtful whether the result carries over so simply. I would truly appreciate a reference to this extension for the case of hypersurfaces or compact manifolds and a study of what kind of spaces are involved here.
PS: I am aware that this type of problem is called the Caffarelli-Silvestre extension, which they did for the $\mathbb{R}^n$ case. I just want to know about the generalisation to hypersurfaces.
