Caracterization of Sobolev spaces on the sphere

We introduce a new characterization of the Sobolev spaces $H^\alpha$ on the unit sphere $\mathbb{S}^{d-1}$, where the smoothness index $\alpha$ is any positive real number and $d\geq 2$. This characterization is in terms of a square function, it avoids the notion of differentiability and it is advanced in response to a comment in \cite{AMV}, where the authors characterize Sobolev spaces on $\mathbb{R}^d$. The result requires a subtle use of special functions and zonal Fourier multipliers.