Sobolev spaces and the Cayley transform

The generalised Cayley transform  $\mathcal{C}$ from an Iwasawa $N$-group into the corresponding real unit sphere  $\mathbb{S}$ induces isomorphisms between suitable Sobolev spaces $\mathcal{H}^\alpha(\mathbb{S})$ and $\mathcal{H}^\alpha(N)$. We study the differential of  $\mathcal{C}$, and we obtain a criterion for a function to be in  $\mathcal{H}^\alpha(\mathbb{S})$.