For a bounded $\Omega\subset \mathbb{R}^n$ with Lipschitz boundary, there are various definitions of fractional Sobolev spaces (a.k.a. Sobolev-Slobodeckij spaces) on $\partial \Omega$, either by using the trace operator or by the completion of smooth functions. In most cases, the involved measure for the integration is the surface measure. It is often difficult to verify whether a given function is in the desired space based on these definitions.
It is also known that if the boundary $\partial \Omega$ can be locally parameterised with a sufficiently regular map $\phi:\mathbb{R}^{n-1}\to \mathbb{R}^n$, then one can also define an equivalent Sobolev norm through the map $\phi$.
Here I want to confirm my understanding of the statement by considering $\partial \Omega$ to be a unit hypersphere $\mathbb{S}^{n-1}$. It is well-known that all points on $\mathbb{S}^{n-1}$ can be uniquely parameterized by the spherical coordinate $\phi:\mathbb{R}^{n-1}\mapsto \mathbb{S}^{n-1}$. Then for each $s\ge 0$ and $p\ge 1$, is it possible to compute (or estimate) the $W^{s,p}(\partial \Omega)$-norm of (a sufficiently regular) $u$ using the spherical coordinate?
From this post, the claim seems to be correct for $s\in[0,1)$. For example, for $p=2$, $$ \int_{\partial \Omega}|u(x) |^2\mathrm{d}f(x) + \int_{\partial \Omega}\int_{\partial \Omega}\frac{|u(x) - u(y)|^2}{|x-y|^{n-1+2s}}\;\mathrm{d}f(x)\mathrm{d}f(y)$$ where $\mathrm{d}f$ denotes the surface measure on $\partial\Omega$. The $\mathrm{d}f(x)$ can be computed based on the Jacobian of the spherical coordinate and the Lebesgue measure on $\mathbb{R}^{n-1}$.
But I am not sure whether similar computation works for $s\ge 1$. In particular, I am not quite clear about the computation of $$ \int_{\partial \Omega}|Du(x) |^2\mathrm{d}f(x), $$ where $D u$ is the weak derivative of (a sufficiently regular) $u$ on the surface.
