For any domain $\Omega \subset \mathbb{R}^{N}$, we define $W^{s}(\Omega)$ to be the space of all the distributions $u$ in $L^{2}(\Omega)$ such that $$ D^{\alpha} u \in L^{2}(\Omega), \quad|\alpha| \leq s $$ where $\alpha$ is a multiindex and $|\alpha|=\alpha_{1}+\cdots+\alpha_{N}$. The standard $W^s$- norm ||.||$_{W^{s}(\Omega)}$ is defined by $$ \|u\|_{W^{s}(\Omega)}^{2}=\sum_{|\alpha| \leq s}\left\|D^{\alpha} u\right\|_{(\Omega)}^{2}<\infty $$
My question is that: When is the standard $W^s$-norm equivalent with $\sum_{|\alpha| = s}\left\|D^{\alpha} u\right\|_{(\Omega)}^{2}$? (considering the constant function, it seems impossible?)
P.S. In my impression, I seem to have seen this equivalence somewhere (or just for $W^1$-norm?), but I haven't found it after looking through a lot of literature
The expression using only derivatives of order $s$ cannot be a norm on $W^s$ (think constant functions). You have to restrict to some subspace, where the subspace is not allowed to contain polynomials of order up to $s-1$. These are precisely the functions for which the weak derivatives of order $s$ vanish. One proof of equivalence can be done by contradiction, much like the proof of the standard Poincare inequality.