Given $s\in (0,1)$ and a measurable function $u:\mathbb{R^n}\to\mathbb{C}$, let us define $$\|u\|_{\dot H^s(\mathbb{R}^n)}^2:=\iint\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy$$ and let $\dot H^s(\mathbb{R}^n)$ denote the completion of $C^\infty_c(\mathbb{R}^n)$ wrt this norm. One can define $H^s(\mathbb{R}^n)$ as the completion of $C^\infty_c(\mathbb{R}^n)$ wrt the stronger norm $\|u\|_{L^2}+\|u\|_{\dot H^s}$, as well.

(1) Clearly, $H^s(\mathbb{R}^n)\subseteq \dot H^s(\mathbb{R}^n)$. Is the inclusion an equality? In light of the open mapping theorem, this is equivalent to ask whether the norms are equivalent on $C^\infty_c(\mathbb{R}^n)$. If not, does $\dot H^s(\mathbb{R}^n)$ contain the constant functions?

(2) If we define instead $\dot H^s(\mathbb{R}^n)$ as the set of measurable functions $u$ such that $\|u\|_{\dot H^s}<\infty$, do we get a larger space?