If $u:\mathbb R^n \to \mathbb R$ satisfies $$\int_{\mathbb R^n}\int_{\mathbb R^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dxdy < \infty,$$ but $u$ is not in $L^2(\mathbb R^n)$, is $(-\Delta)^su$ well-defined?
Also, if we assume $u \in C^\infty(\mathbb R^n) \cap L^\infty(\mathbb R^n)$, does this imply $$\int_{\mathbb R^n}\int_{\mathbb R^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dxdy < \infty \ \ ?$$
The natural domain for $(-\Delta)^s$ is the homogeneous Sobolev space $\dot H^{2s}$, which you may define as the set of tempered distributions $u$ such that $\hat u$ is locally integrable and verifies $$ \int \vert\hat u(\xi)\vert^{2} \vert\xi\vert^{4s} d\xi<+\infty. $$ This is implied by your condition. It could be more convenient to use the homogeneous Besov space $\dot B^{2s}_{p,q}$ which has the same definition as the standard Besov space but where the partition of unity is concerned also with small frequencies: you write $$ 1=\sum_{\nu\in \mathbb Z}\phi_\nu(\xi), $$ where $\phi_\nu$ is supported where $\vert\xi\vert\in(2^{\nu-1},2^{\nu+1}) $ and then you ask $$ \bigl(2^{2s \nu}\Vert\phi_\nu(D) u\Vert_{L^p}\bigr)_\nu\in \ell^q(\mathbb Z). $$ Your last question is asking $C^\infty\cap L^\infty\subset H^s$, which is untrue for $s=0$ (take $u$ smooth equal to $1/\sqrt{x}$ for $x\ge 1$) and more untrue (!) for $s>0$ (take say $s=1$ $u(x)= e^{ie^{x^4}}e^{-x^2}$.
