  We introduce a non-linear criterion which allows us to determine when a
function can be written as a sum of functions belonging to homogeneous
fractional spaces: for $\ell \in \mathbb{N}^*$, $s_i\in (0, 1)$ and $p_i \in
[1, +\infty)$, $u : \Omega \to \mathbb{R}$ can be decomposed as $u =
u_1+\dotsc+u_\ell$ with $u_i \in \dot{W}^{s_i,p_i}(\Omega)$ if and only if $$
\iint\limits_{\Omega \times \Omega} \min_{1 \le i \le \ell} \frac{|u (x) - u
(y)|^{p_i}}{|x - y|^{n+s_ip_i}}\,\mathrm{d}x \,\mathrm{d}y <+\infty. $$
