Metric characterization of the sum of fractional Sobolev spaces

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 $\mathrm{\ell }\in {\mathbb{N}}^{\ast }$, $s_i\in (0,1)$ and $p_i \in [1, +\infty)$,$u : \Omega \to $\mathbb{R}$$canbedecomposedas$u = u_1+\dotsc+u_\ell$with$u_i \in \dot{W}^{s_i,p_i}(\Omega)$ifandonlyif$$ \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. $$