Metric characterization of the sum of fractional Sobolev spaces

Abstract

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 $$ \iint\limits_{\Omega \times \Omega} \min_{1 \le i \le \ell} \frac{|u (x) - u (y)|^{p_i}}{|x - y|^{n+s_ip_i}}\,\mathrm{d}x \,\mathrm{d}y <+\infty. $$