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 $\ell \in {\mathbb{N}}^{\ast }$, $s_i\in (0,1)$ and $p_i\in [1,+\infty )$, $u:\Omega \to \mathbb{R}$ can be decomposed as $u=u_1+\dots +u_{\ell }$ with $u_i\in {\dot{W}}^{s_i,p_i}(\Omega )$ if and only if $$\underset{\Omega \times \Omega }{\iint }\underset{1\le i\le \ell }{\min }\frac{|u(x)-u(y){|}^{p_i}}{|x-y{|}^{n+s_i{p}_i}}\mathrm{d}x\mathrm{d}y<+\infty .$$

19 pages