Metric characterization of the sum of fractional Sobolev spaces
Volume 258 / 2021
Studia Mathematica 258 (2021), 27-51
MSC: Primary 46E35.
DOI: 10.4064/sm190408-21-4
Published online: 3 November 2020
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+\cdots +u_\ell $ with $u_i \in \dot {W}^{s_i,p_i}(\Omega )$ if and only if $$ \iint _{\Omega \times \Omega } \min _{1 \le i \le \ell } \frac {|u (x) - u (y)|^{p_i}}{|x - y|^{n+s_ip_i}}\,\mathrm {d}x \,\mathrm {d}y \lt +\infty . $$
