Sum-intersection property of Sobolev spaces (Q1713254)

Let $1\le p \le \infty$. If $s \ge 0$ is an integer, then $W^s_p = W^s_p (\mathbb{R}^n)$ are the classical Sobolev spaces. If $s>0$ is not an integer, then $W^s_p = W^s_p (\mathbb{R}^n) = B^s_{p,p} (\mathbb{R}^n)$ are the special Besov or Slobodeckij spaces. A triple $(W^{s_1}_{p_1}, W^s_p, W^{s_2}_{p_2})$ is called admissible if \[ s= \theta s_1 + (1-\theta) s_2, \qquad \frac{1}{p} = \frac{\theta}{p_1} + \frac{1-\theta}{p_2}, \] for some $0 \le \theta \le 1$. It is the main aim of the paper to discuss for which admissible triples any $f\in W^s_p$ can be decomposed as \[ f= f_1 + f_2, \qquad f_j \in W^{s_j}_{p_j} \cap W^s_p, \quad j=1,2. \] This is mainly the case (regular triples, Theorem 1). Exceptional limiting cases, called irregular triples, are discussed in detail (Theorem 2) where the nowadays fashionable (though existing for some 40 years) spaces $F^s_{\infty,q}$ occur (with $bmo =F^0_{\infty,2}$). For the entire collection see [Zbl 1403.46004].