Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces
Let \(1\leq s \leq m \in\mathbb{N}\) and \(1<p< \infty\). Let \(f\) be a function on \(\mathbb{R}\) having bounded derivatives up to order \(m\). It is the main aim of this paper to prove that \(\psi \mapsto f(\psi)\) is well-defined and continuous as a map from \(W^s_p (\mathbb{R}^n) \cap W^1_{sp} (\mathbb{R}^n)\) into \(W^s_p (\mathbb{R}^n)\). In addition, the paper deals with compositions and products in spaces of type \(F^s_{pq} (\mathbb{R}^n)\) and \(F^s_{pq} (\mathbb{R}^n) \cap L_\infty (\mathbb{R}^n)\), complementing existing results of this type by Runst and Sickel.
scientific article; zbMATH DE number 1757480