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Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces. (English) Zbl 1023.46031
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.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Keywords:
function spaces
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