## 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:

 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

function spaces
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