An elementary proof of the Brezis and Mironescu theorem on the composition operator in fractional Sobolev spaces (Q1602299)

Let \(1\leq p < \infty\) and \(1<s \not\in\mathbb{N}\). Let \(f\) be a complex-valued function on \({\mathbb R}\) with \(f(0)=0\) and \(f^{(l)} \in L_\infty ({\mathbb R})\) for \(l = 1,\dots,[s]+1\). Then one has for the composition \[ \|f(u) |W^s_p ({\mathbb R}^n) \|\leq c \sum^{[s]+1}_{l=1} \|f^{(l)} |L_\infty ({\mathbb R}) \|\cdot \left( \|u |W^s_p ({\mathbb R}^n)\|+ \|\nabla u |L_{ps} ({\mathbb R}^n) \|^s \right) . \]