Composition in fractional Sobolev spaces.
A classical result about composition in Sobolev spaces asserts that if \(u\in W^{k}_p(\Omega )\cap L_{\infty}(\Omega )\) and \(\Phi\in C^k(\mathbb{R})\), then \(\Phi\circ u\in W^k_p(\Omega )\). Here \(\Omega\) denotes a smooth bounded domain in \(\mathbb{R}^N\), \(k\geq 1\) is an integer and \(1\leq p<\infty\). In particular, if \(u\in W^{k}_p(\Omega )\) with \(kp>N\) and \(\Phi\in C^k(\mathbb{R})\) then \(\Phi\circ u\in W^{k}_p(\Omega )\) since \(W^{k}_p\subset L_{\infty}\) by the Sobolev embedding theorem. When \(kp=N\), the situation is more delicate since \(W^{k}_p\) is not contained in \(L_{\infty}\). In the paper under review, the authors prove: if \(s>1\) is real, \(1<p<\infty\) with \(sp=N\), and if \(\Phi\in C^k(\mathbb{R})\), where \(k=[s]+1\), is such that \[ D^j\Phi \in L_{\infty}(\mathbb{R}) \quad\forall j\leq k, \] and if \(u\in W^s_p(\Omega )\) then \(\Phi\circ u\in W^s_p(\Omega )\). Here \(W^s_p(\Omega )\) denotes the restriction of the Besov space \(B^s_{p,p}(\mathbb{R}^N)\) to \(\Omega\) when \(s\) is not an integer.
scientific article; zbMATH DE number 1882973