Pointwise inequalities and approximation in fractional Sobolev spaces (Q2773416)

Let \(L^{\alpha, p}(\mathbb{R}^n)\) be a fractional Sobolev space (i.e. a Bessel potential space), \(1< p<\infty\), \(\alpha> 0\), with norm \(\|\;\|_{\alpha, p}\), and let \(B_{\alpha, p}(E)\) denote the \((\alpha, p)\)-capacity of a set \(E\subset \mathbb{R}^n\). Suppose \(m\) is an integer such that \(0\leq m\leq \alpha- 1\) and \(0\leq \lambda< 1\). The following theorem is proved:NEWLINENEWLINENEWLINEFor every function \(f\in L^{\alpha, p}(\mathbb{R}^n)\) and every \(\varepsilon> 0\) there exists a function \(g\in C^{m,\lambda}(\mathbb{R}^n)\) and an open set \(\Omega\subset \mathbb{R}^n\) satisfying the conditionsNEWLINENEWLINENEWLINE1) \(B_{\alpha- m-\lambda, p}(\Omega)< \varepsilon\),NEWLINENEWLINENEWLINE2) \(D^\sigma f(x)= D^\sigma g(x)\) for all \(x\in \mathbb{R}^n\setminus\Omega\) and all multiindices \(|\sigma|\leq m\),NEWLINENEWLINENEWLINE3) \(\|f-g\|_{m+ 1,p}< \varepsilon\).NEWLINENEWLINENEWLINEThis result is proved applying the Calderón-Zygmund extension operator, separately for \(\lambda= 0\) and for \(0<\lambda< 1\). It extends a number of earlier results.