A mollifying operator with a variable radius, and an inverse theorem on traces
Let \(\Omega\) be an open subset of \(R^ n\) with smooth boundary \(\Gamma\), \(\alpha\), \(p\in R\), \(1\leq p\leq \infty\) and \(r\in N\). The terminology and notations being as in \textit{S. M. Nikol'skij}, Approximation of functions of several variables and imbedding theorems, (Russian) (1977; Zbl 0496.46020), it is well known that \(W^ r_{p,\alpha}\subset W^{r- \ell}_{p,\alpha +\ell}\) and the embedding operator is continuous provided that \(0<r+\alpha -p^{-1}<r\), \(s_ 0\in N\) is the integer part of \(r+\alpha -p^{-1}\) and \(\ell \in N\), \(1\leq \ell <r-s_ 0\). Obviously, the reverse inclusion is not true, but the author constructs a bounded linear operator T:\(W^ r_{p,\alpha}\to \cap^{\infty}_{\ell =0}W^{r+\ell}_{p,\alpha -\ell}\) such that \(\partial^ sTf/\partial \nu^ s|_{\Gamma}=\partial^ sf/\partial \nu^ s|_{\Gamma}\), \(s=0,...,s_ 0-1\), where \(\nu\) denotes the outer normal to \(\Gamma\). The operator T is an averaging operator with variable radius and an approximately chosen kernel.
scientific article; zbMATH DE number 3962480