Relations between convolution type operators on intervals and on the half-line
The main object of the paper is an operator of the form \({\mathcal W}_{\Phi_{\mathcal A}, \Omega}= r_{\Omega} {\mathcal A}_{|\widetilde{H}^{r,p}(\Omega)} : \widetilde{H}^{r,p}(\Omega) \rightarrow H^{s,p}(\Omega)\), acting between Bessel potential spaces, where \(r,s \in {\mathbb R}\), \(p \in (1,\infty)\), \(\Omega\) is a finite union of intervals on \(\mathbb R\), \({\mathcal A}= {\mathcal F}^{-1} \Phi_{\mathcal A} {\mathcal F}^{-1}\) is a convolution type operator from \(H^{r,p}(\mathbb R)\) into \(H^{s,p}(\mathbb R)\), and \(r_{\Omega}\) is the restriction of distributions from \({\mathcal S}'({\mathbb R})\) to \(\Omega\). The operator \({\mathcal W}_{\Phi_{\mathcal A}, \Omega}\) and a matrix Wiener-Hopf operator \(W_{\Phi}\) on the half axis \({\mathbb R}_+\) are said to be matricially coupled if the following matrix equality holds \[ \begin{pmatrix} {\mathcal W}_{\Phi_{\mathcal A}, \Omega} & 0 \\ 0 & I_Y\end{pmatrix} = E \begin{pmatrix} W_{\Phi} & 0 \\ 0 & I_Z\end{pmatrix} F, \] where \(Y\), \(Z\) are Banach spaces, and \(E\), \(F\) are invertible bounded linear operators. One refers to the above matrix equality as to the equivalence after extension relation. The main result of the paper is a method to construct equivalence after extension relations for arbitrary unions of intervals and orders \(k= r,s \in {\mathbb R}\) which are not critical, i.e., \(k-1/p \notin {\mathbb Z}\). Some applications are given as well.
scientific article; zbMATH DE number 1507906
scientific article