Wiener-Hopf operators with oscillating symbols and convolution operators on a union of intervals
Wiener-Hopf operators associated with matrix-valued symbols containing oscillating functions are studied from the point of view of Fredholm properties. The operators considered act between Sobolev spaces \(H_ \alpha(\mathbb{R})\) and are assumed to be of the general form \[ T_ w=r_ +F^{-1}WFI_ +:H^ +_ \alpha(\mathbb{R})\to H_ \beta(\mathbb{R}^ +), \] where \(H^ +_ \alpha(\mathbb{R})\) is the subspace of \(H_ \alpha(\mathbb{R})\) of elements with support contained in \(\mathbb{R}^ +\), \(H_ \beta(\mathbb{R}^ +)\) is the space of restrictions to \(\mathbb{R}^ +\) of distributions from \(H_ \beta(\mathbb{R})\), \(r_ +\) is the restriction operator to \(\mathbb{R}^ +\), \(I_ +\) is the extension operator to \(\mathbb{R}\), \(W\) is a matrix-valued function containing oscillating entries. The approach followed in this paper is based on a theorem of reduction of the symbol \(W(\xi)\) to a standard symbol \(G_ 0(\xi)\) i.e. a symbol possessing limits as \(\xi\to\pm\infty\) along the real line. The application to Fredholm properties of the convolution operators on a union of intervals is given.
Wiener-Hopf operators associated with matrix-valued symbols containing oscillating functions
scientific article; zbMATH DE number 222050
scientific article