A relation between convolution type operators on intervals in sobolev spaces

We construct an operator relation between convolution type operators with or without a reflection on a union of finite intervals and corresponding Wiener-Hopf operators. This relation is the reult for several other relations between intermediate operators constructed for that purpose. The presented relations are obtained with the help of different extension methods that annulate particular actions of the related operators. All the operators are defined in Bessel potential spaces or Sobolev spaces. In particular, the final relation enables us to derive properties from a Wiener-Hopf operator to the intial one. An example of application of the presented results is done in a differaction problem by a union of two strips