Interplay of Wiener--Hopf and Hankel operators with almost periodic Fourier symbols on standard and variable exponent Lebesgue spaces

Abstract

Wiener--Hopf plus Hankel and Wiener--Hopf minus Hankel operators in both frameworks of standard and variable exponent Lebesgue spaces are considered in this paper. The main aim is to describe certain dependencies between the Fredholm property of some Wiener--Hopf operators acting between variable exponent Lebesgue spaces and the invertibility of Wiener--Hopf plus and minus Hankel operators on all the standard Lebesgue spaces. Different types of Fourier symbols will be used but special focus will be considered on the Wiener subclass of almost periodic matrix functions. In the first part of the paper we will give a survey of investigations on related results. This will be useful at the end of the paper to derive the above mentioned dependencies between the operators under study.