Wiener-Hopf operators with semi-almost periodic matrix symbols on weighted Lebesgue spaces
The paper is devoted to the Fredholm theory of Wiener-Hopf operators \[ W_\Phi : [L^p(\mathbb{R}_+, w)]^N \rightarrow [L^p(\mathbb{R}_+, w)]^N \] with semi-almost periodic \(N\times N\) matrix Fourier symbols \(\Phi\), on weighted Lebesgue spaces with exponent \(p \in (1, +\infty)\), and considering \(w\) in a subclass of the Muckenhoupt weights. A Fredholm criterion and a Fredholm index formula for this \(W_\Phi\) are obtained when considering the almost periodic representatives of \(\Phi\) at \(\pm\infty\) to admit an \(AP\) factorization, or when the almost periodic representatives of \(\Phi\) at \(\pm\infty\) have absolutely convergent Bohr-Fourier series and the weight \(w\) satisfies \[ \lim_{|t|\to\infty} {\underset{x,y\in [t,t+1]}{\text{sup\,ess}}} \big|\ln w(x)-\ln w(y)\big| = 0. \] In addition, for \(N=1\), a (semi-)Fredholm criterion for \(W_\Phi\) is given, and invertibility criteria for almost periodic and ``Wiener almost periodic'' symbols are also deduced. In view of obtaining these fundamental characterizations, several auxiliary results are deduced. Namely, an analogue of the Sarason representation of semi-almost periodic matrix functions, and the inverse closedness of the algebras of almost periodic and semi-almost periodic matrices are proved in the weighted framework under study.
scientific article; zbMATH DE number 5617567