Wiener-Hopf Operators with Slowly Oscillating Matrix Symbols on Weighted Lebesgue Spaces

Abstract

Fredholm conditions and an index formula are obtained for Wiener-Hopf operators W(a) with slowly oscillating matrix symbols a on weighted Lebesgue spaces \(L^p_N({\mathbb{R}}_{+},w)\) where 1 < p < ∞, w is a Muckenhoupt weight on \({\mathbb{R}}\) and \(N \in {\mathbb{N}}\). The entries of matrix symbols belong to a Banach subalgebra of Fourier multipliers on \(L^{p}({\mathbb{R}},w)\) that are continuous on \({\mathbb{R}}\) and have, in general, different slowly oscillating asymptotics at ±∞. To define the Banach algebra SOp, w of corresponding slowly oscillating functions, we apply the theory of pseudodifferential and Calderon-Zygmund operators. Established sufficient conditions become a Fredholm criterion in the case of Muckenhoupt weights with equal indices of powerlikeness, and also for Muckenhoupt weights with different indices of powerlikeness under some additional condition on p, w and a.