Singular Integral Operators with Piecewise Almost Periodic Coefficients and Carleman Shift

Abstract

A Fredholm criterion and an index formula for singular integral operators with fixed singularities and piecewise almost periodic matrix coefficients on the space L ⁿ _p (ℝ) are obtained. The proof is based on the fact that the operators

$$I + \sum\limits_j {{B_j}{H_j} and I + \sum\limits_j {M({B_j}){H_j},} }$$

where B_j are almost periodic matrix functions, M(B_j) are their Bohr mean values and H_j are integral operators with fixed singularities at infinity, are Fredholm on L ⁿ _p (ℝ) only simultaneously, and that in this case their indices coincide. Applying this result, more general singular integral operators with fixed singularities at ∞ and semi-almost periodic coefficients are studied. The Fredholm study of singular integral operators with fixed singularities and piecewise almost periodic coefficients is reduced to the study of singular integral operators with piecewise continuous matrix coefficients and singular integral operators with fixed singularities at ∞ and semi-almost periodic matrix coefficients. Based on a reduction of operators with a shift to singular integral operators without shift, a Fredholm theory for singular integral operators with a Carleman backward shift and oscillating matrix coefficients on Lebesgue spaces over the real line is developed.