On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

Let a be a semi-almost periodic matrix function with the almost periodic representatives al and ar at -∞ and +∞, respectively. Suppose p:R→(1,∞) is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space Lp(.)(R). We prove that if the operator aP+Q with P=(I+S)/2 and Q=(I-S)/2 is Fredholm on the variable Lebesgue space LNp(.)(R), then the operators alP+Q and arP+Q are invertible on standard Lebesgue spaces LNql(R) and LNqr(R) with some exponents ql and qr lying in the segments between the lower and the upper limits of p at -∞ and +∞, respectively.