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

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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.