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 a l and a r 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 L p ( ⋅ ) ( R ) . We prove that if the operator a P + Q with P = ( I + S ) / 2 and Q = ( I − S ) / 2 is Fredholm on the variable Lebesgue space L N p ( ⋅ ) ( R ) , then the operators a l P + Q and a r P + Q are invertible on standard Lebesgue spaces L N q l ( R ) and L N q r ( R ) with some exponents q l and q r lying in the segments between the lower and the upper limits of p at −∞ and +∞, respectively.