On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces
Abstract
Let a be a semi-almost periodic matrix function with the almost periodic representatives a(l) and a(r) at infinity and +infinity, respectively. Suppose p : R - > (1, infinity) 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 aP + 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(ql)(R) and L-N(qr)(R) with some exponents q(l) and q(r) lying in the segments between the lower and the upper limits of p at -infinity and +infinity, respectively. (C) 2011 Elsevier Inc. All rights reserved.
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