Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L^p(·) (Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces L^p(·) (ℝ₊, dμ) where dμ is an invariant measure on multiplicative group ℝ₊ = {r ∈ ℝ : r > 0} . (2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L^p(·) (Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on ℝ₊ and local invertibility of singular integral operators on ℝ (3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L^p(·) (Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities.