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(.)(Gamma, w) on a class of composed Carleson curves Gamma 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+, d mu) where d mu is an invariant measure on multiplicative group R+ = {r is an element of R : r > 0}. (2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L-p(.)(Gamma, 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(.)(Gamma, w) where Gamma 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.