Boundedness and Fredholmness of pseudodifferential operators in variable exponent spaces

We prove a statement on the boundedness of a certain class of singular type operators in the weighted spaces L-p(.)(R-n, w) with variable exponent p(x) and a power type weight w, from which we derive the boundedness of pseudodifferential operators of Hormander class S-1,0(0) in such spaces. This gives us a possibility to obtain a necessary and sufficient condition for pseudodifferential operators of the class OPS1,0m with symbols slowly oscillating at infinity, to be Fredholm within the frameworks of weighted Sobolev spaces H-w(s,p(.))(R-n) with constant smoothness s, variable p(.)-exponent, and exponential weights w.