This paper is devoted to the boundedness and Fredholmness of SGpseudo-di erential operators with non-zero order in the Lebesgue spaces with variable exponent p(x): Moreover, we obtain a necessary and sufficient condition for SGpseudo-di erential operators of the class OPSGm1 ;0 to be Fredholm on Sobolev spaces Hs p(:) with constant smoothness s and variable exponent p(x):
