Abstract. Let M(R n ) be the class of bounded away from one and infinity functions p : R n → [1, ∞] such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space L p(·) (R n ). We show that if a belongs to the Hörmander class S n(ρ−1) ρ,δ with 0 < ρ ≤ 1, 0 ≤ δ < 1, then the pseudodifferential operator Op(a) is bounded on the variable Lebesgue space