Pseudodifferential operators on variable lebesgue spaces

Let M(ℝⁿ) be the class of bounded away from one and infinity functions p: ℝⁿ → [1,∞] such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space L^p(⋅)(ℝⁿ). 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 L^p(⋅)(ℝⁿ) provided that p∈M(ℝⁿ). Let M* (ℝⁿ) be the class of variable exponents p∈M(ℝⁿ) represented as 1/p(x) = θ/p₀ + (1 – θ)/p₁(x) where p₀ ∈ (1,∞), θ ∈(0, 1), and p₁ ∈M(ℝⁿ). We prove that if a ∈ S⁰ _1,0 slowly oscillates at infinity in the first variable, then the condition (Formula presented.) is sufficient for the Fredholmness of Op(a) on L^p(⋅)(ℝⁿ) whenever p∈M* (ℝⁿ). Both theorems generalize pioneering results by Rabinovich and Samko [24] obtained for globally log-Hölder continuous exponents p, constituting a proper subset of M* (ℝⁿ).