Admissible pair of spaces for non-correctly solvable linear differential equations

We consider the differential equation [Presented Equation] Under these conditions, the above equation is not correctly solvable in L_p(ℝ) for any p ϵ [1, ∞). Let q∗(x) be the Otelbaev-type average of the function q(t), t ∞ ℝ, at the point t = x; let θ(x) be a continuous positive function for x ϵ ℝ, and [Presented Equation] We show that if there exists a constant c ϵ [1, ∞) such that the inequality c^-1q∗(x) ≤ θ(x) ≤ cq∗(x) holds for all x ϵ ℝ, then under some additional conditions for q the pair of spaces {L_p,θ(ℝ); L_p(ℝ)} is admissible for the considered equation.