An asymptotic majorant for solutions of Sturm-liouville equations in L _P(ℝ)

Under certain assumptions on g(x), we obtain an asymptotic formula for computing integrals of the form F(x, α) = ∫_-∞ ^∞ g(t)^αexp (- |∫_x^t g(ξ) dξ|) dt, α ∈ ℝ, as |x| → ∞. We use this formula to study the properties (as |x| → ∞) of the solutions of the correctly solvable equations in L_P(ℝ), p ∈ [1, ∞], -y″(x) + q(x)y(x) = f(x), x ∈ ℝ, (1) where 0 ≤ q ∈ L₁^loc(ℝ), and f ∈ L_P(ℝ). (Equation (1) is called correctly solvable in a given space L _p(ℝ) if for any function f ∈ L_P(ℝ) it has a unique solution y ∈ L_p(ℝ) and if the following inequality holds with an absolute constraint c_p ∈ (0, ∞): ||y||L _p,(ℝ) ≤ c(p)||f||L_p(ℝ), ∀f ∈ L_P(ℝ).).