Spaces admissible for the sturm-liouville equation

We consider the equation -y″(x) + q(x)y(x) = f(x); x ∈ R (1) where f ∈ L^loc _p (R); p ∈ [1;1) and 0 q ∈2 L^loc ₁ (R): By a solution of (1) we mean any function y; absolutely continuous together with its derivative and satisfying (1) almost everywhere in R: Let positive and continuous functions μ(x) and θ;(x) for x ∈ R be given. Let us introduce the spaces L_p(R;μ) = {f∈2 ^loc _p (R) : //f//^pL_pp_(R;μ) = /∞-∞ /μ(x)f(x)jpdx < 1∞}. L_p(R;θ) = {f ∈ L^loc _p (R) : kfkp L_p(R;μ) = /∞-∞ /θ(x)f(x)/^pdx <∞} : In the present paper, we obtain requirements to the functions f∈L_p and q under which 1) for every function f ∈ L_p(R;θ) there exists a unique solution (1) y ∈ L_p(R;μ) of (1); 2) there is an absolute constant c(p) ∈ (0;1) such that regardless of the choice of a function f ∈ L_p(R;θ) the solution of (1) satisfies the inequality //y//L_p(R;μ) c(p)kfkL_p(R;θ):.