Integral Inequalities for the Principal Fundamental System of Solutions of a Homogeneous Sturm-Liouville Equation

We consider the equation
−y′′(x)+q(x)y(x)=f(x),x∈R,
where f∈Lp(R), p∈[1,∞] (L∞(R):=C(R)) and
0≤q∈Lloc1(R);∃a>0:infx∈R∫x+ax−aq(t)dt>0,
(Condition (2) guarantees correct solvability of (1) in class Lp(R), p∈[1,∞].) Let y be a solution of (1) in class Lp(R), p∈[1,∞], and θ some non-negative and continuous function in R. We find minimal additional requirements to θ under which for a given p∈[1,∞] there exists an absolute positive constant c(p) such that the following inequality holds:
supx∈Rθ(x)|y(x)|≤c(p)∥f∥Lp(R)  ∀f∈Lp(R).