Weighted estimates for solutions of a Sturm-Liouville equation in the space L ₁(ℝ)

We consider the equation -y″(x)+q(x)y(x)=f(x), x ∈ ℝ, where f ∈ L ₁(ℝ), 0 ≤ q ∈ L ₁ ^loc(ℝ), inf _x∈ℝ ∫ _x-a ^x+a q(t)dt>0 for some a>0. Under these conditions, (1) is correctly solvable in L ₁(ℝ), i.e. (i) for any function f ∈ L ₁(ℝ), there exists a unique solution of (1), y ∈ L ₁(ℝ); (ii) there is an absolute constant c ₁ ∈ (0, ∞) such that the solution of (1), y ∈ L ₁(ℝ), satisfies the inequality ∥y∥ ₁ ≤ c ₁∥f∥ ₁ for all f ∈ L ₁(ℝ). In this work we strengthen the a priori inequality (1). We find minimal requirements for a given weight function θ ∈ L ₁ ^loc(ℝ), under which the solution of (1), y ∈ L ₁(ℝ), satisfies the estimate ∥θy∥ ₁ ≤ c ₂∥f∥ ₁ for all f ∈ L ₁(ℝ), where c ₂ is some absolutely positive constant.