Weighted Estimates for Solutions of the General Sturm-Liouville Equation and the Everitt-Giertz Problem. I

Consider the equation -(r(x))y′(x))′ + q(x)y(x) = f(x), x ∈ ℝ, (∗) where f ∈ Lp(ℝ), p ∈ (1, ∞) and r > 0, q ≥ 0, 1/r ∈ L₁^loc(ℝ), q ∈L₁^loc(ℝ) lim _|d|→∞∫_x-d^x dt/r(t) ∫_x-d^x q(t) dt = ∞ ∀x ∈ ℝ. By a solution of (∗), we mean any function y absolutely continuous together with (ry′) and satisfying (∗) almost everywhere on ℝ. In addition, we assume that (∗) is correctly solvable in the space L_p (ℝ), i.e. (1) for any function f ∈ L_p^loc(ℝ), there exists a unique solution y ∈ L_p(ℝ) of (∗); (2) there exists an absolute constant c₁(p) > 0 such that the solution y ∈ L_p(ℝ) of (∗) satisfies the inequality ∥y∥L_p(ℝ) ≤ c₁(p)∥f∥L_p(ℝ) ∀f ∈ L_p(ℝ). (∗∗) We study the following problem on the strengthening estimate (∗∗). Let a non-negative function θ ∈ L₁^loc(ℝ) be given. We have to find minimal additional restrictions for θ under which the following inequality holds: ∥θy∥L_p(ℝ) ≤ c₂(p)∥f∥L_p(ℝ) ∀f ∈ L_p(ℝ). Here, y is a solution of (∗) from the class L_p(ℝ), and c₂(p) is an absolute positive constant.