An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

We consider the weighted space W ₁ ⁽²⁾(ℝ,q) of Sobolev type Here f ε L ₁(ℝ) and 0 ≥ q ∈ L ₁ ^loc(ℝ). We prove the following:The problems of embedding W ₁ ⁽²⁾(ℝq) {right arrow, hooked} L ₁(ℝ) and of correct solvability of (1) in L ₁(ℝ) are equivalent an embedding W ₁ ⁽²⁾(ℝ,q) {right arrow, hooked} L ₁(ℝ) exists if and only if.