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⁾ (?,q) of Sobolev type $$W_1^{(2)} (mathbb{R},q) = left{ {y in A_{loc}^{(1)} (mathbb{R}):left| {y''} right|_{L_1 (mathbb{R})} + left| {qy} right|_{L_1 (mathbb{R})} < infty } right} $$ and the equation $$ - y''(x) + q(x)y(x) = f(x),x in mathbb{R} $$ Here f ε L ₁(?) and 0 ? q ∈ L ₁ ^loc (?). We prove the following:

1. The problems of embedding W ₁ ⁽²⁾ (?q) ? L ₁(?) and of correct solvability of (1) in L ₁(?) are equivalent
2. an embedding W ₁ ⁽²⁾ (?,q) ? L ₁(?) exists if and only if $$exists a > 0:mathop {inf }limits_{x in R} int_{x - a}^{x + a} {q(t)dt > 0} $$