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

Abstract

We consider the weighted space W ⁽²⁾₁ (ℝ,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(ℝ) and 0 ⩾ q ∈ L ^loc₁ (ℝ).

We prove the following:

• 1) The problems of embedding W ⁽²⁾₁ (ℝq) ↪ L ₁(ℝ) and of correct solvability of (1) in L ₁(ℝ) are equivalent

• 2) an embedding W ⁽²⁾₁ (ℝ,q) ↪ L ₁(ℝ) exists if and only if

  $$\exists a > 0:\mathop {\inf }\limits_{x \in R} \int_{x - a}^{x + a} {q(t)dt > 0} $$