Spaces admissible for the Sturm-Liouville equation

We consider the equation

$-{y}''(x)+q(x)y(x)=f(x),\ \ \ \ x\in \mathbb{R}\text{ }\ \ \ \ \ \ \ \ \ \ \left( 1 \right)$

where $f∈ L_p^{\text{loc}}(\mathbb R),$ $p∈[1,∞)$ and $0≤ q∈ L_1^{\text{loc}}(\mathbb R).$ By a solution of (1) we mean any function $y,$ absolutely continuous together with its derivative and satisfying (1) almost everywhere in $\mathbb R.$ Let positive and continuous functions $μ(x)$ and $θ(x)$ for $x∈\mathbb R$ be given. Let us introduce the spaces

$\begin{align} & {{L}_{p}}(\mathbb{R},\mu )=\left\{ f\in L_{p}^{\text{loc}}(\mathbb{R}):\|f\|_{{{L}_{p}}(\mathbb{R},\mu )}^{p}=\int_{-\infty }^{\infty }{|}\mu (x)f(x){{|}^{p}}dx < \infty \right\}, \\ & {{L}_{p}}(\mathbb{R},\theta )=\left\{ f\in L_{p}^{\text{loc}}(\mathbb{R}):\|f\|_{{{L}_{p}}(\mathbb{R},\theta )}^{p}=\int_{-\infty }^{\infty }{|}\theta (x)f(x){{|}^{p}}dx <\infty \right\}. \\ \end{align}$

In the present paper, we obtain requirements to the functions $μ,θ$ and $q$ under which

1) for every function $f∈ L_p(\mathbb R,θ)$ there exists a unique solution (1) $y∈ L_p(\mathbb R,μ)$ of (1);

2) there is an absolute constant $c(p)∈(0,∞)$ such that regardless of the choice of a function $f∈ L_p(\mathbb R,θ)$ the solution of (1) satisfies the inequality

$\|y\|_{L_p(\mathbb R,μ)}≤ c(p)\|f\|_{L_p(\mathbb R,θ)}.$