Admissible Spaces for the Sturm-Liouville Equation

We consider the equation \begin{equation} -y''(x)+q(x)y(x)=f(x),\quad x\in \mathbb R \end{equation} where $f \in L_p^{loc}(\mathbb R),$ $p \in [1,\infty)$ and $0 < q \in L_1^{loc}(\mathbb R).$ By a solution of this equation we mean any function $y,$ absolutely continuous together with its derivative and satisfying the equation almost everywhere in $\mathbb R.$ Let positive and continuous functions $\mu(x)$ and $\theta(x)$ for $x \in \mathbb R$ be given. Let us introduce the spaces $L_p(\mathbb R,\mu) = \{f \in L_p^{loc}(\mathbb R): ||f||_{L_p(\mathbb R,\mu)}^p =\int_{-\infty}^\infty|\mu(x)f(x)|^p dx < \infty\},$ $L_p(\mathbb R,\theta) = \{f\in L_p^{loc}(\mathbb R):||f||_{L_p(\mathbb R,\theta)}^p = \int_{-\infty}^\infty|\theta(x)f(x)|^p dx < \infty\}.$ In the present paper, we obtain requirements to the functions $\mu,\theta$ and $q$ under which 1) for every function $f \in L_p(\mathbb R,\theta)$ there exists a unique solution of the equation $y \in L_p(\mathbb R,\mu)$ ; 2) there is an absolute constant $c(p) \in (0,\infty)$ such that regardless of he choice of a function $f \in L_p(\mathbb R,\theta)$ the solution of the equation satisfies the inequality $$\|y\|_{L_p(\mathbb R,\mu)} < c(p)\|f\|_{L_p(\mathbb R,\theta)}.$$