Admissible Spaces for the Sturm-Liouville Equation

Abstract

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 $$\Vert y{\Vert }_{L_p(\mathbb{R},\mu )}<c(p)\Vert f{\Vert }_{L_p(\mathbb{R},\theta )}.$$