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 $ μ(x) $ and $ θ(x) $ for $ x \in \mathbb R $ be given. Let us introduce the spaces $ L_p(\mathbb R,μ) = \{f \in L_p^{loc}(\mathbb R): ||f||_{L_p(\mathbb R,μ)}^p =\int_{-\infty}^\infty|μ(x)f(x)|^p dx < \infty\}, $ $ L_p(\mathbb R,θ) = \{f\in L_p^{loc}(\mathbb R):||f||_{L_p(\mathbb R,θ)}^p = \int_{-\infty}^\infty|θ(x)f(x)|^p dx < \infty\}. $ In the present paper, we obtain requirements to the functions $μ,θ$ and $q$ under which 1) for every function $f \in L_p(\mathbb R,θ) $ there exists a unique solution of the equation $y \in L_p(\mathbb R,μ)$ ; 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,θ) $ the solution of the equation satisfies the inequality $$ \|y\|_{L_p(\mathbb R,μ)} < c(p)\|f\|_{L_p(\mathbb R,θ)}. $$