Admissible Spaces for the Sturm-Liouville Equation

TL;DR

This paper characterizes the admissible weight functions for the Sturm-Liouville equation, ensuring existence, uniqueness, and boundedness of solutions within weighted Lp spaces.

Contribution

It provides new criteria for the functions μ, θ, and q that guarantee solution existence, uniqueness, and boundedness in weighted Lp spaces for the Sturm-Liouville equation.

Findings

Criteria for μ, θ, and q ensuring unique solutions.

Conditions under which solutions are bounded in weighted spaces.

Explicit bounds for solutions in terms of data functions.

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