Correct Solvability, Embedding Theorems and Separability for the Sturm-Liouville Equation

TL;DR

This paper investigates the conditions under which the Sturm-Liouville equation is correctly solvable in Lp spaces, establishing a link between solvability and embedding theorems for a specific weighted function space.

Contribution

It provides a characterization of correct solvability for the Sturm-Liouville equation via embedding theorems in weighted function spaces.

Findings

Existence of embedding S_p^{(2)}(R,q) into L_p(R) is equivalent to correct solvability.

Provides necessary and sufficient conditions for correct solvability in Lp spaces.

Connects the properties of the potential function q with the solvability and embedding results.

Abstract

We consider the equation - y"(x)+q(x)y(x)=f(x), x\in R and the weighted function space S_p^{(2)}(R,q)=\{y\in AC_{\loc}^{(1)}(R):\|y"-qy\|_p+\|q^{1/p}y\|_p<\infty\}; p\in[1,\infty), f\in L_p(R)$ and $0\le q\in L_1^{\loc}(R)$. We show that there exists an embedding S_p^{(2)}(R,q)\hookrightarrow L_p(R) if and only if equation above is correctly solvable in $L_p(R).