Geodesic
An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 709-716
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We consider the weighted space $W_1^{(2)}(\mathbb R,q)$ of Sobolev type $$ W_1^{(2)}(\mathbb R,q)=\left \{y\in A_{\rm loc}^{(1)}(\mathbb R)\colon \|y''\|_{L_1(\mathbb R)}+\|qy\|_{L_1(\mathbb R)}\infty \right \} $$ and the equation $$ - y''(x)+q(x)y(x)=f(x),\quad x\in \mathbb R. \leqno (1) $$ Here $f\in L_1(\mathbb R)$ and $0\le q\in L_1^{\rm loc}(\mathbb R).$ \endgraf We prove the following: \item {1)} The problems of embedding $W_1^{(2)}(\mathbb R,q)\hookrightarrow L_1(\mathbb R)$ and of correct solvability of (1) in $L_1(\mathbb R) $ are equivalent; \item {2)} an embedding $W_1^{(2)}(\mathbb R,q)\hookrightarrow L_1(\mathbb R) $ exists if and only if $$\exists a>0\colon \inf _{x\in \mathbb R}\int _{x-a}^{x+a} q(t) {\rm d} t>0.$$
We consider the weighted space $W_1^{(2)}(\mathbb R,q)$ of Sobolev type $$ W_1^{(2)}(\mathbb R,q)=\left \{y\in A_{\rm loc}^{(1)}(\mathbb R)\colon \|y''\|_{L_1(\mathbb R)}+\|qy\|_{L_1(\mathbb R)}\infty \right \} $$ and the equation $$ - y''(x)+q(x)y(x)=f(x),\quad x\in \mathbb R. \leqno (1) $$ Here $f\in L_1(\mathbb R)$ and $0\le q\in L_1^{\rm loc}(\mathbb R).$ \endgraf We prove the following: \item {1)} The problems of embedding $W_1^{(2)}(\mathbb R,q)\hookrightarrow L_1(\mathbb R)$ and of correct solvability of (1) in $L_1(\mathbb R) $ are equivalent; \item {2)} an embedding $W_1^{(2)}(\mathbb R,q)\hookrightarrow L_1(\mathbb R) $ exists if and only if $$\exists a>0\colon \inf _{x\in \mathbb R}\int _{x-a}^{x+a} q(t) {\rm d} t>0.$$
DOI : 10.1007/s10587-012-0041-6
Classification : 34B24, 34B40, 46E35
Keywords: Sobolev space; embedding theorem; Sturm-Liouville equation 
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     title = {An embedding theorem for a weighted space of {Sobolev} type and correct solvability of the {Sturm-Liouville} equation},
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Chernyavskaya, Nina A.; Shuster, Leonid A. An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 709-716. doi: 10.1007/s10587-012-0041-6

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