An embedding theorem

Abstract:

We consider a weighted space $W_1^{(2)}(\mathbb R,q)$ of Sobolev type: \[ W_1^{(2)}(\mathbb R,q)=\left \{y\in AC_{\operatorname {loc}}^{(1)}(\mathbb R): \|y''\|_{L_1(\mathbb R)}+\|qy\|_{L_1(\mathbb R)}<\infty \right \},\] where $0\le q\in L_1^{\operatorname {loc}}(\mathbb R)$ and \[ \|y\|_{W_1^{(2)}(\mathbb R,q)}=\|y''\|_{L_1(\mathbb R)}+\|qy\|_{L_1(\mathbb R)}.\]

We obtain a precise condition which guarantees the embedding \[ W_1^{(2)}(\mathbb R,q)\hookrightarrow L_p(\mathbb R),\ p\in [1,\infty ).\]