We consider a weighted space
W
1
(
2
)
(
R
,
q
)
W_1^{(2)}(\mathbb R,q)
of Sobolev type:
\[
W
1
(
2
)
(
R
,
q
)
=
{
y
∈
A
C
loc
(
1
)
(
R
)
:
‖
y
‖
L
1
(
R
)
+
‖
q
y
‖
L
1
(
R
)
>
∞
}
,
W_1^{(2)}(\mathbb R,q)=\left \{y\in AC_{\operatorname {loc}}^{(1)}(\mathbb R): \|y\|_{L_1(\mathbb R)}+\|qy\|_{L_1(\mathbb R)}>\infty \right \},
\]
where
0
≤
q
∈
L
1
loc
(
R
)
0\le q\in L_1^{\operatorname {loc}}(\mathbb R)
and
\[
‖
y
‖
W
1
(
2
)
(
R
,
q
)
=
‖
y
‖
L
1
(
R
)
+
‖
q
y
‖
L
1
(
R
)
.
\|y\|_{W_1^{(2)}(\mathbb R,q)}=\|y\|_{L_1(\mathbb R)}+\|qy\|_{L_1(\mathbb R)}.
\]
We obtain a precise condition which guarantees the embedding
\[
W
1
(
2
)
(
R
,
q
)
↪
L
p
(
R
)
,
p
∈
[
1
,
∞
)
.
W_1^{(2)}(\mathbb R,q)\hookrightarrow L_p(\mathbb R),\ p\in [1,\infty ).
\]