AN EMBEDDING THEOREM

• Published in: Proceedings of the American Mathematical Society Vol. 141; no. 9; pp. 3213 - 3221
• Main Authors: CHERNYAVSKAYA, N. A., SHUSTER, L. A.
• Format: Journal Article
• Language: English
• Published: American Mathematical Society 01-09-2013
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-2013-11805-8

We consider a weighted space W
1
(2)
(ℝ,q) of Sobolev type:
${\mathrm{W}}_{1}^{\left(2\right)}(\mathrm{\mathbb{R}},\mathrm{q})=\{\mathrm{y}\in \mathrm{A}{\mathrm{C}}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\left(1\right)}(\mathrm{\mathbb{R}}):\Vert \mathrm{y}\prime \prime {\Vert }_{{\mathrm{L}}_{1}\left(\mathrm{\mathbb{R}}\right)}+\Vert \mathrm{q}\mathrm{y}{\Vert }_{{\mathrm{L}}_{1}\left(\mathrm{\mathbb{R}}\right)}<\mathrm{\infty }\}$
, where 0 ≤ q ∈
$0\le \mathrm{q}\in {\mathrm{L}}_{1}^{\mathrm{l}\mathrm{o}\mathrm{c}}\left(\mathrm{\mathbb{R}}\right)$
and
$\Vert \mathrm{y}{\Vert }_{{\mathrm{W}}_{1}^{\left(2\right)}(\mathrm{\mathbb{R}},\mathrm{q})}={\Vert \mathrm{y}\prime \prime \Vert }_{{\mathrm{L}}_{1}\left(\mathrm{\mathbb{R}}\right)}+\Vert \mathrm{q}\mathrm{y}{\Vert }_{{\mathrm{L}}_{1}\left(\mathrm{\mathbb{R}}\right)}$
.
We obtain a precise condition which guarantees the embedding
${\mathrm{W}}_{1}^{\left(2\right)}(\mathrm{\mathbb{R}},\mathrm{q})\hookrightarrow {\mathrm{L}}_{\mathrm{p}}\left(\mathrm{\mathbb{R}}\right), \ \mathrm{p}\in [1,\mathrm{\infty })$
.