Boundedness and compactness of the embedding between spaces with multiweighted derivatives when $1\le q<p<\infty$

Abdikalikova, Zamira, Oinarov, Ryskul, and Persson, Lars-Erik. "Boundedness and compactness of the embedding between spaces with multiweighted derivatives when $1 \le q < p <\infty $." Czechoslovak Mathematical Journal 61.1 (2011): 7-26. <http://eudml.org/doc/196770>.

@article{Abdikalikova2011,
abstract = {We consider a new Sobolev type function space called the space with multiweighted derivatives $W_\{p,\bar\{\alpha \}\}^n$, where $\bar\{\alpha \} = (\alpha _0, \alpha _1, \ldots , \alpha _n)$, $\alpha _i \in \mathbb \{R\}$, $i=0,1, \ldots , n$, and $\Vert f\Vert _\{W_\{p,\{\bar\{\alpha \}\}\}^n\} = \Vert D_\{\{\bar\{\alpha \}\}\}^n f\Vert _p + \sum _\{i=0\}^\{n-1\} |D_\{\bar\{\alpha \}\}^i f(1)|$, \[ D\_\{\{\bar\{\alpha \}\}\}^0 f(t) = t^\{\alpha \_0\} f(t), \quad D\_\{\{\bar\{\alpha \}\}\}^i f(t) = t^\{\alpha \_i\} \frac\{\{\rm d\}\}\{\{\rm d\}t\} D\_\{\{\bar\{\alpha \}\}\}^\{i-1\} f(t), \hspace\{5.0pt\}i=1, 2, \ldots , n. \] We establish necessary and sufficient conditions for the boundedness and compactness of the embedding $W_\{p,\{\bar\{\alpha \}\}\}^n \hookrightarrow W_\{q,\{\bar\{\beta \}\}\}^m $, when $1 \le q < p < \infty $, $0\le m <n$.},
author = {Abdikalikova, Zamira, Oinarov, Ryskul, Persson, Lars-Erik},
journal = {Czechoslovak Mathematical Journal},
keywords = {weighted function space; multiweighted derivative; embedding theorems; compactness; weighted function space; multiweighted derivative; embedding theorem; compactness},
language = {eng},
number = {1},
pages = {7-26},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Boundedness and compactness of the embedding between spaces with multiweighted derivatives when $1 \le q < p <\infty $},
url = {http://eudml.org/doc/196770},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Abdikalikova, Zamira
AU - Oinarov, Ryskul
AU - Persson, Lars-Erik
TI - Boundedness and compactness of the embedding between spaces with multiweighted derivatives when $1 \le q < p <\infty $
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 1
SP - 7
EP - 26
AB - We consider a new Sobolev type function space called the space with multiweighted derivatives $W_{p,\bar{\alpha }}^n$, where $\bar{\alpha } = (\alpha _0, \alpha _1, \ldots , \alpha _n)$, $\alpha _i \in \mathbb {R}$, $i=0,1, \ldots , n$, and $\Vert f\Vert _{W_{p,{\bar{\alpha }}}^n} = \Vert D_{{\bar{\alpha }}}^n f\Vert _p + \sum _{i=0}^{n-1} |D_{\bar{\alpha }}^i f(1)|$, \[ D_{{\bar{\alpha }}}^0 f(t) = t^{\alpha _0} f(t), \quad D_{{\bar{\alpha }}}^i f(t) = t^{\alpha _i} \frac{{\rm d}}{{\rm d}t} D_{{\bar{\alpha }}}^{i-1} f(t), \hspace{5.0pt}i=1, 2, \ldots , n. \] We establish necessary and sufficient conditions for the boundedness and compactness of the embedding $W_{p,{\bar{\alpha }}}^n \hookrightarrow W_{q,{\bar{\beta }}}^m $, when $1 \le q < p < \infty $, $0\le m <n$.
LA - eng
KW - weighted function space; multiweighted derivative; embedding theorems; compactness; weighted function space; multiweighted derivative; embedding theorem; compactness
UR - http://eudml.org/doc/196770
ER -