| Title:
|
Boundedness and compactness of the embedding between spaces with multiweighted derivatives when $1 \leq q < p <\infty $ (English) |
| Author:
|
Abdikalikova, Zamira |
| Author:
|
Oinarov, Ryskul |
| Author:
|
Persson, Lars-Erik |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
61 |
| Issue:
|
1 |
| Year:
|
2011 |
| Pages:
|
7-26 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider a new Sobolev type function space called the space with multiweighted derivatives $W_{p,\bar {\alpha }}^n$, where $\bar {\alpha } = (\alpha _0, \alpha _1, \ldots , \alpha _n)$, $\alpha _i \in \Bbb R$, $i=0,1, \ldots , n$, and $\|f\|_{W_{p,{\bar \alpha }}^n} = \|D_{{\bar \alpha }}^n f\|_p + \sum _{i=0}^{n-1} |D_{\bar \alpha }^i f(1)|$, $$ D_{{\bar \alpha }}^0 f(t) = t^{\alpha _0} f(t), \quad D_{{\bar \alpha }}^i f(t) = t^{\alpha _i} \frac {{\rm d}}{{\rm d}t} D_{{\bar \alpha }}^{i-1} f(t), \enspace i=1, 2, \ldots , n. $$ 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 \leq q < p < \infty $, $0\leq m <n$. (English) |
| Keyword:
|
weighted function space |
| Keyword:
|
multiweighted derivative |
| Keyword:
|
embedding theorems |
| Keyword:
|
compactness. |
| MSC:
|
46E30 |
| MSC:
|
46E35 |
| idZBL:
|
Zbl 1224.46062 |
| idMR:
|
MR2782756 |
| DOI:
|
10.1007/s10587-011-0014-1 |
| . |
| Date available:
|
2011-05-23T12:26:56Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141515 |
| . |
| Reference:
|
[1] Abdikalikova, Z. T., Baiarystanov, A., Oinarov, R.: Compactness of embedding between spaces with multiweighted derivatives -- the case $p\leq q$.Math. Inequal. Appl Submitted. |
| Reference:
|
[2] Abdikalikova, Z. T., Kalybay, A. A.: Summability of a Tchebysheff system of functions.J. Funct. Spaces Appl. 8 (2010), 87-102. Zbl 1189.41013, MR 2648767, 10.1155/2010/405313 |
| Reference:
|
[3] Andô, T.: On compactness of integral operators.Nederl. Akad. Wet., Proc., Ser. A 65 24 (1962), 235-239. MR 0139016 |
| Reference:
|
[4] Kalybay, A. A.: Interrelation of spaces with multiweighted derivatives.Vestnik Karaganda State University (1999), 13-22 Russian. |
| Reference:
|
[5] Kudryavtsev, L. D.: Equivalent norms in weighted spaces.Proc. Steklov Inst. Math. 170 (1987), 185-218. Zbl 0616.46033, MR 0790335 |
| Reference:
|
[6] Nikol'skiĭ, S. M.: Approximation of Functions of Several Variables and Imbedding Theorems, 2nd ed., rev. and suppl.Nauka Moskva (1977), Russian. MR 0506247 |
| Reference:
|
[7] Oinarov, R.: Boundedness and compactness of superposition of fractional integration operators and their applications.In: Function Spaces, Differential Operators and Nonlinear Analysis 2004 Math. Institute, Acad. Sci. Czech Republic (2005), 213-235 (www.math.cas.cz/fsdona2004/oinarov.pdf). |
| . |
