# Article

Full entry | PDF   (0.2 MB)
Keywords:
weighted function space; multiweighted derivative; embedding theorems; compactness.
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 $1 \leq q < p < \infty$, $0\leq m <n$.
References:
[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.
[2] Abdikalikova, Z. T., Kalybay, A. A.: Summability of a Tchebysheff system of functions. J. Funct. Spaces Appl. 8 (2010), 87-102. DOI 10.1155/2010/405313 | MR 2648767 | Zbl 1189.41013
[3] Andô, T.: On compactness of integral operators. Nederl. Akad. Wet., Proc., Ser. A 65 24 (1962), 235-239. MR 0139016
[4] Kalybay, A. A.: Interrelation of spaces with multiweighted derivatives. Vestnik Karaganda State University (1999), 13-22 Russian.
[5] Kudryavtsev, L. D.: Equivalent norms in weighted spaces. Proc. Steklov Inst. Math. 170 (1987), 185-218. MR 0790335 | Zbl 0616.46033
[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
[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).

Partner of