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

Abdikalikova, Zamira ; Oinarov, Ryskul ; Persson, Lars-Erik

Czechoslovak Mathematical Journal, Tome 61 (2011) no. 1, pp. 7-26

Résumé

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 $.

Détail

DOI : 10.1007/s10587-011-0014-1

Classification : 46E30, 46E35
Keywords: weighted function space; multiweighted derivative; embedding theorems; compactness.