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Boundedness and compactness of the embedding between spaces with multiweighted derivatives when 1 ⩽ q < p < ∞

  • Zamira Abdikalikova
  • Ryskul Oinarov
  • Lars-Erik Persson
Article
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Abstract

We consider a new Sobolev type function space called the space with multiweighted derivatives \( W_{p,\bar \alpha }^n \), where \( \bar \alpha \) = (α 0, α 1,…, α n ), α i ∈ ℝ, i = 0, 1,…, n, and \( \left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|} \),
$$ D_{\bar \alpha }^0 f(t) = t^{\alpha _0 } f(t),D_{\bar \alpha }^i f(t) = t^{\alpha _i } \frac{d} {{dt}}D_{\bar \alpha }^{i - 1} f(t),i = 1,2,...,n $$
We establish necessary and sufficient conditions for the boundedness and compactness of the embedding \( W_{p,\bar \alpha }^n \)\( W_{q,\bar \beta }^m \), when 1 ⩽ q < p < ∞, 0 ⩽ m < n.

Keywords

weighted function space multiweighted derivative embedding theorems compactness 

MSC 2010

46E35 46E30 

References

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2011

Authors and Affiliations

  • Zamira Abdikalikova
    • 1
    • 2
  • Ryskul Oinarov
    • 1
    • 2
  • Lars-Erik Persson
    • 3
    • 4
  1. 1.AstanaKazakhstan
  2. 2.L.N. Gumilyev Eurasian National UniversityAstanaKazakhstan
  3. 3.LuleåSweden
  4. 4.Luleå University of TechnologyLuleåSweden

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