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Approximation numbers of Sobolev embeddings

  • David E. Edmunds
  • W. Desmond Evans
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

Suppose |Ω| < ∞ and let W 1 p ,c (Ω) denote the quotient space W 1 p (Ω)/C with norm
$$ \parallel \left[ f \right]\parallel W_{p,C}^1(\Omega )\parallel : = \parallel \nabla f{\parallel _{p,}}\Omega , $$
(6.1.1)
where [•] denotes an equivalence class in W 1 p ,c (Ω) of functions which differ a.e. by a constant. The quotient space L p (Ω)/C will be denoted by Lp,c(Ω) and its norm by
$$ \parallel \left[ f \right]\parallel {L_{_{P,C}}}(\Omega )\parallel \equiv \parallel \left[ f \right]{\parallel _{p,\Omega }}: = \mathop {\lim }\limits_{c \in \mathbb{C}} \parallel f - c{\parallel _{p,\Omega .}} $$
(6.1.2)

Keywords

Quotient Space Approximation Number Sobolev Embedding Hardy Operator Poincare Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • David E. Edmunds
    • 1
  • W. Desmond Evans
    • 2
  1. 1.Department of MathematicsSussex UniversityBrightonUK
  2. 2.School of MathematicsCardiff UniversityCardiffUK

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