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Imbedding Theorems for Different Metrics and Dimensions

  • S. M. Nikol’skii
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 205)

Abstract

We begin by presenting the formulation of the imbedding theorem of S. L. Sobolev [3], with later complements due to V. I. Kondrašov [1] and V. P. Il’in [2]1. As applied to the space ℝ n and to its coordinate subspace ℝ m (1 ≦ m ≦), this theorem reads:If the function ƒ ∈ W p l n ) and (1) $$0{\mathop<\limits_=}\ \varrho = l - {n\over p}+{m\over p^\prime}, \ 1\ {\mathop<\limits_=}\ p.<p^\prime <\infty,$$ P ? then2 (2) $$W_p^l({\rm R}_n) \rightarrow W_{p^\prime}^{\lbrack \varrho \rbrack}({\rm R}_m),$$ where [ϱ] is the integer part of ϱ. This means that there exists a trace of the function ƒ|m =φ, lying in the class Wp′(ϱ)(ℝm), and that the inequality (3) $$\|\varphi\|_{W_{p\prime}^{\lbrack \varrho \rbrack}({\rm R}_m)}\ {\mathop<\limits_=}\ {\cal C}\|f\|_{W_p^l({\rm R}_n)}$$ is satisfied, where c does not depend on ƒ.

Keywords

Entire Function Trigonometric Polynomial Exponential Type Integer Vector Inverse Theorem 
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 1975

Authors and Affiliations

  • S. M. Nikol’skii
    • 1
  1. 1.Sergei Mihailovič Nikol’skii Steklov MathematicalInstitute Academy of SciencesMoscowUSA

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