# Imbedding Theorems for Different Metrics and Dimensions

• S. M. Nikol’skii
Chapter
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
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