View source for Peetre's inequality

In [[mathematics]], '''Peetre's inequality,''' named after [[Jaak Peetre]], says that for any [[real number]] <math>t</math> and any [[Vector space|vector]]s <math>x</math> and <math>y</math> in <math>\Reals^n,</math> the following [[Inequality (mathematics)|inequality]] holds:
<math display=block>\left(\frac{1+|x|^2}{1+|y|^2}\right)^t ~\leq~ 2^{|t|} (1+|x-y|^2)^{|t|}.</math>

The inequality was proved by J. Peetre in 1959 and has founds applications in [[functional analysis]] and [[Sobolev space]]s.

==See also==

* {{annotated link|List of inequalities}}

==References==

{{reflist}}

* {{citation|title=Introduction to the Theory of Linear Partial Differential Equations|series=Studies in Mathematics and its Applications|first1=J.|last1=Chazarain|first2=A.|last2=Piriou|publisher=Elsevier|year=2011|isbn=9780080875354|page=90|url=https://books.google.com/books?id=Gh9XeWnOzagC&pg=PA90}}.
* {{citation|title=Pseudo-Differential Operators and Symmetries: Background Analysis and Advanced Topics|volume=2|series=Pseudo-Differential Operators, Theory and Applications|first1=Michael|last1=Ruzhansky|first2=Ville|last2=Turunen|publisher=Springer|year=2009|isbn=9783764385132|page=321|url=https://books.google.com/books?id=DDpz_MfxZrUC&pg=PA321}}.
* {{citation|title=Elementary Introduction to the Theory of Pseudodifferential Operators|volume=3|series=Studies in Advanced Mathematics|first=Xavier|last=Saint Raymond|publisher=CRC Press|year=1991|isbn=9780849371585|page=21|url=https://books.google.com/books?id=kD5ZCJDIg4oC&pg=PA21}}.

{{PlanetMath attribution|id=4681|title=Peetre's inequality}}

==External links==

* [http://planetmath.org/peetresinequality Planetmath.org: Peetre's inequality]

{{Banach spaces}}

{{mathanalysis-stub}}

[[Category:Functional analysis]]
[[Category:Inequalities]]
[[Category:Linear algebra]]