Peetre's inequality

In mathematics, Peetre's inequality, named after Jaak Peetre, says that for any real number $t$ and any vectors $x$ and $y$ in ${\mathbb{ℝ}}^n,$ the following inequality holds: $${\left(\frac{1+|x{|}^2}{1+|y{|}^2}\right)}^t\le 2^{|t|}(1+|x-y{|}^2{)}^{|t|}.$$

The inequality was proved by J. Peetre in 1959 and has founds applications in functional analysis and Sobolev spaces.

See also

• List of inequalities – None

References

• Chazarain, J.; Piriou, A. (2011), Introduction to the Theory of Linear Partial Differential Equations, Studies in Mathematics and its Applications, Elsevier, p. 90, ISBN 9780080875354, https://books.google.com/books?id=Gh9XeWnOzagC&pg=PA90 .
• Ruzhansky, Michael; Turunen, Ville (2009), Pseudo-Differential Operators and Symmetries: Background Analysis and Advanced Topics, Pseudo-Differential Operators, Theory and Applications, 2, Springer, p. 321, ISBN 9783764385132, https://books.google.com/books?id=DDpz_MfxZrUC&pg=PA321 .
• Saint Raymond, Xavier (1991), Elementary Introduction to the Theory of Pseudodifferential Operators, Studies in Advanced Mathematics, 3, CRC Press, p. 21, ISBN 9780849371585, https://books.google.com/books?id=kD5ZCJDIg4oC&pg=PA21 .

External links

• Planetmath.org: Peetre's inequality