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 $$\Reals^n,$$ the following inequality holds: $$\left(\frac{1+|x|^2}{1+|y|^2}\right)^t ~\leq~ 2^{|t|} (1+|x-y|^2)^{|t|}.$$

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