Peetre's inequality

The Peetre inequality , named after Jaak Peetre , is an inequality from the mathematical subfield of functional analysis , more precisely from the theory of Hilbert spaces .

It is a Hilbert dream. Then the inequality holds for all and for all real numbers ${\displaystyle H}$ ${\displaystyle x,y\in H}$ ${\displaystyle t}$

${\displaystyle {\frac {(1+\|x\|^{2})^{t}}{(1+\|y\|^{2})^{t}}}\leq 2^{|t|}\cdot (1+\|x-y\|^{2})^{|t|}\,.}$

This inequality was proven by J. Peetre in 1959 and is used for numerical and theoretical estimates. If you consider the above inequality

${\displaystyle (1+\|x\|^{2})^{t}\leq 2^{|t|}\cdot (1+\|x-y\|^{2})^{|t|}\cdot (1+\|y\|^{2})^{t}}$

um, one recognizes that this estimation can be helpful in Sobolev spaces of real-valued order , because there even functions of the form occur under an integral . An application of Peetre's inequality in this direction can be found in the textbook given below when examining multiplication operators on Sobolev spaces. ${\displaystyle (1+\|x\|^{2})^{t}}$

Individual evidence

1. ^ J. Heine: Topology and Functional Analysis , Oldenbourg Verlag (2002), ISBN 3-486-24914-2 , sentence 1.1-10
2. ↑ J. Peetre: Une Characterization abstraite the surgeon differentiels , Math Scandinavica, Volume 7 (1959), pages 211-118 (J. Peetre: Rectification à l'article "Une Characterization abstraite the surgeon differentiels" , Math Scandinavica, Volume 8 ( 1960), pages 116-120)
3. ^ Herbert Schröder: functional analysis , Harri Deutsch Verlag (2000), ISBN 3-8171-1623-3 , sentence 6.1.7