PlanetMath: Peetre's inequality

(more info)

Math for the people, by the people.

Main Menu

Peetre's inequality

(Theorem)

Theorem [Peetre's inequality] [1,2] If is a real number and are vectors in $\mathbb{R}^n$ , then

$\displaystyle \Big( \frac{1+\vert x\vert^2}{1+\vert y\vert^2} \Big)^t \le 2^{\vert t\vert} (1+\vert x-y\vert^2)^{\vert t\vert}.$

Proof. (Following [1].) Suppose and are vectors in $\mathbb{R}^n$ . Then, from $(\vert b\vert-\vert c\vert)^2\ge 0$ , we obtain

$\displaystyle 2\vert b\vert \cdot \vert c\vert \le \vert b\vert^2 + \vert c\vert^2.$

$\displaystyle 1+ \vert b-c\vert^2$	$\displaystyle =$	$\displaystyle 1+ \vert b\vert^2 - 2 b\cdot c + \vert c\vert^2$
	$\displaystyle \le$	$\displaystyle 1+ \vert b\vert^2 + 2 \vert b\vert \vert c\vert + \vert c\vert^2$
	$\displaystyle \le$	$\displaystyle 1+ 2\vert b\vert^2 + 2\vert c\vert^2$
	$\displaystyle \le$	$\displaystyle 2\big( 1+\vert b\vert^2+ \vert c\vert^2+\vert b\vert^2 \vert c\vert^2\big)$
	$\displaystyle =$	$\displaystyle 2( 1+\vert b\vert^2)(1+ \vert c\vert^2)$

Let us define

. Then for any vectors

and

, we have

$\displaystyle \frac{1+\vert a\vert^2}{1+\vert b\vert^2} \le 2 (1+\vert a-b\vert^2).$

(1)

Let us now return to the given inequality. If

, the claim is trivially true for all

in $\mathbb{R}^n$ . If

, then raising both sides in inequality 1 to the power of

, using $t=\vert t\vert$ , and setting

yields the result. On the other hand, if

, then raising both sides in inequality 1 to the power to

, using $-t=\vert t\vert$ , and setting

yields the result. $\Box$

Bibliography

1: J. Barros-Neta, An introduction to the theory of distributions, Marcel Dekker, Inc., 1973.
2: F. Treves, Introduction To Pseudodifferential and Fourier Integral Operators, Vol. I, Plenum Press, 1980.

"Peetre's inequality" is owned by Koro. [ full author list (2) | owner history (1) ]

(view preamble)

Log in to rate this entry.
(view current ratings)

Cross-references: power, sides, Cauchy-Schwarz inequality, inequality, proof, vectors, real number
There is 1 reference to this entry.

This is version 7 of Peetre's inequality, born on 2003-09-01, modified 2004-09-04.
Object id is 4681, canonical name is PeetresInequality.
Accessed 2144 times total.

Classification:

AMS MSC:	15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
	15A39 (Linear and multilinear algebra; matrix theory :: Linear inequalities)

Pending Errata and Addenda

None.[ View all 3 ]

Discussion

forum policy

No messages.

Interact