# Proving Bounded operator

1. Apr 12, 2012

### dikmikkel

1. The problem statement, all variables and given/known data
The operator T maps from $L^p(-2,2)\rightarrow L^p(-2,2)$ is defined $(Tf)(x) = f(x) x$
Show that the operator maps from L^p(-2,2) into the same.
2. Relevant equations
p is a natural from 1 to infinity.
Holders inequality
Substitution integrals
3. The attempt at a solution
I look at the following integral
$\int\limits_{-2}^{2} |f(x)x|^pdx = \int\limits_{-2}^{2}|f(x)x|^{p-1}|f(x)x|dx\leq \int\limits_{-2}^{2} |f(x)x|^{p-1}\left[\left(\int\limits_{-2}^{2}|f(x)|^rdx\right)^{1/r}\left(\int\limits_{-2}^{2}|x|^qdx\right)^{1/q}\right] = \\ ||f(x)||_p\int\limits_{-2}^{2} |f(x)x|^{p-1}\left[\left(\int\limits_{-2}^{2}|x|^qdx\right)^{1/q}\right]$
And here im stuck
Edit: maybe a substitution would do? I really need a hint.

Last edited: Apr 12, 2012
2. Apr 14, 2012

### morphism

I don't think you need anything fancy here: just notice that |x|<=2.

3. Apr 14, 2012

### dikmikkel

You are right :) i made it. Tnx