Hi guys, i need to prove that the operator Tf = x*f(x) maps from L^p(-2,2) into L^p(-2,2) where:

$\displaystyle L^p(-2,2) = \lbrace f: [-2,2] \rightarrow \mathbb{C}:\, \int\limits_{-2}^2 |f(x)|^p\,\text{d}x < \infty \rangle$

I think that Holders inequality would be useful so i use it like this

$\displaystyle \int\limits_{-2}^2 |f(x)x|^{p-1}|f(x)x|\,\text{d}x \leq \int\limits_{-2}^2 \left(|f(x)x|^{p-1}\left[\int\limits_{-2}^2|f(x)|^p\, dx\right]^{1/p}\left[\int\limits_{-2}^2|x|^q\right]^{1/q}\right)dx = ||f(x)||_p \int\limits_{-2}^2 \left(|f(x)x|^{p-1} \int\limits_{-2}^2 |x|^q\right]^{1/q}\right) dx

$

By doing that until the integrand of the order p equals 1 i obtain with the relation $\displaystyle 1/p+1/q = 1$:

$\displaystyle 16 ||f(x)||_p^p 2^{p/(p-1)}(2p-1)^{1-p}(p-1)^{p-1}$

This expression converges when p goes to infinity so can i conclude that it maps as wished?

But i cant assume that the norm^P is finite can i?

Is there a smarter way(correct)

Im really lost, fought this problem for almost 6 hours.