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:

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

By doing that until the integrand of the order p equals 1 i obtain with the relation :

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.