# Showing that $||V(f,x)||_{L^2((1,\infty),m)}\le C||f||_{L^p((1,\infty),m)}$ [duplicate]

Possible Duplicate:
How can I give a bound on the $L^2$ norm of this function?

For $f\in L^p((1,\infty),m)$, $2<p<4$,

Want to prove that there exists $C$ which only depends on $p$, such that

$$V(f,x)=\frac{1}{x}\int^{10x}_x\frac{f(t)}{t^{\frac{1}{4}}}dm(t)$$

satisfies

$$||V(f,x)||_{L^2((1,\infty),m)}\le C||f||_{L^p((1,\infty),m)}$$

-
I have seen a similar question on this website if it is not exactly the same. – Mhenni Benghorbal Nov 29 '12 at 22:26

## marked as duplicate by Davide Giraudo, martini, draks ..., Matt Pressland, Martin ArgeramiNov 30 '12 at 14:08

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

## 1 Answer

Hint Use Holder's inequality

$$\|fg\|_1 \le \|f\|_p \|g\|_q \quad \frac{1}{p}+\frac{1}{q} = 1,$$

and note that $q=\frac{p}{p-1}.$

-