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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



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

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marked as duplicate by Davide Giraudo, martini, draks ..., Matthew Pressland, Martin Argerami Nov 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.

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

1 Answer 1

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}. $

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