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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

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

satisfies

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

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

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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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