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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,

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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    I have seen a similar question on this website if it is not exactly the same.2012-11-29

1 Answers 1

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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}. $