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Let $1\leq p<∞$: Suppose that there exists a constant $C>0$ such that for all $f\in S(\mathbb{R})$ with $L^p$ norm one we have $$\biggl|\{x:|H(f)(x)|>1\}\biggr|\leq C.$$ Here $H(f)$ is Hilbert transform, defined via $$H(f)(x)=\frac1{π} \mathop{p.v.}\int \frac{f(x-y)}{y} dy.$$

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What is the question? To find the constant? –  Fabian Apr 30 '11 at 5:59
Notice that f has Lp norm 1,so just do a weak estimate for p=1(if p>1,we know that Hilbert transform is (p,p) style.) –  Strongart May 2 '11 at 7:29

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