Best bounds for the Hilbert transform on \(L^ p(\mathbb{R}^ 1)\)
In this note the author gives a short proof (which avoids the use of conjugate function on the circle) of the following known best bounds result: The Hilbert transform \(H\), \[ Hf(x)=\text{p.v. }\Biggl({1\over\pi}\Biggr) \int_\mathbb{R} {f(x-t)\over t} dt, \] maps \(L^p(\mathbb{R})\) into itself with bound \(\tan(\pi/2p)\) for \(1<p\leq 2\) and \(\text{cot}(\pi/2p)\) for \(2\leq p<\infty\).
scientific article; zbMATH DE number 1082974