Sharp norm inequalities for the truncated Hilbert transform.

We split the classical Hilbert transform into the sum of two convolution integrals, one supported away from the origin and another one supported near the origin. We prove that the exact norms of the outer truncated transform on L^p, for p<1<infinity, coincide wiht the corresponding norms of the full Hilbert transform. We then prove that the norms of the inner transform are strictly bigger and related to the Gibbs phenomenon.