On the $l^p$-norm of the Discrete Hilbert transform

Rodrigo Bañuelos, Mateusz Kwaśnicki

Published 2017-09-21Version 1

Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob $h$-processes, we prove that its $l^p$-norm, $1<p<\infty$, is bounded above by the $L^p$-norm of the continuous Hilbert transform. Together with the already known lower bound, this resolves the long-standing conjecture that the norms of these operators are equal.