Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob -processes, we prove that its -norm, , is bounded above by the -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.
"On the -norm of the discrete Hilbert transform." Duke Math. J. 168 (3) 471 - 504, 15 February 2019. https://doi.org/10.1215/00127094-2018-0047