On the $l^p$-norm of the discrete Hilbert transform
Classical Analysis and ODEs
2019-03-20 v2 Complex Variables
Functional Analysis
Probability
Abstract
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.
Keywords
Cite
@article{arxiv.1709.07427,
title = {On the $l^p$-norm of the discrete Hilbert transform},
author = {Rodrigo Bañuelos and Mateusz Kwaśnicki},
journal= {arXiv preprint arXiv:1709.07427},
year = {2019}
}