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arXiv · 1709.07427

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

Rodrigo Bañuelos·Mateusz Kwaśnicki
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Abstract

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.

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BibTeXRIS

Rodrigo Bañuelos, Mateusz Kwaśnicki. 2017-09-21. On the $l^p$-norm of the discrete Hilbert transform. https://doi.org/10.1215/00127094-2018-0047

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