Vol. 15, No. 3, 2022

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$L^2$ bounds for a maximal directional Hilbert transform

Jongchon Kim and Malabika Pramanik

Vol. 15 (2022), No. 3, 753–794

Given any finite direction set Ω of cardinality N in Euclidean space, we consider the maximal directional Hilbert transform HΩ associated to this direction set. Our main result provides an essentially sharp uniform bound, depending only on N, for the L2 operator norm of HΩ in dimensions 3 and higher. The main ingredients of the proof consist of polynomial partitioning tools from incidence geometry and an almost-orthogonality principle for HΩ. The latter principle can also be used to analyze special direction sets Ω and derive sharp L2 estimates for the corresponding operator HΩ that are typically stronger than the uniform L2 bound mentioned above. A number of such examples are discussed.

directional Hilbert transform, maximal function, almost orthogonality, polynomial partitioning
Mathematical Subject Classification 2010
Primary: 42B20, 42B25
Received: 10 February 2020
Revised: 23 September 2020
Accepted: 30 October 2020
Published: 10 June 2022
Jongchon Kim
Department of Mathematics
University of British Columbia
Vancouver, BC
Department of Mathematics
City University of Hong Kong
Hong Kong
Malabika Pramanik
Department of Mathematics
University of British Columbia
Vancouver, BC