#### 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
##### Abstract

Given any finite direction set $\mathrm{\Omega }$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform ${H}_{\mathrm{\Omega }}$ associated to this direction set. Our main result provides an essentially sharp uniform bound, depending only on $N$, for the ${L}^{2}$ operator norm of ${H}_{\mathrm{\Omega }}$ 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 ${H}_{\mathrm{\Omega }}$. The latter principle can also be used to analyze special direction sets $\mathrm{\Omega }$ and derive sharp ${L}^{2}$ estimates for the corresponding operator ${H}_{\mathrm{\Omega }}$ that are typically stronger than the uniform ${L}^{2}$ bound mentioned above. A number of such examples are discussed.

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