L2 bounds for a maximal directional Hilbert transform

Kim, Jongchon; Pramanik, Malabika

doi:10.2140/apde.2022.15.753

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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

DOI: 10.2140/apde.2022.15.753

Abstract

Given any finite direction set $Ω$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform $H_{Ω}$ 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 $H_{Ω}$ 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 $H_{Ω}$ . The latter principle can also be used to analyze special direction sets $Ω$ and derive sharp $L^{2}$ estimates for the corresponding operator $H_{Ω}$ that are typically stronger than the uniform $L^{2}$ bound mentioned above. A number of such examples are discussed.

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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

Vol. 15, No. 3, 2022