Vol. 310, No. 2, 2021

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Bilinear Hilbert transforms and (sub)bilinear maximal functions along convex curves

Junfeng Li and Haixia Yu

Vol. 310 (2021), No. 2, 375–446
DOI: 10.2140/pjm.2021.310.375
Abstract

We determine the Lp() × Lq() Lr() boundedness of the bilinear Hilbert transform Hγ(f,g) along a convex curve γ:

Hγ(f,g)(x) := p.v.f(x t)g(x γ(t))dt t ,

where p, q, and r satisfy 1p + 1q = 1r, and r > 1 2, p > 1, and q > 1. Moreover, the same Lp() × Lq() Lr() boundedness property holds for the corresponding (sub)bilinear maximal function Mγ(f,g) along a convex curve γ

Mγ(f,g)(x) := sup𝜀>0 1 2𝜀𝜀𝜀|f(x t)g(x γ(t))|dt.

Keywords
bilinear Hilbert transform, (sub)bilinear maximal function, convex curve, time frequency analysis
Mathematical Subject Classification
Primary: 42B20
Secondary: 47B38
Milestones
Received: 7 June 2020
Revised: 2 September 2020
Accepted: 4 September 2020
Published: 8 March 2021
Authors
Junfeng Li
School of Mathematical Sciences
Dalian University of Technology
Dalian
China
Haixia Yu
Department of Mathematics
Shantou University
Shantou
China