On the curved Trilinear Hilbert transform

2023/08/21 by Hu, Bingyang, Lie, Victor · 3 citations
#42A20 #42A24 #42A38 #42A45 #42A50 #42B20 #42B25 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.10706

Abstract

Building on the (Rank I) LGC-methodology introduced by the second author and on the novel perspective employed in the time-frequency discretization of the non-resonant bilinear Hilbert--Carleson operator, we develop a new, versatile method -- referred to as Rank II LGC -- that has as a consequence the resolution of the L^p boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator H_C(f₁, f₂, f₃)(x):= \textrmp.v. ∫_ℝ f₁(x-t)f₂(x+t²)f₃(x+t³) (dt)/(t), x ∈ ℝ , is bounded from L^p₁(ℝ)× L^p₂(ℝ)× L^p₃(ℝ) into L^r(ℝ) within the Banach Hölder range (1)/(p₁)+(1)/(p₂)+(1)/(p₃)=(1)/(r) with 1

Cited by

• On the boundedness of the curved trilinear Hilbert transform and the curved n-linear maximal operator in the quasi-Banach regime
• On Multi-linear Maximal Operators Along Homogeneous Curves
• On Polynomial Progressions Inside Sets of Large Dimension

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