The Bilinear Hilbert-Carleson operator along curves. The purely non-zero curvature case

2025/07/06 by Bényi, Árpád, Hu, Bingyang, Lie, Victor · 1 citation
#42A16 #42A24 #42B15 #42B20 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.04467

Abstract

In this paper, we provide the maximal boundedness range (up to end-points) for the Bilinear Hilbert-Carleson operator along curves in the (purely) non-zero curvature setting. More precisely, we show that the operator BHC_[a,α](f₁,f₂)(x) := sup_λ∈ℝ | p.v. ∫_ℝ f₁(x - a₁ t^α₁) f₂(x - a₂ t^α₂) e^i λ a₃ t^α₃ (dt)/(t)| obeys the bounds ‖BHC_[a,α] (f₁,f₂)‖_L^r \lesssim_a α,r,p₁,p₂ ‖f₁‖_L^p1 ‖f₂‖_L^p2 whenever a=(a₁,a₂,a₃), α=(α₁,α₂,α₃)∈ (ℝ∖\0\)³ with α having pairwise distinct coordinates and for any Hölder range (1)/(p₁)+(1)/(p₂)=(1)/(r) with 1

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