Endpoint bounds for the quartile operator

2012/06/17 by Ciprian Demeter, Demeter, Ciprian, Francesco Di Plinio +1
Mathematics · Medicine · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Pelvic and Acetabular Injuries

paper · pdf · doi:10.48550/arxiv.1206.3798

openalex publication_date 2012/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is a result by Lacey and Thiele that the bilinear Hilbert transform maps L^p₁(R) × L^p₂(R) into L^p₃(R) whenever (p₁,p₂,p₃) is a Holder tuple with p₁,p₂ > 1 and p₃>2/3. We study the behavior of the quartile operator, which is the Walsh model for the bilinear Hilbert transform, when p₃=2/3. We show that the quartile operator maps L^p₁(R) × L^p₂(R) into L^2/3,∞(R) when p₁,p₂>1 and one component is restricted to subindicator functions. As a corollary, we derive that the quartile operator maps L^p₁(R) × L^p₂,2/3(R) into L^2/3,∞(R). We also provide restricted weak-type estimates and boundedness on Orlicz-Lorentz spaces near p₁=1,p₂=2 which improve, in the Walsh case, on results of Bilyk and Grafakos, and Carro-Grafakos-Martell-Soria. Our main tool is the multi-frequency Calderón-Zygmund decomposition first used by Nazarov, Oberlin and Thiele.

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