Endpoint bounds for the quartile operator

TL;DR

This paper investigates the endpoint behavior of the Walsh model for the bilinear Hilbert transform, establishing new boundedness results at the critical exponent p_3=2/3 and improving existing estimates.

Contribution

It provides new endpoint bounds for the quartile operator, a Walsh model of the bilinear Hilbert transform, including weak-type and Orlicz-Lorentz space estimates.

Findings

Quartile operator maps L^{p_1} imes L^{p_2} into L^{2/3, ext{infinity}} when p_1,p_2>1 with one component restricted.

Derived boundedness of the quartile operator from L^{p_1} imes L^{p_2,2/3} into L^{2/3, ext{infinity}}.

Improved weak-type estimates near p_1=1,p_2=2 over previous results.

Abstract

It is a result by Lacey and Thiele that the bilinear Hilbert transform maps L^{p_1}(R) \times L^{p_2}(R) into L^{p_3}(R) whenever (p_1,p_2,p_3) is a Holder tuple with p_1,p_2 > 1 and p_3>2/3. We study the behavior of the quartile operator, which is the Walsh model for the bilinear Hilbert transform, when p_3=2/3. We show that the quartile operator maps L^{p_1}(R) \times L^{p_2}(R) into L^{2/3,\infty}(R) when p_1,p_2>1 and one component is restricted to subindicator functions. As a corollary, we derive that the quartile operator maps L^{p_1}(R) \times L^{p_2,2/3}(R) into L^{2/3,\infty}(R). We also provide restricted weak-type estimates and boundedness on Orlicz-Lorentz spaces near p_1=1,p_2=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 Calder\'on-Zygmund decomposition first used by Nazarov, Oberlin and Thiele.