Endpoint Bounds for the Quartile Operator

It is a result by Lacey and Thiele (Ann. of Math. (2) 146(3):693-724, 1997; ibid. 149(2):475-496, 1999) that the bilinear Hilbert transform maps Lp1(ℝ) × Lp2(ℝ) into Lp1(ℝ) whenever (p1,p2,p3) is a Hölder tuple with p1,p3>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 Lp1(ℝ) × Lp2(ℝ) into L2/3,∞(ℝ) when p1,p2>1 and one component is restricted to subindicator functions. As a corollary, we derive that the quartile operator maps Lp1(ℝ) × Lp2,2/3(ℝ) into L2/3,∞(ℝ). We also provide weak type estimates and boundedness on Orlicz-Lorentz spaces near p1=1,p2=2 which improve, in the Walsh case, the results of Bilyk and Grafakos (J. Geom. Anal. 16 (4):563-584, 2006) and Carro et al. (J. Math. Anal. Appl. 357(2):479-497, 2009). Our main tool is the multi-frequency Calderón-Zygmund decomposition from (Nazarov et al. in Math. Res. Lett. 17(3):529-545, 2010). © 2013 Springer Science+Business Media New York.