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.Comment: 17 pages; update includes referee's suggestions and two improved
results near L^1 x L^2. To appear on Journal of Fourier Analysis and
Application
