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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). 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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). 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