M. Lacey and C. Thiele proved in [26] (Annals of Math. (1997)) and [27] (Annals of Math. (1999)) that the bilinear Hilbert transform maps L1 × L2 → L boundedly when 1 p1 + 1 p2 = 1 p with 1 < p1, p2 ≤ ∞ and 2 3 < p < ∞. Whether the L estimates hold in the range p ∈ (1/2, 2/3] has remained an open problem since then. In this paper, we prove that the bilinear Hilbert transform does not satisfy an...