Se p 20 14 L p ESTIMATES FOR THE BILINEAR HILBERT TRANSFORM FOR 1 / 2 < p ≤ 2 / 3 : A COUNTEREXAMPLE AND GENERALIZATIONS TO NON - SMOOTH SYMBOLS

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 any L estimates for p ∈ (1/2, 2/3) (Theorem 1.2). Nevertheless, we can establish L estimates for the bilinear Fourier multipliers whose symbols are not identical to but arbitrarily close to that of the bilinear Hilbert transform in the full range p ∈ (1/2,∞) (Theorem 1.3).