The disc as a bilinear multiplier

A classical theorem of C. Fefferman says that the characteristic function of the unit disc is not a Fourier multiplier on L ^p(ℝ ²) unless p = 2. In this article we obtain a result that brings a contrast with the previous theorem. We show that the characteristic function of the unit disc in ℝ ² is the Fourier multiplier of a bounded bilinear operator from L ^p1(R) × Lp ^p2(R) into L ^p(R), when 2 ≤ p ₁, p ₂, < ∞ and 1 < p = p ₁p ₂/p ₁+p ₂ ≤ 2. The proof of this result is based on a new decomposition of the unit disc and delicate orthogonality and combinatorial arguments. This result implies norm convergence of bilinear Fourier series and strengthens the uniform boundedness of the bilinear Hilbert transforms, as it yields uniform vector-valued bounds for families of bilinear Hilbert transforms.