Inversion Theorem for Bilinear Hilbert Transform

where f ∈ L2(R) and g ∈ L∞(R), respectively f ∈ Lp1(R) and g ∈ Lp2(R), 1 < p1, p2 <∞. Their main result is the affirmative answer on the Calderon conjecture, first for p1 = 2, p2 = ∞ ([5]), then for p1, p2 ∈ (1,∞). Let 2/3 < p = p1p2 p1+p2 or p1 = 2, p2 = ∞ and p = 2. Then their main result is ||Hα(f, a)||Lp ≤ C||f ||Lp1 ||a||Lp2 , f ∈ L p1, a ∈ Lp2, where C > 0 depends on α, p1, p2. We refer to [7] and the references therein for further reading on multi-linear operators given by singular multipliers. The bilinear Hilbert transform Hα : L 2 × L∞ → L2 respectively, Hα : Lp1×Lp2 → Lp, was extended in [1] to D′ L2×DL∞ → D ′ L2, respectively, D ′ Lq× DLp2 → D ′ q1, (with suitable parameters) as a hypocontinuous, respectively, University of Skopje, R. Macedonia University of Novi Sad, Serbia University of Sarajevo, Bosnia and Herzegovina