L^p estimates for bilinear and multiparameter hilbert transforms

Muscalu, Pipher, Tao and Thiele proved that the standard bilinear and biparameter Hilbert transform does not satisfy any L^p estimates. They also raised a question asking if a bilinear and biparameter multiplier operator defined by satisfies any L^p estimates, where the symbol m satisfies for sufficiently many multi-indices α = (α₁, α₂) and β = (β₁, β₂), Γ_i (i = 1, 2) are subspaces in R² and dim Γ₁ = 0, dim Γ₂ = 1. Silva partially answered this question and proved that Tm maps L^p1 × L^p2 →L^p boundedly when 1/p1 + 1/p2 = 1/p with p1, p2 >1, 1/p1 + 2/p2 <2 and 1/p2 C 2/p1 <2. One notes that the admissible range here for these tuples (p1, p2, p) is a proper subset of the admissible range of the bilinear Hilbert transform (BHT) derived by Lacey and Thiele. We establish the same L^p estimates as BHT in the full range for the bilinear and d-parameter (d ≥ 2) Hilbert transforms with arbitrary symbols satisfying appropriate decay assumptions and having singularity sets Γ₁, . . . , Γ_d with dim Γ_i = 0 for i = 1, . . . , d-1 and dim Γ_d = 1. Moreover, we establish the same L^p estimates as BHT for bilinear and biparameter Fourier multipliers of symbols with dim Γ₁=dim Γ₂=1 and satisfying some appropriate decay estimates. In particular, our results include the L^p estimates as BHT in the full range for certain modified bilinear and biparameter Hilbert transforms of tensor-product type with dim Γ₁ = dim Γ₂ = 1 but with a slightly better logarithmic decay than that of the bilinear and biparameter Hilbert transform BHT⊗BHT.