L Estimates for Bilinear and Multi-parameter Hilbert Transforms

C. Muscalu, J. Pipher, T. Tao and C. Thiele proved in [27] that the standard bilinear and bi-parameter Hilbert transform does not satisfy any L estimates. They also raised a question asking if a bilinear and bi-parameter multiplier operator defined by Tm(f1, f2)(x) := ∫ R m(ξ, η)f̂1(ξ1, η1)f̂2(ξ2, η2)e 1122dξdη satisfies any L estimates, where the symbol m satisfies |∂ ξ ∂ ηm(ξ, η)| . 1 dist(ξ,Γ1) · 1 dist(η,Γ2) for sufficiently many multi-indices α = (α1, α2) and β = (β1, β2), Γi (i = 1, 2) are subspaces in R and dimΓ1 = 0, dimΓ2 = 1. P. Silva answered partially this question in [30] and proved that Tm maps L p1 × L2 → L boundedly when 1 p1 + 1 p2 = 1 p with p1, p2 > 1, 1 p1 + 2 p2 < 2 and 1 p2 + 2 p1 < 2. One observes that the admissible range here for these tuples (p1, p2, p) is a proper subset contained in the admissible range of BHT. In this paper, we establish the same L estimates as BHT in the full range for the bilinear and multi-parameter Hilbert transforms with arbitrary symbols satisfying appropriate decay assumptions (Theorem 1.3). Moreover, we also establish the same L estimates as BHT for certain modified bilinear and bi-parameter Hilbert transforms with dimΓ1 = dimΓ2 = 1 but with a slightly better decay than that for the bilinear and biparameter Hilbert transform (Theorem 1.4).