Abstract
Muscalu, Pipher, Tao and Thiele proved that the standard bilinear and biparameter Hilbert transform does not satisfy any estimates. They also raised a question asking if a bilinear and biparameter multiplier operator defined by
satisfies any estimates, where the symbol satisfies
for sufficiently many multi-indices and , () are subspaces in and , . Silva partially answered this question and proved that maps boundedly when with , , and . One notes that the admissible range here for these tuples is a proper subset of the admissible range of the bilinear Hilbert transform (BHT) derived by Lacey and Thiele.
We establish the same estimates as BHT in the full range for the bilinear and -parameter () Hilbert transforms with arbitrary symbols satisfying appropriate decay assumptions and having singularity sets with for and . Moreover, we establish the same estimates as BHT for bilinear and biparameter Fourier multipliers of symbols with and satisfying some appropriate decay estimates. In particular, our results include the estimates as BHT in the full range for certain modified bilinear and biparameter Hilbert transforms of tensor-product type with but with a slightly better logarithmic decay than that of the bilinear and biparameter Hilbert transform .
Citation
Wei Dai. Guozhen Lu. "$L^{p}$ estimates for bilinear and multiparameter Hilbert transforms." Anal. PDE 8 (3) 675 - 712, 2015. https://doi.org/10.2140/apde.2015.8.675
Information