Abstract
This article proves the Lp-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. The Main Theorem establishes a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies earlier results of Coifman-Meyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth. Using a Whitney decomposition in the Fourier plane, a general bilinear operator is represented as infinite discrete sums of time-frequency paraproducts obtained by associating wave-packets with tiles in phase-plane. Boundedness for the general bilinear operator then follows once the corresponding Lp-boundedness of time-frequency paraproducts has been established. The latter result is the main theorem proved in Part in Part II, our subsequent article [11], using phase-plane analysis.
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Communicated by Y. Meyer
In memory of A.P. Calderón
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Gilbert, J.E., Nahmod, A.R. Bilinear operators with non-smooth symbol, I. The Journal of Fourier Analysis and Applications 7, 435–467 (2001). https://doi.org/10.1007/BF02511220
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DOI: https://doi.org/10.1007/BF02511220