Bilinear operators with non-smooth symbol, I

This article proves the L-P-boundedness of general bilinear operators assoc iated to a symbol or multiplier which need not be smooth. The Main Theorem establishes a general result for multipliers that are allowed to have singu larities along the edges of a cone as well as possibly at its vertex. It th us unifies earlier results of Coifinan-Meyer for smooth multipliers and one s, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplie r is not smooth. Using a Whitney decomposition in the Fourier plane, a gene ral bilinear operator is represented as infinite discrete sums of time-freq uency paraproducts obtained by associating wave-packets with tiles in phase -plane. Boundedness for the general bilinear operator then follows once the corresponding L-P-boundedness of time-frequency paraproducts has been esta blished The latter result is the main theorem proved in Part II, our subseq uent article [II], using phase-plane analysis.