INITIAL L2 x middot middot middot x L2 BOUNDS FOR MULTILINEAR OPERATORS

The Lp boundedness theory of convolution operators is based on an initial L2-+ L2 estimate derived from the Fourier transform. The corresponding theory of multilinear operators lacks such a simple initial estimate in view of the unavailability of Plancherel's identity in this setting, and up to now it has not been clear what a natural initial estimate might be. In this work we obtain initial L2 x center dot center dot center dot x L2-+ L2/m estimates for three types of important multilinear operators: rough singular integrals, multipliers of Ho center dot rmander type, and multipliers whose derivatives satisfy qualitative estimates. These estimates lay the foundation for the derivation of other Lp estimates for such operators.