Initial 𝐿²×⋯×𝐿² bounds for multilinear operators

Abstract

The L p L^p boundedness theory of convolution operators is based on an initial L 2 → L 2 L^2\to L^2 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 L 2 × ⋯ × L 2 → L 2 / m L^2\times \cdots \times L^2\to L^{2/m} estimates for three types of important multilinear operators: rough singular integrals, multipliers of Hörmander type, and multipliers whose derivatives satisfy qualitative estimates. These estimates lay the foundation for the derivation of other L p L^p estimates for such operators.