AMS :: Transactions of the American Mathematical Society

Initial $L^2\times \cdots \times L^2$ bounds for multilinear operators
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by Loukas Grafakos, Danqing He, Petr Honzík and Bae Jun Park;

Trans. Amer. Math. Soc. 376 (2023), 3445-3472

DOI: https://doi.org/10.1090/tran/8877

Published electronically: February 16, 2023

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Abstract:

The $L^p$ boundedness theory of convolution operators is based on an initial $L^2\to L^2$ 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 $L^2\times \cdots \times L^2\to L^{2/m}$ 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 $L^p$ estimates for such operators.

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