Initial $L^2\times\cdots\times L^2 $ bounds for multilinear operators

TL;DR

This paper establishes an initial $L^2 imes imes imes L^2 o L^{2/m}$ boundedness estimate for multilinear operators, filling a gap in the theory and enabling new bounds for various multilinear multipliers.

Contribution

It provides the first natural initial $L^2$-based estimate for multilinear operators, extending the classical $L^2$ theory to multilinear settings.

Findings

Derived initial $L^2 imes imes imes L^2 o L^{2/m}$ estimate for multilinear operators.

Applied the estimate to multilinear rough singular integrals and H"ormander-type multipliers.

Established bounds for multipliers with derivatives satisfying qualitative estimates.

Abstract

The $L^p$ boundedness theory of convolution operators is \linebreak 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 achieve exactly this goal, i.e., obtain an initial $L^2\times\cdots\times L^2\to L^{2/m}$ estimate for general building blocks of $m$-linear multiplier operators. We apply this result to deduce analogous bounds for multilinear rough singular integrals, multipliers of H\"ormander type, and multipliers whose derivatives satisfy qualitative estimates.