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The $L^p$ boundedness theory of convolution operators is \linebreak based onan initial $L^2\to L^2$ estimate derived from the Fourier transform. Thecorresponding theory of multilinear operators lacks such a simple initialestimate in view of the unavailability of Plancherel's identity in thissetting, and up to now it has not been clear what a natural initial estimatemight 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 analogousbounds for multilinear rough singular integrals, multipliers of H\"ormandertype, and multipliers whose derivatives satisfy qualitative estimates.

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