  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.
