  In this article we prove a sharp decay estimate for certain multilinear
oscillatory integral operators of a form inspired by the general framework of
Christ, Li, Tao, and Thiele [6]. A key purpose of this work is to determine
when such estimates are stable under smooth perturbations of both the phase and
corresponding projections, which are typically only assumed to be linear. The
proof is accomplished by a novel decomposition which mixes features of Gabor or
windowed Fourier bases with features of wavelet or Littlewood-Paley
decompositions. This decomposition very nearly diagonalizes the problem and
seems likely to have useful applications to other geometrically-inspired
objects in Fourier analysis.
