Multilinear Oscillatory Integral Operators and Geometric Stability

Abstract

In honor of Guido Weiss, whose generous mentorship and enthusiasm for mathematics have enriched the lives of many students.Abstract. 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.

Higher order transversality in harmonic analysis

“…It is straightforward to see that such data cannot have the power decay property, and the objective is to establish power decay in all other situations. Although this remains a challenging open problem, we refer the reader to [48,51,69,70,73,74,92,106] for substantial recent progress in this direction.…”

On frames of smooth, compactly-supported wave packets adapted to tilings of frequency space

“…The article [6] of the author and E. Urheim establishes stable norm decay inequalities for multilinear oscillatory integral functionals through the use of certain continuous frame decompositions of L 2 (R) which behave like Gabor systems at low frequencies and so-called "wave atoms" at higher frequencies. Here the adjective "continuous" refers to decompositions which involve integrals over all translations of some countable family of window functions, where "discrete" would instead refer to summation over discrete lattices of translations.…”

“…A common approach when formulating such a criterion is to constrain the geometry of tiles to settings which are effectively spaces of homogeneous type and then to assume that the size of each tile is controlled by some slowly-growing function of its distance to the origin; see, for example, [2,4,12,13]. Perhaps the most interesting example of tilings of this sort which is neither Gabor-nor wavelet-like is the case of so-called "wave atoms," in which frequency space is decomposed into isotropic balls of radius ∼ R 1/2 at distance ∼ R to the origin for R ≥ 1 (as is done for the high frequency decomposition in [6]).…”

“…A second sample application of Theorem 2 which directly resolves the frame construction issue raised in [6] is described in the following proposition.…”

“…The tilings B associated with the geometry in [6] all have the property that the boxes B ∈ B have integer side lengths and adjacent boxes have side lengths differing by at most 1, so taking r(t) := max{1, t − 1} and applying the proposition and Theorem 2 provides for the existence of a single window function generating a frame for any possible geometry of this sort with constants that are independent of the particular tiling B.…”