Wave packet decompositions and sharp bilinear estimates for rough Hamiltonian flows

TL;DR

The paper develops sharp bilinear $L^p$ estimates for dispersive evolutions with rough $C^{1,1}$-type coefficients by combining a robust wave packet decomposition with a refined FBI-transform–based parametrix for rough Hamiltonian flows. It formulates two gauge classes of transversality-driven bilinear bounds (non-degenerate and $1$-homogeneous) and proves them via an induction-on-scales framework that leverages energy/momentum conservation and a detailed combinatorial analysis of wave packets. The key innovations are (i) a phase-space–adapted wave packet construction with time-frequency localization under low regularity, (ii) a precise control of bicharacteristics and their interactions through transversality, and (iii) a parametrix built from the FBI transform that yields tight localization and propagation estimates for rough coefficients. Together these yield sharp bilinear estimates that extend classical cone and paraboloid results to dispersive PDEs with rough, uniformly elliptic or partially degenerate symbols, with implications for Strichartz-type bounds and PDE well-posedness in rough media.

Abstract

The goal of this paper is to prove bilinear $L^p$ estimates for rough dispersive evolutions satisfying non-degeneracy and transversality assumptions. The estimates generalize the sharp Fourier extension estimates for the cone and the paraboloid. To this end, we require a wave packet decomposition with localization properties in space-time and space-time frequencies. Secondly, we construct a refined wave packet parametrix for dispersive equations with $C^{1,1}$-coefficients by using the FBI transform. As a consequence, we obtain bilinear estimates for solutions to dispersive equations with $C^{1,1}$ coefficients provided that the solutions interact transversely.

Figures (1)

• Figure 1: On the left: two transversely interacting wavepackets for Fourier extension operators. The bottoms of the cylinders intersect the $x$-plane in a circle of radius $R^{\frac{1}{2}}$. The direction of the long-side with length $R$ is determined by the support $\theta$ in Fourier space. On the right: the transverse interaction of curved wave packets. The wave packets retain their size, but the cores become curved.

Theorems & Definitions (84)

• Theorem 1.1: Bilinear estimates for rough wave equations
• Theorem 1.2
• Lemma 2.2: Bi-Lipschitz property of bicharacteristics
• proof
• Definition 2.4: Thickenings of sets in phase space
• Remark 2.6
• Lemma 2.7
• Remark 2.9
• Remark 2.11: Basic transversality
• Lemma 2.12