Wave packet decompositions and sharp bilinear estimates for rough Hamiltonian flows

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.

💡 Research Summary

The paper “Wave packet decompositions and sharp bilinear estimates for rough Hamiltonian flows” develops a comprehensive theory of bilinear (L^{p}) estimates for dispersive equations whose coefficients are only (C^{1,1}) regular. The authors extend the celebrated sharp bilinear Fourier extension estimates for the cone and the paraboloid—originally proved for constant‑coefficient operators—to a much broader class of variable‑coefficient Hamiltonian flows that satisfy non‑degeneracy and a quantitative transversality condition.

The work is organized around two main technical achievements. First, the authors construct a wave‑packet decomposition that is simultaneously localized in space‑time, frequency, and time‑frequency. This is accomplished by employing the FBI (Fourier–Bros–Iagolnitzer) transform, which writes a function as a superposition of Gaussian coherent states. Each coherent state is then propagated along the Hamiltonian flow generated by the principal symbol (a(t,x,\xi)). Under the assumption that the symbol belongs to a class with bounds \