Hörmander oscillatory integral operators: A revisit

In this paper, we present new proofs for both the sharp L^p estimate and the decoupling theorem for the Hörmander oscillatory integral operator. The sharp L^p estimate was previously obtained by Stein (1986) and Bourgain and Guth (2011) via the TT* and multilinear methods, respectively. We provide a unified proof based on the bilinear method for both odd and even dimensions. The strategy is inspired by Barron’s work (2022) on the restriction problem. The decoupling theorem for the Hörmander oscillatory integral operator can be obtained by the approach in Beltran et al. (2020), where the key observation can be roughly formulated as follows: in a physical space of sufficiently small scale, the variable setting can be essentially viewed as translation-invariant. In contrast, we reprove the decoupling theorem for the Hörmander oscillatory integral operator through the Pramanik-Seeger approximation approach (Pramanik and Seeger (2007)). Both proofs rely on a scale-dependent induction argument, which can be used to deal with perturbation terms in the phase function.