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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2505.03330 |
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Sommario:
- 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\;\cite{stein1} and Bourgain-Guth \cite{BG} via the $TT^\ast$ 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 \cite{Bar} on the restriction problem. The decoupling theorem for the Hörmander oscillatory integral operator can be obtained by the approach in \cite{BHS}, 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 \cite{PS}. Both proofs rely on a scale-dependent induction argument, which can be used to deal with perturbation terms in the phase function.