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Sharp $L^2$ decay rate for (1+2)-dimensional oscillatory integral operators with cubic polynomial phases

Jayden Lang, Wan Tang

TL;DR

The paper addresses the sharp $L^2$ decay rate for $(1+2)$-dimensional oscillatory integral operators with cubic polynomial phases, focusing on the degenerate case where $P_{2}$ has multiple factors. Building on prior results that yield a $\lambda^{-3/8}$ bound (up to logarithmic factors) for the degenerate format, it proves that this rate is in fact sharp by a constructive lower-bound argument. The method combines a rescaling analysis and a carefully chosen test function to demonstrate near-constant phase on a shrinking region, leading to a quantitative lower bound $\|T_{\lambda}\| \gtrsim \lambda^{-3/8}$. This resolves the sharpness question for the degenerate cubic-phase operator and clarifies the precise decay behavior in this regime.

Abstract

In this paper, we consider the (1+2)-dimensional oscillatory integral with degenerate cubic homogeneous polynomial phase. We prove that the $L^{2}$ decay rate of 3/8 given in (Archiv der Mathematik, 122: 437-447, 2024) is sharp.

Sharp $L^2$ decay rate for (1+2)-dimensional oscillatory integral operators with cubic polynomial phases

TL;DR

The paper addresses the sharp decay rate for -dimensional oscillatory integral operators with cubic polynomial phases, focusing on the degenerate case where has multiple factors. Building on prior results that yield a bound (up to logarithmic factors) for the degenerate format, it proves that this rate is in fact sharp by a constructive lower-bound argument. The method combines a rescaling analysis and a carefully chosen test function to demonstrate near-constant phase on a shrinking region, leading to a quantitative lower bound . This resolves the sharpness question for the degenerate cubic-phase operator and clarifies the precise decay behavior in this regime.

Abstract

In this paper, we consider the (1+2)-dimensional oscillatory integral with degenerate cubic homogeneous polynomial phase. We prove that the decay rate of 3/8 given in (Archiv der Mathematik, 122: 437-447, 2024) is sharp.
Paper Structure (2 sections, 1 theorem, 17 equations)

This paper contains 2 sections, 1 theorem, 17 equations.

Table of Contents

  1. Introduction
  2. Sharpness of the decay rate

Key Result

Theorem 1

Let $T_{\lambda}$ be the oscillatory integral operator defined in c. Then for any amplitude functions $\Phi$ with $\Phi \left( 0,0,0\right) \neq0$, we have

Theorems & Definitions (2)

  • Theorem 1: Sharpness of the decay rate
  • proof