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

TL;DR

This paper establishes the sharp $L^2$ decay rate of 3/8 for a class of (1+2)-dimensional oscillatory integrals with degenerate cubic polynomial phases, confirming previous conjectures about their decay behavior.

Contribution

It proves the sharpness of the $L^2$ decay rate for specific oscillatory integral operators with cubic phases in (1+2) dimensions.

Findings

The $L^2$ decay rate of 3/8 is proven to be sharp.

The decay rate applies to degenerate cubic polynomial phases.

The result confirms previous conjectures about decay behavior.

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.