Some new decay estimates for $(2+1)$-dimensional degenerate oscillatory integral operators

TL;DR

This paper improves decay estimates for certain degenerate oscillatory integral operators in 2+1 dimensions, establishing sharper $L^2$ decay and a new $L^6$ estimate using fractional integration.

Contribution

It introduces improved decay rates for degenerate oscillatory integrals with cubic phases and establishes a sharp $L^2$ to $L^6$ decay estimate.

Findings

Enhanced $L^2$ decay rate to 3/8

Established sharp $L^2$ to $L^6$ decay estimate

Applied fractional integration method for proofs

Abstract

In this paper, we consider the $(2+1)-$dimensional oscillatory integral operators with cubic homogeneous polynomial phases, which are degenerate in the sense of \cite{Tan06}. We improve the previously known $L^2\to L^2$ decay rate to $3/8$ and also establish a sharp $L^2\to L^6$ decay estimate based on fractional integration method.