A sharp decay estimate for degnerate oscillatory integral operators using broad-narrow method

TL;DR

This paper establishes a sharp $L^4$ decay estimate for a class of degenerate oscillatory integral operators in 2+1 dimensions using the broad-narrow method, focusing on a specific cubic phase function.

Contribution

It introduces a novel application of the broad-narrow method to obtain sharp decay estimates for degenerate oscillatory integrals with a specific cubic phase.

Findings

Proves sharp $L^4$ decay estimate for the model phase function

Demonstrates effectiveness of broad-narrow method in degenerate cases

Provides insights into decay behavior of degenerate oscillatory operators

Abstract

We use broad-narrow method to estabish the sharp $L^4$ decay estimate for a class of degenerate oscillatory integral operators in $(2+1)$ dimensions. Especially, the model phase function is \[xt^2+y^2t,\] a cubic homogeneous polynomial which is degenerate in the sense of [28].