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

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 (Forum Math. 18:427–444, 2006). We improve the previously known \(L^2\rightarrow L^2\) decay rate to 3/8 and also establish a sharp \(L^2\rightarrow L^6\) decay estimate based on the fractional integration method.