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

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 $$x{t}^2+y^{2}t,$$ a cubic homogeneous polynomial which is degenerate in the sense of [28].