Uniform estimates for oscillatory integrals with homogeneous polynomial phases of degree 4

In this paper we consider the uniform estimates for oscillatory integrals with homogeneous polynomial phases of degree 4 in two variables. The obtained estimate is sharp and the result is an analogue of the more general theorem of Karpushkin (Proc I.G.Petrovsky Seminar 9:3-39, 1983) for sufficiently smooth functions, thus, in particular, removing the analyticity assumption.