Sharp estimates for convolution operators associated to hypersurfaces in R³ with height h≤2

In this article, we study the convolution operator Mₖ with oscillatory kernel, which is related with solutions to the Cauchy problem for the strictly hyperbolic equations. The operator Mₖ is associated to the characteristic hypersurface Σ⊂ R³ of the equation and the smooth amplitude function, which is homogeneous of order $-k$ for large values of the argument. We study the convolution operators assuming that the support of the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point v∈ Σ at which the height of the surface is less or equal to two. Such class contains surfaces related to simple and the X₉, \, J₁₀ type singularities in the sense of Arnol'd's classification. Denoting by kₚ the minimal exponent such that Mₖ is Lᵖ↦ Lp'-bounded for k>kₚ, we show that the number kₚ depends on some discrete characteristics of the Newton polygon of a smooth function constructed in an appropriate coordinate system.