In this paper, we study the convolution operators \(M_k\) with oscillatory kernel, which are related with solutions to the Cauchy problem for strictly hyperbolic equations. The operator \(M_k\) is associated to the characteristic hypersurface \(\Sigma \subset {\mathbb {R}}^3\) of the equation and the smooth amplitude function, which is homogeneous of order \(-k\) for large values of the argument. By localizing arguments one can 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 \(v\in \Sigma \) at which the height of the surface is less than or equal to two. Such a class contains surfaces related to simple and the \(X_9, \, J_{10}\) type singularities in the sense of Arnol’d’s classification. Denoting by \(k_p\) be the minimum exponent such that the operator \(M_k: L^p \rightarrow L^{p'}\) is bounded for \(k>k_p\) . We show that the value of \(k_p\) depends on the discrete characteristics of the Newton polygon of a smooth function, defined in an appropriate coordinate system.
