  In this article, we study the convolution operator $M_k$ with oscillatory
kernel, which is related with solutions to the Cauchy problem for the 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. 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 $v\in \Sigma$ at which the height of the surface
is less or equal to two. Such class contains surfaces related to simple and the
$X_9, \, J_{10}$ type singularities in the sense of Arnol'd's classification.
Denoting by $k_p$ the minimal exponent such that $M_k$ is $L^p\mapsto
L^{p'}$-bounded for $k>k_p,$ we show that the number $k_p$ depends on some
discrete characteristics of the Newton polygon of a smooth function constructed
in an appropriate coordinate system.
