On the sharp estimates for convolution operators with oscillatory kernel

In this talk, we discuss the $L^p\mapsto L^{p'}$-boundedness problem for the convolution operator $M_k$ (where $k$-means that the smooth amplitude function is homogeneous of order $-k$ for large values of the argument) with oscillatory kernel. We study the convolution operators, assuming that the characteristic surface $S\subset \mathbb{R}^3$ is contained in a sufficiently small neighborhood of a given point $x^0\in S$ at which exactly one of the principal curvatures of $S$ does not vanish. Such surfaces exhibit singularities of type $A$ 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 surface.