On the sharp estimates for convolution operators with oscillatory kernel

In this talk, we the consider the $L^p\mapsto L^{p'}$-boundedness problem for convolution operators $M_k$ with oscillatory kernel. We study the convolution operators assuming that $S$ 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 Arnold’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 given as the graph of function having singularities of type $A$.