On the sharp estimates for convolution operators with oscillatory kernel

Abstract

In this article, we study the convolution operators $M_k$ with oscillatory kernel, which are related to solutions to the Cauchy problem for the strictly hyperbolic equations. The operator $M_k$ is associated to the characteristic hypersurfaces $\Sigma\subset \mathbb{R}^3$ of a hyperbolic equation and smooth amplitude function, which is homogeneous of order $-k$ for large values of the argument. We study the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point $v\in \Sigma$ at which exactly one of the principal curvatures of the surface $\Sigma$ does not vanish. Such surfaces exhibit singularities of type $A$ in the sense of Arnol'd's classification. Denoting by $k_p$ the minimal number 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 $\Sigma$.