On the Sharp Estimates for Convolution Operators with Oscillatory Kernel

In this article, we studied the convolution operators M_k with oscillatory kernel, which are related to the solutions of the Cauchy problem for the strictly hyperbolic equations. The operator M_k is associated to the characteristic hypersurfaces Σ⊂ℝ^3 of a hyperbolic equation and smooth amplitude function, which is homogeneous of the order -k for large values of the argument. We investigated the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point v∈Σ at which, exactly one of the principal curvatures of the surface Σ does not vanish. Such surfaces exhibit singularities of the type A in the sense of Arnold’s classification. Denoting by k_p the minimal number such that M_k is L^p↦ L^p' -bounded for k>k_p, we showed that the number k_p depends on some discrete characteristics of the surface Σ .