On the Sharp Estimates for Convolution Operators with Oscillatory Kernel

Abstract

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\(\Sigma \subset {\mathbb {R}}^3\) 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 \(v\in \Sigma \) at which, exactly one of the principal curvatures of the surface \(\Sigma \) 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 \(L^p\mapsto L^{p'}\)-bounded for \(k>k_p,\) we showed that the number \(k_p\) depends on some discrete characteristics of the surface \(\Sigma \).