On the sharp estimates for convolution operators related to some non-convex hypersurfaces

In this article, we study the convolution operators Mk with oscillatory kernel, which are related to solutions to the Cauchy problem for the strictly hyperbolic equations. The operator Mk is associated to the characteristic hypersurfaces Σ ⊂ Rn+1 of a strictly 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 2 Σ at which at least n − 1 principal curvatures do not vanish. Such hyper-surfaces exhibit singularities of type A in the sense of Arnol’d’s classification. Denoting by kp the minimal number such that Mk is Lp 7! Lp0-bounded for any k > kp; we show that the number kp depends on some discrete characteristics of the hyper-surface Σ.