On the sharp estimates for convolution operators with oscillatory kernel

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 equat...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2023-03
Main Authors: Ikromov, Isroil A, Ikromova, Dildora I
Format: Article
Language:English
Subjects:
Amplitudes
Cauchy problems
Convolution
Hyperspaces
Kernels
Operators
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 \(\Sigma\).
ISSN:2331-8422