On the sharp estimates for convolution operators with oscillatory kernel

2023/03/11 by Isroil A. Ikromov, Ikromov, Isroil A., Dildora I. Ikromova +1
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2303.06446

openalex publication_date 2023/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

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 Σ⊂ ℝ³ 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∈ Σ at which exactly one of the principal curvatures of the surface Σ 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↦ L^p'-bounded for k>k_p, we show that the number k_p depends on some discrete characteristics of the surface Σ.

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