On the Sharp Estimates for Convolution Operators with Oscillatory Kernel

• Published in: The Journal of fourier analysis and applications 2024-06, Vol.30 (3), Article 29
• Main Authors: Ikromov, Isroil A., Ikromova, Dildora I.
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Amplitudes; Approximations and Expansions; Cauchy problems; Convolution; Fourier Analysis; Hyperspaces; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Operators; Partial Differential Equations; Signal,Image and Speech Processing
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-024-10085-z

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 .