E Type Singularities for Sufficiently Smooth Functions

2025/05/20 by Ibrokhimbek Akramov, Akramov, Ibrokhimbek, Dildora I. Ikromova +1
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Mathematical Analysis and Transform Methods #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2505.13939

openalex publication_date 2025/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we will consider E-type singularities which are Arnol'd type. We provide invariant conditions for a sufficiently smooth functions to have singularities of type E_k (6≤ k≤ 8). We show the functions can be reduced to E_k, k=6, 7, 8 type normal form under some certain conditions. Moreover, we show that result on normal form for sufficiently smooth functions can be showed by the use of Implicit Function Theorem. The results can be utilized to investigate oscillatory integrals with sufficiently smooth phase function.

Citations

• On the sharp estimates for convolution operators with oscillatory kernel
• Estimates for maximal functions associated to hypersurfaces in \Bbb R³ with height h<2: Part I

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