Sharp $$L^p$$ -Estimates for the Fourier Transform of Surface Measures

IF 0.6 4区数学 Q3 MATHEMATICS

Mathematical Notes Pub Date : 2024-04-22 DOI:10.1134/s000143462401005x

I. A. Ikromov, D. I. Ikromova

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Abstract

We consider estimates for the Fourier transform of measures concentrated on smooth surfaces $S\subset \mathbb{R}^3$ given by the graph of a smooth function with simple Arnold singularities such that both principal curvatures of the surface vanish at some point. We prove that if the multiplicity of the critical point of the function does not exceed $7$, then the Fourier transforms of the corresponding surface measures belong to $L^{p}(\mathbb{ R}^3)$ for any $p>3$. Note that for any smooth surface the Fourier transform of a nontrivial surface measure with compact support does not belong to $L^3(\mathbb{R}^3)$; i.e., the $L^p(\mathbb{R}^3)$-estimate obtained is sharp. Moreover, there exists a function with an $E_8$ singularity (the multiplicity of the critical point of the function is equal to $8$) such that the Fourier transform of the corresponding surface measure does not belong to $L^{22/7}(\mathbb{R}^3)$, which shows the sharpness of the results for the multiplicity of the critical point.

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表面测量的傅立叶变换的尖锐 $$L^p$$ 估计值

摘要我们考虑了集中在光滑表面 $S\subset \mathbb{R}^3$上的度量的傅里叶变换的估计值，该光滑表面由具有简单阿诺德奇点的光滑函数的图给出，使得表面的两个主曲率在某一点上消失。我们证明，如果函数临界点的多重性不超过 $7$，那么对于任意 $p>3$，相应曲面度量的傅立叶变换属于 $L^{p}(\mathbb{R}^3)$。需要注意的是，对于任何光滑表面，具有紧凑支撑的非难表面度量的傅里叶变换不属于$L^3(\mathbb{R}^3)$；也就是说，得到的$L^p(\mathbb{R}^3)$估计值是尖锐的。此外，存在一个具有 $E_8$ 奇异性（函数临界点的多重性等于 $8$）的函数，使得相应曲面度量的傅里叶变换不属于 $L^{22/7}(\mathbb{R}^3)$，这表明了临界点多重性结果的尖锐性。

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来源期刊

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.