A structure theorem for fundamental solutions of analytic multipliers in $${\mathbb {R}}^n$$

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Pseudo-Differential Operators and Applications Pub Date : 2024-02-26 DOI:10.1007/s11868-024-00586-2

引用次数: 0

Abstract

Using a version of Hironaka’s resolution of singularities for real-analytic functions, any elliptic multiplier $\text {Op}(p)$ of order $d>0$ , real-analytic near $p^{-1}(0)$ , has a fundamental solution $\mu _0$ . We give an integral representation of $\mu _0$ in terms of the resolutions supplied by Hironaka’s theorem. This $\mu _0$ is weakly approximated in $H^t_{\text {loc}}({\mathbb {R}}^n)$ for $t<d-\frac{n}{2}$ by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute fundamental solutions for two non-polynomial symbols.

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$${mathbb {R}}^n$ 中解析乘数基本解的结构定理

摘要使用 Hironaka 的实解析函数奇点解析的一个版本，任何阶 $d>0$ 的椭圆乘法器 $text {Op}(p)$ ，在 $p^{-1}(0)$ 附近的实解析，有基本解 $\mu _0$ 。有一个基本解。我们根据 Hironaka 定理提供的决议给出了 $\mu _0$ 的积分表示。对于 $t<d-\frac{n}{2}$ 来说，这个 $\mu _0$ 在 $H^t_{\text {loc}}({\mathbb {R}}^n)$ 中被帕利-维纳空间的序列弱逼近。在全局对称的特殊情况下，所得到的积分表示可以是完全显式的，我们用它来计算两个非多项式符号的基本解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Pseudo-Differential Operators and Applications MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

2.20

自引率

9.10%

发文量

期刊介绍： The Journal of Pseudo-Differential Operators and Applications is a forum for high quality papers in the mathematics, applications and numerical analysis of pseudo-differential operators. Pseudo-differential operators are understood in a very broad sense embracing but not limited to harmonic analysis, functional analysis, operator theory and algebras, partial differential equations, geometry, mathematical physics and novel applications in engineering, geophysics and medical sciences.