Many phenomena in geometry and analysis can be explained via the theory of $D$ -modules, but this theory explains close to nothing in the non-archimedean case, by the absence of integration by parts. Hence there is a need to look for alternatives. A central example of a notion based on the theory of $D$ -modules is the notion of holonomic distributions. We study two recent alternatives of this notion in the context of distributions on non-archimedean local fields, namely $\mathscr{C}^{\text{exp}}$ -class distributions from Cluckers et al. [‘Distributions and wave front sets in the uniform nonarchimedean setting’, Trans. Lond. Math. Soc.5(1) (2018), 97–131] and WF-holonomicity from Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)]. We answer a question from Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)] by showing that each distribution of the $\mathscr{C}^{\text{exp}}$ -class is WF-holonomic and thus provides a framework of WF-holonomic distributions, which is stable under taking Fourier transforms. This is interesting because the $\mathscr{C}^{\text{exp}}$ -class contains many natural distributions, in particular, the distributions studied by Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)]. We show also another stability result of this class, namely, one can regularize distributions without leaving the $\mathscr{C}^{\text{exp}}$ -class. We strengthen a link from Cluckers et al. [‘Distributions and wave front sets in the uniform nonarchimedean setting’, Trans. Lond. Math. Soc.5(1) (2018), 97–131] between zero loci and smooth loci for functions and distributions of the $\mathscr{C}^{\text{exp}}$ -class. A key ingredient is a new resolution result for subanalytic functions (by alterations), based on embedded resolution for analytic functions and model theory.

中文翻译：

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类分布在非阿基米德局部场上的波前完整度

几何和分析中的许多现象都可以通过 $D$ -modules，但这个理论在非阿基米德情况下几乎没有解释，因为没有部分集成。因此，需要寻找替代方案。一个基于理论的概念的中心例子 $D$ -modules 是完整分布的概念。我们在非阿基米德局部场分布的背景下研究了该概念的两个最新替代方案，即 $\mathscr{C}^{\text{exp}}$ 来自 Cluckers 的类分布等。['均匀非阿基米德设置中的分布和波前集'，反式。伦敦。数学。社会党。5(1) (2018), 97–131] 和来自 Aizenbud 和 Drinfeld 的 WF-holonomicity ['代数度量的傅里叶变换的波前集'，以色列 J. 数学。207(2) (2015), 527–580 (英文)]。我们回答了来自 Aizenbud 和 Drinfeld 的问题 ['代数度量的傅里叶变换的波前集'，以色列 J. 数学。207(2) (2015), 527–580 (English)] $\mathscr{C}^{\text{exp}}$ -class 是WF-完整的，因此提供了WF-完整分布的框架，在进行傅里叶变换时它是稳定的。这很有趣，因为 $\mathscr{C}^{\text{exp}}$ - 类包含许多自然分布，特别是 Aizenbud 和 Drinfeld 研究的分布 ['代数度量的傅里叶变换的波前集'，以色列 J. 数学。207(2) (2015), 527–580 (英文)]。我们还展示了该类的另一个稳定性结果，即可以在不离开 $\mathscr{C}^{\text{exp}}$ -班级。我们加强了来自 Cluckers 的链接等。['均匀非阿基米德设置中的分布和波前集'，反式。伦敦。数学。社会党。5(1) (2018), 97–131] 在零位点和平滑位点之间的函数和分布 $\mathscr{C}^{\text{exp}}$ -班级。一个关键因素是子解析函数的新解析结果（通过更改），基于解析函数和模型理论的嵌入式解析。