Wave front holonomicity of-class distributions on non-Archimedean local fields

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 C-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 C-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 C-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 C-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 C-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.