WAVE FRONT HOLONOMICITY OF $\text{C}^{\text{exp}}$-CLASS DISTRIBUTIONS ON NON-ARCHIMEDEAN LOCAL FIELDS
Forum of Mathematics, Sigma, Tome 8 (2020)
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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.
@article{10_1017_fms_2020_27,
author = {AVRAHAM AIZENBUD and RAF CLUCKERS},
title = {WAVE {FRONT} {HOLONOMICITY} {OF} $\text{C}^{\text{exp}}${-CLASS} {DISTRIBUTIONS} {ON} {NON-ARCHIMEDEAN} {LOCAL} {FIELDS}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {8},
year = {2020},
doi = {10.1017/fms.2020.27},
language = {en},
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AVRAHAM AIZENBUD; RAF CLUCKERS. WAVE FRONT HOLONOMICITY OF $\text{C}^{\text{exp}}$-CLASS DISTRIBUTIONS ON NON-ARCHIMEDEAN LOCAL FIELDS. Forum of Mathematics, Sigma, Tome 8 (2020). doi: 10.1017/fms.2020.27
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