WAVE FRONT HOLONOMICITY OF $\text{C}^{\text{exp}}$-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...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2020-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509420000274/type/journal_article |
| _version_ | 1811156201149300736 |
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| author | AVRAHAM AIZENBUD RAF CLUCKERS |
| author_facet | AVRAHAM AIZENBUD RAF CLUCKERS |
| author_sort | AVRAHAM AIZENBUD |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-10T04:47:33Z |
| format | Article |
| id | doaj.art-c989d68dc69445b0ab9d0bbf97db31f9 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:33Z |
| publishDate | 2020-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-c989d68dc69445b0ab9d0bbf97db31f92023-03-09T12:34:47ZengCambridge University PressForum of Mathematics, Sigma2050-50942020-01-01810.1017/fms.2020.27WAVE FRONT HOLONOMICITY OF $\text{C}^{\text{exp}}$-CLASS DISTRIBUTIONS ON NON-ARCHIMEDEAN LOCAL FIELDSAVRAHAM AIZENBUD0RAF CLUCKERS1https://orcid.org/0000-0002-3056-3280Faculty of Mathematical Sciences, Weizmann Institute of Science, Rehovot, Israel; http://aizenbud.orgUniv. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, F-59000Lille, France KU Leuven, Department of Mathematics, B-3001Leuven, Belgium; http://rcluckers.perso.math.cnrs.fr/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.https://www.cambridge.org/core/product/identifier/S2050509420000274/type/journal_article14E1822E5003C1011S8011U09 |
| spellingShingle | AVRAHAM AIZENBUD RAF CLUCKERS WAVE FRONT HOLONOMICITY OF $\text{C}^{\text{exp}}$-CLASS DISTRIBUTIONS ON NON-ARCHIMEDEAN LOCAL FIELDS Forum of Mathematics, Sigma 14E18 22E50 03C10 11S80 11U09 |
| title | WAVE FRONT HOLONOMICITY OF $\text{C}^{\text{exp}}$-CLASS DISTRIBUTIONS ON NON-ARCHIMEDEAN LOCAL FIELDS |
| title_full | WAVE FRONT HOLONOMICITY OF $\text{C}^{\text{exp}}$-CLASS DISTRIBUTIONS ON NON-ARCHIMEDEAN LOCAL FIELDS |
| title_fullStr | WAVE FRONT HOLONOMICITY OF $\text{C}^{\text{exp}}$-CLASS DISTRIBUTIONS ON NON-ARCHIMEDEAN LOCAL FIELDS |
| title_full_unstemmed | WAVE FRONT HOLONOMICITY OF $\text{C}^{\text{exp}}$-CLASS DISTRIBUTIONS ON NON-ARCHIMEDEAN LOCAL FIELDS |
| title_short | WAVE FRONT HOLONOMICITY OF $\text{C}^{\text{exp}}$-CLASS DISTRIBUTIONS ON NON-ARCHIMEDEAN LOCAL FIELDS |
| title_sort | wave front holonomicity of text c text exp class distributions on non archimedean local fields |
| topic | 14E18 22E50 03C10 11S80 11U09 |
| url | https://www.cambridge.org/core/product/identifier/S2050509420000274/type/journal_article |
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