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Bergman spaces of natural <i>G</i>-manifolds.

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The Borel map in locally integrable structures.

Giuseppe Della Sala1, Paulo D Cordaro2, Bernhard Lamel3

  • 1American University of Beirut, Beirut, Lebanon.

Mathematische Annalen
|August 9, 2020
PubMed
Summary

This study investigates the Borel map, which links smooth solutions of differential equations to their Taylor series. New conditions for the Borel map

Area of Science:

  • Differential Geometry
  • Analysis on Manifolds

Background:

  • The Borel map associates germs of smooth solutions of differential equations with their formal Taylor series.
  • Previous work established foundational results on the Borel map.

Purpose of the Study:

  • To present new results concerning the Borel map.
  • To establish necessary conditions for the surjectivity of the Borel map.
  • To investigate specific classes of CR structures where surjectivity holds.

Main Methods:

  • Definition and analysis of the Borel map for locally integrable structures.
  • Development of new analytical devices.
  • Study of CR structures and their properties related to the Borel map.

Main Results:

Keywords:
.

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  • A general necessary condition for the surjectivity of the Borel map is proven.
  • Certain classes of CR structures are identified for which the Borel map is surjective.
  • The application of the Borel map to the algebra of solution germs is demonstrated.

Conclusions:

  • The study advances the understanding of the Borel map's properties, particularly its surjectivity.
  • New insights into the behavior of the Borel map on specific geometric structures are provided.
  • The utility of the Borel map as a tool for analyzing solution algebras is highlighted.