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Subelliptic estimates for complexes.

V Guillemin1, S Sternberg

  • 1MASSACHUSETTS INSTITUTE OF TECHNOLOGY, HARVARD UNIVERSITY.

Proceedings of the National Academy of Sciences of the United States of America
|September 1, 1970
PubMed
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New research links differential complex properties to test estimates, introducing a normal form near Cohen-MacCauley points. This work establishes conditions for satisfying (1/2)-estimates on manifolds with boundaries.

Area of Science:

  • Differential geometry
  • Complex analysis
  • Partial differential equations

Background:

  • Hörmander's work on (1/2)-estimates is foundational in the analysis of differential operators.
  • Differential complexes and their symbol modules are key structures in geometric analysis.
  • Cohen-MacCauley points represent specific singularities in algebraic and analytic geometry.

Purpose of the Study:

  • To establish a connection between the algebraic properties of differential complexes and analytic estimates.
  • To introduce a normal form for differential complexes near singular points.
  • To derive conditions for satisfying (1/2)-estimates for boundary value problems.

Main Methods:

  • Analysis of the symbol module and characteristic variety of differential complexes.

Related Experiment Videos

  • Utilizing pseudo-differential changes of coordinates to establish invariance.
  • Introducing a Hermitian form based on Poisson brackets to analyze the geometry of the characteristic variety.
  • Investigating properties of boundary complexes on manifolds with smooth boundaries.

    • Main Results:

    • Demonstrated invariance of test estimates under coordinate transformations.
    • Introduced a normal form for differential complexes near Cohen-MacCauley points.
    • Established conditions involving the signature and rank of an invariant Hermitian form for satisfying test estimates.
    • Provided criteria for boundary complexes to satisfy the (1/2)-estimate.

    Conclusions:

    • The study provides a deeper understanding of the interplay between algebraic and analytic properties of differential complexes.
    • The introduced normal form and invariant Hermitian form offer new tools for analyzing differential complexes.
    • The derived conditions for (1/2)-estimates have implications for the well-posedness of boundary value problems on manifolds.