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On the lowest eigenvalue of a pseudo-differential operator.

C Fefferman1, D H Phong

  • 1Department of Mathematics, Princeton University, Princeton, New Jersey 08544.

Proceedings of the National Academy of Sciences of the United States of America
|December 1, 1979
PubMed
Summary

This study derives positive lower bounds for pseudo-differential operators, providing crucial subelliptic estimates for specific operator types. These findings advance the understanding of operators formed by sums of squares of vector fields.

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Area of Science:

  • Mathematics
  • Analysis
  • Partial Differential Equations

Background:

  • Pseudo-differential operators are fundamental in analyzing partial differential equations.
  • Nonnegative symbols are essential for establishing well-posedness and regularity properties.
  • Subelliptic estimates are key to understanding the behavior of solutions near lower-order terms.

Purpose of the Study:

  • To derive novel positive lower bounds for pseudo-differential operators with nonnegative symbols.
  • To establish subelliptic estimates for operators that are sums of squares of vector fields.
  • To contribute to the theoretical framework of partial differential equations and harmonic analysis.

Main Methods:

  • Analysis of pseudo-differential operators using symbol calculus.

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  • Development of techniques for establishing lower bounds on operator norms.
  • Application of functional analysis to derive subelliptic estimates.

    • Main Results:

    • Established positive lower bounds for a class of pseudo-differential operators.
    • Demonstrated that these bounds yield subelliptic estimates.
    • The results are applicable to operators constructed as sums of squares of vector fields.

    Conclusions:

    • The derived lower bounds provide a powerful tool for analyzing pseudo-differential operators.
    • Subelliptic estimates are obtained, offering insights into the regularity of solutions.
    • This work advances the study of operators related to sums of squares of vector fields.