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Oscillatory singular integrals and harmonic analysis on nilpotent groups.

F Ricci1, E M Stein

  • 1Dipartimento di Matematica, Politecnico di Torino, 10129 Torino, Italy.

Proceedings of the National Academy of Sciences of the United States of America
|January 1, 1986
PubMed
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This study examines operators on nilpotent Lie groups, featuring oscillatory factors and singular convolutions. The research provides a general framework applicable even without specific dilation properties.

Area of Science:

  • Harmonic Analysis
  • Lie Group Theory
  • Functional Analysis

Background:

  • Nilpotent Lie groups are fundamental in mathematics and physics.
  • Operators on these groups are crucial for understanding their structure and representations.
  • Existing theories often rely on specific properties like dilations, limiting their applicability.

Purpose of the Study:

  • To introduce and analyze novel classes of operators on nilpotent Lie groups.
  • To develop a more general framework for studying these operators, removing restrictive assumptions.
  • To explore the interplay between oscillatory factors, singular convolutions, and group structure.

Main Methods:

  • Analysis of operators involving oscillatory factors (exponentials of imaginary polynomials).

Related Experiment Videos

  • Investigation of convolutions with singular kernels supported on submanifolds.
  • Development of techniques valid in a general setting, not requiring automorphism dilations.

    • Main Results:

    • Characterization of new operator classes on nilpotent Lie groups.
    • Demonstration of the applicability of these operators under relaxed conditions.
    • Insights into the behavior of singular integral operators in a generalized context.

    Conclusions:

    • The study expands the toolkit for analyzing operators on nilpotent Lie groups.
    • The generalized framework enhances the scope of harmonic analysis on these groups.
    • The findings are relevant for areas employing Fourier analysis and geometric methods on Lie groups.