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L(p) estimates for the bilinear Hilbert transform.

M Lacey1, C Thiele

  • 1School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA.

Proceedings of the National Academy of Sciences of the United States of America
|January 7, 1997
PubMed
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This study establishes a new inequality for the bilinear Hilbert transform, providing bounds for its operator norm. The findings are crucial for understanding the behavior of this transform in harmonic analysis.

Area of Science:

  • Harmonic Analysis
  • Functional Analysis

Background:

  • The bilinear Hilbert transform is a fundamental operator in harmonic analysis.
  • Understanding its boundedness properties is essential for various mathematical applications.

Purpose of the Study:

  • To establish a new inequality for the bilinear Hilbert transform.
  • To determine bounds for the operator norm of the bilinear Hilbert transform.

Main Methods:

  • The study employs techniques from harmonic analysis.
  • The core of the method involves establishing an inequality for the bilinear Hilbert transform.

Main Results:

  • An inequality for the bilinear Hilbert transform is announced: ||Hfg||_{p(3)} <= K(p(1), p(2)) ||f||_{p(1)} ||g||_{p(2)}.

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  • This inequality holds under specific conditions for the parameters p(1), p(2), and p(3).

    • Conclusions:

    • The established inequality provides crucial insights into the boundedness of the bilinear Hilbert transform.
    • The results contribute to the ongoing research in the field of harmonic analysis and operator theory.