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Maximal functions and h spaces defined by ergodic transformations.

R R Coifman1, G Weiss

  • 1Department of Mathematics, Washington University, St. Louis, Missouri 63130.

Proceedings of the National Academy of Sciences of the United States of America
|June 1, 1973
PubMed
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This study introduces the Ergodic Hilbert transform within ergodic flow probability spaces. Researchers characterize the H(1) function space using maximal functions and identify its dual space.

Area of Science:

  • Ergodic Theory
  • Functional Analysis
  • Harmonic Analysis

Background:

  • Ergodic flows on probability spaces provide a framework for defining transformations like the Ergodic Hilbert transform.
  • The L(p) spaces and the H(1) function space are fundamental in harmonic analysis and operator theory.

Purpose of the Study:

  • To introduce and analyze the Ergodic Hilbert transform for functions in L(p) spaces under ergodic flows.
  • To characterize the H(1) function space in relation to maximal functions.
  • To identify the dual space of H(1) within the context of ergodic theory.

Main Methods:

  • Definition of the Ergodic Hilbert transform for functions acted upon by an ergodic flow.
  • Characterization of the H(1) function space using a specific class of maximal functions.

Related Experiment Videos

  • Identification of the dual space of H(1) through its relationship with functions of bounded mean oscillation.

    • Main Results:

    • The Ergodic Hilbert transform is successfully defined for an ergodic flow on a probability space.
    • The H(1) function space is shown to be characterized by a class of maximal functions.
    • The dual space of H(1) is identified as a space of functions of bounded mean oscillation, intrinsically linked to the ergodic flow.

    Conclusions:

    • The study provides a novel characterization of the H(1) function space using maximal functions in the context of ergodic theory.
    • The identification of the dual space offers new insights into the structure of function spaces associated with ergodic flows.
    • These findings contribute to the understanding of harmonic analysis on probability spaces with dynamical systems.