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A class of bounded pseudo-differential operators.

A P Calderón1, R Vaillancourt

  • 1Department of Mathematics, University of Chicago, Chicago, Ill. 60637.

Proceedings of the National Academy of Sciences of the United States of America
|May 1, 1972
PubMed
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This study demonstrates that pseudo-differential operators are bounded in L(2) spaces under specific parameter conditions. These findings are crucial for understanding operator theory and its applications.

Area of Science:

  • Mathematical Analysis
  • Functional Analysis
  • Operator Theory

Background:

  • Pseudo-differential operators are essential tools in analyzing partial differential equations.
  • Understanding the boundedness of these operators in function spaces like L(2) is fundamental.

Purpose of the Study:

  • To establish the boundedness of pseudo-differential operators of order -M and type rho, delta(1), delta(2) in L(2) spaces.
  • To define the precise conditions on the parameters rho, delta(1), and delta(2) for this boundedness.

Main Methods:

  • The study employs techniques from harmonic analysis and functional analysis.
  • Analysis involves characterizing the operators based on their order and type parameters.

Main Results:

Related Experiment Videos

  • Pseudo-differential operators are proven to be bounded in L(2) spaces.
  • The conditions for boundedness are established as 0 <= rho <= delta(1) < 1 and 0 <= rho <= delta(2) < 1, along with a specific formula relating these parameters.

Conclusions:

  • The research provides a significant result on the L(2) boundedness of a class of pseudo-differential operators.
  • These results contribute to the theoretical understanding of operator properties and their parameter dependencies.