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Pseudo-Differential Operators with Discontinuous Symbols: Widom’s Conjecture
 
A. V. Sobolev University College London, London, United Kingdom
Front Cover for Pseudo-Differential Operators with Discontinuous Symbols: Widom's Conjecture
Available Formats:
Electronic ISBN: 978-0-8218-9509-2
Product Code: MEMO/222/1043.E
104 pp 
List Price: $72.00
MAA Member Price: $64.80
AMS Member Price: $43.20
Front Cover for Pseudo-Differential Operators with Discontinuous Symbols: Widom's Conjecture
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  • Front Cover for Pseudo-Differential Operators with Discontinuous Symbols: Widom's Conjecture
  • Back Cover for Pseudo-Differential Operators with Discontinuous Symbols: Widom's Conjecture
Pseudo-Differential Operators with Discontinuous Symbols: Widom’s Conjecture
A. V. Sobolev University College London, London, United Kingdom
Available Formats:
Electronic ISBN:  978-0-8218-9509-2
Product Code:  MEMO/222/1043.E
104 pp 
List Price: $72.00
MAA Member Price: $64.80
AMS Member Price: $43.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2222013
    MSC: Primary 47; Secondary 35;

    Relying on the known two-term quasiclassical asymptotic formula for the trace of the function \(f(A)\) of a Wiener-Hopf type operator \(A\) in dimension one, in 1982 H. Widom conjectured a multi-dimensional generalization of that formula for a pseudo-differential operator \(A\) with a symbol \(a(\mathbf{x}, \boldsymbol{\xi})\) having jump discontinuities in both variables. In 1990 he proved the conjecture for the special case when the jump in any of the two variables occurs on a hyperplane. The present paper provides a proof of Widom's Conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Main result
    • 3. Estimates for PDO’s with smooth symbols
    • 4. Trace-class estimates for operators with non-smooth symbols
    • 5. Further trace-class estimates for operators with non-smooth symbols
    • 6. A Hilbert-Schmidt class estimate
    • 7. Localisation
    • 8. Model problem in dimension one
    • 9. Partitions of unity, and a reduction to the flat boundary
    • 10. Asymptotics of the trace
    • 11. Proof of Theorem
    • 12. Closing the asymptotics: Proof of Theorems and
    • 13. Appendix 1: A lemma by H. Widom
    • 14. Appendix 2: Change of variables
    • 15. Appendix 3: A trace-class formula
    • 16. Appendix 4: Invariance with respect to the affine change of variables
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Volume: 2222013
MSC: Primary 47; Secondary 35;

Relying on the known two-term quasiclassical asymptotic formula for the trace of the function \(f(A)\) of a Wiener-Hopf type operator \(A\) in dimension one, in 1982 H. Widom conjectured a multi-dimensional generalization of that formula for a pseudo-differential operator \(A\) with a symbol \(a(\mathbf{x}, \boldsymbol{\xi})\) having jump discontinuities in both variables. In 1990 he proved the conjecture for the special case when the jump in any of the two variables occurs on a hyperplane. The present paper provides a proof of Widom's Conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces.

  • Chapters
  • 1. Introduction
  • 2. Main result
  • 3. Estimates for PDO’s with smooth symbols
  • 4. Trace-class estimates for operators with non-smooth symbols
  • 5. Further trace-class estimates for operators with non-smooth symbols
  • 6. A Hilbert-Schmidt class estimate
  • 7. Localisation
  • 8. Model problem in dimension one
  • 9. Partitions of unity, and a reduction to the flat boundary
  • 10. Asymptotics of the trace
  • 11. Proof of Theorem
  • 12. Closing the asymptotics: Proof of Theorems and
  • 13. Appendix 1: A lemma by H. Widom
  • 14. Appendix 2: Change of variables
  • 15. Appendix 3: A trace-class formula
  • 16. Appendix 4: Invariance with respect to the affine change of variables
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