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Pseudo-differential operators with discontinuous symbols: Widom’s Conjecture

About this Title

A. V. Sobolev, Department of Mathematics, University College London, Gower Street, London, WC1E 6BT UK

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 222, Number 1043
ISBNs: 978-0-8218-8487-4 (print); 978-0-8218-9509-2 (online)
Published electronically: June 6, 2012
Keywords:Pseudo-differential operators with discontinuous symbols, quasi-classical asymptotics, Szegő formula
MSC: Primary 47G30; Secondary 35S05, 47B10, 47B35

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Main result
  • Chapter 3. Estimates for PDO’s with smooth symbols
  • Chapter 4. Trace-class estimates for operators with non-smooth symbols
  • Chapter 5. Further trace-class estimates for operators with non-smooth symbols
  • Chapter 6. A Hilbert-Schmidt class estimate
  • Chapter 7. Localisation
  • Chapter 8. Model problem in dimension one
  • Chapter 9. Partitions of unity, and a reduction to the flat boundary
  • Chapter 10. Asymptotics of the trace (9.1)
  • Chapter 11. Proof of Theorem 2.9
  • Chapter 12. Closing the asymptotics: Proof of Theorems 2.3 and 2.4
  • Chapter 13. Appendix 1: A lemma by H. Widom
  • Chapter 14. Appendix 2: Change of variables
  • Chapter 15. Appendix 3: A trace-class formula
  • Chapter 16. Appendix 4: Invariance with respect to the affine change of variables

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