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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)
DOI: https://doi.org/10.1090/S0065-9266-2012-00670-8
Published electronically: June 6, 2012
Keywords: Pseudo-differential operators with discontinuous symbols,
quasi-classical asymptotics,
Szegő formula
MSC: Primary 47G30; Secondary 35S05, 47B10, 47B35
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
Abstract
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.- Gruia Arsu, On Schatten-von Neumann class properties of pseudodifferential operators. The Cordes-Kato method, J. Operator Theory 59 (2008), no. 1, 81–114. MR 2404466
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