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# memo_has_moved_text(); Pseudo-differential operators with discontinuous symbols: Widom's Conjecture

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: http://dx.doi.org/10.1090/S0065-9266-2012-00670-8
Posted: June 6, 2012
Keywords: Pseudo-differential operators with discontinuous symbols, quasi-classical asymptotics, Szegő formula
MSC (2010): Primary 47G30; Secondary 35S05, 47B10, 47B35

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Chapters

• 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

### 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.