**Memoirs of the American Mathematical Society**

2013;
104 pp;
Softcover

MSC: Primary 47;
Secondary 35

Print ISBN: 978-0-8218-8487-4

Product Code: MEMO/222/1043

List Price: $72.00

Individual Member Price: $43.20

**Electronic ISBN: 978-0-8218-9509-2
Product Code: MEMO/222/1043.E**

List Price: $72.00

Individual Member Price: $43.20

# Pseudo-Differential Operators with Discontinuous Symbols: Widom’s Conjecture

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*Alexander V Sobolev*

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

# Table of Contents

## Pseudo-Differential Operators with Discontinuous Symbols: Widom's Conjecture

- Chapter 1. Introduction 18 free
- Chapter 2. Main result 512 free
- Chapter 3. Estimates for PDO’s with smooth symbols 1320
- Chapter 4. Trace-class estimates for operators with non-smooth symbols 2330
- Chapter 5. Further trace-class estimates for operators with non-smooth symbols 2936
- Chapter 6. A Hilbert-Schmidt class estimate 3744
- Chapter 7. Localisation 4148
- Chapter 8. Model problem in dimension one 4754
- Chapter 9. Partitions of unity, and a reduction to the flat boundary 5764
- Chapter 10. Asymptotics of the trace (9.1) 6572
- Chapter 11. Proof of Theorem 2.9 7582
- Chapter 12. Closing the asymptotics: Proof of Theorems 2.3 and 2.4 7986
- Chapter 13. Appendix 1: A lemma by H. Widom 8794
- Chapter 14. Appendix 2: Change of variables 93100
- Chapter 15. Appendix 3: A trace-class formula 99106
- Chapter 16. Appendix 4: Invariance with respect to the affine change of variables 101108
- Bibliography 103110