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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 Click above image for expanded view 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.

• 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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