eBook ISBN: | 978-0-8218-9509-2 |
Product Code: | MEMO/222/1043.E |
List Price: | $72.00 |
MAA Member Price: | $64.80 |
AMS Member Price: | $43.20 |
eBook ISBN: | 978-0-8218-9509-2 |
Product Code: | MEMO/222/1043.E |
List Price: | $72.00 |
MAA Member Price: | $64.80 |
AMS Member Price: | $43.20 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 222; 2013; 104 ppMSC: 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.
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Table of Contents
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Chapters
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1. Introduction
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2. Main result
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3. Estimates for PDO’s with smooth symbols
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4. Trace-class estimates for operators with non-smooth symbols
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5. Further trace-class estimates for operators with non-smooth symbols
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6. A Hilbert-Schmidt class estimate
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7. Localisation
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8. Model problem in dimension one
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9. Partitions of unity, and a reduction to the flat boundary
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10. Asymptotics of the trace
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11. Proof of Theorem
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12. Closing the asymptotics: Proof of Theorems and
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13. Appendix 1: A lemma by H. Widom
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14. Appendix 2: Change of variables
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15. Appendix 3: A trace-class formula
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16. Appendix 4: Invariance with respect to the affine change of variables
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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