Electronic ISBN:  9780821895092 
Product Code:  MEMO/222/1043.E 
104 pp 
List Price:  $72.00 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 222; 2013MSC: Primary 47; Secondary 35;
Relying on the known twoterm quasiclassical asymptotic formula for the trace of the function \(f(A)\) of a WienerHopf type operator \(A\) in dimension one, in 1982 H. Widom conjectured a multidimensional generalization of that formula for a pseudodifferential 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

Chapters

1. Introduction

2. Main result

3. Estimates for PDO’s with smooth symbols

4. Traceclass estimates for operators with nonsmooth symbols

5. Further traceclass estimates for operators with nonsmooth symbols

6. A HilbertSchmidt 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 traceclass formula

16. Appendix 4: Invariance with respect to the affine change of variables


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Relying on the known twoterm quasiclassical asymptotic formula for the trace of the function \(f(A)\) of a WienerHopf type operator \(A\) in dimension one, in 1982 H. Widom conjectured a multidimensional generalization of that formula for a pseudodifferential 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. Traceclass estimates for operators with nonsmooth symbols

5. Further traceclass estimates for operators with nonsmooth symbols

6. A HilbertSchmidt 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 traceclass formula

16. Appendix 4: Invariance with respect to the affine change of variables