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Pseudo-Differential Operators with Discontinuous Symbols: Widom's Conjecture
A. V. Sobolev, University College London, United Kingdom

Memoirs of the American Mathematical Society
2013; 104 pp; softcover
Volume: 222
ISBN-10: 0-8218-8487-5
ISBN-13: 978-0-8218-8487-4
List Price: US$72
Individual Members: US$43.20
Institutional Members: US$57.60
Order Code: MEMO/222/1043
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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.

Table of Contents

  • Introduction
  • Main result
  • Estimates for PDO's with smooth symbols
  • Trace-class estimates for operators with non-smooth symbols}
  • Further trace-class estimates for operators with non-smooth symbols
  • A Hilbert-Schmidt class estimate
  • Localisation
  • Model problem in dimension one
  • Partitions of unity, and a reduction to the flat boundary
  • Asymptotics of the trace (9.1)
  • Proof of Theorem 2.9
  • Closing the asymptotics: Proof of Theorems 2.3 and 2.4
  • Appendix 1: A lemma by H. Widom
  • Appendix 2: Change of variables
  • Appendix 3: A trace-class formula
  • Appendix 4: Invariance with respect to the affine change of variables
  • Bibliography
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