Memoirs of the American Mathematical Society 2013; 104 pp; softcover Volume: 222 ISBN10: 0821884875 ISBN13: 9780821884874 List Price: US$72 Individual Members: US$43.20 Institutional Members: US$57.60 Order Code: MEMO/222/1043
 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  Introduction
 Main result
 Estimates for PDO's with smooth symbols
 Traceclass estimates for operators with nonsmooth symbols}
 Further traceclass estimates for operators with nonsmooth symbols
 A HilbertSchmidt 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 traceclass formula
 Appendix 4: Invariance with respect to the affine change of variables
 Bibliography
