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