Quasi-Classical Asymptotics for Pseudodifferential Operators with Discontinuous Symbols: Widom's Conjecture

In 1982 H. Widom conjectured a multi-dimensional generalization of a well-known two-term quasi-classical asymptotic formula for the trace of the function $f(A)$ of a Wiener–Hopf-type operator $A$ in dimension $1$ for a pseudodifferential operator $A$ with 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 in a hyperplane.
This note announces a proof of Widom's conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces.