Pseudo-Differential Operators with Discontinuous Symbols

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 generalisation of that formula for a pseudo-differential operator A with a symbol a(x,) 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.