Pseudo-Differential Operators with Discontinuous Symbols : Widom's Conjecture

ISBN-13: 9780821884874 / Angielski

A V Sobolev

Opis

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.

Spis treści

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