Functions of self-adjoint operators in ideals of compact operators

For self-adjoint operators A,B, a bounded operator J , and a function f : R → C, we obtain bounds in quasi-normed ideals of compact operators for the difference f(A)J − Jf(B) in terms of the operator AJ − JB. The focus is on functions f that are smooth everywhere except for finitely many points. A typical example is the function f(t) = |t|γ with γ ∈ (0, 1). The obtained results are applied to derive a two-term quasi-classical asymptotic formula for the trace trf(S) with S being a Wiener–Hopf operator with a discontinuous symbol.